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0. 1.3 By the differentiability assumption (ii) the partial derivatives of the right hand side of (3.1) are continuous in Rnm þ ; hence, this right hand . This implies, by Rouche and Mawhin side is locally Lipschitz in Rnm þ (1973), Theorem 5.6, p. 83 that there exists a unique solution x ( ), defined on some interval, which cannot be continued on the right. 1.4 By Lemma 1, this solution x ( ), defined on some interval [0, b), maps into U(x0 ), which is a compact subset of Rmn þ , and so has at . If b < 1, then by the second part of least one accumulation point, say x ) must belong to the Rouche and Mawhin (1973), Theorem 5.6, (b, x frontier of R Rmn . Since by step 1.1, x > 0, for all i and h, we can ih þ only have b ¼ þ1.
Ch.4 Commodity Exchanges as Gradient Processes
87
(2) Convergence. 2.1 To prove the second part of the proposition we note as an immediate corollary to Theorem 6.1 in Champsaur, Dre`ze, and Henry 0 2 X , (1977), that if a dynamic system has a unique solution for every x 0 0 say x (x ; t), which varies continuously with x in X and which remains in X for all t, and if there is a Lyapunov function,4 then the system is quasithe system (3.1), existence of a unique solution x (x0 ; ): stable.5 From [0, 1) ! X has just been established; moreover, by Rouche and Mawhin 0 (1973), Theorem 3.1, p. 105, x ( ; t) is continuous in x on X , since the right hand side of (3.1) is locally Lipschitz; finally, the collective monotonicity (Lemma 1) and the boundedness of W imply that we can take L ¼ W as a Lyapunov function for (3.1). Hence (3.1) is quasi-stable. 2.2 Since by (2.4) any equilibrium of the system (3.1) is a Pareto efficient allocation, it remains only to show that limt!1 x (x0 ;t) exists for 0 any x 2 X. But this is immediate since, by our convexity assumptions, there is only one Pareto efficient allocation x such that L(x ) ¼ limt!1 L (x(t)) ¼ limt!1 W (x(t)). &
Corollary 1: For every x0 2 X , the unique solution x ( ): [0, 1] ! X is such that W ( limt!1 x(t)) ¼ maxx2X W (x).
4.
The ‘‘M70’’ Process as a Gradient Process
In footnote 4 (p. 215) of his first paper on public goods, Malinvaud (1970–71) sketches out a dynamic process which we call here the ‘‘M70 Process,’’ and which in the case of our private goods pure exchange economy may be written as follows (all variables xi are kept assumed to be functions of time). M70 Process:6 x_ ih ¼
uih 1 X ujh krui k n j2N kruj k
8i 2 N,
8h 2 M:
(4:1)
A Lyapunov function for a dynamic system x_ (x) is any continuous function L from X to R such that for any x0 2 X , and any solution x( ) starting at x0 , limt!1 L(x(t)) exists and, whenever x(t) is constant for all t 2 [0, t] [0, 1) with t > 0, then x0 is an equilibrium. An equilibrium of a dynamic system is a state x 2 X of the system such that x_ (x) ¼ 0. 5 A dynamic system is said to be quasi-stable iff any accumulation point of any solution is an equilibrium. 6 It should also be pointed out that in fact, Malinvaud (1970–71) uses for the second term of (4.1) a weighted average, instead of the simple average we use here. 4
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Public Goods, Environmental Externalities and Fiscal Competition
Comparing (3.1) with (4.1), the latter is immediately seen to be a simple modification of the former, in which every coefficient li is now a particular function of xi , namely: li (xi ) ¼ (krui k)1 ,
8i 2 N:
(4:2)
This choice of the vector l simply corresponds to a normalization of the gradient vector (ui1 , . . . , uih , . . . , uim ) of the utility function of each agent i. Referring to the original welfare maximization problem (2.1)–(2.3), and to the reasoning that led to the formulation of the WMP process (3.1) associated with it, the M70 process appears to be also a ‘‘gradient projection process,’’ but not exactly for the problem (2.1)–(2.3) though. Instead, it is such a process for a continuous sequence of such problems, each of which is redefined at each x (t) in terms of coefficients l(x(t)) defined by (4.2). It is interesting to note that, while the WMP process (3.1) was formulated by means of the form (2.4) of the first order conditions for a maximum, it is the use of the alternative form (2.6) of these conditions that yields the M70 process (4.1), which is the gradient projection of a modified optimization problem. Another way of viewing the M70 dynamic system is to consider it as the system which, for each x 2 X , determines the direction of the ‘‘locally’’ best improvement among all the feasible ones; more precisely,7 this direction is proportional to the optimal solution of the following ‘‘local’’ programming problem: X X X 1 _ max (kru k) u subject to (4:3) x x_ ih ¼ 0, 8h 2 M, i ih ih nm x_ 2R
i2N
h2M
kx_ k ¼ 1:
i2N
(4:4)
Turning now to the properties of the M70 process, we note first the remarkable fact that the introduction of a sequence of welfare maximization problems by using l’s which are functions of x, allows for a stronger monotonicity property: Lemma 2: The M70 process (4.1) is individually monotone in the sense that, for any solution x: T ! Rnm of the associated system of differential equations and for any t 2 T, if x(t) 2 X then X uih (xi (t))x_ ih (x(t)) > 0, 8i 2 N, u_ i (x(t)) ¼ h2M
7
See Luenberger (1974), Exercise 9, p. 274.
Ch.4 Commodity Exchanges as Gradient Processes
89
unless x(t) is a Pareto efficient allocation, in which case 8i 2 N, u_ i (x(t)) ¼ 0. Proof: To simplify the notation, let, 8i 2 N, 8h 2 M, vih ¼ uih = krui k. Clearly, u_ i $0 iff u_ i =krui k$0 (recall that krui k > 0). Since kvi k ¼ kvj k ¼ 1, we get: ! ! X 1X 1X X 2 X u_ i vih vih vjh ¼ v vih vjh ¼ krui k h2M n j2N n j2N h2M ih h2M ¼
1X 1X (kvi k2 jvi vj j) ¼ (kvi kkvj k jvi vj j)$0 n j2N n j2N
by Cauchy-Schwarz inequality. Moreover, since for any j 2 N, kvi k2 ¼ jvi vj j ¼ kvj k2 ¼ 1 iff vi ¼ vj , we must have, by conditions (2.5), u_ i =krui k > 0; unless x is Pareto efficient. & With this result, we may now prove the following proposition, just as in Proposition 1 for the WMP process.
Proposition 2: For the M70 process (4.1) and for every x0 2 X , there exists a unique solution x ( ): [0, 1) ! X, which is such that limt!1 x(t) exists and is a Pareto efficient allocation. Proof: The first part of the proof is analogous to the argumentation in Proposition 1, redefining U(x0 ) as
{x 2 X jui (xi ) $ ui (x0i )8i 2 N} and replacing Lemma 1 by Lemma 2 in paragraphs 1.1 and 1.4, and using in paragraph 1.3 the fact that li is continuously differentiable in Rnm . þ For the second part, we may again use Theorem 6.1 Pof Champsaur, Dre`ze, and Henry (1977), taking as Lyapunov function i2N ui (xi ). &
5.
The ‘‘MDP’’ Process as a Gradient Process
In Chapter 8 of his microeconomics textbook [(1972), pp. 190–192], Malinvaud develops another dynamic process for a pure exchange
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Public Goods, Environmental Externalities and Fiscal Competition
economy, which has come to be known as the ‘‘MDP process’’ in its version with public goods.8 In our present notation, this process reads: MDP Process: 1X pjh x_ ih ¼ pih n j2N x_ i1 ¼
X
8h 2 M, h 6¼ 1, 8i 2 N,
pih x_ ih þ di
h2M h6¼1
XX h2M h6¼1
j2N
"
1X pjh pkh n k2N
(5:1)
#2 8i 2 N,
(5:2)
P where the constants di satisfy di $ 0 8i 2 N and i2N di ¼ 1. (Note that here again we use simple arithmetic averages instead of weighted ones.) Comparing this system with (4.1) and with (3.1) above, and remembering the efficiency conditions (2.5), one sees that as far as all commodities h 6¼ 1 are concerned, this is again a process formulated on the basis of the first order conditions for a Pareto efficient allocation, the version (2.5) being used this time. However, the equations (5.2) of the process do not bear much resemblance with these conditions. Because it is known (as recalled below; see Proposition 3) that for this process too the unique solution converges to a Pareto efficient allocation, it is natural to ask oneself which further modification of the original welfare maximization problem yields the MDP process as a gradient process. In the same spirit as our preceding development regarding the M70 process, a sequence of changing problems is involved; but in addition, the original maximization problem undergoes a substantial transformation that we presently develop. , and the associated Consider some arbitrary allocation in X , say x xi ) 8i 2 N. To the welfare utility vector v ¼ (v1 , . . . , vn ) given by vi ¼ ui ( maximization problem (2.1)–(2.3), let a set of n additional constraints be added, namely: ui (xi1 , . . . , xim )$ yi
8i 2 N:
(5:3)
is a feasible solution for the augmented problem By construction, x (2.1)–(2.3) and (5.3), with strict equality holding for each one of the new constraints (5.3). Note at this point that for every solution x 2 X for which such equalities hold, each one of the latter can be made explicit in xi1 rather than in y i , using the monotonicity assumption, and be written as xi1 ¼ fi (xi2 , . . . , xim ; yi ) 8i 2 N:
(5:4)
On the other hand, for every other solution of the augmented problem for which ui (xi ) > yi holds for some i, there is for each such i an amount of commodity 1, say yi1 $0, such that 8
See Champsaur (1976), p. 293.
Ch.4 Commodity Exchanges as Gradient Processes
91
xi1 ¼ fi (xi2 , . . . , xim ; yi ) þ yi1 : (5:5) P Now, since i2N xi1 ¼ v1 must hold for any solution which belongs to X , we may write that every feasible solution x of the augmented problem satisfies X X X fi (xi2 , . . . , xim ; yi ) þ xi1 ¼ yi1 ¼ v1 , i2N
or X
i2N
i2N
X fi (xi2 , . . . , xim ; yi ) : yi1 ¼ v1
i2N
(5:6)
i2N
This in turn implies Pthat there exists a vector d ¼ (d1 , . . . , di , . . . , dn ) with di $0, 8i 2 N, and i2N di ¼ 1 such that X fj (xj2 , . . . , xjm ; yj ) 8i 2 N: (5:7) yi1 ¼ di v1 j2N
P 2 X , and for every d 2 Rnþ such that i2N di ¼ 1, Finally, to any x one may associate the following nonlinear optimization problem: X max li ui (xi ) (5:8) nm
x2R
i2N
subject to
X fj (xj2 , . . . , xjm ; yj ) , 8i 2 N, xi1 ¼ fi (xi2 , . . . , xim ; yi ) þ di v1 X
(5:9)
j2N
xih ¼ vh , 8h 2 M ,
(5:10)
i2N
xih > 0, 8i 2 N,
8h 2 M,
(5:11)
where M denotes the set Mn{1}. It is now possible to use on this problem a combination of both the reduced gradient and the gradient projection techniques. Indeed we may reduce first the problem by eliminating the xi1 ’s in the following way: X def li ui fi (xi2 , . . . , xim ; yi ) max W (x; l, y) ¼ x2Rn(m1)
þ di v1
i2N
X j2N
subject to
fj (xj2 , . . . , xjm ; yj ) , xi2 , . . . , xim
(5:12)
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Public Goods, Environmental Externalities and Fiscal Competition
X
xih ¼ vh
8h 2 M ,
(5:13)
i2N
xih > 0
8i 2 N,
8h 2 M :
(5:14)
, whereby the indifference functions fi ( ) Clearly, the allocation x appearing in (5.12) have been defined, is a feasible solution for this problem. as Next, let us compute the gradient of the objective function W . Its typical element is of the form: given by (5.12), at the allocation x X @W @ fi ¼ li ui1 @ fi di @ fi þ uih l u d i 2 N, h 2 M : (5:15) j j1 j @xih @xih @xih @xih x j6¼i Upon noticing that by construction of (5.4) @ fi uih ¼ ¼ pih ( xi ), @xih ui1 xi (5.15) reduces to, after rearranging terms, X @W ¼ li ( uih þ ui1 di pih þ uih ) þ lj uj1 dj pih ¼ a( x)pih ( xi ), @x ih x
(5:16)
j6¼i
where a( x) ¼
X
lj uj1 dj :
(5:17)
j2N
If we compute the projection of this gradient onto the tangent , by the constraints (5.13), by means of the subspace determined, at x matrix operator P defined in Section 2 above (which is of dimension n(m 1) n(m 1) in this case), we obtain the feasible direction for all h 2 M as: 1X x) ¼ a( x) pih ( xi ) pjh ( xj ) 8i 2 N, 8h 2 M : (5:18) x_ ih ( n j2N Differentiating (5.9), expression (5.18) yields in addition the direction for h ¼ 1 as X X X X x) ¼ pih ( xi )x_ ih ( x) þ d i pjh ( x)x_ jh ( x) ¼ pih ( xi )x_ ih ( x) x_ i1 ( h2M
x) þ di a(
X X
h2M j2N
j2N h2M
(pjh ( xj )
1X n k2N
pkh ( xk ))2 8i 2 N:
h2M
(5:19)
Comparing (5.18) and (5.19) thus obtained with (5.1) and (5.2), , say, of its the MDP process appears to determine, at each point x
Ch.4 Commodity Exchanges as Gradient Processes
93
trajectory, the direction of the gradient projection of the transformed welfare maximization problem (5.12)–(5.14), with the coefficient li chosen so that li ( x) ¼ [ui1 ( xi )]1
8i 2 N:
(5:20)
Indeed, (5.20) implies, through (5.17), that a( x) ¼ 1 in (5.18) and (5.19). Note however that if the li ’s were taken as arbitrary positive constants in (5.12), the direction of the vector x_ ( x) defined by (5.18) and (5.19) would not be modified; only its length (which is constant when a( x) ¼ 1, 8( x)) would be modified by the variable factor a( x). The economic significance of the MDP dynamic system is further brought to light by considering it in an alternative (although related) way, as we did above, in (4.3)–(4.4), for the M70 process. Indeed the process 2 X the feasible direction may be viewed as one that determines at each x that solves the following ‘‘local’’ problem, derived from (5.12)–(5.14): X X @W def _ x_ ih (5:21) max W ¼ @xih x x_ 2Rn(m1) i2N h2M
subject to X x_ ih ¼ 0 8h 2 M ,
(5:22)
i2N
kx_ k ¼ 1:
(5:23)
In view of (5.16), the maximand (5.21) reduces to X X _ ¼ a( x)pih ( xi )x_ ih , W i2N h2M
and an easy computation shows that the solution of this problem yields for each component x_ ih ,i 2 N and h 2 M , an expression proportional to (5.18). Using then (5.20) for determining a( x) through (5.17), the value of the maximand at this solution is proportional to: 2 X X 1X pih pjh : n j2N i2N h2M
This magnitude, which is expressed in units of commodity h ¼ 1, is simply the well known ‘‘surplus’’ that characterizes all processes of the ‘‘MDP’’ type. The gradient direction determined by such a process is thus the one which, by maximizing the rate of increase of (5.12), as specified by (5.21)–(5.23), in fact maximizes the amount of commodity 1 that can be generated at each point x of the solution, under the condition of preserving both feasibility and the utility level yi of each agent.
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Public Goods, Environmental Externalities and Fiscal Competition
Other intrinsic properties of the MDP dynamic process (5.1)–(5.2) are rather well known (see, e.g., Malinvaud (1972), Chapter 8 or Dre`ze and de la Valle´e Poussin (1971), Section 2). For the sake of completeness in our comparison, we just restate them here, in the same terms as our previous lemmas and propositions: Lemma 3: For di > 0 8i 2 N, the MDP process (5.1)–(5.2) is individually monotone. Proof: See, e.g., Malinvaud [(1972), p. 192].
Proposition 3: For the MDP process (5.1)–(5.2) and for every x0 2 X , there exists a unique solution x ( ): [0, 1) ! X , which is such that limt!1 x (t) exists and is a Pareto efficient allocation. Proof: Analogous to the proof of Proposition 2.
6.
Concluding remarks
The preceding developments call naturally for some comparative assessment of the three processes involved. Leaving aside computational issues, such as those relating, for example, to the convergence rate, we shall consider instead two points which are perhaps more directly relevant for the social scientist. The essential difference between the WMP process and the M70 process lies in the property of individual monotonicity of the latter. As our approach has made apparent, this results from a simple, but quite specific, modification of the former, viz. a normalization of the gradient vector of each utility function. The social weights represented by coefficients li are in a sense made endogenous, and linked to preference characteristics of each agent. In the spirit of the usual planning interpretation of the above processes, individual monotonicity may be seen as a desirable property because it enhances the acceptability of the ‘‘plan’’ by the agents of the economy. By the same token, however, it reduces the distributional choices available to the planning authority. Actually, in the M70 process there is no such choice left whatsoever, since the directions of utility increase are unique for each i; on the other hand, the MDP process offers more variety from this point of view, thanks to the vector parameters di : by his ‘‘neutrality’’ theorem, Champsaur (1976) has exhibited the range of such choices. There is however another aspect whereby both the M70 and the MDP processes distinguish themselves from the WMP process, that we
Ch.4 Commodity Exchanges as Gradient Processes
95
might call operationality. The economic meaning of any dynamic process is much conditioned by the observability of the variables and parameters that govern its behavior. From this point of view, the marginal utilities whereby the WMP process is defined offers a rather poor basis. On the contrary, the marginal rates of substitution used in the MDP process are independent of any direct measurement of utility: they are theoretically observable, and in practice at least susceptible of being expressed (or ‘‘revealed’’) in unambiguous numerical terms. The MDP process therefore could be considered for implementation. This is also true9 of the M70 process, in view of (2.7). Finally, it should be pointed out that for both the M70 and the MDP processes presented there, their (unique) solutions verify the definition of an exchange curve as defined by Smale (1976).
References Champsaur, P., 1976. Neutrality of planning procedures in an economy with public goods, Review of Economic Studies, 43, 293–300. Champsaur, P., Dreze, J. H. and Henry, C., 1977. Stability theorems with economic applications, Econometrica, 45, 273–294. Dre`ze, J. H. and de la Valle´e Poussin, D., 1971. A taˆtonnement process for public goods, Review of Economic Studies, 38, 133–150. Luenberger, D. G., 1973. Introduction to linear and nonlinear programming. Reading, Massachusetts: Addison-Wesley. Malinvaud, E., 1970–1971. Procedures for the determination of a program of collective consumption, European Economic Review, 2, 187–217. Malinvaud, E., 1972. Lectures on Microeconomic Theory. Amsterdam: North-Holland. Milleron, J. C., 1974. Procedures to a Lindahl Equilibrium corresponding to a given distribution of income, paper presented at the European Meeting of the Econometric Society in Grenoble, September, 1974. Rouche, N. and Mawhin, J., 1973. Equations diffe´rentielles ordinaires, Tome 1: The´orie ge´ne´rale. Paris: Masson. Smale, S., 1976. Exchange processes with price adjustment, Journal of Mathematical Economics, 3, 211–226.
9
We owe this point to Paul Champsaur.
PART II ENVIRONMENT, PUBLIC GOODS AND EXTERNALITIES
Introduction Parkash Chander, National University of Singapore
101
IIa Dynamic processes for achieving optimality in international environmental problems Chapter 5 An economic model of international negotiations relating to transfrontier pollution Henry Tulkens 1 Motivation and plan of the paper 2 The basic model 3 Properties of the process 4 Concluding remarks References Chapter 6 Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems Parkash Chander, Henry Tulkens 1 Introduction: The general setting 2 A simple model of transfrontier pollution 3 Modelling negotiations 4 Cost sharing and strategic considerations 5 Summary and conclusion References Chapter 7 The acid rain game as a resource allocation process, with application to negotiations between Finland, Russia and Estonia Veijo Kaitala, Karl Go¨ran Ma¨ler, Henry Tulkens 1 Introduction 2 Achieving optimality in the acid rain problem 3 Individual rationality, group rationality and cooperation
107
123
135
4 Increasing marginal damages 5 Conclusion References
IIb The « core » as a solution concept for international environmental agreements Chapter 8 The core of an economy with multilateral environmental externalities 153 Parkash Chander, Henry Tulkens 1 Introduction 2 The model of an economy with multilateral environmental externalities and its efficiency characterization 3 Noncooperative games associated with the economy 4 Cooperative games associated with the economy 5 A strategy in the g-core with linear payoff functions 6 g-Core property of the strategy under nonlinear payoff functions 7 Concluding remarks References Chapter 9 A core-theoretic solution for the design of cooperative agreements on transfrontier pollution 177 Parkash Chander, Henry Tulkens 1 Introduction 2 Transfrontier pollution: the economic model and its associated games 3 The noncooperative game and its Nash equilibria 4 The cooperative game: imputations, the core, and alternative characteristic functions 5 An imputation in the g-core (and in the a-core) of the cooperative game 6 Conclusion: economic interpretation of the proposed g-core solution References Chapter 10 The Kyoto Protocol: an economic and game theoretic interpretation 195 Parkash Chander, Henry Tulkens, Jean-Pascal van Ypersele, Stephane Willems 1 Introduction: cooperation at the world level, from Rio to Kyoto 2 Main features of the protocol
3 Economics of the issues at stake 4 Noncooperative characterizations of the pre-agreement stage 5 Kyoto quotas and worldwide trading: efficiency and coalitional stability of the scheme 6 Beyond the Kyoto quotas: the possibility of a scheme achieving a world coalitionally stable optimum References
IIc Dynamic cooperative games and stock pollutants Chapter 11 Simulating coalitionally stable burden sharing agreements for the climate change problem Johan Eyckmans, Henry Tulkens 1 Introduction and summary 2 CWS, an integrated eonomy–climate world model 3 Alternative scenarios as to cooperation in the CWS model 4 Transfers ensuring individual and coalitional rationality 5 Simulations 6 Conclusion Appendix A References
217
Chapter 12 Transfers to sustain dynamic core-theoretic cooperation in international stock pollutant control 251 Marc Germain, Philippe Toint, Henry Tulkens, Aart de Zeeuw 1 Introduction 2 International optimality and noncooperative equilibrium 3 Transfers to sustain individual rationality at the international optimum 4 Transfers to sustain coalitional rationality at the international optimum 5 Computability issues 6 Conclusion Appendix. Equations and parameters used in the numerical example References
INTRODUCTION
Parkash Chander
Ambient pollution, in general, has all the classical characteristics of a pure public good (or more appropriately a public ‘‘bad’’) such as nonrivalry in consumption and non-excludability. The economics of transnational pollution problems, such as global warming or acid rain, thus belongs to the theory of public goods. However, the transnational pollution problems bring with them at least two additional issues that require an extension of the public good analysis. First, there is no supranational authority, endowed with sufficient powers to enforce a program of public good production (or abatement of pollution). Thus any participation in the public good provision must be essentially voluntary or contractual. Second, and more importantly, an agent may not only fail to contribute towards the public good provision; it may even reduce the existing provision made possible by the contributions of others. For example, a country may increase its own pollution to levels that more than offset the abatements done by the other countries. Thus, unlike the standard public good model, there is no minimum payoff that an agent or coalition of agents can assure for itself by producing the public good on its own. For this reason, as noted by Ma¨ler (1989), the usual concept of the core of a public good economy (Foley (1979)) when applied to transnational pollution problems results in too large a set of outcomes, actually the whole set of Pareto optima. This second issue, however, has little bearing on the type of dynamic processes analysed in Part I. This is so because the agents are concerned only with their instant payoffs and thus the focus is on the stability of the instant allocations in terms of the local surplus-sharing games, and not on the game theoretic properties of the first-best allocations to which these processes converge. The dynamic processes can thus be applied to transnational pollution problems independently of the second issue. Indeed, the first three chapters in this part of the volume address the first issue only.
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Public Goods, Environmental Externalities and Fiscal Competition
Chapter 5 introduces a dynamic process which is driven by a sequence of Pareto improving cost-benefit analyses using only local information. Since each country taken individually is better-off at each instant, it is presented as a negotiation process (instead of as a planning procedure) among countries that pollute each other in a common water resource. The problem of strategic (i.e. coalitional) stability of negotiations in terms of sharing schemes for the ‘‘ecological surplus’’ generated at each instant of the process is raised but not resolved. This task is completed in Chapter 6, co-authored with me, which resolves the stability problem in terms of local surplus-sharing games. A simple sharing scheme is proposed and shown to have the ‘core’ property of not being possibly improved upon by any coalition of countries that would attempt to adopt another abatement strategy. Chapter 7 presents the first numerical application of dynamic processes to an actual transnational pollution problem, namely the acid rain. Numerical simulations are performed on actual data relating to the acid rain negotiations between Finland, Russia and Estonia. This problem had been studied earlier by Ma¨ler (1989) who formulated it as a game and showed that the usual concept of the core of a public good economy is useless when applied to the acid rain model, because any first-best allocation can be shown to belong to the core. Thus the analysis in this paper points to an additional important role of dynamic processes, namely that of a meaningful mechanism for selecting allocations from among the numerous core allocations. Chapters 8 and 9, co-authored with me, introduce an alternative concept of a characteristic function, namely the g-characteristic function that addresses the second issue as outlined above. It is based on an assumption concerning the behaviour of the non-members which is more plausible than those considered previously. When a coalition deviates it neither takes as given the strategies of its complement, as in the case of strong and coalition proof Nash equilibriums, nor presumes that the complement would follow minimax or maximin strategies, as in the case of a- and b-characteristic functions. Instead, it is assumed that the nonmembers will not form any coalition but will simply adopt their individually best reply strategies. This results into a non-cooperative equilibrium between the deviating coalition S and the non-members acting individually. The members of S play their joint best reply strategies to the individually best reply strategies of the non-members. The justification for the assumption that the non-members act individually comes from the fact that if the non-members form one or more non-singleton coalitions, then the payoff of S defined by the resulting Nash equilibrium between the coalitions would only be higher. The assumption that the non-members act individually is thus equivalent to granting S a certain degree of
Introduction
103
pessimism.1 In other words, the assumption is not concerning which coalitions will form after the deviation, but rather which coalitions the deviating coalition S thinks will or will not form, and in fact S presumes that it is the worst possible coalition structure that will form. The uncertainty regarding the coalition structure subsequent to a deviation is thus resolved by assuming that the deviating coalition presumes that coalition structure that is worst from its point of view. The following simple example illustrates the g-characteristic function vis-a`-vis the a- and b-characteristic functions. Consider an economy consisting of three identical agents, i.e., N ¼ {1,2,3}. Let X 1 yi ¼ e2i , i 2 N, z ¼ ei , i2N
and 1 ui ( yi , z) ¼ yi z þ , i 2 N, 4 where e denotes emissions, y denotes output and z pollution. Let Ti ¼ {ei : ei $0}, T ¼ T1 T2 T3 , and u ¼ (u1 , u2 , u3 ). Standard arguments show that the strategic form game [N, T, u] has a unique Nash equilibrium which induces the following state of the economy 1 1 3 ei ¼ ; yi ¼ , i 2 N; z ¼ ; and ui ¼ 0, i 2 N, 4 2 4 where (e1 , e2 , e3 ) 2 T are the Nash equilibrium strategies. It is easily seen that a Pareto efficient state of the economy is given by ei ¼
1 1 1 1 , yi ¼ , i 2 N; z ¼ ; and ui ¼ ; i 2 N, 36 6 12 3
and that the emission levels are the same for all Pareto efficient states. We claim that the payoffs corresponding to this Pareto efficient state belong to the g-core. Since the players are identical, we need to consider only two types of deviations, namely: a deviation by a coalition of any two players, say {1,2} and a deviation by a coalition of any single player, say {3}. 1 1 Define (~e1 , ~e2 , ~e3 ) such that ~e1 , ~e2 ¼ argmax (e21 þ e22 2e1 2e2 1 2~e3 ) and ~e3 ¼ argmax (e23 ~e1 ~e2 e3 ). Then, ~e1 ¼ ~e2 ¼
1 1 1 3 u2 ¼ and ~ , ~e3 ¼ , ~ u1 ¼ ~ u3 ¼ : 16 4 8 8
1 This is pessimism of a different sort: it is not concerning the strategies that will be adopted by the non-members (as in the case of a- and b-characteristic functions), but about the coalition structures that will be formed.
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The strategies (~e1 , ~e2 , ~e3 ) represent the Nash equilibrium between the coalitions {1,2} and {3}. By comparing the payoffs of the coalition {1,2} under the strategies (~e1 , ~e2 , ~e3 ) and (e1 , e2 , e3 ), it is seen that the coalition {1,2} will not gain from the deviation. Note that alternatively the a- and b-strategies require ~e3 ! 1 and ~ u1 , ~u2 ! 1, even though player 3 has a dominant strategy ~e3 ¼1⁄4 . Consider now the deviation by {3} which illustrates another important aspect of the g-characteristic function. When S ¼ {3} deviates it presumes that NnS ¼ {1,2} will break up into singletons and the resulting equilibrium will be the Nash equilibrium between {3}, {1} and {2}. It is seen that the payoffs of the deviating coalition are in each case less than those corresponding to the said Pareto efficient state. A game theoretic interpretation of the Kyoto Protocol on climate change is put forward in Chapter 10. The analysis uses key numerical information provided by Denny Ellerman at MIT and is supportive of the Protocol. Arguments in favor of worldwide competitive emissions trading are presented. Game theory is used at two different levels. First, ideas from the g-theory are borrowed to justify the Kyoto Protocol and then classical cooperative game theory is applied to claim a core stability property of the market equilibrium of the competitive emissions trading. Two policy experts, who have been participants on behalf of the Belgian Government in the international negotiations on climate change, namely Jean-Pascal van Ypersele and Stephane Willems, come in as additional co-authors for this paper. The remaining two papers in this part of the volume deal with environmental externalities that arise from stock pollutants that accumulate (and possibly decay) and not from flow pollutants, as assumed until now. With this the problem acquires an intertemporal dimension. The main issue, however, remains that of voluntary implementation of the international optimum. Open-loop dynamic game theory and numerical simulations are used in Chapter 11 to show that a certain allocation satisfies the g-core property in a discrete time model with a long but finite horizon. Chapter 12, however, uses closed-loop (or feedback) dynamic games. A novel backward induction method that applies the g-theory to solve the problem at each time period is used to find a solution of the closed-loop dynamic game. In this method cooperation is negotiated in each period assuming cooperation in the subsequent periods. It starts with a solution of the game in the final period T. Since the final period game is like a static game, it has a cooperative solution that satisfies the g-core property. Next, the game in period T 1 is similarly solved by assuming cooperation in period T, and so on. . . . This leads to a solution with the subgame property.
Introduction
105
References Foley, D. (1970), ‘‘Lindahl solution and the core of an economy with public goods’’, Econometrica, 38, 66 –72. Ma¨ler, K.-G. (1989), ‘‘The acid rain game’’, in H. Folmer and E. van Ierland (eds), Valuation Methods and Policy Making in Environmental Economics, 231–252, Elsevier, Amsterdam.
Chapter 5 AN ECONOMIC MODEL OF INTERNATIONAL NEGOTIATIONS RELATING TO TRANSFRONTIER POLLUTION
Henry Tulkens In Communication and Control in Society, K. Krippendorff (ed.), 1979, New York: Gordon and Breach, Chapter 16, 199–212.
This paper formulates a mathematical model of negotiations taking place between delegates of countries whose citizens share a common resource, viz. the ‘‘Southern Bight’’ (i.e. the southern part of the North Sea) and are concerned about its quality for various and possibly conflicting reasons. The negotiations process, proved to converge to a Pareto optimum of the international economy, is formulated in terms of gradual domestic abatements of pollutants, with a hydrodynamic and ecological model called upon for expressing the effects of these abatements on the quality of the sea. In addition, compensatory payments among countries are defined which are shown to cover all abatement costs, to break even and to be such that for each country individually, benefits are larger than the net costs they incur. The concluding section raises the free riding issue in this context, and considers the relation between reaching an optimum in this way and the ‘‘polluters pay principle’’.
1.
Motivation and plan of the paper1
Let us briefly describe the problem: in each of the countries involved (U.K., France, Belgium, Netherlands, Germany, Denmark and Norway) 1 This is a revised version of a paper presented at the Conference on ‘‘Communications and Control in Social Processes’’, sponsored by the American Society for Cybernetics and the University of Pennsylvania, Philadelphia, October 31 – November 2, 1974, and based on reports CB1/5 and CB1/7 written under contract with the Belgian Ministry of Scientific Policy under its ‘‘Programme National de Recherche et de De´veloppement sur l’Environnement’’. This revision has benefitted from support of the Ford Foundation (Program of Research on International Economic Order). Thanks are due to Professors Theodore Bergstrom, Franc¸ois de Donnea, Jacques Nihoul, Robert Rosenthal, Yves Smeers, Robert Wilson, and especially Maurice Marchand.
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various industries and municipalities use the North Sea for discharging residuals, both off-shore and near their national coasts; such discharges have increased considerably over the last decades. Today, however, ecologists warn that the amount of discharges in that region has reached a point which is beyond that of the so-called assimilative capacity of the waters; even the ecologically unsophisticated vacationers along the coasts complain increasingly about the quality of the North Sea waters. Thus, if we approach the situation from the individualistic viewpoint of each of the seven countries involved, it appears that each of them both benefits and suffers from the prevailing state of affairs: the discharges of residuals take place in the sea, quite probably because this is a (nationally) cheaper way to dispose of them than any kind of in-land treatment; on the other hand, each country has to put up with the nuisance effects of the aggregate amount of the discharges, mainly from the scenic and health points of view. Of course, within each country, beneficiaries of discharges and sufferers from pollution are not necessarily the same persons; but this is only a side issue in the present investigation. More important is the fact that the relative advantages from discharging on the one hand, and disadvantages from pollution, on the other hand, vary considerably from one country to another (e.g. discharges from the industrial area of Dunquerque are carried northeastwards by marine currents, and therefore affect French waters only for a short time, while they affect Belgium and the Netherlands more permanently). The paper addresses itself to the following question: if it is felt (or shown) that the water quality in the Southern Bight is nonoptimal (in a sense to be made precise below), can a collective decision mechanism be defined that would (a) determine changes in the countries’ discharges at sea such that the sea’s quality level be brought to an ‘‘optimal’’ level; and (b) be specified in such a way that for each country taken individually, benefits from improvements in the quality of the sea would be larger (or at least equal to) the costs imputed to it by the decision mechanism? The propositions 1 and 2 proved below (Section 3) establish that the dynamic process presented in Section 2 of this paper indeed possesses these two properties. The process is formulated in terms of gradual reductions over time of the physical discharges that each country allows its national economic agents to make; a hydrodynamic and ecological model is then called upon for expressing the estimated effects on the quality of the sea of such gradual reductions; finally, compensatory transfer payments among countries are defined, which are supposed to accompany the discharge
Ch.5 An Economic Model of International Negotiations
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reductions; these are shown to cover the costs of the reductions, and at the same time to give the process property (b) defined above. In particular, if such compensatory transfer payments were managed by a specially designed international agency, it is shown that, under the proposed process, the budget of the agency would break even. In concluding Section 4, the question is approached (4.1) whether the model proposed here also enjoys another and much stronger property, namely assuming that each country has the choice between either participating, or not participating in the negotiation (as formalized here), does participation always benefit all participants more than nonparticipation? Some conditions for a positive answer are suggested, but the idea remains a conjectural one at this stage. Finally (4.2), the compensatory payments scheme implied by the process is confronted with the currently popular ‘‘polluters pay’’ principle.
2.
The basic model
2.1
Notation; agents and commodities; ecological and behavioral relations
Let i ¼ {iji ¼ 1, . . . , I} be the set of countries involved in the pollution problems of the Southern Bight, either as polluters, or as victims of its polluted waters, or both. Let b ¼ {bjb ¼ 1, . . . , B} denote the set of pollutants (i.e. material substances characterized by their physical and chemical properties) subject to negotiations between countries i (or a subset of the latter). Let d ¼ {djd ¼ 1, . . . , D} be the set of geographical points, inland or on the sea surface, where pollutant discharges or dumpings take (or may take) place; let finally l ¼ {‘j‘ ¼ 1, . . . , L} be the set of geographical points on the sea surface where water is sampled and its quality technically evaluated, to serve as a basis for sanitary and ecological policy decisions. All the above sets are assumed to be finite. Environmental variables (typically the amounts per unit of time of the various pollutants present in the waters of the Southern Bight) are written in vector form R ¼ (R1 , . . . , Ra , . . . , RA ), where a 2 b l ¼ {aja ¼ 1, 2, . . . , A; A ¼ BL} a; thus, each component Ra refers to an amount of some pollutant b observed at some sampling location ‘ in the sea waters. Similarly, discharge variables (i.e. amounts of each pollutant discharged by each country per unit of time) are written as vectors Si ¼ (Si1 , . . . , Sig , . . . , SiG ), i 2 i, where g 2 b d ¼ {gjg ¼ 1, 2, . . . , G; G ¼ BD} g. By convention, Ra % 0 and Sig % 0 for all a, i and g.
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We shall assume that the relations between environmental and discharge variables can be determined by a predictive model such as, e.g. the one called ‘‘Math. Modelsea’’ [see Nihoul (1971)], whose general form reads: with
Ra ¼ fa (S1 , . . . , SG ) a 2 a X Sig g 2 g: Sg ¼
(1) (2)
i2i
The functions fa , assumed nondecreasing in all their arguments, are called transfer functions. Only their first-order partial derivatives @fa =@Sg are supposed to be known in the negotiation model developed below. As far as the economic variables are concerned, we shall denote by two vectors xi and yi the quantities of all commodities respectively consumed and produced per unit of time in country i. For simplification purposes, we shall consider that each vector xi has only two components xi1 ^ 0, denoting a physical aggregate of all commodities consumed by households in country i, and xi2 % 0 denoting an aggregate of all factors of production supplied by households of country i (including, typically, labor). Similarly, let yi ¼ (yi1 , yi2 ) where yi1 ^ 0 denotes the aggregate (physical) output of commodities in country i and yi2 % 0 the aggregate factors used in production (including labor). Moreover we shall assume that outputs from production (yi1 ) can be traded among countries, whereas factors of production (yi2 ) cannot. This last qualification implies that xi2 ¼ yi2 for every country i. The reduction to R2 of the space of economic variables, together with the assumption of nontradability of commodity 2, amounts to exclude from the analysis a detailed account of international trade phenomena (except for the possibility of transfers of commodity 1 between countries), and especially of price effects resulting from anti-pollution decisions eventually reached by the countries involved. On the one hand, such exclusion allows a substantial expository simplification for the pollution negotiation process. On the other hand, the consideration of a model embodying international trade effects raises rather serious additional problems of both theoretical and applied character, which would deserve separate treatment. The presently limited model nevertheless keeps its relevance from the environmental point of view. With every country i, let us finally associate two basic concepts whose purpose is to describe economic behaviors, namely an (aggregate) preference function Wi (xi , R), defined on a consumption set xi r2þA , and assumed to be monotonously and strictly increasing in all its arguments, strictly quasi-concave, continuous and twice differentiable, and an (aggregate) production function Fi ( yi , Si ) ¼ 0, defined on a bounded production set yi r2þG , and assumed to be strictly increasing in all its arguments, convex, continuous and twice differentiable.
Ch.5 An Economic Model of International Negotiations
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Notice that each production set yi is determined only by the variables yi and Si , which are specific to country i; production conditions in i are thus assumed to be independent of the discharges Sj made by the other countries j 6¼ i, and of the quality parameters of the sea, Ra , a 2 A, as well. This assumption amounts to ignore in our model the inter-country externalities that arise at the production level; we concentrate only on the ‘‘producer-consumer’’ externality aspect of the problem. The reason for this limitation comes from the nonconvexity problems that may arise when production externalities are taken into account. The mathematical methodology that will be used below for converging to an optimum is no longer applicable in such cases, and other methods should be sought for. 2.2
States of the international economy and behavioral assumptions
Definition 1 The feasible states of the economic-ecological system constituted by the Southern Bight and the surrounding countries are P P the vectors {xi , yi , R, Si ; i 2 i} such that (1) and (2) hold, xi1 ¼ yi1 , i i and for every i, xi2 ¼ yi2 , (xi , R) 2 xi and (yi , Si ) 2 yi . Let V denote the set of feasible states. Definition 2 Optimal states of the system are the feasible states þ þ þ {xþ i , yi , R , Si ; i 2 i} such that there exists no alternative feasible 0 0 þ state {xi , yi , R0 , Si0 } for which Wi (x0i , R0 ) ^ Wi (xþ i , R ), i 2 i, with a strict inequality for at least one i. Using standard Lagrangean techniques, it can be shown that, given our assumptions (and ignoring corner solutions), necessary and sufficient conditions for a feasible state to be an optimum are: X X pja hag fig ¼ 0 i 2 i, g 2 g; (3) j2i a2a
pi2 fi2 ¼ 0
i 2 i,
(4)
where the notation is as follows: @Wi =@Ra def @Fi =@Sig ; fig ¼ ; @Wi =@xi1 @Fi =@yi1 def @Wi =@xi2 def @Fi =@yi2 def @fa pi2 ¼ ; fi2 ¼ ; and hag ¼ : @Wi =@xi1 @Fi =@yi1 @Sg def
pia ¼
The marginal rate of substitution pia measures the amount of commodity 1 (the numeraire) that country i is willing to contribute to obtain a unit reduction of pollutant a (at some location ‘); similarly, the
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Public Goods, Environmental Externalities and Fiscal Competition
marginal rate of transformation fig represents the cost for country i, measured in terms of foregone production of commodity 1, of a unit reduction of its discharges of pollutant g (at some location d). Thus by (3), an optimum is characterized by the fact that the cost of reducing any discharge Sig should be equal to the sum of what all countries are together willing to contribute to avoid the effects across the ecosystem of such discharge. The mere definition of an optimum does not imply that such a situation is likely to occur. On the contrary, the nature of pollution phenomena is precisely such that the individual behaviors of each economic agent (in our case, of each country i) lead the economic-ecological system to a state of affairs which is not an optimum in the sense defined above. This is why current concern about ‘‘transfrontier’’ pollution is so relevant, and why the latter deserves special corrective action, organized through specifically designed cooperation schemes. In order to make this claim more precise, we need to state explicitly at this point how one assumes that each country ‘‘behaves’’ individually, or, in other words, we need to specify the economic (xi and yi ) and environmental (Si , R) choices that each country is likely to make in absence of environmental cooperation with the other countries. This we shall do by means of the following concept and assumption: Definition 3 An ‘‘environmentally nationalistic equilibrium (E.N.E.) for j , j 2 i, j 6¼ i, of the country i, relative to given environmental behaviors S ~i ~i , ~yi , S other countries’’ is a set of economic and environmental choices x ~ ~ ~ and R such that (~ xi , ~ yi , Si , R) maximizes Wi (xi , R) on xi subject to y ; (ii) F (y , S ) ¼ 0; and (iii) Ra ¼ fa (S1 , . . . ,SG ), a 2 a, where (i) xi ¼ i i i i P Sg ¼ S þ S , g 2 g. jg ig j2i j6¼i
The justification for the term ‘‘equilibrium’’ lies, as usual in economics, in the notion of (constrained) individual maximizing behavior implied by this concept. On the other hand, the reason for the adjectives ‘‘environmentally nationalistic’’ is best seen by noting the following property of such an equilibrium [which is easily derived from the first order conditions for a maximum of Wi ( )]: X pia hag fig ¼ 0 g 2 g (5) a
Thus, at an E.N.E., each country i makes choices such that its own cost of unit reduction in discharges Sig is equal to what itself is willing to contribute to avoid the effects of those discharges on the ecosystem; in other words, no account is taken by country i of the effects of its discharges on other countries j 2 i, j 6¼ i.
Ch.5 An Economic Model of International Negotiations
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Behavioral assumption 1 In absence of international agreement on pollution, each country is supposed to make its own economic and environmental choices in such a way as to find itself at an E.N.E., taking as given the behavior of the other countries. If we now apply this assumption to all countries, a precise description of the economic-environmental situation prevailing in the Southern Bight and in the surrounding countries when no cooperation is taking place among the latter, is provided by the following concept: Definition 4 A ‘‘noncooperative equilibrium’’ (N.C.E.) of the system is a ~i , R ~ } for set of national economic and environmental choices {~ xi , ~yi , S ~1, . . . , S ~i , . . . , S ~ G ), a 2 a, and (ii) ~ a ¼ fa (S every i 2 i, such that (i) R every country i is at an ‘‘environmentally nationalistic equilibrium relative to the behaviors of the other countries’’. It is easy to see that a N.C.E. is a feasible state, and that (5) and (4) hold for every i at such an equilibrium. Comparing this with (3), it is now clear that actions taken in isolation by the various countries lead the system to a state that cannot be an optimum. This is so perfectly in agreement with common sense that it might seem trivial; what is less so is that we now have an explicit description of the nonoptimal outcome of the absence of international cooperation (and thus a well-defined starting point for an analysis of negotiations), as well as a precise expression for the discrepancy between this starting point and the optimum we want to reach through negotiations. 2.3
Statement of a negotiation process
From now on, all variables in the model are functions of time, i.e. Ra (t), Sig (t), xih (t), yih (t); most often, however, the argument t is omitted. A dot on top of a variable denotes the operator d/dt. Our interest lies in analyzing the move of the economic-ecological system from a state of absence of cooperation to an optimum. Given behavioral assumption 1, it is natural to consider that, at time t ¼ 0, the system is at a N.C.E., to be denoted as {xi (0), yi (0), Si (0), R(0)}, i 2 i; these values of the variables are to be called later on ‘‘initial values of the solution’’. Now, the main effect of the negotiation process is to impose gradually on each country i various upper bounds on their allowed discharges Sig , and hence on the quality variables Ra of the ecosystem, say S ig (t) and Ra (t), for any t > 0. When this is the case, the countries can no longer be assumed to behave as defined by the E.N.E. concept. We need to characterize otherwise their remaining economic choices; to this effect we introduce:
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Public Goods, Environmental Externalities and Fiscal Competition
Definition 5 A ‘‘conditional equilibrium for country i’’ (i.e. conditional upon given environmental choices internationally made, and upon the economic behaviors of the other countries) is a set of economic ~ ~i , ~ ), x j1 , yj1 ( j 2P ~ yi , and a set of constants S j ( j 2 i i; j 6¼ i)Psuch choices x ~ ~ i1 ; ~i , ~ that (x yi ) maximizes Wi (xi , R) subject to (i) xi1 yi1 ¼ yi1 x j6¼i
j6¼i
i ) ¼ 0; and (iii) fa (S 1, . . . , S i , S j , . . . , S I ) R a ¼ 0, a 2 a. (ii) Fi ( yi , S Such a conditional equilibrium is characterized by the conditions pi2 fi2 ¼ 0, i 2 i. Accordingly, we also introduce: Behavioral assumption 2 Every country i participating in the negotiation process defined below is assumed to make its economic choices in such a way as to find itself, at all t, at a conditional equilibrium in the sense of definition 5. Consider now the following system of differential equations2: X X a _Sig ¼ a (6) pja hg fig i 2 i, g 2 g, R_ a ¼
X
j
a
X
S_ ig
a2a
(7)
i2i X X XX 2 pia R_ a þ fig S_ ig þ di a S_ jg x_ i1 ¼ y_ i1
(8)
hag
g
i
x_ i2 ¼ y_ i2
a
g
j2i
i 2 i,
(9)
g
where a is an arbitrary positive scalar (assumed ¼ 1 below, and henceforth ignored without loss of generality) and X 0 < di < 1, di ¼ 1: (10) i
A solution for this system consists of a vector-valued function Z: [0, þ 1] ! V; t ! Z[t; Z(0)], whose derivative with respect to time exists and is equal to (6)–(9), and defined by Z[t; Z(0)] ¼ {S1 [Z(0)] , . . . , SI [Z(0)] ; R1 [Z(0)] , . . . , RA [Z(0)] ; x1 [Z(0)] , . . . , xI [Z(0)] ; y1 [Z(0)] , . . . , yI [Z(0)] } where Z(0) ¼ [S1 (0), . . . , SI (0); R1 (0), . . . , RA (0); x1 (0), . . . , xI (0); y1 (0), . . . , yI (0)] is the initial value of the solution. We shall assume, by conven2
The reader who is familiar with the theory of public goods will have recognized that we are dealing here with an economy with ‘‘international public goods’’; the present dynamic process is an adaptation to our environmental problem of similar procedures recently proposed in Dre`ze and de la Valle´e Poussin (1971) and Malinvaud (1972).
Ch.5 An Economic Model of International Negotiations
115
tion, that such initial value is a noncooperative equilibrium of the economic-ecological system, in the sense of definition 4. Equilibrium values of the solution are values of the variables Si , R, xi and yi , 8i 2 i at some time tþ such that the right-hand side of the system (6)–(9) is equal to zero. The system is said to converge if there exists a solution for the equations such that for t ! 1 the values of the variables tend to equilibrium values of the solution. It is easily seen that equilibrium values of a solution constitute an optimum in the sense defined in Section II above, since the necessary and sufficient conditions (3) are met for such values. Thus, if the process so defined converges at all, it converges to an optimum. 2.4
Interpretation
The interpretation of the system (6)–(9) as a ‘‘negotiation process’’ among countries concerning their discharges Sig goes as follows: each country is supposed to be represented by one delegate, who essentially determines at every t with his colleagues the magnitudes S_ ig (t) and x_ i1 (t) y_ i1 (t), according to their respective definitions in (6) and (9); this may be seen as taking place in a sequence of steps: (i) each delegate makes known to the conference the coefficients pia (t) and fig (t), as they are evaluated within his own country i; (ii) ecological experts provide the conference with the matrix of transfer coefficients hag (t); (iii) then, reductions of each country’s discharges of the various pollutants b at all locations d, S_ ig (t), are specified according to formula (6), and their resulting effects across the entire ecosystem are estimated as magnitudes R_ a (t), computed according to formula (7); (iv) finally, net transfers x_ i1 (t) y_ i1 (t) of commodity 1, as specified by equation (9), are collected from – or distributed to – each of the I countries. To be more specific, let us imagine that an international agency has been set up by the I countries, prior to the negotiation, which is in charge of managing these transfers as they are decided by the negotiators, as the process develops over time. The net transfer of each country i may then be seen as the algebraic sum of three successive payments (for clarity, we consider here the case where S_ ig and R_ a > 0 for all a’s and g’s; however, since the model does not imply any sign restriction on these variables, the payments described _ _ below are reversed when P S and R are negative): first, country i pays to the agency an amount pia R_ a < 0, i.e. a payment based on what country i a
has announced it is willing to contribute (pia ) per unit change in each of the Ra ’s times the actual changes R_ a ; second, the agency pays to country
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Public Goods, Environmental Externalities and Fiscal Competition
P
fig S_ ig > 0, i.e. the cost to that country of reducing its discharges by the specified amounts S_ ig , given that it has announced of them; third, the agency pays to country i a unit cost figPfor P each 2 an amount di > 0, which can be shown (see Proposition 1(a) S_ jg i an amount
g
j2i g
in Section 3) to be a fraction of the surplus that would remain in the hands of the agency if only the first two payments just mentioned were made. For each i, the algebraic sum of these payments may be positive, negative, or zero, depending upon the relative magnitudes of the coefficients pia and fig , on the one hand, and of the variables S_ ig and R_ a on the other. In the aggregate however, the sum of all payments and receipts made by the agency can be shown to break even (see proposition 1(b) in Section 3). Thus, the negotiation process appears to be driven by repeated cost-benefit analyses, which determine the direction of decisions at each step. The key information on which it rests is the set of coefficients pia , fig and hag , all duly updated as the process develops.
3.
Properties of the process
3.1
Individual rationality and feasibility
_ Proposition 1 (a) P Along the Pprocess (6)–(9), Wi > 0 for every i 2 i; (b) x_ i1 ¼ y_ i1 for all t ^ 0. i
i
Proof:
(a) Along the negotiation process, one has at every t, and for every l: X @Wi _ i ¼ @Wi x_ i1 þ @Wi x_ i2 þ W R_ a @xi1 @xi2 @R a a X @Wi ¼ [x_ i1 þ pi2 x_ i2 þ pia R_ a ]: @xi1 a
Replacing x_ i1 in this expression by its value in (9) leads to: X XX 2 _ i ¼ @Wi [ y_ i1 þ fig S_ ig þ pi2 x_ i2 þ di ]; W S_ jg @xi1 g g j in addition, by (8), x_ i2 ¼ y_ i2 along the process, and pi2 ¼ fi2 in view of behavioral assumption 2; hence, X XX 2 _ i ¼ @Wi [ y_ i1 þ fig S_ ig þ fi2 y_ i2 þ di ]: (11) S_ jg W @xi1 g g j Now, differentiating the production function Fi ( ) ¼ 0 yields
Ch.5 An Economic Model of International Negotiations
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X @Fi @Fi @Fi y_ i1 þ y_ i2 þ S_ ig ¼ 0, @yi1 @yi2 @Si g or, since @Fi =@yi1 > 0, X fig S_ ig : y_ i1 ¼ fi2 y_ i2 g
Replacing y_ i1 in (11) by this value gives @Wi X X _ 2 [di Sjg ] > 0, @xi1 g j which is positive since Wi is a strictly increasing function of all its arguments, di > 0, and all terms within brackets are positive squares. (b) Starting from equation (9) of the process, we have " !# X X X X X XX 2 : fig S_ ig pia R_ a þ di S_ x_ i1 ¼ y_ i1 þ ig
i
i
g
i
a
g
i
We have to show that the expression within brackets is equal to zero, or equivalently that ! XX X X X fig S_ ig pia R_ a ¼ S_ 2 , ig
i
g
a
g
i
P where use has been made of the fact that di ¼ 1. Using now (7) we have i " !# X X X X X XX fig S_ ig pia ha S_ 2 ¼ S_ jg ig
i
and
XX i
g
g
i 2 S_ ig
¼
g
"
a
X X " X X
fig S_ ig
i
g
S_ ig
" XX i
j
pia hag
X
a
XX g
XX
g
XX g
g
i
¼
fig S_ ig
g
i
¼
g
# S_ jg
j
S_ jg
j
pia hag
X a
#
pia hag #
fig ,
a
where the term within brackets equals S_ ig in view of (6). & The main implication of part (a) of this proposition is that all countries do benefit from the procedure, no matter whether they are polluting, polluted, or both. The relationship between this ‘‘individual rationality’’ property and the ‘‘polluters pay’’ principle is considered in
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_ i for Section 4 below. Note also that the importance of the ‘‘benefit’’ W each country i is proportional toPthe di which determines the Pcoefficient 2 fraction it receives of the surplus distributed by the agency as the S_ jg j
g
‘‘third’’ payment mentioned above. On the other hand, together with equation (8) of the process, part (b) of this proposition shows that along the procedure, the economic-ecological system is led through a sequence of feasible states (in the sense of definition 1 above), from the initial N.C.E. to the final optimum. In our terminology, a deficit or a surplus of the agency managing the transfers would imply that the negotiation process leads the system into an unfeasible state, as it happens to be the case with other schemes proposed in the literature [see Smets (1973)]. 3.2
Existence of a solution and convergence to an optimum
Assumption 3 The system (6)–(9) is Lipschitzian. While this could be proved as a lemma, given our previous assumptions on the functions Wi and Fi , on the sets xi and yi and given Proposition 1, we state it here as an assumption in order to avoid lengthy technical developments in this essentially expository paper. For an explicit treatment of the Lipschitzian property of processes of this type, see Tulkens and Zamir (1978, Section IV). Proposition 2 Under assumption 3 and those stated earlier on the functions Wi , Fi and fa , the process (6)–(9) has a unique solution which is globally stable; it converges to an optimum. Proof: (1) We use first the quasi-stability theory of Uzawa (1961), pp. 619– 620: (a) In view of assumption 3, existence, uniqueness and continuity of a solution Z [t; Z(0)] is guaranteed; (b) moreover, the solution is contained in the compact set of feasible states (see definition 1); (c) a ‘‘modified P Lyapunov function’’ is provided by W (t) ¼ i Wi (t), which is shown to be monotonous and nonincreasing by proposition 1. Hence, by theorem 1 of Uzawa (1961) the process (6)–(9) is quasi-stable. (2) Every limit point of the process is an optimum, since it verifies conditions (3). (3) Given continuity and monotonicity of Wi , and compactness of the set of feasible states, one has that 8i, lim Wi [z (tv ); z(0)] ¼ Wiþ v!1
is unique. This in turn implies that the limit point zþ (an optimum in view of (2) above) is also unique: indeed, if there were two limit points, say zþ and zþþ , strict quasi-concavity of Wi would imply, for every i,
Ch.5 An Economic Model of International Negotiations
119
Wi [(zþ þ zþþ )=2] > Wi (zþ ) ¼ Wi (zþþ ), contradicting the optimality of zþ and zþþ . Hence, the set of equilibrium values of the solution is finite, implying equivalence between quasi-stability and stability (see Uzawa (1961) p. 619). &
4.
Concluding remarks
4.1
Is it worth participating in such a process for every country?
If for some reason some country j decided not to join in the negotiation, knowing that the other ones will nevertheless proceed according to the above model, would such an attitude be beneficial or detrimental to such country? The answer is easily provided by computing ^_ ¼ @Wj x_ þ @Wj x_ þ X @Wj R ^_ , W j j1 j2 a @xj1 @xj2 @R a a where, under such ‘‘isolationist’’ circumstances, 0 1 X X B C ^_ ¼ hag @ R S_ ig þ S_ jg A, x_ jh ¼ y_ jh , h ¼ 1, 2, a g
(12)
i2i i6¼j
and X @Fj @Fj @Fj y_ j1 þ y_ j2 þ S_ jg ¼ 0: @yj1 @yj2 @Sjg g As in part (a) of proposition 1 above, we may rewrite 2 0 13 ^_ ¼ @Wj 6x_ þ p x_ þ X p X ha BX S_ þ S_ C7, W 4 j1 j j2 j2 ja ig jg A5 g@ @xj1 i2i a g i6¼j
and derive, from the differentiation of Fj ( _ ) ¼ 0, X fjg S_ jg : y_ j1 ¼ fj2 y_ j2 g
Using (12) and recalling the first order conditions of an E.N.E. for j, ^_ reduces to the expression for W j " # X XX @W j ^ _ j¼ (13) pja hag S_ ig ; in{ j} , W @xj1 a g i6¼j
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where S_ ig ;
in{ j}
¼
X
pia hag fig :
i2i i6¼j
If the factor within brackets in (13) is strictly positive, i.e. if the ^_ is positive. Clearly other countries do take anti-pollution decisions, W j there is thus an incentive for (any) country j not to participate in a transfrontier pollution agreement, as soon as the other ones start doing something. This is not a phenomenon genuine to the process developed above: rather, it is an intrinsic characteristic of the public good character of our transfrontier pollution problem. It has been shown above, however, that if country j were to partici_ j would depend upon the value of pate in the process, its benefit W coefficient dj assigned to it for distributing the surplus emerging from the transfer payments scheme. Thus, the question whether for a given country j it is worth participating in the above negotiation process boils down to the following: is the surplus large enough to be shared among participants in such a way that each of them benefits more from participating than from not participating? Formally, the problem is to find values dþ j such that for every i 2 i, XX ^_ ¼ @Wj X p X X ha S_ ; 2 _ j ¼ @Wj dþ W $W S_ ig j ja j g ig in{ j} : @xj1 @xj1 a i2i g i6¼j g
(14)
Preliminary investigation of this problem by means of game theoretic tools leads to the conclusion P þ that there may not exist, in general, such values dþ satisfying also j j dj ¼ 1 (feasibility). In special cases, however (which depend upon the structure of the preference functions Wi ) they may exist. On the other hand, if (14) cannot be satisfied, one may want to ^_ _ look for dþþ j ’s which would minimize W j Wj , i.e. the incentive not to participate. 4.2
Do the ‘‘Polluters Pay’’ under this Process?
It is currently impossible to propose any decision process in the pollution field without taking a stand with respect to the so-called ‘‘polluters pay’’ principle, which has much popularity in both official and unofficial circles. Clearly, the present process does not derive from that rule; on the contrary one could say that its characteristic is that, to every polluter (viz. each country i) all countries (including i itself) do pay a transfer the total of which exceeds the cost of reducing its discharges. On
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the other hand, there is also the important feature that all countries do benefit from the outcome. Essentially, there appears to be here a shift in emphasis, from the question of ‘‘who should pay’’? to that of ‘‘is it worthwhile for everybody to combat pollution’’? A rather strong rationale can be called upon for such a shift. Indeed, the essence of the ‘‘principle’’ is that it is a normative one, whose application requires an authority empowered to impose burdens on some economic agents, without (or with only little) compensa_ i < 0). Now, at the international level such tion (thus, actions such that W an authority is non existent; and it seems quite unlikely that any country would voluntarily sign an agreement establishing such an authority. Therefore, if the ‘‘principle’’ may deserve consideration at the national level (for an analysis of its various forms, see Smets (1973)), it cannot have much relevance as regards the substance of international environmental decisions; here, the only policy proposals susceptible of entailing voluntary agreement seem to be those which are shown to be clearly beneficial to each party taken individually. Thus, if the objective of the analysis (and of the negotiations) is to help reaching an optimum, agreements motivated by and based on the self-interest of the parties involved are probably – but also alas! – a stronger means to get things done, than quoting ‘‘principles’’, no matter how attractive they may be from other viewpoints.
References Dre`ze, J. and de la Valle´e Poussin, D., 1971. A taˆtonnement process for public goods, Review of Economic Studies, 38, 133–150. Malinvaud, E., 1972. Prices for individual consumption, quantity indicators for collective consumption, Review of Economic Studies, 39, 385–406. Nihoul, J. C. J., 1971. Mathematical model of continental seas: preliminary results concerning the Southern Bight. Liege: Institut de Mathe´matique, Universite´ de Lie`ge, mimeo. Smets, H., 1973. Le principe de la compensation re´ciproque: un instrument e´conomique pour la solution de certains proble`mes de pollution transfrontie`re. Paris: Note du Secre´tariat, Comite´ de l’Environnement, O.C.D.E., September, 1973. Tulkens, H. and Schoumaker, F., 1975. Stability analysis of an effluent charge and the ‘polluters pay’ principle, Journal of Public Economics, 4, 245–269. Tulkens, H. and Zamir, S., 1979. Surplus-sharing local games in dynamic exchange processes, Review of Economic Studies, XLVI (2) (Symposium on Incentive Compatibility), 305–314. (Chapter 2 in this volume). Uzawa, M., 1961. The stability of dynamic processes, Econometrica, 29, 617–631.
Chapter 6 THEORETICAL FOUNDATIONS OF NEGOTIATIONS AND COST SHARING IN TRANSFRONTIER POLLUTION PROBLEMS
Parkash Chander, Henry Tulkens European Economic Review, 36 (2/3), 388–399 (1992).
In this paper, a mathematical model is formulated of negotiations taking place between agents who use a common resource, and are also concerned with the quality of the latter, for various (and possibly conflicting) reasons. The model makes explicit an ‘‘ecological surplus’’ that summarizes the benefits from cooperation among the agents in situations of this type. The negotiations model is described by a dynamic process bearing on a multilateral externality, that converges to an individually rational Pareto optimum. The process embodies a cost sharing rule for pollutant abatement in the countries involved, derived from a sharing rule of the surplus, that is also shown to have ‘‘strategic stability’’ in the game theoretic sense of an imputation in the core of some cooperative game associated with the negotiation process. The connections are explored between this cost-sharing rule and the ‘‘free rider problem’’ in public goods theory.
1.
Introduction: The general setting
From the point of view of resource allocation analysis, the economics of transfrontier pollution problems belongs to the theory of voluntary provision of public goods. The public goods component of the problem stems from the fact that ambient pollution, in general, has all the classical characteristics of a public good (perhaps more appropriately called a public ‘bad’) such as non rivalry in consumption and non excludability.1 1
The issue may alternatively be said to be one of ‘voluntary internalization of a multirecipient externality’. Because of the heaviness of this terminology, we tend to avoid it, without ignoring, however, the valid distinctions that have sometimes to be made between externalities and public goods. But this is not the place to enter into these.
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As to the ‘voluntary provision’ aspect, it refers to the decision process to be considered, which is characterized by the fact that no single public authority can be assumed to exist at a supranational level, endowed with sufficient powers to enforce by authority a program of production of public goods (or for that matter of reduction of public bads). The independence of national sovereignties is an essential component of the issues involved. The post-1960 economic theories of public goods (and externalities), with the help of parallel developments in the theories of resource allocation processes, as well as of both cooperative and noncooperative games, have gradually provided a rich and well structured logical framework for dealing with the basic issues involved. What are these issues? On the one hand, the ignorance of social costs imposed on others by polluting behaviors is an expression of noncooperation in individual maximization; on the other hand, dissatisfaction with the overall outcome of such behaviors prompts various forms of collective action to regulate them. Perhaps the deepest contribution that economics has brought to the analysis and understanding of both of these facts is to have noted the nonoptimality of the former, and to have proposed efficiency as an objective for the latter. One thus finds in many analyses [e.g. by Ma¨ler (1989, 1990), Hoel (1990), or Newbery (1990), to name only a few recent ones] a similar general framework of thought, which seems to be now well received, if not part of common wisdom in this area. It consists in (i) characterizing as noncooperative equilibria the laissez-faire situations, (ii) showing their inefficiency, and (iii) proposing courses of action that would achieve efficiency, that is, not necessarily zero amounts of pollution, but rather amounts that, taken together with the other characteristics of the economy, would satisfy the Pareto criterion. The purpose of the present paper is to present an explicit elaboration of these ideas in a simple setting, general enough to provide a unifying formulation for dealing with more specific issues such as the role of costbenefit analyses, the reasons for cooperation, the logic of cost sharing in pollution abatement, and the rationale for possible international transfers in cooperative agreements. Technicalities and limited writing space force us to deal only with flow-type of pollution and to ignore here stock-type pollutants, whose accumulation requires more complex modelling [see e.g. Ma¨ler (1990, Section IV(ii)) and again Hoel (1990)].
2.
A simple model of transfrontier pollution
Consider a set N of countries, indexed by i ¼ 1, 2, . . . , n, which share a common environmental resource such as e.g. a sea or a lake of
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which they all are riverains, or just the atmosphere above them. Let the function ui (xi , z), assumed to be increasing in both its arguments, denote country i’s preferences over the consumption of a single private good (xi ^ 0) and a single,2 quantitatively measurable ambient characteristic (z % 0)3 of the environmental resource. Define pi ¼ (@ui =@z=@ui =@xi ) ^ 0 as country i’s marginal willingness to pay (in commodity x) for an improvement in environmental quality. Let furthermore fi (yi , pi ) ¼ 0 be the same country’s production function, linking its output yi ^ 0 of the private good with its discharges pi ^ 0 of pollutant in the environment, and such that @fi =@yi > 0 and @fi =@pi % 0; labor, capital and the other inputs are taken as constant and subsumed in the functional symbol fi . Define g i ¼ (@fi =@pi = @fi =@yi ) ^ 0 to be the country’s marginal cost (in yi ) of reducing4 its discharges. In this setting, countries can interact in two ways: (i) through the private good, which we assume to be transferable between the countries in amounts Ti (< 0 if given away by country i; > 0 if received by it). Thus for every country i the feasible consumption is xi ¼ yi þ Ti , and taking all countries together we have the feasibility condition X X X X yi þ Ti , with Ti ¼ 0: (1a, b) xi ¼ (ii) Interactions also take place through the ‘transfer function’ z ¼ F ( p1 , . . . , pi , . . . , pn ) that specifies how the pollutant discharges of all countries are diffused and transformed by ecological processes into the ambient quantity z. For simplicity, we assume here the function F to be of the simple additive form X z¼ pi : (2) For this economic-ecological system – an expression we use to emphasize the presence and the role of the transfer function5 in this otherwise standard specification of an Arrow–Debreu economy with a 2 Extensions of this model to several (quantifiable) characteristics of the environment raises no particular problems. Such is not the case, however, with extending the number of private goods, as this would bring in all foreign trade activities and require modelling their adjustments to the environmental changes designed below. Attempts made in that direction will not be reported here. 3 The absence of subscript attached to this variable reflects its public good character; and with the convention of measuring the ambient characteristic in non-positive amounts, our assumption @ui =@z ^ 0 implies that z is felt by the consumers as a public bad. 4 The derivatives are taken here to the left. 5 This points to the fact that other forms of transfer functions may be imposed by ecological phenomena. In Tulkens (1979) functions that allow for the spatial distribution of discharge and ambient pollutants in sea waters are considered, prompted by a model pertaining to the North Sea.
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public good – a noncooperative equilibrium (NCE) of the Nash type is defined as a state of the system – i.e. a specification of the consumptions, productions and discharges of each country i ¼ 1, . . . , n, and of the resulting characteristic of the environment through (2) – such that each country maximizes its utility subject to its production function, with xi ¼ yi (implying Ti ¼ 0), and taking as given the other countries’ discharges pj ( j 6¼ i). At such an equilibrium, the first-order conditions of individual behaviors also imply: pi ¼ gi , i ¼ 1, 2, . . . , n:
(3)
By contrast, an optimum in the Pareto sense is defined in the usual way [with some appropriate lower (negative) bounds for the transfer variables Ti ]. Every optimum is characterized by Samuelson-type firstorder conditions of the form ! n X pN pj ¼ gi , i ¼ 1, 2, . . . , n: (4) j¼1
The noncooperative equilibrium is thus clearly not an optimum. However, only a subset of the Pareto optima are relevant for our purposes, since we limit our interest to those that can be achieved by voluntary cooperation: such optima are those unanimously preferred to the NCE, and called technically individually rational with respect to the NCE.
3.
Modelling negotiations
For implementing the idea of moving the economic-ecological system from a prevailing noncooperative equilibrium to a unanimously preferred optimum, one can conceive of two alternative methods: a ‘one shot’ (or ‘global’) method, and a ‘gradual’ (or ‘sequential’) one. With the one shot method, the preferred optimum is directly computed, in physical terms, from the system’s model. This requires a complete knowledge of all the preference, production (or cost) and transfer functions. The obtained numerical results are then to be directly implemented by appropriate command and/or incentive schemes. A good example of this, applied to acid rains problems in Europe, is provided in Ma¨ler’s (1990) paper [Section IV(i)]. With the sequential method, computations are needed too, which are simpler, but are also to be repeated over time. They bear at each step on small size changes in the economic-ecological variables of the system and are based only on local properties of the functions that related them. The changes are designed in such a way that repetition entails a sequence of states that converges to the desired optimum.
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The introduction of the gradual method in environmental economic issues6 was directly inspired from one of the models in the then newly developed ‘planning’ or ‘taˆtonnement’ approaches to public goods,7 which was itself just a branch of what Arrow and Hurwicz (1977) aptly call the theory of resource allocation processes. However, both terms of planning and taˆtonnement are more restrictive than necessary, because the model in question may just as well describe actions taken spontaneously by the agents themselves, without a planner or an auctioneer. This is why this model may also serve as a framework for analyzing negotiations conducted on a voluntary basis.8 This point seems to have remained unnoticed in subsequent contributions on the voluntary provision of public goods such as e.g. Cornes and Sandler (1986) [except, to some extent, for Mueller (1976)]. 3.1
Statement of the process
Applying to the simple economic-ecological system described above the resource allocation process used in Tulkens (1979) in a more complex model yields the following system of differential equations (all variables are henceforth considered as functions of time): p_ i ¼ (pN gi ), X z_ ¼ p_ i ,
i ¼ 1, . . . , n
y_ i ¼ g i p_ i ,
i ¼ 1, . . . , n
(7)
x_ i ¼ y_ i þ T_ i ,
i ¼ 1, . . . , n
(8)
(pN gj )2 , i ¼ 1, . . . , n
(9)
(5) (6)
T_ i ¼ g i p_ i pi z_ þ di
n X j¼1
where 0 % di % 1 8i and
n X
di ¼ 1,
(10)
i¼1
with the noncooperative equilibrium as initial conditions.
6 In Tulkens and Schoumaker (1975) for a (domestic) effluent charge and in Tulkens (1979) for international negotiations. 7 See Dre`ze and de la Valle´e Poussin (1971) and Malinvaud (1970 –1971). See also Chander and Parikh (1990) for a recent survey of the planning branch of that literature. 8 In fact, the model is more akin to the non-taˆtonnement theory of price formation, and in the spirit of some parts of Buchanan (1968), as explained in Tulkens (1978, Section 4).
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Public Goods, Environmental Externalities and Fiscal Competition
Main formal properties
The theory of resource allocation processes provides formal proofs of the existence of a solution to this system of equations as well as of its convergence to a Pareto optimum, unanimously preferred to the noncooperative equilibrium. The driving force of the process (5)–(10) lies in eq. (5), that specifies for each country a direction and a size of change for its pollutant discharges, the changes being different from zero as long as the state of the system does not satisfy the n conditions (4), i.e. is not a Pareto optimal one. For their part, eqs. (6), (7) and (8) respectively ensure ecological, technological and consumption feasibility of the reallocations of commodities specified by the process. Finally, eq. (9) specifies net transfers to or from each country, whose components can formally be shown to play the following roles: the first two terms are such that if the transfer were limited to their sum, for some i, then the process would induce u_ i ¼ 0 for this i along the solution; by contrast the third term, which is always nonnegative, ensures u_ i ^ 0 for each i and even u_ i > 0 if di > 0. It is this property of the process, called individual monotonicity, that guarantees unanimous preference of its convergence point vis-a`-vis the initial noncooperative equilibrium. P The sum of squares j (pN gj )2 appearing in this third term plays a crucial role. It is similar to the ‘gain from trade’ that characterizes voluntary exchange processes of private goods, and may be called in the present context the ‘ecological surplus’ emerging from the Pareto improving character of the reallocations specified by the process. As suggested by (10), the constant parameters di , i ¼ 1, 2, . . . , n determine the proportions in which that surplus is shared between the countries i. 3.3
Interpretation
The interpretation of this process as one of the negotiations goes as follows: (the description made below bears on one step of the process; it is formulated as if the process was expressed in discrete time, rather than in continuous time): (i) At every point in time where a negotiations round takes place, the delegate of each country i is supposed to provide two numerical informations: (18) the amount of his country’s willingness pi (t) to contribute for a marginal improvement in the prevailing quality z(t) of the environment, and (28) the marginal cost gi (t) of reducing the discharges pi (t) currently made by his own country. (ii) A change of each country’s discharges is then computed according to formula (5), which amounts to a benefit minus cost calculation on the
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basis of the just provided information. While the direction of such changes is thereby unambiguously determined, their magnitude requires further precisions, considered below. (iii) The sum of such changes then determines the resulting variation in environmental quality, as stated by (7). This need not be calculated – except for the sake of information – as it corresponds to spontaneous ecological adjustments to (5). (iv) Adjustments in domestic productions of private goods are similarly assumed to take place according to (8); they need not be calculated either, unless again for information, as these are supposed to describe the spontaneous adjustments of the economies to the decisions (5). (v) Finally, the net transfers T_ i specified by eq. (9) are collected from, or distributed to each of the n countries. For performing these transfer operations, one might think of an international clearing house operating as follows, on the basis of the three terms that make up expression (9): first, a payment is made to country i of an amount gi p_ i that covers the cost of abatement of its own discharges (or, an equivalent payment of its cost savings is made by the country if p_ i happens to be positive); next, an amount pi z_ is paid to or received from the fund by country i (depending upon the sign of z_ ), proportional to the change z_ in the quality of the environment; and finally, a payment is made to the country of a fraction di of the ‘ecological surplus’ mentioned earlier (this surplus being automatically available to the fund as soon as all countries do make and/or receive the payments specified by the first two elements just described).9 After all these ecological as well as economic adjustments have taken place, another step comprising these five stages may be taken. 3.4
Further remarks
Each step, or negotiations round, described above achieves Pareto improving changes (upon the prevailing situation), rather than a full optimum. This is to be seen as a virtue rather than a limitation, since the only ‘local’ information used (which often is also the only one available) may not be sufficient for computing where an optimum lies. However, with successive rounds, the system gets arbitrarily close to an optimum. As to the actual size of each step, and for each country, these are in fact infinitesimal ones with the above continuous time formulation – which leaves some arbitrariness. Only an explicit discrete time version of 9
With the surplus sharing rule that will be advocated in Section IV below, and that leads to the simpler expression (12) for the transfers, it will be seen that the second item disappears in the above description, and the third one is replaced by a share in the aggregate cost of pollution abatement of all countries.
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(5)–(9) can permit a rigorous discussion of this issue, including related ones such as whether the abatements p_ i should be uniform or differentiated ones [as analyzed in Newbery (1990)]. Next, the changes in (5) are typically negative ones, at least at a NCE (because in general pi ¼ gi there, implying thus pN > gi ). However they need not always be so along a solution of the process: this all depends upon the evolution of the marginal costs gi as well as of the aggregate preferences pN along this solution. For this reason at least, one may conjecture that the Newbery thesis of non-desirability of uniform reductions might well be confirmed within the present model. As to the international transfers, finally, one might wonder whether and why they are at all necessary. On this issue the model (5)–(9) can easily be used to show that their necessity is essentially related with the requirement of individual rationality for all parties involved, i.e. u_ i > 0 for every i, rather than with the requirement of optimality. For this reason, constraining T_ i to remain zero in the system (5)–(9) – which amounts to each country bearing its own abatement cost – or simply ignoring the possibility of transfers in the analysis may not prevent convergence to some optimum,10 but this optimum would not be, in general, a unanimously preferred one to the NCE. Note on this issue that while it is in general difficult, if not impossible, to say anything about the sign and the global amount of the individual transfers between the NCE and a unanimously preferred optimum, formal conditions can be stated on this sign for the local transfers specified by (9) as well as by (12) below.
4.
Cost sharing and strategic considerations
While the process stated above clearly identifies, with the ecological surplus, what are the gains to be obtained from cooperation, its formulation does not provide any justification for the proportions d ¼ (d1 , . . . , di , . . . , dn ) according to which this surplus is to be shared, nor does it make very clear how the costs of pollution abatement are actually borne by the countries involved. We now report on some recent results on these issues. As to the cost burden, if it is considered in terms of foregone output y_ i of private good, eqs. (7) state what is borne by each country. However, the foregone consumptions x_ i , as stated by eqs. (8), may be different, due to the transfers. Since these are determined in part by the share of the ecological surplus accruing to each country, the choice of a distribution 10
Of the type of point T in case no. 5 analyzed on pp. 258–259 in Tulkens and Schoumaker (1975).
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profile for the surplus also in fact determines the importance of the total cost burden borne by each country. 4.1
A proposal
In Chander and Tulkens (1991), it is proposed to adopt the distribution profile d defined by di (t) ¼ pi (t)=pN (t),
i ¼ 1, . . . , n:
(11)
The authors show that this choice gives the transfers (9), after some algebraic manipulations, the form n pi (t) X g p_ j , T_ i ¼ g i p_ i þ pN (t) j¼1 j
i ¼ 1, . . . , n:
(12)
Using this as well as (7) then yields for (8): x_ i ¼
n pi (t) X g p_ j , pN (t) j¼1 j
i ¼ 1, . . . , n,
(13)
an expression that clearly shows that under the surplus-sharing rule (11) the consumption cost of the total pollution abatement is borne by each country in a proportion equal to the intensity of its own preferences for environmental quality, pi , relative to the intensity of the total such preferences, pN , of the countries involved. Notice from (12) that this burden is incurred by each country after compensation is made for the cost of its own abatement. Thus, to a country that does not care about the environment (alleging pi ¼ 0), no share of the total cost is imposed, and the cost of its own abatement is compensated (as just mentioned); but no share of the surplus is attributed either, implying u_ i ¼ 0. By contrast, for every country that does care (revealing pi > 0), the cost of its own abatement is similarly compensated, but a share of the total cost is imposed on it; as this share turns out to correspond to an amount that it is less than what this country reveals it is willing to pay for total abatement, the scheme implies11 u_ i > 0 unless pi ¼ 0. 4.2
Cooperative game-theoretic justification
Beyond these intuitive considerations, the sharing rule (11) may be given a stronger justification by showing that it actually derives from a 11
Formally, from (5) – (8) and (12), which now describe the process, it can be seen that the expression x_ i þ pi z_ is non-negative, which establishes the claim.
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game-theoretic analysis of some form of free riding, formulated after considering alternative ways to participate in the negotiations. Various attitudes of the individual countries, when faced with the procedure described in the previous section, can indeed be distinguished: (i) some country, or several of them, may choose to provide incorrect information on its (resp. their) preferences pi and/or cost gi , so as to induce the process to operate more favorably for itself (resp. themselves); (ii) some or several countries may consider that, in spite of their individual utility increases, they do not receive a share of the surplus that is large enough and decide, as a result, not to participate any more and to take another course of action; (iii) finally some or several countries, while accepting the procedure and its sharing rule, may wish that the direction of change of the pollutant discharges, p_ i , be more directly determined by their own interest. For analyzing these behaviors in the framework of the dynamic process stated above, a concept of ‘local games’ was introduced in the seventies, that led to results of the following nature. On the first type of attitude, that gives rise to a form of free riding that bears on information revelation on preferences and/or cost, Yen (1987), extending Roberts (1979) as far as costs are concerned, shows that information corresponding to Nash equilibria of ‘preference and cost revelation’ local (noncooperative) games is sufficient for convergence to a Pareto optimum, in spite of the fact, surprisingly enough, that it is not correct information in general. As to the second type of attitude, no result was obtained, at least to our knowledge, due to formal difficulties in specifying appropriately, for each ‘alternative course of action’ chosen any coalition of countries, the actions chosen by the countries not in the coalition. The linear structure of the strategy spaces on which the local games are defined raises unboundedness problems that are unresolved so far. In the more restricted framework of the assumed third attitude, ‘surplus-sharing’ cooperative local games of a type formulated in Chander (1987) for general public goods economies are shown there to have at least one imputation that belongs to the core of such games. In Chander and Tulkens (1991–2001, Proposition 1), it is further shown that when applied to the present dynamic process, this core imputation precisely induces the surplus (and cost) sharing rule (11). This gives the rule a property of ‘strategic stability’, in the core-theoretic sense that for no coalition the own strategy can provide a higher utility change than the one induced by the process with the rule (11).12 12
The rule can also be shown to correspond to the Shapley value of the local games, which turn out to be convex games.
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For a fuller understanding of the issue, let us make precise here that for each coalition S N, the assumed possible strategy is defined by a triplet [( p_ Si )i2S , (T_ iS )i2S , (T_ jS )j62S ],
(14)
whose first two elements respectively specify a direction of change in the discharges of the members of the coalition13 and transfers to or from these members, and the third element specifies contributions from the countries not in coalition S if p_ Si is negative, or to these countries if p_ Si is positive.14 This last element is somewhat unusual as it involves actions by players that are not members of the coalition under consideration. It does correspond, however, to the assumed situation in which all countries have made the commitment to accept the procedure15 as well as the sharing rule, and only bargain on the direction of change in the pollutant discharges. More importantly, this feature in the specification of the game is precisely one that permits to circumvent the difficulties alluded to above about the form (ii) of free riding, namely those of specifying the behavior of the agents not in the coalition. The retained strategies have indeed the property of being beneficial not only to the coalition members but also the non-members. This may then serve as an argument to justify the (implicitly assumed) non-malevolent behavior of the latter.
5.
Summary and conclusion
This paper presents international negotiations on pollution abatement as based on ‘gradual’ benefit-cost analyses and involving international transfers, derived from the theory of resource allocation processes. In so doing, it identifies as an ‘ecological surplus’ the gains to be obtained from the cooperation required for implementing the decisions suggested by such analyses. Furthermore, from a game-theoretic analysis of the problems raised by the issue of how to share that surplus, a simple sharing rule is obtained that also applies to the total cost of pollution abatement. This rule has the ‘core’ property of not being possibly improved upon by any coalition of countries that would attempt to adopt another abatement strategy. The simplicity and intuitive appeal of both the sharing rule and the cost-benefit framework advocated here may seem to be in sharp contrast P Specifically, p_ Si ¼ (pS dS gi ), i 2 S, with dS ¼ i2S di and di as in (11). 14 All these transfers are designed so as to ensure feasibility of the reallocation that would be induced by the coalition’s strategy. 15 Because of the gains they are sure to obtain from it. 13
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Public Goods, Environmental Externalities and Fiscal Competition
with the somewhat sophisticated formal apparatus used for justifying them. It is however the latter that permitted to formulate the former, illustrating once more how theory can be, as it should, at the service of applied issues.
References Arrow, K. and Hurwicz, L., (eds.), 1977. Studies in resource allocation processes. Cambridge: Cambridge University Press. Buchanan, J., 1968. The demand and supply of public goods Chicago, IL: Rand McNally. Chander, P., 1987. Cost-sharing local games in dynamic processes for public goods, in B. Dutta and D. Ray, (eds.). Theoretical issues in development economics. India: Oxford University Press. Chander, P. and Parikh, A., 1990. Theory and practice of decentralized planning procedures, Journal of Economic Surveys, 4(1), 19–58. Chander, P. and Tulkens, H., 1991–2001. Strategically stable cost sharing in an economicecological negotiation process, CORE discussion paper no. 9135, Louvain-la-Neuve: Universite´ catholique de Louvain, also in A. Ulph (ed.). Environmental policy, International agreements, and International trade. Oxford: Oxford University Press, 2001, chapter 4, 66–80. Cornes, R. and Sandler, T., 1986. The theory of externalities, public goods and club goods. Cambridge: Cambridge University Press. Dre`ze, J. and de la Valle´e Poussin, D., 1971. A taˆtonnement process for public goods, Review of Economic Studies 38, 133–150. Hoel, M., 1990. Emission taxes in a dynamic game of CO2 emissions, in R. Pethig, (ed.). Conflicts and cooperation in managing environmental resources. Berlin: SpringerVerlag. Malinvaud, E., 1970–71. Procedures for the determination of a programme of collective consumption, European Economic Review, 2, 187–217. Ma¨ler, K.-G., 1989. The acid rain game, in H. Folmer and E. Van Ierland, (eds.). Valuation methods and policy making in environmental economics. Amsterdam: Elsevier, chapter 12, 231–252. Ma¨ler, K.-G., 1990. International environmental problems, Oxford Review of Economic Policy, 6(1), 80–108. Mueller, D., 1976. Public choice: a survey, Journal of Economic Literature, 14(2), 396–433. Newbery, D., 1990. Acid rain, Economic Policy, 5(2), 297–346. Roberts, J., 1979. Incentives in planning procedures for the provision of public goods, Review of Economic Studies, 46(2) (Symposium on Incentive Compatibility), 283–292. Tulkens, H., 1978. Dynamic processes for public goods: an institution-oriented survey, Journal of Public Economics, 9, 163–201. (Chapter 1 in this volume). Tulkens, H., 1979. An economic model of international negotiations relating to transfrontier pollution, in K. Krippendorff, (ed.), Communication and control in society. New York: Gordon and Breach, chapter 16, 199–212. (Chapter 5 in this volume). Tulkens, H. and Schoumaker, F., 1975. Stability analysis of an effluent charge and the polluters pay principle, Journal of Public Economics, 4, 245–269. Yen, P.Y., 1987. Incentives in an economic-ecological negotiation process, Jingi Yanjiu (Economics Research, Journal published by the Department of Economics of ChungSing University, Taipei) 27, 27–40.
Chapter 7 THE ACID RAIN GAME AS A RESOURCE ALLOCATION PROCESS, WITH APPLICATION TO NEGOTIATIONS BETWEEN FINLAND, RUSSIA AND ESTONIA
Veijo Kaitala, Karl Go¨ran Ma¨ler, Henry Tulkens Scandinavian Journal of Economics, 97(2), 325–342, 1995.
We consider optimal cooperation in transboundary air pollution abatement among several countries under incomplete information, i.e., local information only on marginal emission abatement costs and damage costs. Directions of emission abatement in each country are determined that generate a succession of emissions programs shown to converge to an economic optimum. A cost sharing scheme, that results from appropriately designed international transfers, guarantees that the individual costs of all parties are nonincreasing along the path towards the optimum. A version of Ma¨ler’s (1989) ‘‘acid rain game’’ is used for a numerical application.
1.
Introduction
The problem of international cooperation on transboundary air pollution abatement is considered for the case of sulphur emissions using the ‘‘acid rain game’’ formulation introduced by Ma¨ler (1989). Our purpose is twofold. First, in contrast to the full information approach used by Ma¨ler and several followers quoted below, we wish to adopt the alternative formulation suggested by Tulkens (1979), in which the countries make use of only ‘‘local’’ information – that is, only the current values of their marginal emission abatement and depositions damage
This paper is part of Task Force 3 of the European Science Foundation Program on Environment, Science and Society. We are grateful to Matti Pohjola, Jacques Thisse and Alistair Ulph for constructive and critical comments. The first and third authors also express their thanks to the Beijer Institute for its hospitality.
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costs – when they negotiate on establishing cooperative emission reduction programs. Such local information is indeed sufficient to determine the cooperative international economic optimum assumed to be the purpose of the negotiations. This methodology has its origin in the theory of resource allocation processes initiated in 1960 by Arrow and Hurwicz,1 applied later on to public goods by e.g. Dre`ze and de la Valle´e Poussin (1971) and Malinvaud (1971)2 and recently further extended by Chander (1993). In an application, we also make use of a cost sharing scheme for abatement due to Chander and Tulkens (1991–2001) which ensures, by means of international transfers, that the total of abatement and damage costs is non-increasing for each of the parties involved, along the path to the cooperative optimum, as well as that no subset of the participants has an interest in following another course of action. Our second purpose is to demonstrate the applicability of the method, by using it in numerical simulations performed on actual data relating to the acid rain negotiations between Finland, Russia and Estonia. This issue has been dealt with in several previous works: thus, Kaitala, Pohjola and Tahvonen (1992a) presented a first basic analysis of the acid rain game between Finland and neighboring areas; these authors then extended their analysis to deal with two countries with several different regions (1991). Later, after the collapse of the USSR, Kaitala and Pohjola (mimeo) applied this approach to three independent countries.3 In all these contributions, simulations are carried out under the assumption of full information, and they compute the optima directly. Here, using the same data as in the last paper quoted, we proceed with the local information methodology announced above and identify whole paths of emission abatement, of deposition reduction as well as of transfers ensuring total cost improvement, for each of the regions involved. While the optimum we find is the same, the interest of this exercise lies in the information it provides on various characteristics of the directions that lead to it.
2.
Achieving optimality in the acid rain problem
We begin by characterizing optimality, assuming that full information is available for the players. Next, we formulate a scheme to approach 1
See Arrow and Hurwicz (1977). See Tulkens (1978) for a survey. 3 Two further contributions, in a somewhat different spirit however, are Kaitala, Pohjola and Tahvonen (1992b), who present a dynamic game model in which the damage function is related to the acidification of soil, as well as Tahvonen, Kaitala and Pohjola (1993) who analyze cost efficiency in the case where the countries agree on target deposition levels. 2
Ch.7 The Acid Rain Game as a Resource Allocation Process
137
the optimal solution by using local information only. The scheme is then illustrated by an example of the sulphur negotiations between seven regions of Finland, Russia and Estonia. Cooperation issues are dealt with in Section 3. Optimality Let E ¼ ((E1 , . . . , En )t denote emissions, Q ¼ (Q1 , . . . , Qn )T depositions, and B ¼ (B1 , . . . , Bn )T background depositions in n regions, indexed by i ¼ 1, . . . , n. Furthermore, let A ¼ (aij )i, j ¼ 1, . . . , n, denote a transportation matrix, each element aij of which denotes the fraction of emission Ej that deposits in region i. As in Ma¨ler (1989) let the emissiondeposition interactions be presented as Qi ¼
n X
aij Ej þ Bi
(1)
j¼1
for all i, or in matrix form Q ¼ AE þ B: Furthermore, let Ci (Ei ) denote a function that associates the total costs of aggregate productive activity in region i with the level of emissions Ei taking place there. For this function we assume that Ci0 (Ei ) < 0 in the relevant range, reflecting the fact that increased emissions allow for lower production costs or, taking the derivative to the left, that reducing emissions increases production costs. Let also the function Di (Qi ) denote the total costs entailed in country i by the damages caused there by the depositions Qi . The natural assumption here is that D0i (Qi ) > 0. Let finally Ji ¼ Ci (Ei ) þ Di (E))
(2)
denote the aggregate for region i of the two categories of costs just defined. All these costs are assumed to be measured in some common units. Consider now the optimization problem min J ¼ E1 ,..., En
n X
(Ci (Ei ) þ Di (Qi (E)) )
(3)
i¼1
such that (1) be satisfied. The optimality conditions are Ci0 (Ei )
þ
n X j¼1
aji D0j (Qj (E)) ¼ 0, i ¼ 1, . . . , n:
(4)
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Public Goods, Environmental Externalities and Fiscal Competition
We assume that the solution to problem (3) exists and is unique. The process of finding such a solution, in a way that in addition satisfies all regions, is what Ma¨ler (1989) refers to as the ‘acid rain game’. In this section we deal only with the optimality aspect; game theoretical aspects are introduced in Section 3. Path to the optimal solution Computing the solution that meets the optimality conditions above requires that the regions have full information about their entire emission abatement cost functions as well as about their entire deposition damage cost functions. However, when the actual environmental conditions that prevail in the regions are far from the level that can be considered optimal, then these costs may be very difficult to estimate, since the emission and deposition levels would change widely. This lack of information might make it impossible for the regions to negotiate cooperative environmental agreements that could satisfy the optimality condition (4). To overcome this difficulty Tulkens (1979) proposed that the regions use only local information to establish a gradual emission abatement program towards the optimal emission levels implied by (4). He showed that in spite of the absence of full information, local information is sufficient to guarantee that the changes in the emissions are towards the optimal levels. Specifically, assume as in Ma¨ler (1989) that the pollution policies of the regions are initially selfish and such that their emission levels correspond to a noncooperative Nash equilibrium emissions level i . Assume further that local information on the cost functions – that is, E on the marginal emission abatement costs Ci0 (Ei ) and the marginal damage costs D0j (Qj (E)) – are available at any emissions level E. The value n X
aji D0j (Qj (E)) þ Ci0 (Ei )
j¼1
may be seen as the aggregate cost savings – thus the willingness to pay – of all regions j for a marginal reduction of country i’s emissions Ei , net of the cost C 0 (Ei ) of this reduction (as regards reductions, given the signs of D0i and Ci0 defined above, the first term in the last expression is negative and the second one is positive). On that basis, an emissions reduction program of the Tulkens (1979) type can be formulated as follows, in a continuous time setting: ! n X dEi i, E_ i ¼ K Ci0 (Ei ) þ aji D0j (Qj (E)) , Ei (0) ¼ E (5) dt j¼1
Ch.7 The Acid Rain Game as a Resource Allocation Process
139
n n X X dQi _i ¼ j þ Bj , aij E_ j , Qi (0) ¼ aij E Q dt j¼1 j¼1
(6)
i ¼ 1, . . . , n,
is the initial Nash noncooperative where K > 0 is a constant, and E emission level vector mentioned above. Thus, the solution of the optimality problem is obtained by integrating (5)–(6) in time. Note that changing the value of K corresponds to changing the time scale such that increasing K makes the speed of emission changes increase. Example: Finland, Russia and Estonia As an application of the above approach, consider now the problem of optimal cooperation on transboundary air pollution abatement among Finland, Russia and Estonia analyzed in detail by Kaitala and Pohjola (mimeo). In this model both Finland and Russia have been divided into three regions, whereas Estonia has been treated as one region (for the indexes, see Table 1). The sulphur emissions, E, the total depositions, Q, and the background depositions, B in 1987 are given in Table 1, while the sulphur transportation matrix is given in Table 2 (both tables are taken from the paper just quoted, where details and further references are given). The authors use the following functional forms for the cost and damage functions: i Ei ) þ bi (E i Ei )2 þ g i Ci (Ei ) ¼ ai (E
(7)
and Di (Qi ) ¼ pi Qi
(8)
whose parameter values are given in Table 3. The estimates of the parameters of the annual emission abatement cost functions (a, b and g in Table 3) are based on extensive work carried out by the Finnish Integrated Acidification Assessment (HAKOMA) project at the Technical Research Center of Finland for the purpose of evaluating abatement Table 1. Sulphur emissions, depositions and background depositions in 1987 (1000 tons per year) Regions
i
Northern Finland Central Finland Southern Finland
1 2 3
Ei 5 60 97
Qi 46 98 66
26 59 35
Bi
Kola Karelia St Petersburg
4 5 6
350 85 112
131 95 88
27 50 46
Estonia
7
104
60
32
140 Table 2.
Public Goods, Environmental Externalities and Fiscal Competition Sulphur transportation matrix A for the year 1987 NFin 1
CFin 2
0.200 0.000 0.000 0.000 0.000 0.000 0.000
0.017 0.300 0.017 0.017 0.033 0.017 0.000
Emitting region ( j): SFin 3 Kol 4 Kar 5
StPtg 6
Est 7
0.000 0.036 0.027 0.009 0.045 0.268 0.018
0.000 0.029 0.038 0.000 0.019 0.058 0.221
Receiving region (i): Northern Finland Central Finland Southern Finland Kola Karelia St Petersburg Estonia
1 2 3 4 5 6 7
0.010 0.062 0.227 0.000 0.031 0.031 0.031
0.046 0.011 0.003 0.286 0.017 0.003 0.000
0.012 0.047 0.000 0.023 0.318 0.012 0.000
costs emission sources that are relevant to the acid rain problem in Finland; see Johansson, Ta¨htinen and Amann (1991). Less reliable estimates are available on the costs of damage (p in Table 3) to forestries caused by sulphur depositions. A rough attempt has been made to estimate directly the losses to forestries caused by soil acidification; see Kaitala, Pohjola and Tahvonen (1991). These direct estimates may fail, however, to capture other essential damages that also affect the willingness to pay. For that reason, the damage function parameters that we use are obtained using an indirect method of ‘‘revealed preference’’ proposed by Ma¨ler (1989), and applied here as follows. In 1987 the governments of Finland and the USSR signed an agreement aimed at limiting their sulphur emissions and reducing depositions on a cooperative basis. Thus, 1987 was a year that could be characterized as a Nash equilibrium between Finland and the USSR. At a Nash equilibrium, each country emits sulphur until the country’s marginal emission abatement costs are equal to its marginal damages caused by sulphur depositions. This observation suggests that at this equilibrium at least, marginal damage costs (pi ) in regions i of each country are revealed simply by observing their respective marginal abatement costs. These are the values reported for p in Table 3. Table 3. Emission abatement cost function parameters a (FIM/kg S), b(FIM=ton2 ), and g(106 FIM), and marginal damage costs p (FIM/kg S) Northern Finland Central Finland Southern Finland Kola - for 98 < E4 #350: - for 0 < E4 #98: Karelia St Petersburg Estonia
a
b
10.0 3.8 4.6
2.093 0.172 0.068
g 5.9 33.0 53.6
50.0 8.9 15.6
p
1.0 1.0 4.0 6.0 2.0
0.0 0.077 0.045 0.051 0.015
0.0 252.0 0.0 0.0 0.0
2.6 11.6 20.2 2.8
Ch.7 The Acid Rain Game as a Resource Allocation Process
141
However, in 1987 the USSR was still in existence, with Estonia belonging to it. Estonia has now regained its independence and, unlike Russia, does not recognize the agreements signed by the former USSR. Consequently, in this new political situation, environmental policy would change in Estonia if its own marginal willingness to pay were found to be different from what it was under the Soviet regime. As we have no indication so far of any major changes in the environmental policy of Estonia, we think it justified to continue using the damage function estimate derived for the socialist Estonia for our illustrative purposes. Applying the functional forms (7) and (8) in (5) we get ! n X : i Ei ) þ E_ i ¼ K ai 2bi (E E(0) ¼ E (9) aji pj , j¼1
Figures 1 and 2 illustrate the values Et (t) and Qi (t) along the solution of the system (9), thus describing the ‘‘path’’ over time towards an environmental optimum among the seven regions. The simulations are carried out assuming that the parameter estimates given in Table 3 are correct. Thus, the simulations represent a single possible outcome of the cooperative game. Deviations from this solution would occur if the marginal willingness to pay were to deviate from the one estimated. In the figures the time horizon is 150 years. Note, however, that taking K ¼ 10 we get the same results on a time scale of 15 years. Sizeable emission reductions occur in Kola and Estonia (76.6 and 44.2 percent, respectively), and minor emission reductions are also carried out in the remaining regions (3.3 –12.9 percent). The exception is Northern EMISSIONS (THOUSAND TONNES/YEAR) 350 300 250 200 150 100 50 0
0
10
20
30
40
50
60
70
80
90
100 110 120 130 140 150
TIME (YEARS) (K = 1)
NFi
CFi
SFi
Figure 1. Paths of sulphur emissions.
Kola
Kar
SPb
Est
142
Public Goods, Environmental Externalities and Fiscal Competition DEPOSITIONS (THOUSAND TONNES/YEAR)
150
100
50
0
0
10
20
30
40
50
60
70
80
90
100 110 120 130 140 150
TIME (YEARS) NFi
CFi
SFi
Kola
Kar
SPb
Est
Figure 2. Paths of sulphur depositions.
Finland where no reductions are observed (see Figure 1). The main reason for the high emission reductions in Kola is that the estimated marginal damage value of neighboring Northern Finland is about twenty times higher than that of the Kola region. Moreover, the marginal emission abatement costs are lowest in Kola and highest in Northern Finland. Similarly, notable amounts of Estonian emissions are transported to St. Petersburg and Southern Finland, whose marginal damage values are 5 –7 times higher than that of Estonia and the emission abatement costs of Estonia are lower than those in St. Petersburg and Southern Finland. Finally, the reason for the absence of emission abatement in Northern Finland is that this region does not contribute to the depositions in any other region. Thus, the noncooperative and cooperative emissions of that region are equal. (This property may fail to be true, however, with nonlinear damage functions). Depositions are illustrated in Figure 2. The reductions are strongest in Kola (59 percent), Northern Finland (26 percent), and Estonia (18 percent). In Karelia the reductions are 10 percent, and in the three remaining regions 6 –7 percent. The final depositions in regions 1–7 stabilize at the levels 0.33, 0.54, 1.16, 0.37, 0.50, 0.95, and 1:10 Sg=m2 respectively.
3.
Individual rationality, group rationality and cooperation
We now turn to the study of the gains — that is, the aggregate cost reductions — accruing to the regions when achieving the optimum. Let Ji
Ch.7 The Acid Rain Game as a Resource Allocation Process
143
denote the aggregate abatement and damage costs for region i at the Nash equilibrium and let Ji (t) denote these same costs at time t. The gain for region i at time t is then defined as Ji (t) ¼ J~i (t) J~i . It is the difference in aggregate cost that region i enjoys as a result of cooperation up to time t as compared to the original Nash situation. Since the costs at the Nash ~ equilibrium are constant Pwe have dJi (t)=dt ¼ dJi (t)=dt. For all regions the total gains read J(t) ¼ i Ji (t). Recalling (2), we then have dJ(t)=dt j ¼
n X
_ i) Ci0 (Ei )E_ i þ D0i (Qi )Q
(10)
i¼1
or, using (6), j¼
n X
Ci0 (Ei )E_ i þ D0i (Qi )
i¼1
¼
n X
n X
! aij E_ j
j¼1
Ci0 (Ei )E_ i
i¼1
þ
n X
!
aji D0j (Qj )E_ i
j¼1
or still, using (5), j¼
n 1 X E_ 2 #0 K i¼1 i
(11)
with J(0) ¼ 0. Thus, the direction is towards the global minimum at each point in time. 3.1
Individual rationality
Two problems arise, however. First, (11) does not imply that J_i < 0 for each i. Thus it may happen that the individual aggregate costs for some regions increase along the solution and end up being higher at the optimum than they are at the Nash equilibrium. That this is the case in the present numerical application is illustrated in Figure 3, where one sees that indeed Kola and Estonia do incur, at the optimum, annual costs of FIM 87 and 64 million, respectively, above the cost associated with the initial Nash equilibrium solution. Such an outcome is likely to prevent Kola and Estonia from joining in the abatement program specified by the solution to (9). To induce cooperation, the use of transfer payments between the regions involved has often been proposed in the literature; see e.g. Tulkens (1979), Ma¨ler (1989) and Kaitala, Pohjola and Tahvonen (1992a,b). This will be done in our model by adding to each aggregate cost function (2) a
144
Public Goods, Environmental Externalities and Fiscal Competition COST CHANGES (MILLION FIM/YEAR) 200 100 0
(100) (200) (300) (400) (500) (600) 0
10
20
30
40
50
60
70
80
90
100 110 120 130 140 150
TIME (YEARS) NFi
CFi
SFi
Kola
Kar
SPb
Est
Figure 3. Aggregate cost changes with no transfer payments (Ji ¼ Ci þ Di , i ¼ 1, . . . , 7).
transfer payment variable Ti (> 0 if the transfer is paid out by i, and < 0 if it is received by i). For each region, the aggregate cost defined in (2) now reads Ji þ Ti ¼ Ci (Ei ) þ Di (Qi (E)) þ Ti :
(12)
Imposing the constraint n X
Ti ¼ 0
(13)
i¼1
ensures that these transfer payments are feasible, i.e., that the sum of positive and negative transfers cancel out so as to balance the budget of the international agency conceivably established to implement them. The condition also leaves unaffected the solution to (3), that is, the optimal joint emissions policy, as well as its first-order conditions (4). How large should each transfer Ti be? The purpose of inducing cooperation from all parties suggests that for each i, the total cost Ji at the optimum, augmented by the transfer Ti , be lower than what Ji alone is at the initial Nash equilibrium. The resulting cost level incurred by each region may then be called ‘‘individually rational’’ with respect to the initial Nash equilibrium. Here the second problem arises. If only local information on the cost and damage functions is available, it is not possible to compute in advance the transfer payment scheme that achieves such individual rationality for all parties. However, the problem can be solved if, setting
Ch.7 The Acid Rain Game as a Resource Allocation Process
145
Ti (0) ¼ 0 so as to keep Ji (0) ¼ 0 for all i, one can find T_i depending on local information only and such that J_i #0 for all i. This is provided by specifying n X _ i) 1 dij E_ j2 T_i ¼ (Ci0 (Ei )E_ i þ D0i (Qi )Q K j¼1
(14)
for each i, with 0 # dij # 1 n X
dij ¼ 1
for all i, j
(15)
for all j,
(16)
i¼1
where according to our sign convention, T_i < 0 means region i receives an additional transfer and T_i > 0 means i pays an additional transfer. It follows from (12) and (14) that n 1 X dij E_ j2 # 0, J_i þ T_ i ¼ K j¼1
i ¼ 1, . . . , n:
(17)
Clearly, the transfer payment program defined by (14) guarantees that the individual aggregate costs of all parties are nonincreasing along the solution, a property whereby the system (5),(6) and (14) is given individual rationality. Group rationality and cost sharing A stronger form of cooperation can be obtained, however, by making use of the cost sharing rule proposed by Chander and Tulkens (1991, 1992). It implies in the present model that the parameters dij in (15)–(16) be chosen as dij ¼
aij D0i
n P
k¼1
akj D0k
,
i, j ¼ 1, . . . , n:
(18)
P This choice guarantees feasibility, that is i Ti (t) ¼ 0 for all t (as is easily computed from (14), using (18)), as well as individual rationality (since the inequality in (17) is preserved with (18)). It further possesses – as shown in the 1991 paper just quoted – the cooperative game theoretical property of inducing an imputation in the core of ‘‘local’’ games associated with each point of the solution of (5), (6) and (14). The cooperation thus achieved is claimed to be stronger because it is not only individually rational but also ‘‘coalitionally’’ rational or ‘‘group rational’’.
146
Public Goods, Environmental Externalities and Fiscal Competition
It is also of some interest to note that with (5), (16) and (18), expression (14) for the transfers reduces to n X a D0 _ i) 1 Pn ij i 0 E_ j2 T_i ¼ (Ci0 E_ i þ D0i Q K j¼1 k¼1 akj Dk ! n n X a D0 1 X 0 _ 0 Pn ij i 0 E_ j2 ¼ Ci E i þ Di aij E_ j K j¼1 k¼1 akj Dk j¼1 ! ! n n n 0 X X X a D ij i Pn ¼ Ci0 E_ i0 þ D0i aij E_ j Cj0 þ akj D0k E_ j 0 a D kj k¼1 k j¼1 j¼1 k¼1
¼ Ci0 E_ i þ
n X j¼1
aij D0i 0 _ 0 Cj Ej : k¼1 akj Dk
Pn
(19)
In this form, the transfer paid out or received by a region i appears to be the algebraic sum of two parts: (i) a first part received by the region to cover the cost of its own emissions abatement E_ i (first term in (19), which is positive if E_ i < 0); and (ii) a second part paid out by i to each of the other regions j P to share the cost of their emissions abatement E_ j , in the 0 proportion aij Di = nk¼1 akj D0k for each j. With (19) thus substituted for (18), it becomes clear that our process also implements a rule for sharing the cost of emissions abatement in the regions involved. 3.3
The cooperative solution in the example
We now introduce the transfer payments (14), with (18), in our example of acid rain cooperation among the three countries. Table 4 gives the cost sharing coefficients dij as defined by (18). The numbers in each column should add up to 1, as they each represent the fraction of the unit abatement cost in country j that is borne by country i. Reference to the elements aij of the transport matrix in Table 2, as well as to the values
Table 4.
Chander-Tulkens (1991) abatement cost sharing parameters dij j
NF 1
CF 2
SF 3
Kl 4
Kr 5
SP 6
Est 7
1 2 3 4 5 6 7
1.000 0.000 0.000 0.000 0.000 0.000 0.000
0.187 0.586 0.058 0.010 0.084 0.075 0.000
0.088 0.097 0.625 0.000 0.063 0.111 0.015
0.667 0.028 0.014 0.216 0.057 0.018 0.000
0.120 0.084 0.000 0.012 0.736 0.048 0.000
0.000 0.047 0.062 0.003 0.077 0.802 0.007
0.000 0.090 0.207 0.000 0.077 0.409 0.216
i NF CF SF KL Kr SP Est
Ch.7 The Acid Rain Game as a Resource Allocation Process
147
pi of the marginal damage costs in Table 3 explains the structure of the numbers in Table 4. For instance d11 ¼ 1:000 because a11 ¼ 0 for all i’s not equal to 1 in Table 2. On the other hand, diagonal elements dominate (except in two cases) because they also dominated in Table 2. Considering the exceptions, the case of d14 – that is, the fraction of abatement costs in Kola ( j ¼ 4) borne by Northern Finland (i ¼ 1) – is striking: it is equal to 0.667, i.e., 2/3 of the total and more than three times what Kola bears itself (i.e., 0.216). This results from the combination, in formula (18), of a relatively large value of a14 (0.046 in Table 2), a very large value of p1 (50.0 in Table 3) and a very low value of p4 (2.6 in Table 3). A similar argument explains d67 . Moving now to the adjustment process (5), (6), (14) and (18), Figure 4 illustrates the results in terms of the transfer payments (T), and Figure 5 shows the evolution of each Ji (t) in (13), which are the costs after the side payments are paid/received. According to the solution, Kola and Estonia both receive international environmental support from Finland. It should be noted that the transfer payments paid by St. Petersburg and Karelia are not high enough to cover the transfer payments received by Kola, so that Russia as a whole is a net receiver of the monetary environmental aid. According to Table 5, where the cost figures at the limit are recorded, cooperation shows an aggregate gain of about FIM 688 million which, if shared according to the transfers programs (14) with (18), yields the overall cost reductions offered in the third column of the table. The
TRANSFER PAYMENTS (MILLION FIM/YEAR) 200 100 0 (100) (200) (300) (400) (500) (600) 0
10
20
30
40
50
60
70
80
90
100 110 120 130 140 150
TIME (YEARS) NFi
CFi
Figure 4. Transfer payments.
SFi
Kola
Kar
SPb
Est
148
Public Goods, Environmental Externalities and Fiscal Competition COST CHANGES (MILLION FIM/YEAR) 200 100 0 (100) (200) (300) (400) (500) (600) 0
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150
TIME (YEARS) NFi
CFi
SFi
Kola
Kar
SPb
Est
Figure 5. Aggregate cost changes with transfer payments (Ji ¼ Ci þ Di þ Ti ).
costs reductions are distributed very unevenly among the regions (18.5, 2.6, 1.8, 40.3, 4.0, 1.6 and 4.7 percent of the Nash equilibrium noncooperative costs, respectively). Equal overall cost reductions can also be computed, as in the fourth and fifth columns of the table. Apart from the fact that the outcome is then much less favorable for Finland (in particular, the transfers it makes to Russia and Estonia are much larger), this latter scheme may also not enjoy the cooperative game theoretic properties of the former. Table 5. Cost variations between the initial Nash equilibrium and the optimum (106 FIM) with no transfers Ji (eq. (10)) Northern Finland Central Finland Southern Finland Finland Kola Karelia St Petersburg Nearby Russia Estonia Total Total þ Total
with Chander-Tulkens transfer scheme Ti Ji þ Ti
with cost-Egalitarian transfer scheme TiE JiE
629 46 32 707
þ203 þ23 þ12 þ238
426 23 20 469
þ530.7 52.3 66.3 þ412.3
98.3 98.3 98.3 294.9
þ88 61 72 45
225 þ17 þ42 166
137 44 30 211
186.3 37.3 26.3 249.9
98.3 98.3 98.3 294.9
þ64 840 þ151 688
72 297 þ297 0
8 688 .0 688
162.3
98.3
0
688
Ch.7 The Acid Rain Game as a Resource Allocation Process
4.
149
Increasing marginal damages
It is of particular interest to consider the consequences of different possible formulations of the damage functions, that is, of the willingness to pay in each region. Indeed, this information will actually be revealed only with time, as emission abatement proceeds and depositions in each region decrease. Simulations with alternative specifications, however, may give some idea of how emissions abatement policies are influenced by damage functions. In this section we deviate from the linearity assumption (8) and study as an alternative the possibility of increasing marginal damages. In particular, assume that the realization of the damage function is of the quadratic form4 1 Di (Qi ) ¼ pi Q2i , 2
(20)
with the marginal damage of the linear form pi Qi . While the structure of the process (5), (6) and (14) is, of course, independent of the form of the cost and damage functions, one may wonder by how much its realization differs depending on whether the damage functions (8) or (20) are used. To answer this question, we must first reevaluate the local preferences of the countries, assuming again that year 1987 describes the noncooperative Nash equilibrium situation. The new values for the parameters pi , i ¼ 1, . . . , 7, corresponding to that assumption, are p ¼ (1:09, 0:09, 0:24, 0:02, 0:12, 0:22, 0:05) 106 FIM=(103 ton)2 :
(21)
Realization of the process can now be simulated by using, instead of (9), the equation ! n X : i Ei ) þ E_ i ¼ K ai 2bi (E (22) aji pj Qj , E(0) ¼ E j¼1
The parameters dij in expression (14) for the transfers now depend explicitly on the depositions, and thus on time; they are given as aij pi Qi (t) : k akj p k Qk (t)
dij (t) ¼ P
(23)
4 One may argue that with nonlinear damage functions such as (20) it would be more realistic to include the background depositions Bi in the value of the variables, because they influence the optimality conditions (which is not the case under linearity). This modification, however, would not alter the nature of the results that we obtain in the exercise presented in this section.
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In the emissions abatement program obtained when marginal damages are increasing, the total reductions are less important than those obtained with linear damage functions, as one could expect. In the present model the difference is not large, however. Kola and Estonia still retain the lion’s share in total abatement (with 74.2 and 43.0 percent reduction, respectively, from their Nash equilibrium emission levels). On the other hand, almost no reduction takes place any more in the Central Finland and St Petersburg regions. A new and interesting phenomenon occurring here is that for Northern Finland the emissions increase along the solution, and reach in the limit a level that is about 12 percent above their Nash equilibrium level. Moreover, since at the Nash equilibrium E_ i is necessarily negative for all i’s, it is also the case that along the solution the Northern Finland emissions Ei (t) follow a path such that they first decrease and then increase. This can be explained as follows: observe that at the Nash equilibrium, the marginal abatement costs Ci0 in the various regions are quite different from one another; some may be very high, as is precisely the case with PnNorthern Finland. Observe also that the sum of the marginal damages, j¼1 aji p j Qj , now an increasing function of the depositions, decreases with the reduction of the latter. Therefore, if at some point t and for some i, the sum of the marginal damages becomes equal to its marginal abatement cost (imply E_ t (t) ¼ 0 for this i), while such equality does not (yet) hold for the other regions j 6¼ i, the reduction in depositions P achieved by these other regions induces a lowering of nj¼1 aji pj Qj while Ci0 does not change, hence a reversal of sign in E_ i (t). As far as costs are concerned, the figures in Table 6 indicates that the total benefit to be obtained from achieving an optimum decreases to FIM 534 million per year (from 688 million with linear damages). As to Table 6. Cost variation figures (106 FIM) with increasing marginal damages under Chander–Tulkens transfer scheme Ji (eq.(10))
Ti
Ji þ Ti
Northern Finland Central Finland Southern Finland Finland
528 40 33 601
þ189 þ20 þ16 þ225
339 20 17 376
Kola Karelia St Petersburg Nearby Russia
þ127 57 67 þ3
218 þ21 þ43 154
91 36 24 153
þ64
71 289 þ289 0
7
Estonia Total Total þ Total
534
534
Ch.7 The Acid Rain Game as a Resource Allocation Process
151
the way this aggregate benefit gets shared among the countries involved, it appears that without transfers, Russia is almost indifferent between this optimum and the noncooperative Nash equilibrium, whereas Estonia would be a loser. Note also that the aggregate cost increases in Kola and Estonia, when moving from the Nash equilibrium to the optimum without transfers, are now much larger (þ127 and þ64, respectively) than with linear damages (þ88, resp. þ38). Finally, the Chander–Tulkens transfers, that are designed to ensure individual as well as group rationality, appear to be much more favorable to Russia than to Estonia. This may be explained by formula (23), by taking into account the numerical values of the elements of the transport matrix, the values of the parameters p in the damage functions as reestimated by (21) and the deposition levels Qi . It is impossible, however, to disentangle the exact role of each one of these factors in the determination of the transfers that we obtain. Nevertheless, one may note that while Estonia has a rather low value for its pi , namely 0:05 106 in (21), at least two of the three Russian regions have substantially higher ones, namely Karelia (0:12 106 ) and St. Petersburg (0:22 106 ). Moreover, although p is very small for Kola, the large depositions Qi that occur there (see Table 1) weight significantly in (23). These two elements explain that a larger share of the surplus generated by the process goes to Russia rather than to Estonia. The spirit of formula (23) is indeed that the surplus be shared according to the intensity of the marginal damage costs. The game theoretic ‘‘core’’ property of the Chander–Tulkens cost sharing rule (as claimed by its authors) may also be called on to explain the structure of the transfers in the following terms: the amounts of the latter may be seen as reflecting how much is needed to prevent a country or group of countries from leaving the process and acting on their own in the hope of achieving for themselves still larger cost reductions. The figures obtained in the present simulation reflect quite a difference between Estonia and Russia in this respect.
5.
Conclusion
We have studied the problem of cooperation on transboundary air pollution abatement among several countries under the assumption of incomplete information. In particular, it is assumed that the countries have only local information, that is, current values of the marginal emission abatement costs and of the marginal deposition damage costs. The main result of the paper is that such local information is sufficient for determining a succession of cooperative emissions abatement programs that converge to the international economic Pareto optimum — an optimum that the countries cannot know beforehand under
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our assumption. The study also shows that the lack of more information does not prevent international transfers from being devised whereby the resulting environmental cooperation is beneficial for all parties.
References Arrow, K. and Hurwicz, L., (eds.), 1977. Studies in resource allocation processes. Cambridge: Cambridge University Press. Chander, P., 1993. Dynamic procedures and incentives in public good economies, Econometrica 61, 1341–54. Chander, P. and Tulkens, H., 1991–2001. Strategically stable cost sharing in an economicecological negotiation process, CORE DP 9135. Louvain-la-Neuve: Center for Operations Research and Econometrics, Universite´ catholique de Louvain, also in A. Ulph (ed.), 2001. Environmental policy, International agreements, and International trade. Oxford: Oxford University Press, chapter 4, 66–80. Chander, P. and Tulkens, H., 1992. Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems, European Economic Review 36 (2/3), 388–99. Dre`ze, J. and de la Valle´e Poussin, D., 1971. A taˆtonnement process for public goods, Review of Economic Studies 38, 133–50. Johansson, M., Ta¨htinen, M. and Amann, M., 1991. Optimal strategies to achieve critical loads in Finland, in proceedings of the International Symposium on Energy and Environment held in Espoo, Finland. Kaitala, V. and Pohjola, M., Acid rain and international environmental aid: a case study of transboundary pollution between Finland, Russia, and Estonia, mimeo. Kaitala, V., Pohjola, M. and Tahvonen, O., 1991. Transboundary air pollution between Finland and the USSR –a dynamic acid rain game, in R. P. Ha¨ma¨la¨inen and H. Ehtamo (eds.). Dynamic games in economic analysis. Lecture Notes in Control and Information Sciences, 157, Springer, 183–92. Kaitala, V., Pohjola, M. and Tahvonen, O., 1992a. An economic analysis of transboundary air pollution between Finland and the Soviet Union, Scandinavian Journal of Economics 94(3), 409–24. Kaitala, V., Pohjola, M. and Tahvonen, O., 1992b. Transboundary air pollution and soil acidification: a dynamic analysis of an acid rain game between Finland and the USSR, Environmental and Resource Economics 2, 161–81. Ma¨ler, K.-G., 1989. The acid rain game, in H. Folmer and E. van Ierland (eds.). Valuation methods and policy making in environmental economics. Amsterdam: Elsevier, 231–52. Ma¨ler, K.-G., 1990. International economic problems, Oxford Review of Economic Policy 6, 80–108. Malinvaud, E., 1971. A planning approach to the public good problem, Swedish Journal of Economics 73(1), 96–112. Tahvonen, O., Kaitala, V. and Pohjola, M., 1993. A Finnish-Soviet acid rain game: noncooperative equilibria, cost efficiency and sulphur agreements, Journal of Environmental Economics and Management 24, 87–100. Tulkens, H., 1978. Dynamic processes for public goods: an institution oriented survey, Journal of Public Economics 9(2), 163–201. (Chapter 1 in this volume). Tulkens, H., 1979. An economic model of international negotiations relating to transfrontier pollution, in K. Krippendorff (ed.), Communication and control in society. New York: Gordon and Breach, 199–212. (Chapter 5 in this volume).
Chapter 8 THE CORE OF AN ECONOMY WITH MULTILATERAL ENVIRONMENTAL EXTERNALITIES
Parkash Chander, Henry Tulkens International Journal of Game Theory 26, 379–401 (1997).
When environmental externalities are international – i.e. transfrontier – they most often are multilateral and embody public good characteristics. Improving upon inefficient laissez-faire equilibria requires voluntary cooperation for which the game-theoretic core concept provides optimal outcomes that have interesting properties against free riding. To define the core, however, the characteristic function of the game associated with the economy (which specifies the payoff achievable by each possible coalition of players– here, the countries) must also reflect in each case the behavior of the players which are not members of the coalition. This has been for a long time a disputed issue in the theory of the core of economies with externalities. Among the several assumptions that can be made as to this behavior, a plausible one is defined in this paper, for which it is shown that the core of the game is nonempty. The proof is constructive in the sense that it exhibits a strategy (specifying an explicit coordinated abatement policy and including financial transfers) that has the desired property of nondomination by any proper coalition of countries, given the assumed behavior of the other countries. This strategy is also shown to have an equilibrium interpretation in the economic model.
1.
Introduction
We deal in this paper with an economy with several agents, whose productive activities generate ‘‘multilateral’’ externalities, i.e., externalities Acknowledgements are due to Karl Go¨ran Ma¨ler for numerous fruitful discussions and his hospitality at the Beijer International Institute for Ecological Economics, Royal Swedish Academy of Sciences, Stockholm, during May-June 1992, as well as to Francis Bloch and Jacques Dre`ze for their comments on earlier versions of this paper. We also benefitted from discussions at seminar presentations made at CORE, the Copenhagen
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Public Goods, Environmental Externalities and Fiscal Competition
that each one of them can both generate and be a recipient of. We have in mind externalities that are detrimental for the recipients. We also call these externalities ‘‘environmental’’ because we assume that they exhibit public goods (actually public ‘‘bads’’) characteristics in the sense that when they are generated, they affect all agents in the economy. We should also call them ‘‘additive’’ because we further assume that what is received by any recipient is simply the sum of what is emitted by the various generators. For such an economy we call upon the concept of the core of a cooperative game associated with it, in order to identify joint actions that would (a) improve upon inefficient laissez-faire equilibria, (b) achieve Pareto efficiency, and (c) induce this outcome in such a way that it deters both individual and coalitional free riding behaviors. Being motivated by an interest of long standing1 for the sources of cooperation between countries on issues of transfrontier pollution (a currently prominent example of multilateral environmental externalities), we shall interpret our model below as one of an international economy, where the countries are also the players in the games associated with it.2 While other interpretations are conceivable, in our present interpretation the core outcome we are seeking for may be seen as a ‘‘strategically stable’’ result of negotiations on joint pollution abatement. To define the core, the characteristic function of the game – which specifies the payoff achievable by each possible coalition of players – must also reflect, in the case of games with externalities, the behavior of the players which are not members to the coalition. This is a disputed issue, and alternative assumptions in this respect lead to alternative core concepts such as, e.g. the a- and b-cores. These are studied and contrasted to each other in various externalities contexts by Scarf (1971), Starrett (1972) and Laffont (1977 (chapter V)).3 However, when using one of these assumptions in analyzing international environmental negotiations by Business School, the Se´minaire Cournot (Paris I), the 1994 FEEM-IMOP workshop ‘‘Environment: Policy and Market Structure’’ in Athens, and the 1994 Maastricht meeting of the Econometric Society. This work was initially stimulated by the European Science Foundation programme ‘‘Environment, Science and Society’’. The first author is grateful to the California Institute of Technology for providing an adequate environment for completing this research. The second author thanks the Fonds de la Recherche Fondamentale Collective, Brussels (convention n82.4589.92) and the Commission of the European Communities (DG XII: Project ‘‘Environmental Policy, International Agreements and International Trade’’) for their support. 1 A first version of the international environmental model alluded to here was given in Tulkens (1979), and elaborated upon from a game theoretic point of view in Chander and Tulkens (1991), (1992a) and (1992b.) 2 The ‘‘acid rain game’’ of Ma¨ler (1989) is a similar international environmental model, to which we apply, in Chander and Tulkens (1995), the concepts and results developed 3 The issue also arises in games associated with economies with public goods, as in Foley (1970), Champsaur (1975), Moulin (1987) and Chander (1993).
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
155
means of his acid rain game, (actually one implying a particular version of the a-core, as we shall see below), Ma¨ler (1989) comes to the conclusion that the core concept is useless because it results in too large a set of outcomes, actually the whole set of Pareto optima. In this paper, we adopt an alternative assumption on the behavior of players outside each coalition that to our knowledge has not been explored in the literature, and that we think to be a more plausible one in the context of environmental externalities. We use this assumption to formulate what we call the g-characteristic function of the game, and for the ensuing g-core concept we prove nonemptiness. The proof is constructive in the sense that it exhibits a joint strategy (i.e. in our pollution interpretation an explicit coordinated abatement policy accompanied by financial transfers) that has the desired property of nondomination by any proper coalition of the players. Furthermore, this strategy is shown to have an equilibrium interpretation which is conceptually analogous to that of ratio equilibrium (Kaneko (1977); Mas Colell and Silvestre (1989)) in the context of economies with public goods. Our analysis assumes full information throughout: this is because our primary concern is with a satisfactory definition and an existence proof – which are lacking – of the concept of core allocation for the kind of economy we are considering. This should provide firm ground for further work where other information structures are allowed for.
2.
The model of an economy with multilateral environmental externalities and its efficiency characterization
The agents of the economy (countries in our international interpretation) are denoted by the index i, with N ¼ {iji ¼ 1,2, . . . , n}. Three categories of commodities are considered: (i) a standard private good, whose quantities are denoted by x$0 if they are consumed, and by y$0 if they are produced; (ii) pollutant discharges, the quantities of which are denoted by p$0; and (iii) ambient pollutant quantities, denoted by z#0. Each agent i’s preferences are represented by a utility function ui (xi , z) satisfying: Assumption 1: ui (xi z) ¼ xi þ vi (z) i.e. quasi-linearity; and Assumption 2: vi (z) concave, differentiable and such that dvi =dz pi (z) > 0 for all z#0.
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Public Goods, Environmental Externalities and Fiscal Competition
With each agent i there is furthermore associated a technology, described by the production function yi ¼ gi ( pi ), satisfying: Assumption 3: gi ( pi ) strictly concave and differentiable over an interval; and Assumption 4: There exists p0i > 0 such that 8 > 0 if pi < p0i (i) > < dyi g i ( pi ) ¼ 0 if pi $ p0i (ii) > dpi : ¼ 1 if pi ¼ 0: (iii) Inputs, which are not explicitly mentioned in the production functions, are subsumed in the functional symbols gi . Although this amounts to treat them as fixed in the analysis that follows, our results will not rest in an essential way on that assumption, which is made here mostly for expositional convenience. All possible behaviors within the economy, in terms of the consumption, production and pollutant discharge decisions taken by its agents, as well as in terms of the resulting values of ambient pollutant are formally described by feasible states: Definition 1: Feasible states of the economy (or ‘‘allocations’’) are vectors (x,p,z) (x1 , . . . ,xn ; p1 , . . . ,pn ; z) such that X X xi # gi ( pi ) i2N
and z¼
(1)
i2N
X
pi :
(2)
i2N
Among feasible states, Definition 2: A Pareto efficient state of the economy is a feasible state (x, p, z) such that there exists no other feasible state (x0 , p0 , z0 ) for which 0 ui (xi , z0 )$ui (xi , z) for all i 2 N with strict inequality for at least one i. To characterize efficient states, the usual first order conditions take in this case the form of the following system of equalities: X pj (z) ¼ gi ( pi ), i ¼ 1, 2, . . . ,n: (3) j2N
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
157
P Henceforth, we shall always write pN for j2N pj . Existence of (several) Pareto efficient states easily follows from our assumptions and, due to Assumption 4(iii) in particular, one has pi > 0 for all i in any efficient state. A further property of efficient states in our model, important for our purposes, is the following: Proposition 1: Assumptions 1–3 imply that in all Pareto efficient states, the vector of emission levels (p1 , . . . , pn ) is the same. , z) and (~ Proof: Suppose not, and let ( x, p x, p~, ~z) be two states, both Pareto efficient, with pi 6¼ ~ pi for some i. On the one hand, if ~z ¼ z, then pi ) ¼ g i ( ~ pi )8i 2 N; but this, by strict pN (~z) ¼ pN (z) implies by (3) that g( concavity of all functions gi ( pi ), is only possible if ~pi ¼ pi 8i. If, on the other hand, ~z 6¼ z, say ~z < z without loss of generality, the fact that pN (~z) $ pN (z) now implies by (3) that gi ( ~ pi ) $ gi ( pi )8i, which P in turn impliesPby concavity that ~ pi # pi 8i. However, since ~z ¼ i2N ~pi and z ¼ i2N pi , one would have ~z $ ~z, which is a contradiction.
3.
Noncooperative games associated with the economy
In this section we examine various noncooperative equilibrium concepts – the Nash equilibrium, the strong Nash equilibrium and the coalition proof Nash equilibrium – that might be called upon to describe the state of the economy in the absence of cooperation. We conclude that in the case of detrimental environmental externalities studied here only the Nash equilibrium is appropriate for that purpose. We consider now each agent i of the economy as a player in an nperson noncooperative game. To that effect, let Ti ¼ {(xi , pi )j0 # pi # p0i ; 0 # xi # gi ( p0i )}, be the strategy set of each player i, and T(S) ¼ {(xi , pi )i2S j0 # pi # p0i 8i2S and 0 #
i2N X i2S
xi #
X
gi ( p0i )}
i2S
be the space of joint strategies of players in S. Clearly, T(S) Xi2S Ti . Let T denote the space of joint strategies of all players, i.e., T T(N). Any joint strategy choice [(x1 , p1 ), . . . ,(xn , pn )] 2 TPof the players induces a feasible state (x, p, z) of the economy if z ¼ pi . Then, if for each i ¼ 1, . . . , n and any P [(x1 , p1 ), . . . , (xn , pn )] 2 T we choose ui (xi , z) ¼ xi þ vi (z) with z ¼ j2Npj as the payoff for player i and write u ¼ (u1 , . . . ,un ), we have defined a noncooperative game [N, T, u], associated with the economy.
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Public Goods, Environmental Externalities and Fiscal Competition
Definition 3: For the noncooperative game [N, T, u], the joint strategy p1 ), . . . ,( xn , pn )] is a Nash equilibrium if for all i 2 N, choice [( x1 , ( xi , pi ) maximizes xi þ vi (z) subject to xi # gi ( pi ) and pi þ z ¼
X
pj :
j2N j6¼1
Definition 4: For the economy, a disagreement equilibrium is the state P ( x1 , . . . , xn ; p1 , . . . , pn ; z) with z ¼ i2N pi induced by the Nash p1 ), . . . ,( xn , pn )] of the game [N, T, u]. equilibrium [( x1 , To characterize a Nash equilibrium or a disagreement equilibrium, the first order conditions of the maximization problem in Definition 3 yield the well known system of equalities: pi (z) ¼ g i ( pi ),
i ¼ 1, . . . , n:
(4)
From the fact that the system (4) differs from (3), the standard statement is derived that a disagreement equilibrium is not an efficient state of the economy. Furthermore, it will prove useful to have the following two properties established: Proposition 2: For the game [N, T, u] there exists a Nash equilibrium; this equilibrium is unique. Proof: The existence of a Nash equilibrium follows from standard theorems (see, e.g., Friedman (1990)), since each player’s strategy set is compact and convex, and each player’s payoff function is concave, and therefore continuous, and bounded. All these conditions are obviously met in our model. To prove uniqueness of the equilibrium [( x1 , p1 ), . . . , ( xn , pn )], suppose contrary to the assertion that there exists another Nash equilibrium [(^ x1 , ^ p1 ), . . . , (^ xn , ^ pn )] 6¼ [( x1 , p1 ), . . . , ( xn , pn )]: Without loss of generality assume that S^pi # Spi , entailing ^z ¼ S^pi $ z ¼ S pi . From the characterization of the disagreement equilibrium and the concavity of the functions vi (z), we would then have pi (^z) # pi (z) as well as g i ( ^ pi ) # g i ( pi ) for each i 2 N. But given the concavity of the production functions, this last inequality would imply that ^ pi $ pi for each i 2 N, which contradicts the assumption that S^ pi # Spi , and [(^ x1 , ^ p1 ), . . . ,(^ xn , ^ pn )] 6¼ [(^ x1 , p1 ), . . . ,(^ xn , pn )]. &
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
159
Turning to the economy, the existence and uniqueness of a disagreement equilibrium state are also established by Proposition 2. Let us now examine the relevance for our externalities model of two alternative equilibrium concepts which are offered by the noncooperative game theoretic literature, namely the strong Nash equilibrium and the coalition proof Nash equilibrium. Definition 5: For the noncooperative game [N, T, u], the joint strategy ~ ~ ~1 , ~ ~ ~n , ~ ~ choice [(x p1 ), . . . ,(x pn )] is a strong Nash equilibrium if for all S N, X ~ ~ ~i , ~ pi )i2S ] maximizes [xi þ vi (z)] (5:a) [(x i2S
subject to
X
xi #
i2S
and
X i2S
gi ( pi )
(5:b)
i2S
pi þ z ¼
X
0 P ~ j2NnS ~ pj
if S ¼ N if S ¼ 6 N.
(5:c)
For the analysis of economies with detrimental environmental externalities, this concept is unfortunately of no use because of the following result. Proposition 3: For the game [N, T, u], there does not exist a strong Nash equilibrium. Proof: Definition 5 implies both that a strong Nash equilibrium is also a Nash equilibrium, and that it induces a Pareto efficient state of the economy. However, as was observed earlier, the Nash equilibrium cannot induce such a Pareto efficient state. It follows from this contradiction that a strong Nash equilibrium cannot exist for the game [N, T, u].4 & Having noted that strong Nash equilibria almost never exist in general. Bernheim, Peleg, and Whinston (1987) comment that the strong Nash concept is ‘‘too strong’’. They therefore propose as an alternative the coalition proof Nash equilibrium. For the externalities model we are dealing with here, however, this refinement cannot bring much conceptual progress for the following reason: a coalition proof Nash equilibrium is always also a Nash equilibrium, and in our [N, T, u] game the Nash equilibrium is unique; therefore, any coalition proof Nash 4
Ma¨ler (1989) shows that for his ‘‘acid rain game’’ also there exists no strong Nash equilibrium.
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Public Goods, Environmental Externalities and Fiscal Competition
equilibrium – if one exists at all – cannot be different from the Nash equilibrium characterized above. The coalition proof Nash equilibrium concept might be used, however, not for describing the state of the economy in the absence of cooperation, as we have done so far, but instead for predicting and/or characterizing the outcome of negotiations. In this perspective, the fact that a coalition proof equilibrium is not a Pareto optimum in general prevents us to use it, as we are searching for efficient outcomes. Moreover the concept of coalition proof Nash equilibrium involves, as in the case of strong Nash equilibrium, the assumption that when a coalition deviates, it takes as given the strategies of the complement – an assumption that we do not find justified, as we shall argue in the following section.
4.
Cooperative games associated with the economy
4.1
Inappropriateness of the a-characteristic function
Turning now to the cooperative part of our analysis, let us associate, with every coalition S, the number w(S), called the ‘‘worth’’ of the coalition S and defined as the highest aggregate payoff Si2S ui that the members of the coalition can achieve using some strategy. Thus, the pair [N, w()] consisting of the players set N and the characteristic function w() defines a cooperative game (with transferable utilities) associated with our economy. Stated in full, the characteristic function reads X w(S) ¼ max [xi þ vi (z)] (6) {(xi , pi )i2S } i2S For the relation between this function and our economy to be meaningful – i.e. to correspond to feasible states, the variable z should satisfy condition (2). However, this equality involves strategic choices made by players who are not members of S. Thus, the worth of a coalition in our game is not only a function of actions taken by its members, but also of actions of players outside the coalition. This typical feature of cooperative games associated with economies with externalities and/or public goods requires the characteristic function to specify explicitly what the actions are both of the members of S and of the other players. A familiar way to get around this problem has been to assume that the players outside the coalition adopt those strategies that are least favorable to the coalition. That is, the characteristic function is defined as
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
wa (S) ¼
max {(xi , pi )i2S }
subject to
X
i2S
( pi þ z
¼ ¼0
[xi þ vi (z)]
(7:a)
i2S
xi #
i2S
X
X
161
X
gi ( pi )
(7:b)
i2S
P j2NnS
p0j
if S 6¼ N
(7:c)
if S ¼ N.
Games with a characteristic function of this type are used in most of the studies dealing with the core of economies with public goods (see, e.g., Foley (1970), Champsaur (1975), Moulin (1987) and Chander (1993)). This is also the case with the game used by Scarf (1971) in his study of cores of economies with externalities: his ‘‘a-characteristic function’’ is similar to the one defined above5. And more recently, pushing this ‘‘pessimistic’’ view to the extreme, Ma¨ler (1989) assumes in his analysis of the acid rain game that there is no upper bound such as our p0i for the individual pollutant discharges; thus, coalitions can be hurt by up to infinite amounts of pollutants emitted by players outside the coalition. This extreme form of the assumption behind the a-characteristic function suggests that it is actually ill-adapted to the issues at stake with environmental externalities. Indeed there are at least two grounds for considering that assuming such strategies on the part of the players outside the coalition is questionable. On the one hand, these strategies may not maximize the individual payoffs of these players. On the other hand, they rest on the presumption that when a coalition forms, its payoff is what it would get when members of the complementary coalition act so as to minimax this coalition’s payoff. But why should they behave in this extreme fashion, and why should the coalition formed presume that they would follow such strategies? Searching for alternative assumptions in the spirit of the strong Nash equilibrium or of the coalition proof Nash equilibrium is not satisfactory either: they assume for their part that when a coalition deviates from the equilibrium it takes as given the strategies of its complement at that equilibrium. But as before, this may not maximize the payoff of the complement’s members; and in addition, why should their strategies remain unchanged when S forms? 5
Laffont (1977) shows that the a-and the b-characteristic functions are identical for cooperative games associated with economies exhibiting detrimental externalities of the type dealt with here. Hence, all our subsequent developments on a-cores will also be valid for b-cores.
162
4.2
Public Goods, Environmental Externalities and Fiscal Competition
Equilibrium with respect to a coalition
These considerations lead us to introduce here a concept in which it is assumed that when a coalition forms it neither takes as given the strategies of its complement, nor does it assume that the complement would follow minimax strategies: instead, it looks forward to the best reply payoff corresponding to the equilibrium that its actions would induce. More specifically, we propose to assume that when S forms players outside S do not take particular coalitional actions against S, but adopt only individually best reply strategies. This results in a Nash equilibrium between S and the remaining players, with the members of S thus playing their joint best response to the individual strategies of the others.6 In terms of our economic model and of its associated game, this is formalized as follows. Definition 6: For the noncooperative game [N, T, u], given a coalition S N, a Nash equilibrium with respect to S is the joint strategy choice [(~ x1 , ~ p1 ), . . . , (~ xn , ~ pn )] 2 T where X pi )i2S ] maximizes [xi þ vi (z)] (9:a) (i) [(~ xi , ~ subject to and
X
X
xi #
i2S
X
i2S
gi ( pi )
(9:b)
i2S
pi þ z ¼
i2S
X
~ pj ,
(9:c)
j2NnS
pj ) maximizes xj þ vj (z) (ii) 8j 2 NnS, (~ xj , ~
(9:d)
subject to xj # gj ( pj ) X ~ and pj þ z ¼ pj :
(9:e) (9:f)
i2N i6¼j
As suggested by our choice of terminology, this equilibrium amounts to a Nash equilibrium between the coalition S acting as one individual player and the other players acting alone. Definition 7: For the economy, given a coalition S N, a partial agree~n ; ~p1 , . . . , ~pn ; ~z) ment equilibrium with respect to S is the state (~ x1 , . . . , x 6 Our analysis might be extended to the case when N\S may also form, but we do not pursue this possibility in the present paper. It may be noted however that if the players outside S form one or more non-singleton coalitions then the worth of S is generally higher. That players outside S act individually is therefore equivalent to granting S a certain degree of pessimism.
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
163
P with ~z ¼ i2N ~ x1 , ~ p1 ), . . . , (~ xn , ~pn )], the Nash equilibrium pi induced by [(~ with respect to S of Definition 6. In terms of first order conditions, a partial agreement equilibrium with respect to S so defined is characterized by the system of n equalities: X pj (~z) ¼ g i ( ~ pi ), i 2 S j2S
and pj ), pj (~z) ¼ gj ( ~
j 2 NnS:
Note that inequality (9.b) in Definition 6 allows for some members of S possibly to consume more, or less, than what they produce. The strategy thus includes the possibility of transfers of private goods among the members of the coalition. The inequality also implies that the algebraic sum of these transfers be zero within the coalition. Further properties of a partial agreement equilibrium with respect to a coalition S that are important for the sequel are collected hereafter. Proposition 4: For any coalition S N, (i) there exists a partial agreement equilibrium with respect to S; (ii) the individual emission levels corresponding to this equilibrium are unique; (iii) the individual emission levels of the players outside of S are not lower than those at the disagreement equilibrium, (iv) although the total emissions are not higher. Proof: The existence proof follows from similar arguments as in Proposition 2. The uniqueness of individual emission levels follows from similar arguments as in Proposition 1. We therefore only formally prove here the remaining two parts P of the proposition. P Let ~z ¼ ~ pi and z ¼ pi , where ( ~ p1 , . . . , ~pn ) and ( p1 , . . . , pn ) are the emission levels corresponding to a partial agreement equilibrium with respect to S and to the disagreement equilibrium, respectively. We first show that ~z $ z. Suppose contrary to the assertion that ~z < z. We must then have pi (~z) $ pi (z) for each i 2 N. From the characterizations of a partial agreement equilibrium and of the disagreement equilibrium it follows that X pj (~z) ¼ g i ( ~ pi ) $ g i ( pi ) ¼ pi (z), 8i 2 S j2S
and pj ) $ g j ( pj ) ¼ pj (z), pj (~z) ¼ gj ( ~
8j 2 NnS:
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From the strict concavity of each function gi , it follows that ~pi # pi for each i 2 N. But this contradicts our supposition that ~z < z. Hence, we must have ~z $ z. Finally, since ~z $ z, pj (~z) # pj (z) for each j 2 NnS. The inequalities above and strict concavity of gj imply that ~ pj $ pj for each j 2 NnS: & Note that in view of Assumption 2, the total emissions corresponding to a partial agreement equilibrium with respect to a coalition of two or more players are strictly lower than those corresponding to the disagreement equilibrium. This means that as compared to the disagreement equilibrium, the players outside a coalition of two or more players are strictly better-off at a partial agreement equilibrium with respect to that coalition – which is actually a form of free riding on the part of those players. 4.3
The g-characteristic function
We are now equipped to introduce a new characteristic function, that embodies the behavioral assumptions we found desirable at the beginning of the previous subsection. We call it the ‘‘partial agreement characteristic function’’ or, for short, the g-characteristic function, and denote it by wg . It is defined by: X [xi þ vi (z)] (10:a) wg (S) ¼ max {(xi , pi )i2S } i2S X X xi # gi ( pi ) (10:b) subject to and
X
i2S
i2S
pi þ z ¼
i2S
X
pj ,
(10:c)
j2NnS
where 8j 2 NnS, (xj , pj ) maximizes xj þ vj (z)
(10:d)
subject to xj # gj ( pj ) X pi : and pj þ z ¼
(10:e) (10:f)
i2N i6¼j
The worth of coalition S is thus determined by an equilibrium concept, namely that of partial agreement equilibrium with respect to S. It is not assumed that the players outside the coalition S do the worst, as with the a-characteristic function; nor is it assumed, as in the concepts of strong and coalition proof Nash equilibria, that they do not react to the actions of S. Let us note however that for each S, the value assigned by the partial agreement characteristic function wg is at least as much as that assigned by the a-characteristic function, i.e. wg (S) $ wa (S) for each
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
165
S N. In fact, examples can be constructed such that wg (S) > wa (S) for all S N. This implies that the core of the game [N, wg ], i.e. the ‘‘g-core’’, is, if nonempty, contained in the ‘‘a-core’’, and possibly smaller. Moreover, if we had assumed that when a coalition S forms, the complementary coalition N\S also forms and chooses the best response strategy for its members given what S does, then the core of the so-defined game would most likely be smaller than that of the game [N, wg ], and even perhaps empty in which case a stable strategy for the grand coalition N would be impossible to find.7
5.
A strategy in the g-core with linear payoff functions Let us recall:
Definition 8: For any cooperative game [N, w], a strategy of the all player coalition N is said to belong to the core of the game if for any subset S N the payoff it yields to the members of S is larger than w(S), i.e. the payoff that S can achieve by itself. Emptiness vs. nonemptiness of the core typically depends upon the form of the characteristic function, which reflects the power of each coalition in the game. In the presence of externalities, this power is crucially affected by the assumed behavior of the players outside the coalition. Thus, with the a-characteristic function, coalitions are weakened by the presumed minimax behavior, letting hope that the corresponding a-core be nonempty. This is indeed the case in Scarf (1971), as well as in the version given by Laffont (1977) of the Shapley and Shubik (1969) ‘‘garbage game’’,8 and also in Ma¨ler’s (1989) acid rain game. The former two results, however, do not bear on an economy with externalities of the environmental type we are dealing with here, and Ma¨ler’s argument is only an informal one. Furthermore, no results are available, to our knowledge, on cores of games with the characteristic function wg we have introduced above.
7 This remark points to the work of Carraro and Siniscalco (1993) who in a model with identical agents assume that when S forms and achieves the aggregate payoff w(S), if some i 2 S leaves S, the coalition S \{i} remains formed. They show that then it may be better for i to leave S. As this advantage grows with the size of coalitions, they conclude that only small coalitions can prevail, and that N will never form. More recently, in a model where coalition formation is endogeneous, Ray and Vohra (1996) also conclude that in general N cannot form. 8 This is also the case in the many studies of economies with public goods alluded to above of Foley (1970), Champsaur (1975), Moulin (1987) and Chander (1993), where the unfavorable behavior consists in producing no public good at all; these have typically large cores.
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To find out whether or not the core of a cooperative game is nonempty various approaches can be used. Two qualitative ones are offered by Shapley (1967) and Shapley (1971) who respectively show nonemptiness if the game is balanced or convex. None of these approaches proved succesful in our case. We therefore turn to another, constructive, approach, which is to exhibit a strategy for which we can show that it satisfies Definition 8. Specifically, let (x , p , z ) ¼ (x1 , . . . , xn ; p1 , . . . , pn ; z ) be the Pareto efficient state defined by ! X pj X pi ) gj ( p) gj ( pj ) , i 2 N, xi ¼ g i ( pN j2N (11) j2N X pi , z ¼ where pi ¼ pi (z ) for each i 2 N and (p1 , . . . , pn ) and ( p1 , . . . , pn ) are the (unique) individual emission levels corresponding to the Pareto efficient states and to the disagreement equilibrium, respectively.9 By Pareto efficiency pN ¼ gi ( pi ) for each i 2 N. As can be easily seen, this state of the economy10 implies that for each i 2 N, (xi , pi , z ) maximizes xi þ vi (z) subject to 0 1 xi # gi ( pi ) pi þ z ¼
X pi BX C g ( p ) gj ( pj ) gi ( pi )A, @ j j pN j2N j2N
X
j6¼i
pj
j6¼i
and 0 # pi # p0i : which means that the state (x1 , . . . , xn ; p1 , . . . , pn ; z ) is an equilibrium concept. As it can be given an interpretation analogous to that of the ratio 9 Chander (1993) analyzes an instantaneous analog of this cost sharing rule in a public good context. 10 Expression (11) implies transfers Ti ( > 0 if received. < 0 if paid) of the private good, such that 8i 2 N, xi ¼ gi ( pi ) þ Ti , where
Ti ¼
(gi ( pi )
p gi ( pi )) þ i pN
with the property that
P
i2N
Ti ¼ 0
X j2N
gj ( pj )
X j2N
!
gj ( pj ) :
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
167
equilibrium (see Kaneko (1977), and Mas-colell and Silvestre (1989),) we shall refer to it as the ratio equilibrium with respect to the disagreement equilibrium.11 We now prove the nonemptiness of the g-core by showing that the ratio equilibrium with respect to the disagreement equilibrium belongs to the core of the game [N, wg ]. We first consider a special case of our general model, namely the one where it is assumed that the payoff functions are linear, i.e.: Assumption 10 : ui (xi , z) ¼ xi þ p i z,
p i > 0.
This is actually the case for which Ma¨ler (1989) proves the nonexistence of a strong Nash equilibrium. In Shapley and Shubik’s (1969) garbage game also, the payoff functions are linear (unlike here, however, their externalities are directional and involve no diseconomies of scale). As can be easily seen from the characterizations of the Nash equilibrium and of a partial agreement equilibrium, an important consequence of the linearity assumption is that in a partial agreement equilibrium with respect to a coalition, the emission levels of the players outside the coalition are the same as in the Nash equilibrium. In fact, under the linearity assumption the Nash equilibrium is a dominant strategy equilibrium. Theorem 1: Under the linearity Assumption 1’, the strategy [(x1 , p1 ), . . . , (xn , pn )] of the grand coalition N that induces the ratio equilibrium (x ,p ,z ) of the economy belongs to the core of the game [N, wg ]. Proof: Suppose contrary to the assertion, that the strategy inducing (x1 , . . . , xn ; p1 , . . . , pn ; z ) is not in the core of the game [N, wg ]. Then there exists a coalition S N and a strategy for S inducing the feasible ~n ; ~ ~n ; ~p1 , . . . , ~pn ; ~z) is a parstate (~ x1 , . . . , x p1 , . . . , ~ pn ; ~z) such that (~ x1 , . . . , x ~i þ p tial agreement equilibrium with respect to S, and x i~z > xi þ p i z for all i 2 S. From the characterization of a partial agreement equilibrium with respect to a coalition, itPfollows that pi ¼ pi for all i 2 NnS, that P ~ ~i ¼ i2S gi ( ~pi ). ~ pi $ pi for all i 2 S, and that i2S x ^n ; p1 , . . . , pn ; Consider now the alternative efficient state (^ x1 , . . . , x z ), defined as: 11
Note that one can also consider the Pareto efficient state defined by:
p xj ¼ gi ( p0i ) i pN
X j2N
gj ( p0j )
X
! gj ( pj ) , i 2 N:
j2N
But we are unable to establish the same properties for this Pareto efficient state. It seems that the reference point matters.
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p i ^ i ¼ gi ( ~ pi ) x p N X pi : z ¼
X
gi ( ~ pi )
i2N
X
! gi ( pi )
i2N
, i 2 N, (12)
i2N
We show below that, as far as the members of S are concerned, one has X
^i þ x
X
p i z >
X
~i þ x
X
p i~z,
(13)
which implies X X X X ^i þ p i z > xi þ p i z x
(14)
i2S
i2S
i2S
i2S
i2S
i2S
i2S
i2S
~i þ p if S is able to achieve x i~z > xi þ p i z for all i 2 S as it is supposed to do. We further show that as far as the other players are concerned, ^i þ p i z $ xi þ p i z for all i 2 NnS: x
(15)
As together the inequalities (14) and (15) imply that the state ^n ; p1 , . . . , pn ; z ) is Pareto superior to the Pareto efficient alloca(^ x1 , . . . , x tion (x1 , . . . , xn ; p1 , . . . , pn ; z ), we get an impossibility. Proving (13) and (15) will thus establish the theorem. To show (13), the definition (12) allows one to write ! P X X X X X X p i ^i þ p i z ¼ gi ( ~ pi ) i2S gi ( ~ pi ) gi ( pi ) þ p i z x p N i2N i2N i2S i2S i2S i2S ! P X X X X X p i X ~i i2S ¼ gi ( ~ pi ) gi ( pi ) þ p i z p i~z þ p i~z x p N i2N i2N i2S i2S i2S i2S " !# P X X X X p i ~i þ i2S ¼ pN (z ~z) gi ( ~ pi ) gi ( pi ) þ p i ~z: x p N i2N i2N i2S i2S (16) From the respective characterizations of a Pareto efficient state, and of a partial agreement equilibrium, we have for all i 2 N, p N ¼ g i ( pi ) and ~ pi $ pi . Hence, the strict concavity of each function gi implies p N >
gi ( ~ pi ) gi ( pi ) , for all i 2 N, ~ pi pi
which in turn implies that
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
p N (z ~z) >
X i2N
gi ( ~ pi )
X
169
gi ( pi ):
i2N
Then (13) follows from (16). On the other hand, from the respective characterizations of the disagreement equilibrium and of a partial agreement equilibrium with respect to a coalition, we haveP~ pi # ~ pi , for all i 2 N, ~pi ¼ pi for all P pi ) # i2N gi ( pi ). Therefore, i 2 NnS, and thus i2N gi ( ~ ! X p i X ^i ¼ gi ( ~ pi ) gi ( ~ pi ) gi ( pi ) x p N i2N i2N ! X p i X pi ) gi ( pi ) gi ( pi ) ¼ xi for all i 2 NnS: $ gi ( p N i2N i2N These inequalities establish (15).
&
Note that the domination used in the proof of Theorem 1 is more demanding than the usual one in that it requires any dominating payoff vector to correspond to an equilibrium strategy combination. Thus, the a-core includes the g-core, just as the coalition proof Nash equilibria include the strong Nash equilibria if they are both nonempty. In other words, the g-core seems to satisfy a consistency requirement.12
6.
g-Core property of the strategy under nonlinear payoff functions
Doing away with the linearity of payoff functions makes the interaction between the strategic variables more complex. To see this, note that the Nash equilibrium is no longer a dominant strategy equilibrium. Moreover, it need not be true that ~ pi # pi for all i 2 S, where ~pi , i 2 N, are the emission levels corresponding to the partial agreement equilibrium with respect to S. As these inequalities play a crucial role in the proof of Theorem 1, we do extend our result to the more general case by imposing a condition which is sufficient to ensure that these inequalities continue to hold. Consider the following: P Assumption 1’’: For all S N, S 6¼ N, jSj $ 2, i2S pi (z ) $ pj (z), j 2 S, where z and z correspond to the disagreement equilibrium and a Pareto efficient state, respectively. Clearly, this assumption is satisfied when the payoff functions are linear. In words, the assumption says that the (now non-constant) marginal 12
We are grateful to the referee who made that point.
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utilities of z should not fall ‘‘too much’’ between the disagreement equilibrium and the Pareto optimum. It covers a large class of quadratic utility functions, among others. Proposition 5: Under Assumption 1’’, the emission level of each player in the coalition of a partial agreement equilibrium is not higher than the one corresponding to the Nash equilibrium. Proof: Let S N be some coalition, and let ((~ x1 , p~1 ), . . . , (~ xn , p~n )) be a partial agreement equilibrium with respect to S. Using an argument which is in the proof of Proposition 4, we first show that P analogous P to that ~ $ p , where pi , i 2 N, are the emission levels corresponding p i2N i i2N i to a Pareto optimum. P P Indeed, suppose pi < i2N pi instead, implying ~z > z . By i2N ~ concavity of the functions vi (z), one would have pj (~z) # pj (z )8j 2 N, and from this, together with the first order characterizations of a partial agreement equilibrium and of an optimum, it would follow that, for each j 2 S. X X X pk (~z) ¼ g j ( ~ pj ) # pk (z ) # pk (z ) ¼ g j ( pj ): k2S
k2S
k2N
and for each j 2 NnS, pj ) # pj (z ) # g j ( pj ): pj (~z) ¼ gj ( ~ Since gj ( pj ) is decreasing in pj , these two expressions imply that ~ pj $ pj 8j 2 N, contradicting the supposition. P since ~z # z , concavity again implies that z) $ i2S pi (~ P Now, i2S pi (z ) for all S N. From the characterizations of the Nash equilibrium and of the partial agreement equilibrium with respect to S, it then follows that gj ( ~ pj ) $ g j ( pj ) for all j 2 S, that is ~pi # pj for all j 2 S. & We can now state: Theorem 2: Under Assumption 1’’, the strategy [(x1 , p1 ), . . . , (xn , pn )] of the grand coalition N that induces the ratio equilibrium (x , p , z ) of the economy belongs to the core of the game [N, wg ]. Proof: Suppose, contrary to the assertion, that the strategy inducing the above state (x , p , z ) is not in the core of the game [N, wg ]. Then there exists a coalition S N, S 6¼ N, and a Nash equilibrium with respect to S, [(~ x1 , ~ p1 ), . . . , (~ xn , ~ pn )], such that ~i þ vi (~z) > xi þ vi (z ) for all i 2 S, x
(17)
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171
P where ~z ¼ i2N ~ pi . Define a new Pareto efficient state of the economy (^ x1 , . . . , ^n ; p1 , . . . , pn ; z ) as follows: x ! X pi X ^ i ¼ gi ( ~ pi ) gj ( ~ pj ) gj ( pj ) , i ¼ 1, . . . , n: x pN j2N j2N We claim that inequality (17) implies that X X X X ^i þ ~i þ vi (z ) > vi (~z) x x i2S
i2S
and X i2NnS
i2S
X
^i > x
(18)
i2S
xi :
(19)
i2NnS
As these two inequalities together with (17) clearly contradict the Pareto efficiency of (x1 , . . . , xn ; p1 , . . . , pn ; z ), our theorem is proved if we establish them. We first prove (18). By the definition of the new allocation, ! ! X X X X X ^i ¼ gi ( ~ pi ) pi =pN gj ( ~ pj ) gj ( pj ) x i2S
i2S
i2S
X
$
~i x
i2S
X
j2N
!
pi =pN
pN
j2N
X j2N
i2S
~ pj
X
!! pj
,
j2N
using the concavity of the functions gi , as well as the Pareto efficiency condition (3); but the last expression is equal to: ! X X ~i pi =pN pN (z ~z): x i2S
X i2S
i2S
Thus, X X X ^i þ ~i þ pi z $ pi ~z, x x i2S
i2S
i2S
that is X X X X ^i þ ~i þ vi (z ) $ vi (~z) x x i2S
i2S
þ
i2S
X i2S
vi (z )
i2S
X i2S
vi (~z)
X i2S
! pi (z ~z) $
X i2S
~i þ x
X i2S
vi (~z),
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Public Goods, Environmental Externalities and Fiscal Competition
(i.e. (18)), since from concavity vi (z ) vi (~z) $ pi (z ~z) for all i. Next we prove inequality (19). By definition, 0 1 ! X X X X X ^i ¼ gi ( ~ pi ) @ pi =pN A gi ( ~pj ) gj ( pj ) x i2NnS
i2NnS
¼
X
i2NnS
0 gi ( pi ) @
i2NnS
0 þ@
þ@
X
þ@
¼
@
@
gj ( pj )
X
20
X
X
1 pi =pN A
gj ( ~ pj ) 1
gi ( pi )A
gj ( pj )
X
j2N
pi =pN A@
i2NnS
j2N
i2NnS
gi ( ~ pi )
X
gi ( ~ pi )
i2NnS
i2S
gj ( ~ pj ) 1
gi ( pi )A
i2NnS
X
X
!
j2N
i2NnS
X
gj ( pj )
! gj ( pj )
!
j2N
gi ( pi )
10
i2NnS
0
X
xi þ 4@
X
X
X
j2N
pi =pN A
i2NnS
0
X
gi ( pi ) A
X
1
i2NnS
j2N
j2N
1
i2NnS
X
X
0
xi þ @
i2NnS
0
pi =pN A
j2N
i2NnS
1 pi =pN A
i2NnS
¼
X
gi ( ~ pi )
X
X
X
i2NnS
i2NnS
0
1
gi ( ~ pi )
X
13 gi ( pi )A5
i2NnS
X
! gi ( pi ) :
i2S
As Propositions P 4 and 5 implyPthat ~ pi $ pi for all i 2 NnS and ~pi # pi ^i > i2Nn xi , i.e. (19). for all i 2 S, we have i2NnS x & We had referred earlier to an international environmental externalities literature that concludes negatively on the issue of full cooperation. These authors (Ma¨ler (1989), Barrett (1990), Carraro and Siniscalco (1993)) all establish their claim under the alternative assumption of identical players–thus, identical countries. It is therefore of interest that the opposite result can be obtained here under the same assumption, as we now show.
Ch.8 The Core of An Economy With Multilateral Environmental Externalities
173
Assumption 5: For all i, j 2 N, ui ¼ uj , i.e., the utility13 functions of the countries are identical. Proposition 6: Proposition 5 holds under Assumption 5 instead of Assumption 1’’. Proof: When the payoff functions are identical, it follows from the characterizations of the Nash equilibrium and of the partial agreement equilibrium with respect P to S that P for all i, j 2 S, gi ( pi ) ¼ gj ( pj ) and gi ( ~ pi ) ¼ g j ( ~ pj ). Since i2S ~ pi # i2S pi by Proposition 4, it follows that pi ) $ g i ( pi ) for all i 2 S, that is, ~ pi # pi for all i 2 S: & gi ( ~ Corollary: Theorem 2 holds under Assumption 5 instead of Assumption 1’’. Proof: Identical to the proof of Theorem 2.
7.
&
Concluding remarks
For the cooperative game associated with an economy with detrimental environmental externalities, we have exhibited a strategy in the gcore, that is, Pareto optimal joint actions of the all-players set N such that no coalition S N can do better for its members. The essence of our contribution lies (i) in identifying this fully cooperative strategy and (ii) in showing that to deter deviating behavior of any S N against it – that is, to deter free riding by any S, the breaking up of the players not in S into singletons acting rationally is sufficient. We interpret our result as an argument supporting the view that, on logical grounds, full cooperation and efficiency can prevail in economies of this type, in spite of the fact that neither a strong Nash equilibrium nor a Pareto efficient coalition proof Nash equilibrium can be shown to exist for them. The proposed allocation has an equilibrium interpretation, nevertheless: it is analogous to that of ratio and therefore of Lindahl equilibrium in economies with public goods. For such economies, it is known that the ratio equilibrium belongs to the core (actually, the a-core). Since the g-core is a smaller set of allocations, our result establishes an additional property of the ratio equilibrium. Note also that the constructive nature of our result has the virtue of allowing one to compute the g-core allocation in applied problems, as is
13
and not the production functions, as assumed in earlier versions of this paper.
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Public Goods, Environmental Externalities and Fiscal Competition
done, for instance, in Germain, Toint and Tulkens (1996) for the international acid rains model of Ma¨ler’s (1989). Finally, while the restrictive assumptions made have permitted us to introduce and illustrate the basic concepts in a more transparent way, extending our analysis to a wider class of utility functions and other information structures are obvious directions for further research.
References Barrett, S., 1990. The problem of global environmental protection, Oxford Review of Economic Policy, 6(1), 68–79. Bernheim B.D, Peleg, B. and Whinston, M.D., 1987. Coalition proof Nash equilibria 1 concepts, Journal of Economic Theory, 42, 1–12. Carraro, C. and Siniscalco, D., 1993. Strategies for the international protection of the environment, Journal of Public Economics, 52, 309–328. Champsaur, P., 1975. How to share the cost of a public good, International Journal of Game Theory, 4, 113–129. Chander, P., 1993. Dynamic procedures and incentives in public good economies, Econometrica, 61, 1341–1354. Chander, P. and Tulkens, H., 1991–2001. Strategically stable cost sharing in an economicecological negotiation process. CORE Discussion Paper n8 9135, also in A. Ulph (ed.), Environmental policy, International agreements, and International trade. Oxford: Oxford University Press 2001, chapter 4, 66–80. Chander, P. and Tulkens, H., 1992a. Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems, European Economic Review, 36(2/3), 288– 299. (Chapter 6 in this volume). Chander, P. and Tulkens, H., 1992b. Aspects strate´giques des ne´gociations internationales sur les pollutions transfrontie`res et du partage des couˆts de l’e´puration, Revue Economique, 43(4), 769–781. Chander, P. and Tulkens, H., 1995. A core-theoretical solution for the design of cooperative agreements on transfrontier pollution, International Tax and Public Finance, 2(2), 279–294. (Chapter 9 in this volume). Foley, D., 1970. Lindahl solution and the core of an economy with public goods, Econometrica, 38, 66–72. Friedman, J., 1990. Game theory with applications to economics (2nd ed.). Oxford: Oxford University Press. Germain, M., Toint, P. and Tulkens, H., 1996. Calcul e´conomique ite´ratif pour les ne´gociations internationales sur les pluies acides entre la Finlande, la Russie et l’Estonie, Annales d’Economie et de Statistique (Paris), 43, 101–127. Kaneko, M., 1977. The ratio equilibria and the core of the voting game G(N, W) in a public goods economy, Econometrica, 45, 1589–1594. Laffont, J.J., 1977. Effets externes et the´orie e´conomique. Monographies du Se´minaire d’Econome´trie. Paris: Editions du Centre national de la Recherche Scientifique (CNRS). Ma¨ler, K.-G., 1989. The acid rain game, in H. Folmer and E. Van Ierland (eds.). Valuation methods and policy making in environmental economics. Amsterdam: Elsevier, 231–252. Mas-Colell, A. and Silvestre, J., 1989. Cost sharing equilibria: a Lindahlian approach, Journal of Economic Theory, 47, 329–256. Moulin, H., 1987. Egalitarian-equivalent cost sharing of a public good, Econometrica, 55, 963–976. Ray, D. and Vohra, R., 1997. Journal of Economic Theory, 73, 30–78.
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Scarf, H., 1971. On the existence of a cooperative solution for a general class of N-person games, Journal of Economic Theory, 3, 169–181. Shapley, L., 1967. On balanced sets and cores, International Journal of Game Theory Quarterly, 14, 453–460. Shapley, L., 1971. Cores of convex games, International Journal of Game Theory, 1, 11–26. Shapley, L. and Shubik, M., 1969. On the core of an economic system with externalities, American Economic Review, LIX, 678–684. Starrett, D., 1972. A note on externalities and the core, Econometrica, 41(1), 179–183. Tulkens, H., 1979. An economic model of international negotiations relating to transfrontier pollution, in K. Krippendorff (ed.). Communication and control in society. New York: Gordon and Breach, 199–212. (Chapter 5 in this volume).
Chapter 9 A CORE-THEORETIC SOLUTION FOR THE DESIGN OF COOPERATIVE AGREEMENTS ON TRANSFRONTIER POLLUTION
Parkash Chander, Henry Tulkens International Tax and Public Finance 2 (2), 279–294, 1995.
For a simple economic model of transfrontier pollution, widely used in theoretical studies of international treaties bearing on joint abatement, we offer in this paper a scheme for sharing national abatement costs through international financial transfers that is inspired by a classical solution concept from the theory of cooperative games—namely, the core of a game. The scheme has the following properties: total damage and abatement costs in all countries are minimized (optimality property), and no coalition or subset of countries can achieve lower total costs for its members by taking another course of action in terms of emissions or transfers, under some reasonable assumption about the reactions of those not in the coalition (core property). In the concluding section economic interpretations of the scheme are proposed, including its connection with the free riding problem.
1.
Introduction
This paper highlights the relevance of the game-theoretic concept of the core of a cooperative game for the design of international treaties on This paper was prepared for the Fiftieth Congress on Public Finance, Environment and Natural Resources of the International Institute of Public Finance, held at Cambridge, Massachusetts. Thanks are due to Karl Go¨ran Ma¨ler for numerous fruitful discussions and his hospitality at the Beijer International Institute for Ecological Economics, Royal Swedish Academy of Sciences, Stockholm in May 1993, to Tito Cordella for a very careful reading, as well as to Jack Mintz and Carlo Carraro, discussants at the Congress. The first author is also grateful to the California Institute of Technology for providing a stimulating environment for the completion of this work. This research is part of the Commission of the European Communities (DG XII) project on Environmental Policy, International Agreements and International Trade administered by Professor Alistair Ulph through CEPR, London.
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transfrontier pollution. Specifically, a formula is offered (in Section 5) for allocating abatement costs between the countries involved for which the justification is of core-theoretic nature. We examine this concept because of the need to ensure more than mere optimality in the outcomes of international negotiations leading to a treaty. On the one hand, the optimum sought for should be a voluntary one because of the nonexistence of a supranational authority endowed with sufficient coercive power to impose any emissions policy on countries, even if optimal. On the other hand, the optimum should be robust against the temptation of free riding by some of the countries (or groups of countries) involved, given the public good (actually public bad) nature of transfrontier pollution. We claim in this paper that a core-theoretically based argument enjoys the two properties just stated and also offer with formula (13) an explicit policy that implements them. We develop our arguments in the framework of the simplest model traditionally used for the economic analysis of transfrontier pollution agreements (as, for example, in Ma¨ler (1989–1993); Hoel (1992); Barrett (1992); Carraro and Siniscalco (1993); d’Aspremont and Ge´rard-Varet (1992)). Our claims in this paper proceed from results that are presented with full technical details in a companion theoretical paper (Chander and Tulkens (1997)), where use is made of a more general model expressed in terms of an Arrow-Debreu economy with public goods, initially formulated in Tulkens (1979) and also used in Chander and Tulkens (1992). The present paper shows that the Chander and Tulkens (1997) results are readily applicable to the more common simple environmental model just evoked.
2.
Transfrontier pollution: the economic model and its associated games
2.1
The basic economic model
Currently, the model most commonly used for the economic analysis of international agreements on transfrontier pollution is based on the following components: 1. N ¼ {iji ¼ 1, . . . , n}, the set of countries concerned by the analysis, the countries being n in number, each indexed by i; 2. For each country i: . Quantities Ei $0 of some pollutant emitted by the economic agents of country i, per unit of time. Ei is a scalar as the analysis bears on one pollutant only. An extension to several pollutants would require Ei to be a vector.
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179
. Quantities Qi $0 of ambient pollutant present in country i’s environment, per unit of time. As for the emissions, Qi is a scalar; with several pollutants it would be a vector. . A transfer function Qi ¼ Fi (E), (1) where E ¼ (E1 , . . . , En ), that describes the physical, chemical, or biological processes whereby the amounts E of pollutants emitted in all countries get transformed into the quantities Qi of ambient pollutants present in country i. This function is assumed to be nondecreasing in each of its arguments. The fact that Qi is formulated here as being dependent on current emissions only restricts the analysis to flow pollutants, as opposed to stock pollutants. . An abatement-cost function, Ci (Ei ), expressing the costs (monetary and possibly non-monetary) incurred by the polluting agents of country i when their aggregate emissions are restricted to the amount Ei . This function is assumed to be decreasing (Ci0 < 0), a property reflecting the natural assumption that reducing (that is, abating) emissions is costly. . A damage-cost function Di (Qi ), expressing the costs (monetary as well as nonmonetary) incurred by the economic agents of country i as a result of the ambient pollutants Qi they are exposed to. This function is assumed to be nondecreasing (Di0 $0). . The total of abatement and damage costs Ji (E) Ci (Ei ) þ Di (Qi ) ¼ Ci (Ei ) þ Di [Fi (E)],
(2)
incurred by the country as a result of the joint pollutant emissions E of all countries. Notice that the variables Ej , j 6¼ i, that appear as arguments of the function Ji are of the nature of an externality, exerted on country i by each one of the countries j. 3. Finally, the description of this international economy with pollution— henceforth summarily designated by the pair [N, (Ci , Di , Fi )e2N ]—is completed by identifying a vector E ¼ (E1 , . . . , En ) of optimal joint emissions by the n countries, optimality being taken in the sense of minimizing the sum over all countries of both abatement and damage national costs. E is thus the solution of the optimization problem Min
{E1 ;...; En }
where J(E)
J(E),
X i2N
Ji (E):
(3)
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2.2
Public Goods, Environmental Externalities and Fiscal Competition
The associated games
As stated in the introduction, the optimum just defined is only likely to be implemented on a voluntary basis—that is, in terms of joint actions that suit the interests of each one of the countries involved. This argument, which is by now classical, motivates the recourse to game-theoretic concepts. Indeed, the formulation of alternative games associated with the economic model introduces behavioral assumptions on the basis of which voluntary actions can be characterized. In this respect, the distinction offered by classical game theory between noncooperative and cooperative games is particularly relevant. This paper builds explicitly on such distinction (1) by characterizing as equilibria of a noncooperative game associated with the above economic model, national emission policies that satisfy only the objectives of each country; and (2) by identifying with some solution concept for cooperative games (also associated with the economic model) policies that reflect actions taken in a coordinated way by either all the parties, or subsets of them. Formally, . A noncooperative game, defined by its players set N ¼ {iji ¼ 1, . . . , n}, by the sets Ti , i ¼ 1, . . . , n of strategies accessible to each of the players i, and by the payoffs ui that the latter achieve—and henceforth denoted by the triplet [N, (Ti )i2N , (ui )i2N ]—is associated with the economic model [N, (Ci , Di , Fi )i2N ] by identifying the players set with the set of countries, by defining the strategy set of each country as Ti ¼ {Ei jEi $0} (with possibly an upper bound Ei0 to be defined below), and by defining each player’s payoff ui as the value Ji of the function specified in (2) above. . A cooperative game (in characteristic function form and with transferable utility), defined by its players set N ¼ {iji ¼ 1, . . . , n} and the function w(S) that associates with every subset S of N a number called the worth of S (or the payoff to coalition S)—and henceforth denoted by the pair [N, w]—is similarly associated with the economic model [N, (Ci , Di , Fi )i2N ] by identifying again the players set with the set of countries, and by defining the characteristic function as1 X w(S) ¼ Min [Ci (Ei ) þ Di (Qi )]: (4) {Ei }i2S
1
i2S
In standard game theoretic models, payoffs, whether individual as in ui or for coalitions as in w(S), are usually supposed to be maximized by the players. As the economic model used here associates costs only with its agents, maximizing payoffs for them amounts to minimizing costs. The Arrow-Debreu type of model used in Chander and Tulkens (1994) allows for payoffs to be directly defined on the utilities of the economic agents, more in line with usual practice.
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Thus, the worth of each coalition S is determined by some strategy vector (Ei )i2S adopted by the members of the coalition. However, remembering (1), one notices that when S 6¼ N, this worth also depends on the strategies (Ej )j2S adopted by the countries that are not members of S. As those have been left unspecified in our formulation of the function (4), we shall have to return to this issue below when we deal in more detail with the cooperative game. 2.3
Assumptions on the economic and ecological components of the model and characterization of optimality
Precise results on voluntary behavior in this economic model can be obtained when some further assumptions are introduced on its components. Those we shall use—most of which are standard but would deserve critical discussion—are the following: Assumption 1. For every (decreasing) abatement cost function Ci (Ei ), i ¼ 1, . . . , n, there exists Ei0 > 0 such that 8 < ¼ 1 if Ei ¼ 0, if Ei < E 0i , Ci0 (Ei ) < 0 (5) : ¼0 if Ei $ E 0i . Assumption 2. For all i, the function Ci (Ei ) is strictly convex (i.e., Ci00 > 0) over the range ]0, Ei0 [. Assumption 3. For all i, the transfer function Qi ¼ Fi (E) is of the linear additive form Qi ¼
n X
Ej :
(6)
j¼1
Notice that this assumption implies that Qi ¼ Qj for all i, j 2 N, thus making the ambient quantities of pollutant to have the characteristics of an international public good (actually, of a public ‘‘bad’’) for the countries involved.2 In view of Assumption 3, we shall often write, with some notational inconsistency, Di (Qi ) as Di (E). 2
In Ma¨ler’s (1989–1993) acid rain game, where the linear transfer function is of the form
Qi ¼
n X j¼1
aji Ej ,
(7)
P with 0#aji #1 and i2N aji ¼ 1, the externalities are directional and do not have the public good property.
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Assumption 4. For all i, the (nondecreasing) damage cost function Di (Qi ) is convex (Di00 $0); it is strictly increasing (Di0 > 0) for at least some i. Together, Assumptions 2 to 4 imply that for all i, the total cost function Ji (E) is convex. One can then prove, as in Chander and Tulkens (1994, Sec. 3), the following: Proposition 1. Under the Assumptions 1 through 4, the optimal joint emissions vector E is unique and in the range ]0, Ei0 [ for all i. The optimum so defined is usefully characterized by the well-known first-order conditions n X
D0j (E ) þ Ci0 (Ei ) ¼ 0,
i ¼ 1, . . . , n:
(8)
j¼1
3.
The noncooperative game and its Nash equilibria
3.1
The Nash equilibrium
A first form of voluntary behavior in our economic model is the one described by the familiar Nash equilibrium concept of the associated noncooperative game: Definition 1. For the noncooperative game [N, (Ti )i2N , (ui )i2N ], a Nash equilibrium is a joint strategy choice E ¼ (E1 , . . . , En ) such that 8i, Ei minimizes Ji (E), where for each j 2 N, j 6¼ i, Ej ¼ Ej : Existence of this equilibrium follows from standard theorems (see, for example, Friedman (1990)). It is characterized by the first-order conditions Di0 (E) þ Ci0 (Ei ) ¼ 0,
i ¼ 1, . . . , n:
(9)
As these differ from the conditions in (8), whereby the optimum was characterized, a Nash equilibrium is not an optimum for the economy, revealing thus that this form of voluntary behavior is incompatible with international optimality. Notice that the condition (9) imply ( < Ei0 if Di0 > 0, (10) 8i, Ei ¼ Ei0 if Di0 ¼ 0: Furthermore, if Di (Qi ) is linear, Ei is a dominant strategy for i since the first term of (9) is independent of Qi and the vector E in that case.
Ch.9 A Core-Theoretic Solution for the Design of Cooperative Agreements
183
A final property, formally important for our purposes below, is the uniqueness of the Nash equilibrium vector E in this game, as shown in Proposition 2 of Chander and Tulkens (1994). 3.2
Strong and coalition proof equilibria
Ma¨ler (1989–1993) has considered stronger concepts of voluntary behavior in the framework of noncooperative games—namely, the strong Nash equilibrium and the coalition proof Nash equilibrium.3 However, the former is shown by Ma¨ler not to exist in general in the case of the game associated with our economic model, and the latter is not Pareto-efficient, if it exists at all. They are therefore of little use in our enquiry. 3.3
The partial agreement Nash equilibrium with respect to a coalition
Another aspect of noncooperative behavior may be considered— namely, the one adopted by the players outside a coalition when a coalition forms. This is described by the following concept: Definition 2. Given some coalition S N, a partial agreement equilibrium with respect to S in the P game [N, (Ti )i2N , (ui )i2N ], is a joint strategy E~ such that ~ 1. (Ei )i2S minimizes i2S Ji (E), where for every j 2 N, j 62 S, Ej ¼ E~j as defined in 2 below, and 2. 8j 2 NnS, E~j minimizes Jj (E), where for every i 2 S, Ei ¼ E~i as defined in 1 above. In Chander and Tulkens (1994, sec. 3.3), we prove the following: Proposition 2. For any proper coalition S N in the game [N, (Ti )i2N , (ui )i2N ], 1. There exists a partial agreement equilibrium with respect to S; 2. The vector of individual emission levels at such an equilibrium is unique; 3. The individual emissions of the players outside the coalition are not lower than those at a Nash equilibrium; 4. The total emissions level is not higher than at a Nash equilibrium. The equilibrium so defined is also characterized by the first-order conditions X D0j (E~) þ Ci0 (E~i ) ¼ 0, i 2 S j2S (11) D0j (E~) þ Cj0 (E~j ) ¼ 0, j 2 NnS: 3
Both concepts have been discussed in detail in Bernheim, Peleg, and Whinston (1987).
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4.
The cooperative game: imputations, the core, and alternative characteristic functions
4.1
Imputations and the core
Turning now to the cooperative part of our analysis, we first recall some terminology. For a cooperative game P [N, w] in general, an imputation is a vector y ¼ ( y1 , . . . , yn ) such that i2N yi ¼ w(N). Recall that for the game associated with our economic model, the worth w(N) of the grand coalition, as defined in (4), is a total cost: more precisely, it is the minimum of the aggregate total abatement and damage cost over all countries. Here, an imputation is thus a way to share among all players the amount of this cost. In this setting, an imputation y is said to belong to the core of the game if it satisfies the conditions X yi # w(S), 8S N: (12) i2S
The core of our cooperative game is thus the set of imputations having the property that to every conceivable coalition they offer to bear a share of the aggregate cost w(N) lower than the cost w(S) it would bear by itself. To study the core of this game, we therefore shall consider in more detail, in the next two subsections, what its imputations are, as well as how its characteristic function is precisely defined. 4.2
Imputations in the game and monetary transfers in the economic model
As was noted with Proposition 1, the minimum aggregate cost obtained with w(N) is determined by a unique joint strategy vector P E ¼ (E1 , . . . , En ), yielding Ji (E ) for each i and of course i2N Ji (E ) ¼ w(N). The vector J(E ) ¼ [J1 (E ), . . . , Jn (E )] is thus an imputation where each country bears itself the abatement and damage costs E entails for it. Other imputations, associated with the same optimal joint strategy, can be conceived of, however, if monetary transfers between countries are introduced. Let us denote such transfers by Pi (> 0 if the transfer is paid by i, < 0 if it is received by it). Then imputations in the cooperative game associated with our economy can be written as vectors y p ¼ (y1p , . . . , ynp ) defined by yip Ji (E ) þ Pi , and the condition
i ¼ 1, . . . , n,
Ch.9 A Core-Theoretic Solution for the Design of Cooperative Agreements
X
Pi ¼ 0:
i2N
With the condition just stated, we have indeed that 4.3
185
P i2N
yip ¼ w(N).
Alternative characteristic functions
It was observed at the end of Section 2.2 that for arguments S 6¼ N of the characteristic function (4) associated with our economic model, the function involves variables that represent strategic choices made by players who are not members of S. Because of this feature—a typical one when a cooperative game is associated with economies with externalities, such as ours—the characteristic function (4) should specify explicitly what the actions are both of the member of S and of the other players. To this effect, we shall consider the following two alternatives: 1. The cooperative game [N, wa ], defined by the characteristic function of the form X wa (S) ¼Min Ji (E) where, if S 6¼ N, Ej ¼ Ej0 8j 2 NnS: (Ei )i2S
i2S
Ej0
was defined in Assumption 1.) (Recall that This function reflects the assumption that when a coalition forms, its worth is what it gets when the players outside the coalition choose the strategy that is worst for it—that is, pollute up to Ej0 in our model. This form has been often used in economic models with beneficial externalities or with public goods production (see, for example, Foley (1970); Scarf (1971); and recently Chander (1993)) where it is a natural one because the worst strategy of nonmember of S is simply no action in such cases. With detrimental externalities as we have here, it is less natural to assume such an attitude: why should the nonmembers of S act in this way, and for the members of S, why should they necessarily expect the worst and behave in a minimax way? The first of these questions is also raised by Ma¨ler (1989–1993) in his discussion of cooperative games of transfrontier pollution, all the more rightly so that he does not assume in the economic model an upper bound such as our Ej0 for the individual emissions. The worst then becomes infinite amounts, which is hardly credible. While Ma¨ler concludes by dismissing the tool of the characteristic function, and as a consequence the core concept that is built on it,4 we choose to propose instead to consider the following alternative: 4
Another argument made is that with no bounds on the behaviors of players not in S, the worth of coalitions different from N can be reduced to zero, which renders them
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Public Goods, Environmental Externalities and Fiscal Competition
2. The cooperative game [N, wg ], defined by the characteristic function of the form X wg (S) ¼Min Ji (E) where, if S 6¼ N, Ej ¼ E~j 8j 2 NnS: (Ei )i2S
i2S
(Recall that E~j was defined in part 2 of Definition 2—definition of a partial agreement equilibrium with respect to a coalition.) The function wg is to be called the partial agreement equilibrium characteristic function. We assume here that when S forms, the other players break up into singletons, and act noncooperatively so as to reach an equilibrium in their best individual interest, given S. It is thus not assumed that they do the worst; nor is it assumed, as in the concepts of strong and coalition proof equilibria, that they do not react5 to the actions of S. In view of property (10) and of Proposition 2, one has that for each S, wg (S)#wa (S). This implies that the core of the game [N, wg ]— that is, the g-core—is, if nonempty, contained in the a-core and possibly smaller.
5.
An imputation in the g-core (and in the a-core) of the cooperative game
As it is well known that many cooperative games may have an empty core, the concept of a g-core imputation is only useful if we can establish its existence, at least for the cooperative game [N, wg ] that we have associated with the economic model. As far as the a-core is concerned, it was shown to be nonempty in games with externalities by Scarf (1971) as well as in the version given by Laffont (1977, p. 102) of the Shapley and Shubik (1969) well known garbage game.6 We proceed in this section in a constructive way—that is, by exhibiting an imputation for which we show that it has the property of belonging to the g-core. Economic interpretations are given in the next section. powerless. The core then becomes equal to the set of imputations, and the concept brings no more information than optimality. 5 Carraro and Siniscalco (1993), in a model with identical agents, assume instead that when S forms and achieves the aggregate payoff w(S), if some i 2 S leaves S, the coalition Sn{i} remains formed. They show that then, it may be better for i to leave S, and as this advantage grows with the size of coalitions, they conclude that only small coalitions can prevail, and N will never form. 6 Laffont also shows that for the garbage game, the emptiness claimed by Shapley and Shubik applies in fact to the b-core. For a game like ours, the a-core and the b-core coincide.
Ch.9 A Core-Theoretic Solution for the Design of Cooperative Agreements
187
Our result is not fully general, though, as we obtain it only under two alternative additional assumptions; either linearity of the damagecost functions Di or identical abatement-cost functions Ci for all countries i. We limit outselves here to the first case and refer the reader to Chander and Tulkens (1994) for the second one. Theorem. Let E ¼ (E1 , . . . , En ) be the (unique) optimal joint-emissions policy. Under Assumptions 1 through 3 and the linearity of all the damagecost functions Di , the imputation y ¼ (y1 , . . . , yn ) defined by yi ¼ Ji (E ) þ Pi , i ¼ 1, . . . , n, where
" # 0 X X D i Pi ¼ [Ci (Ei ) Ci (Ei )] þ 0 Ci (Ei ) Ci (Ei ) DN i2N i2N and D0N
X
(13)
Di0 ,
i2N
belongs to the core of the game [N, wg ]. Proof. (a) It is easily verified that y is an imputation—that is, X X yi ¼ wg (N) [Ci (Ei ) þ Di (E )], i2N
i2N
since X Pi ¼ 0: i2N
(b) Suppose now that the imputation y be not in the core. Then, there would exist a coalition S and partial-agreement equilibrium with respect to S, E~ ¼ (E~1 , . . . , E~n ), such that X X Ji (E~) < yi : (14) wg (S) ¼ i2S
i2S
Notice first that 8i 2 N S, E~i ¼ Ei in this partial-agreement equilibrium because Ei is a dominant strategy under linearity of the damage-cost functions. Moreover, from the first-order conditions that characterize a partial-agreement equilibrium one has 8i 2 S, E~i $Ei . Consider now the alternative imputation ^y defined by ^ i, ^ yi Ji (E ) þ P
i ¼ 1, . . . , n,
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Public Goods, Environmental Externalities and Fiscal Competition
where the transfers are of the form " # 0 X X D i ^ i ¼ [Ci (E ) Ci (E~i )] þ Ci (Ei ) Ci (E~i ) : P i D0N i2N i2N X
If we can show that X ^ Ji (E~) yi <
i2S
(15)
i2S
and that X X ^ yi , yi # i2NnS
(16)
i2NnS
then, given (14), the imputation yˆ induces an aggregate cost for all countries that is lower than w(N), the solution of (3)—an impossibility that proves the theorem. To show (15), let us write X
^ yi ¼
i2S
¼ ¼
X
Ji (E ) þ
X
i2S
i2S
X
X
Ci (E~i ) þ
i2S
i2S
X
X
i2S
Ci (E~i ) þ
^i P P
Di (E ) þ Di (E~) þ
i2S
P
0 i2S Di D0N
0 i2S Di D0N
" X
"
Ci (Ei )
i2N
D0N
X
# ~ Ci (Ei )
i2N
~)þ (Q Q
X i2N
Ci (Ei )
X
!# ~ Ci (Ei )
i2N
where the last line has been obtained by adding and subtracting P D (E~) to the previous one, and use has been made of the linearity i i2S of the functions Di as well as of the form (6) of the transfer functions. In this expression, a negative value of the term within square brackets can be derived from properties of the optimum E , of a partial agreement equilibrium w.r.t. a coalition and from the strict convexity of the abatement cost functions Ci . To show (16) starting from the fact that ^i ^ yi ¼ Ci (Ei ) þ Di (E ) þ P
" # 0 X X D i ¼ Ci (E~i ) þ Di (E ) þ 0 Ci (Ei ) Ci (E~i ) DN i2N i2N
and yi ¼ Ci (Ei ) þ Di (E ) þ Pi
" # 0 X X D i ¼ Ci (Ei ) þ Di (E ) þ 0 Ci (Ei ) Ci (Ei ) , DN i2N i2N
Ch.9 A Core-Theoretic Solution for the Design of Cooperative Agreements
189
it is sufficient to show that " # 0 X X D i Ci (Ei ) Ci (E~i ) Ci (E~i ) þ 0 DN i2N i2N " # 0 X X D i Ci (Ei ) Ci (Ei ) , 8i 2 NnS: #Ci (Ei ) þ 0 DN i2N i2N But this derives from the characterization of a partial-agreement ~ equilibrium w.r.t. a coalition with P linear functions PDi —namely, Ei ¼ Ei , a ~ dominant strategy 8i 2 NnS, and i2N Ci (Ei )$ i2N Ci (Ei ) according to Proposition 2. & Finally, the remark made at the end of Section 4 allows us to further state: Corollary. The imputation y also belongs to the core of the game [N, wa ].
6.
Conclusion: economic interpretation of the proposed g-core solution
6.1
The cost-sharing formula
As announced in the introduction, the essence of the paper is formula (13), which specifies a (net) monetary transfer for each country. Its core virtue lies in the fact that, given the international optimum E , and the cost Ji (E ) (both of abatement and of damage) that each country has to bear to implement it, the transfers yield to each country an effective net cost lower than the one they would bear under any partial agreement, including under strictly autarkic Nash equilibrium. Each individual transfer consists of two parts: a payment to each country i that covers its increase in cost between the Nash equilibrium and the optimum (first squared bracket of formula (13)), and a payment by each country i of a proportion Di0 =D0N of the total of these differences across all countries (second squared bracket of formula (13)). We already have proposed elsewhere (Chander and Tulkens (1992)) that an international agency be set up to handle these computations and payments. Recall that as they are specified, the transfers break even. Notice that if Di0 ¼ 0 for some country (because it would not be concerned as a recipient of the transfrontier pollution under discussion— or would allege not to be) while its abatement cost Ci0 is positive, that country then only receives the first component of the transfer, based on the abatement it does, and it pays nothing to the others; this leaves the country at an effective cost level equal to the one at the Nash equilibrium.
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In general, though, according to the second component of the formula, the contribution made by a country to the others is a fraction of the aggregate abatement costs (of moving from the Nash equilibrium to the optimum), the fraction being determined by Di0 =D0N —that is, its relative marginal damage cost to the sum of all countries’ marginal damage costs. In other words, each country’s contribution is determined by the relative intensity of its preferences for the public good component of the problem. 6.2
Information requirements
The core-theoretic property of the proposed solution was established in the framework of games of complete information. To put it in practice, the computation of the optimum E as well as of the associated transfers Pi stated in (13) requires full knowledge of the abatement- and damage-cost functions in all countries. As this information has to be provided by the countries themselves, there may be strong incentives for them to give false information. While theoretical analyses of this incentives problem are numerous in the literature,7 extending the present analysis in that direction is beyond the scope of the present paper. Let us observe, however, that in practice the problem can be eased to some extent by international inspections and audits. Ma¨ler (1989–1993) and Kaitala, Ma¨ler, and Tulkens (1995), for instance, use abatement-cost functions estimated on the basis of some plots produced by the Acid Rain project at IIASA. They also show how damage-cost functions can be calibrated from abatement-cost functions. 6.3
On cooperation and robustness against free riding
A core imputation is to be interpreted as a proposal made to all players for sharing w(N), having the property that no coalition S can improve on it for the benefit of its members by means of the payoff it can secure them. Because of this property, it is claimed that no coalition S is in a position to object to it. In the present context of an international externality, objecting against a proposed agreement may be seen as attempting to free ride against it. Our claim of robustness against free riding for the g-core solution is in the following sense: given the solution proposed to N, the all players set, if some coalition S envisages to free ride by seeking an arrangement of its own, the breaking up of the players not in S into singletons acting 7
Kwerel (1977), and Dasgupta, Hammond, and Maskin (1980) propose schemes in which truthful revelation is a dominant strategy but that do not balance the budget.
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rationally is sufficient to make this free riding less attractive to the members of S than the proposed solution. The threat we assume on the part of the non-free riders is thus the source of the deterrence to free ride; it is also what induces full cooperation. The essence of our contribution is in identifying that rational behavior on the part of singletons is sufficient for cooperation in that sense. 6.4
On the role of international transfers
The strategic role of monetary transfers appears clearly as soon as one realizes that polluting countries with nonzero abatement cost and weak preferences for removal of ambient pollutants never have an interest in cooperating toward abatement (let alone an optimal one) because the costs are higher to them than the benefits. This strategic aspect is also reflected in the two following remarks: (1) in our game, the core is empty in general if transfers are not allowed, and (2) when all players are identical, the (unique) core imputation is the one without transfers. It is thus, in a sense, the diversity of the agents that commands transfers when strategic considerations are at stake. Apart from strategic considerations, the payments involved by these transfers (the second bracketed term in formula (13)) are more in the spirit of the ‘‘victim pays’’ principle than of the ‘‘polluter pays’’ principle. This only reflects the fact that the ethical values that inspire the latter are here in opposition to the self-interest considerations that are called on to ensure voluntariness in cooperation and deter free riding. Finally, we show in Chander and Tulkens (1994) that the solution y can also be interpreted as an equilibrium concept, analogous to the one of ratio equilibrium and therefore of Lindahl equilibrium in public goods economies. Elaborating on this requires, however, the Arrow-Debreu setting used in that paper. 6.5
On the linearity assumption on the damage-cost function
While restrictive at a general level, the linearity assumption on preferences (that is, damage costs) may be seen as a mild one in the specific context of transfrontier pollution. On the one hand, at the empirical level, these functions are indeed extremely difficult to estimate. Mainly for that reason, Ma¨ler (1989–1993) and several of his followers have been satisfied with that assumption, all the more that, as he shows, it can be given a useful role in a Nash equilibrium situation. Another argument is that the optimum may lie far away from the situation prevailing at the time the negotiations begin, thereby increasing uncertainties. Techniques of economic computation have therefore been
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devised to move toward the optimum in successive steps (a recent example is given in Germain, Toint, and Tulkens (1994)). For the local information required to apply these techniques, linear functions (with parameters possibly varying over time) are surely not inappropriate.
References Barrett, S., 1992. International environmental agreements as games, in R. Pethig,(ed.). Conflicts and cooperation in managing environmental resources. Berlin: Springer-Verlag, 11–36. Bernheim, D., Peleg, B. and Whinston, M.D., 1986. Coalition proof Nash equilibria. I: concepts, Journal of Economic Theory, 42(1), 1–12. Carraro, C., and Siniscalco, D., 1993. Strategies for the international protection of the environment, Journal of Public Economics, 52, 309–328. Chander, P., 1993. Dynamic procedures and incentives in public good economies, Econometrica, 1341–1354. Chander, P. and Tulkens, H., 1992. Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems, European Review, 36(2/3), 288–299. (Chapter 6 in this volume). Chander, P. and Tulkens, H., 1997. The core of an economy with multilateral environmental externalities, International Journal of Game Theory, 26, 379–401. (Chapter 8 in this volume). Dasgupta, P., Hammond, P. and Maskin, E., 1980. On imperfect information and optimal pollution control, Review of Economic Studies, 47, 857–860. D’Aspremont, C. and Gerard-Varet, L., 1992. Incentive theory applied to negotiations on environmental issues: national strategies and international cooperation. Louvain-laNeuve: Center for Operations Research and Econometrics, Universite´ catholique de Louvain, mimeo. Dre`ze, J.H. and de la Valle´e Poussin, D., 1971. A taˆtonnement process for public goods, Review of Economic Studies, 38, 133–150. Foley, D., 1970. Lindahl solution and the core of an economy with public goods, Econometrica, 38, 66–72. Friedman, J., 1990. Game theory with applications to economics (2nd ed.). Oxford: Oxford University Press. Germain, M., Toint, Ph. and Tulkens, H., 1996. Calcul e´conomique ite´ratif pour les ne´gociations internationales sur les pluies acides entre la Finlande, la Russie et l’Estonie, Annales d’Economie et de Statistique (Paris), 43, 101–127. Hoel, M., 1992. Emission taxes in a dynamic international game of CO2 emissions, in R. Pethig (ed.). Conflicts and cooperation in managing environmental resources. Berlin: Springer-Verlag, 39–68. Kaitala, V., Maler, K-G. and Tulkens, H., 1992. The acid rain game as a resource allocation process, with application to negotiations between Finland, Russia and Estonia, The Scandinavian Journal of Economics, 97(2) 325–343. Reprinted in Carlo Carraro, (ed.), 2003. Governing the global environment: the globalization of the world economy. Cheltenham: Elgar Reference Collection. (Chapter 7 in this volume). Kwerel, E., 1977. To tell the truth: imperfect information and optimal pollution control, Review of Economic Studies, 44, 595–601. Laffont, J.J., 1977. Effets externes et the´orie e´conomique. Monographies du Se´minaire d-Econome´trie. Paris: Editions du Centre National de la Recherche Scientifique (CNRS).
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Ma¨ler, K.-G., 1989–1993. The acid rain game, in H. Folmer and E. Van Ierland (eds.). Valuation methods and policy making in environmental economics. Amsterdam: Elsevier, chapter 12, 231–252. Ma¨ler, K.-G., 1993. The acid rain game II, Beijer Discussion Paper Series 32. Stockholm: Beijer International Institute of Ecological Economics, Royal Swedish Academy of Sciences. Scarf, H., 1971. On the existence of a cooperative solution for a general class of n-person games, Journal of Economic Theory, 3, 169–181. Shapley, L. and Shubik, M., 1969. On the core of an economic system with externalities, American Economic Review, 59, 678–684. Tulkens, H., 1979. An economic model of international negotiations relating to transfrontier pollution, in K. Krippendorff (ed.), Communication and control in society. New York: Gordon and Breach, 199–212. (Chapter 5 in this volume).
Chapter 10 THE KYOTO PROTOCOL: AN ECONOMIC AND GAME THEORETIC INTERPRETATION
Parkash Chander, Henry Tulkens, Jean-Pascal van Ypersele and Stephane Willems In Economic Theory for the Environment. Essays in Honor of Karl-Go¨ran Ma¨ler. B. Kristro¨m, P. Dasgupta, and K.-G. Lo¨fgren (eds.), 2002. Cheltenham: Edward Elgar, chapter 6, 98–117.
Using both positive and normative economics, we examine the issues at stake in the current international negotiations on climatic change. After reviewing the main features of the Protocol, an elementary economic model serves to identify the main concepts involved: optimality, noncooperation, coalitional stability. Alternative concepts are reviewed that characterize the noncooperative equilibrium before the negotiations. Data suggest that the prevailing situation is a mixed one. Turning to the Protocol, we claim that the emission quotas are only a step in the direction of optimality; that for achieving these quotas, trading ensures efficiency, as well as coalitional stability, if it is competitive and adopted at the largest scale. Beyond, we show that worldwide coalitionally stable optimality is a real possibility with quotas and trading, subject to future agreement on appropriate reference emission levels, especially for developing countries.
1.
Introduction: cooperation at the world level, from Rio to Kyoto
Our central theme in this chapter is the one of cooperation at the world level on the issue of climatic change. We start from the facts and The authors are especially indebted to their Massachusetts Institute of Technology (MIT) colleague Denny Ellerman who kindly informed them of the state of his work and provided key numerical information. They also thank their colleague Jean Gabszewicz for a careful reading of an earlier version. Seminar presentations and discussions of this study have been made in 1999 at the third Fondazione ENI Enrico Mattei-Center for Operations Research and Econometrics (FEEM/CORE) workshop on coalition formation held at
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then try to enlighten them by means of ideas provided by economics and game theory. The negotiations on climate change that have been taking place since the late 1980s within the United Nations institutions1 are obviously a quasi-worldwide process, judging by the length of the list of countries2 participating. But these negotiations, prior to the Kyoto meeting, had led only to a ‘framework convention’, signed in 1992 in Rio de Janeiro, that was little more than a declaration of intentions. The real issue was then: are the continuing negotiations eventually going to lead to a sustainable agreement bearing on effective actions that is also worldwide? Or will they lead to a breaking up of the countries into separate blocs, each acting to the best of its own interests? The Kyoto Protocol, signed in December 1997, is the major development in the post-Rio evolution of these negotiations. Its importance lies mainly in the fact that it bears on effective actions to be taken by countries, actions that are recognized as binding commitments by them. However, according to the Protocol, not all countries have to take specific actions. As our summary presentation will report more in detail below, commitments to quantified emissions reduction or limitation are mentioned only for the so-called ‘Annex I’ parties.3 The role of the other countries in the agreement, while not ignored, is much less precisely specified. The natural question that then arises is whether the Kyoto Protocol is to be considered as just an ‘Annex I’ Protocol; or is it to be seen, after further thought and beyond the appearances, as a worldwide Protocol? Below, we defend the second thesis, first on the basis of our own conviction, but also because we think we can support it by means of wellestablished conceptual tools of economics and game theory. GREQAM, Aix en Provence (January 1999), at the United Nations Capacity Building Workshop, New Delhi (March 1999), at the Coventry conference on Environmental Economics (May 1999), as well as at the third Toulouse conference on Environmental and resource economics (June 1999). This paper was circulated earlier as CLIMNEG working paper No. 12, April 1999. The research on which it is reported here is part of the program ‘Changements climatiques, Ne´gociations internationales et Strate´gies de la Belgique’ (CLIMNEG), supported by the Belgian State’s Services du Premier Ministre, Services fe´de´raux des Affaires scientifiques, techniques et culturelles (SSTC), Brussels. See www.core.ucl.ac.be/ climneg. 1 For a thorough account of the scientific evidence on the state of the problem, the reader is referred to the work of the Intergovernmental Panel on Climate Change (IPCC), and in particular to the contribution of its Working Group III (IPCC 1995). 2 There were 178 in Rio, 159 in Kyoto and 161 in Buenos Aires. 3 ‘Annex I’ (to the Rio Convention text) countries are the Organization for Economic Cooperation and Development (OECD) countries, those of the former Soviet Union and those of the Eastern European economies in transition.
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For the applied economist, the Kyoto Protocol offers an exceptionally rich combination of opportunities to put theory into use, both in a positive and a normative way – that is, for explaining events as well as for advising on decision making. Indeed, several strands of theory are involved as we shall show: externalities of the Samuelsonian public good type, Nash noncooperative equilibria, worldwide Pareto optimality, cooperative solution concepts, and finally Walrasian market equilibria with their Edgeworthian coalitional stability. All these are involved!
2.
Main features of the Protocol
We briefly state here what the main features of the Protocol are, from the point of view of our arguments to follow: 1. Dated emission quotas, expressed in percentages of 1990 emissions, are established for Annex I countries, to be met around 2010. 2. The principles of (a) emissions trading by countries (or by their nationals) and of (b) joint implementation are established for Annex I countries. 3. A clean development mechanism (CDM) is established as a way to involve non-Annex I countries (especially developing ones) in some particular form of joint implementation and emissions trading. No explicit provision in the Protocol mentions the introduction of targets for non-Annex I countries. But it is expected that this will take place in the future through the general review clauses of the Protocol and of the Convention. Trade in emissions will be allowed only among countries that do ratify. It is also expected that it will not be allowed with the countries that would not fulfill their obligations under the Protocol. Finally, the Protocol comes into force automatically4 only if (i) at least 55 parties to the Convention have ratified it, and (ii) these 55 parties include a number of Annex I parties accounting for at least 55 percent of the base year CO2 emissions of all Annex I parties to the Convention. While parties are committing themselves to proceed to enforcement within their country, no sanctions are specified if a ratifying country does not fulfill its obligations under the Protocol, except for the above provision on being excluded from emissions trading. A compliance regime, including possible sanctions for non-compliance, is yet to be specified in the process of future negotiations.
4 In Kyoto, the text of the Protocol was adopted unanimously by the delegates of the 159 countries that participated in the negotiations. Signature of the text by governments and ratification by parliaments are the following stages of the process.
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Public Goods, Environmental Externalities and Fiscal Competition
Economics of the issues at stake
Consider the n countries of the world (indexed below i ¼ 1, . . . , n), who each enjoy an aggregate consumption level xi , equal to the aggregate value of their production activities yi , minus the damage Di consisting in lost production that results from global pollution. Each country i’s productive activity entails some amount of polluting emissions ei , which are related to production according to the increasing and strictly concave production function5 yi ¼ gi (eiP ). Damages in each country are generated by the total of such emissions, ni¼1 ei ; they affect production possibilities in each country6 in a way that P is usually represented by an increasing damage cost function Di ¼ di ( nj¼1 ej ) which for simplicity we assume to be linear. Each country’s consumption possibilities are thus given by the expression: n X xi ¼ gi (ei ) di ej , (6:1) j¼1
where di > 0 is thus the damage cost per unit of emission or, equivalently, the benefit per unit of abatement (for decreasing Sej ). 3.1
World optimality
Ignoring distributional issues, world consumption optimality can P then be represented by the consumption levels that maximize ni¼1 xi with respect to the n variables e1 , . . . , en . Let (e1 , . . . , en ) be the vector of emission levels in the n countries that achieve such a world optimum. First-order conditions for a maximum are given by the following system of equations expressing equality between the marginal cost of global damages and the marginal abatement cost of each party i: g0i (ei )
n X
dj ¼ 0, i ¼ 1, . . . , n:
(6:2)
j¼1
We shall develop our arguments below under the assumption that climate change negotiations are aiming at achieving such a world optimum. Attaining it requires coordination among the countries, so as to ensure that each one of them does take into account the effect of its emissions on the other countries as reflected by their damage cost functions. 5
We may think of ei either as the energy input in the production, or as the pollution emission, assuming that a unit of energy generates a unit of pollutant as a byproduct. Accordingly, gi0 (ei )( ¼ dgi (ei )=dei ) may be interpreted either as the marginal product of energy or (for decreasing ei ) the marginal cost of abatement, depending upon the context. 6 Numerical estimates of damages in some regions of the world are given in Table 6.1 below.
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3.2
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Noncooperative equilibrium
It is indeed often argued7 that in the absence of coordination, countries choose emission policies that best suit their own interest, taking as given what the other countries do. This leads us to consider that a noncooperative equilibrium of some sort prevails between countries if no negotiations take place. How would a country determine its best emissions levels? The answer is not immediate since imposing low emissions on itself implies low net production according to the function gi , whereas allowing for high emissions entails high damage costs according to the function Di . Classical economic reasoning suggests that a rational domestic optimum for each country would be one that best balances these two aspects; it is achieved by maximizing its own consumption level xi with respect to ei as defined in (6.1), taking as given all variable ej with j 6¼ i. If all countries adopt such a behavior, a Nash-type equilibrium between countries prevails, which we represent by the vector of emissions8 (e1 , . . . , en ). For each country i, the first-order condition of its maximizing behavior is given by the equation: g0i (ei ) di ¼ 0
(6:3a)
while its achievable consumption level is: i ¼ gi (ei ) di x
n X
ej :
(6:3b)
j¼1
Two characteristics of the noncooperative equilibrium so defined are essential for our purposes: (i) the equilibrium emissions (e1 , . . . , en ) are clearly not a world optimum as can be seen from comparing (6.2) and (6.3a and b): that is why negotiations are necessary; and (ii) ei > ei Pn for each i since gi is concave and j¼1 dj > di for all i’s; thus, world optimal emissions are lower than those prevailing at the noncooperative equilibrium. 3.3
Coalitional stability for the treaty
The basic reason for the nonoptimality of the Nash equilibrium is a well-known externality argument. Each country decides its emission level ei without concern for the effects on other countries: it thus equates its marginal abatement cost, g0i (ei ), to its own marginal damage cost, di , 7
See, for example, Chander and Tulkens (1992). Uniqueness of this vector is ensured under our assumptions of concavity of the functions gi and of linearity of the functions Di . 8
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whereas a world optimum requires each country to equate itsPmarginal abatement cost to the aggregate world marginal damage cost, ni¼1 di . Furthermore, a world optimum may require from the various countries different levels of abatement ei ei , entailing costs and benefits that are a priori by no means identical across them: some may have high abatement costs while having only small damage costs to avoid, whereas other countries may have low abatement costs while facing high damage costs. To have the world optimum voluntarily agreed upon by all countries requires in addition that for each country and for each group of countries the benefits exceed the costs of abatement. Because of the asymmetries just mentioned, this can be achieved only by means of appropriately designed resource transfers from the net gainers to the net losers. To that effect Chander and Tulkens (1997) have proposed that the optimal emission levels (e1 , . . . , en ) specified in the treaty be accompanied by a scheme of transfers (T1 , . . . , Tn ) which are of the form: " # n n X X d i ej ) Ti ¼ [gi (ei ) gi (ei )] n gj ( gj (ej ) , P j¼1 (6:4) dj j¼1 j¼1
i ¼ 1, . . . , n, where Ti > 0 if the transfer is received by country i, while Ti < 0 if the transfer is paid.9 The first expression within square brackets is equal to the abatement cost borne by country i from moving from its Nash equilibrium level of emissions ei to the level ei prescribed by world optimality. As this amount is positive in (6.4), the role of this part of the transfer appears to be to cover that cost increase for i. The second expression within square brackets is the world total over all countries of their emissions abatement cost from the Nash equilibrium levels to Pthe world optimal ones – also a positive magnitude. With the ratio di = nj¼1 dj and taking account of the negative sign, the second term in (6.4) thus determines a contribution of country i, which Pnis specified as a fraction of the aggregate abatement cost. Clearly i¼1 Ti ¼ 0, so that these transfers would ensure a balanced budget if an international agency were established for implementing them. Note also the role played by the reference emission levels ei in the design of the transfers – a feature whose importance will be highlighted below. The ‘coalitional stability’ property claimed above for the Chander– Tulkens proposal of optimality with transfers is that, in addition to 9 The transfers are expressed here in units of physical goods. The issue whether it is preferable that such transfers be financial rather than in real terms is an important one, but we cannot deal with it here.
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making each country individually better off compared to the Nash equilibrium, it also makes every group of countries better off, compared to what they could get by adopting any alternative arrangement among themselves, be it in terms of emissions, or transfers, or both. For further reference in our arguments below, let us be more precise on this property. Let W ¼ {i ¼ 1, . . . , n} denote the set of all countries of the world and S W be any subset, or ‘coalition’ of countries. Then the best outcome that the members of S could obtain by making arrangements among themselves only – to be called a ‘partial agreement Nash equilibrium with respect to S’ (PANE w.r.t. S) – is the one resulting from the emissions policy (~e1 , . . . , ~en ) defined by, for the members of S, 2 3 X X X X ~ej )5, (~ei )i2S ¼ arg max4 gi (ei ) ( di )( ei þ i2S
i2S
i2S
j2W nS
and for the countries not in S: " # n X ~ej ¼ arg max gj (ej ) dj (ej þ ~ei ) , j 2 W nS: i6¼j
A PANE w.r.t. S is thus a Nash equilibrium between the countries in S acting jointly and the remaining countries acting individually. It can be characterized by the first-order conditions: X g0i (~ei ) ¼ dj , i 2 S j2S
and g0i (~ei ) ¼ di , i 2 W nS:
P A comparison of these conditions with (6.2) implies ei # i2S ~ ~ i2S ei and ej ¼ ej , j 2 W nS. Since in a PANE w.r.t. S the countries within the coalition coordinate their emissions so as to take into account their effect on each other, their emissions are lower compared to the Nash equilibrium. The emissions of the countries outside of the coalition are, however, not lower. In fact, they might be higher if the damage function is convex but not linear.10 Moreover, since total world emissions are lower in a PANE w.r.t. S, the countries outside the coalition are better off, although that is not the intention of the coalition. P
10 This might also happen when the countries outside the coalition are not acting rationally but following the business-as-usual policy, since the abatement by the coalition S might result in lower energy prices in the rest of the world. Ellerman and Decaux (1998) observe this phenomenon in their computable general equilibrium model, and call it ‘leakages’.
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Public Goods, Environmental Externalities and Fiscal Competition
Statics versus dynamics
Thus far, and for most of the sequel, the above quantities xi and ei are considered to be flows per unit of time. The damages from climate change are, however, induced less by the flow of greenhouse gas emissions than by the increase11 DS in their accumulated stock S in the atmosphere. At each time t, DSt is thus determined by a relation of the form: DSt ¼ dSt1 þ k
n X
eit , t ¼ 1, 2 . . .
i¼1
where according to climatic science common wisdom d ffi 0:01 and k is of the order of 0.5 (and slowly increasing over time). The issues at stake have thereby an inherently dynamic component that is by no means ignored in the economics literature on climate change; see, for example, Nordhaus and Yang (1996), or, for our part, Germain, Toint, Tulkens and de Zeeuw (2003). One might therefore consider that world optimality is not to be defined in terms of just one-period emission, production and consumption levels as we have done but, instead, of multiperiod emission trajectories {(e1t , . . . , ent )t¼1, 2,... } and similarly for production and consumption. While this more elaborate modeling has its merits, it turns out to be unnecessary for our purposes. Indeed, one may have noted that the specific object of the Kyoto Protocol is not trajectories of emissions: it is emissions levels at some point in time (around 201012). We therefore feel justified in working, in the present chapter, with the usual ‘static’ or oneperiod model.13
4.
NonCooperative characterizations of the pre-agreement stage
The noncooperative behavior described in Section 3.2 is not the only one conceivable of this kind. Indeed, the fulfillment of conditions (6.3a) and (6.3b) which characterize it requires domestic policies to be designed and implemented, involving an energy tax or appropriately priced pollution permits, so that the energy price including the tax or the permit unit price is equal to the domestic marginal damage cost di . These belong to the class of what is often called ‘no-regrets policies’. 11
Usually taken with respect to pre-industrial times. Actually an average over the years 2008–12. 13 Nash equilibrium and optimal trajectories are determined and discussed in Germain et al. (1998). 12
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However, not all countries can be said to have adopted such a nationally rational course of action. For instance, industrial firms in some countries may have strong lobbying power and use it so as to obtain low energy prices. While still choosing, as profit maximizers, energy use and emissions so that g0i is equal to the price of energy (denoted henceforth as pi ), this results into emissions ei higher than ei and such that g0i (ei ) < di , thus successfully preventing the nationally rational policy to be adopted. If this behavior is assumed to prevail in all countries, a different equilibrium – equally noncooperative – results, called by Nordhaus and Yang (1996) the ‘market solution’ (or ‘business as usual’, according to others). Alternatively, energy-importing countries facing balance of payments problems may have introduced high taxes and domestic prices of energy: their emission levels ei are then likely to be such that g0i (ei ) > di . Finally, another reason why a nationally rational policy may not come about is that firms in a country may simply not be profit maximizers, as is particularly the case with large public sector enterprises of non-market economies. In such cases, the domestic equilibria are neither of the ‘market’ nor of the ‘nationally rational’ type, and energy prices do not induce any well-defined emissions policy – except for the fact of a generally low concern for economical use of energy. Our point in this section is that in the situation prevailing at the pre-negotiation stage, all three types of country behavior are likely to be present, and we wish to illustrate this empirically with the data in Table 6.1. We first note a similarity between the structure, across some major countries, of the average energy prices for three kinds of fossil fuels (first three columns) on the one hand, and of the marginal abatement costs (fourth column) on the other hand. In particular, it is seen that the energy prices in the US are systematically lower and so is the marginal cost of abatement.14 Moreover, for the three developed regions the USA, the European Union (EU) and Japan which are also market economies, the higher the energy prices, the higher the marginal abatement costs.15 For the other countries we cannot say much, not only because of lack of data but also because they are either nonmarket or less developed, or both. Second, we have an opportunity to characterize some domestic policies by using equations (6.3a) and (6.3b) – according to which in countries that choose their emission levels rationally, that is, in the ‘noregrets’ sense, the marginal damage cost from emissions must be equal to 14 In the case of Japan, the marginal cost of abatement may look exceptionally high, but this is because of its large dependence on nuclear energy and natural gas. 15 Coal in Japan is a noticeable exception; but its use there is considerably lower.
Table 6.1 Retail prices (in US$ per unit) of industrial fossil fuels, marginal abatement cost and damage cost in selected countries or regions
USA EU Japan India FSU China
Heavy fuel oil for industry (per ton)
Steam coal for industry (per ton)
Natural gas for industry (per 10 kcalGCV)
Marginal abatement cost/ton of carbon for first 100 Mton reduction ($)
Annual damage cost as % of GDP
138.00 187.40 172.86 191.15 na na
35.27 76.00 49.90 19.36 na na
136.62 182.00 423.12 na na na
12 40 350 22 22 3.5
1.3 1.4 1.4 na 0.7 4.7
Sources: Energy Prices and Taxes (1996); Ellerman and Decaux (1998);
Fankhauser (1995).
Type of domestic equilibrium 0
gi (ei ) ¼ pi < di 0 gi (ei ) ¼ pi $di 0 gi (ei ) ¼ pi $di ? ? ?
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the marginal abatement cost and also to the average energy prices. Indeed, the data in the table reveal that marginal abatement costs are lower in the USA compared to the EU and Japan (they are even lower than those of a developing country like India). Now, it can hardly be the case that the marginal damage cost for the USA, the largest economy, is lower than that of the EU or of Japan. Therefore, we have an indication that in the USA, decisions regarding emissions are determined by the businessas-usual policy rather than optimized at the national level. We indicate in the last column the type of pre-negotiation domestic equilibrium we conjecture from the data to prevail in each region. What is the relevance of these observations for our purposes in this chapter? While the optimum emissions (el , . . . , en ) are, as seen from (6.2), independent of those at the pre-negotiation stage, the transfers Ti defined in (6.4) may have to be modified with the ei ’s substituted by the actual emission levels of each country i as they are described here. Does such a substitution affect the coalition stability property of the transfers? The answer is no,16 as long as one can assume that countries do adopt the same behavior at the pre- agreement stage and at a PANE when not in the coalition. For the sake of simplicity, however, we shall continue to consider the ei ’s as Nash equilibrium emission levels.
5.
Kyoto quotas and worldwide trading: efficiency and coalitional stability of the scheme
5.1
The Kyoto quotas: not optimality, but a step in the right direction
While it is straightforward to define and characterize a world optimum in theory, as we have done in Section 3, implementing it is undoubtedly difficult in practice, for several reasons among which we identify four. First, determining optimal emissions at the world level requires knowledge of, and agreement on, what the aggregate marginal damage P costs ni¼1 di are, as well as the countries’ marginal abatement costs g0i (ei ). While ‘objective’ technical studies can provide some of that information, one can expect that, due to the huge interests at stake in many segments of all concerned economies, pressures are exerted for either concealing or simply not collecting the statistical material required. Second, because the achievement of a stable world optimum may require, as noted earlier, resource transfers between the countries to 16 In technical terms, this is because levels of the ei ’s higher than those of a Nash equilibrium induce a larger core for the game whereby Chander and Tulkens (1997) establish the coalitional stability property of the transfers (6.4).
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Public Goods, Environmental Externalities and Fiscal Competition
compensate those for which net benefits, that is, benefits minus costs, are low or negative, institutions or mechanisms that hardly exist today are needed to implement such transfers. Third, the reference emission levels ei , which play a role in the design of the transfers (6.4) ensuring coalitional stability, may themselves be considered unfair, typically by those countries that are in the early stages of their economic development: they currently have comparatively low emission levels, while developed countries have high ones. In the future, when they will be developed, currently developing countries will have higher emissions and they might argue that these should be used as reference levels instead of those of today. Finally, if reductions in emissions ei ei , are very large (as proposed by some countries), they are simply not politically feasible, at least in the short run. In fact, the Kyoto Protocol requires only relatively small reductions for the immediate future (the next 15 years), leaving further reductions for later periods. For all these reasons it is difficult to assess whether the emissions reduction chosen by the Kyoto signatories correspond to world optimal emissions. Yet, countries in Kyoto have agreed upon some scheme of quotas on their emissions. Denote this scheme by the vectorP (^e1 , . . . , ^en ) where ^ei is the the so-induced quota on emissions of country i and write ^e ¼ ni¼1 ^ei for P aggregate reduced emissions.17 Because ^e is lower than e ¼ ni¼1 ei , that is, the total sum of emissions in 1990, these aggregate reduced emissions are certainly a step in the right direction since irrespective of whether 1990 emissions are business-as-usual or no-regrets policies, both imply too large emissions with respect to the world optimum. 5.2
Emissions trading: efficiency of equilibrium
If the Kyoto aggregate emissions reduction to ^e is achieved by letting each country abide by its emissions quota and simply emit up Pnto ^ ei ¼ ^eP , the ensuing aggregate gross world production, y ¼ i def i¼1 ^ yi ¼ ni¼1 gi (^ei ), may not be the highest achievable level. If so, the national policies ei ¼ ^ei for each i would be inefficient. Alternative specifications of the countries’ emissions ei all achieving ^e, are conceivable. In fact, recalling (6.1), the highest possible world consumption levels compatible with ^e would be those given by the vector ^e ¼ (^e1 , . . . , ^en ), which solves the problem:
17
Note that for all non-Annex I countries, we have e^i ¼ ei .
Ch.10 The Kyoto Protocol: An Economic and Game Theoretic Interpretation
max
n X
xi ¼
i¼1
n X
[gi (ei ) di
i¼1
subject to
n X i¼1
ei ¼
n X
^ej ]
207
(6:5)
j¼1 n X
^ei :
(6:6)
i¼1
How are these efficient emission levels to be determined? With appropriate information on the production (or abatement cost) functions gi , this could be done by computation. However, having argued above that such information is hard to come by, it is likely that strong opposition would arise against the computed emission levels, and in particular against those that would be larger than ^ei , which is indeed a possibility! We want to show presently that the desired efficient emission levels are precisely those that a competitive market equilibrium in tradable emission quotas would determine; in other words, that tradability of quotas automatically solves the problem (6.5)–(6.6). To this effect, note first that the vector e^ ¼ (^ e1 , . . . , e^n ) we are interested in identically solves the problem of maximizing aggregate gross production: max
n X
[gi (ei )] subject to
i¼1
n X
ei ¼
i¼1
n X
e^i ,
(6:7)
i¼1
P because the dropped term di nj¼1 e^j is a constant in (6.5). Next, define a competitive emissions trading equilibrium with respect to (^ e1 , . . . , e^n ) as a vector of national emissions (^ e01 , . . . , e^0n ) and a price g^ > 0 for CO2 (expressed in units of consumption goods per unit of CO2 emission) such that for each country i ¼ 1, . . . , n, ei ei )], e^i0 ¼ arg max [gi (ei ) þ g^(^
(6:8)
and n X i¼1
e^i0
¼
n X
e^i :
(6:9)
i¼1
In such a competitive emissions trading equilibrium, the countries (typically their firms, but conceivably also other economic agents) freely trade in their pollution rights, equal to their emissions quotas (^ e1 , . . . , e^n ), at the given price g^, and at that price, demand and supply of pollution rights are equal.18 The magnitudes g^(^ ei e^i0 ) represent the value, in 18 Existence and uniqueness of a competitive emissions trading equilibrium follow from concavity of the functions gi and continuity arguments.
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Public Goods, Environmental Externalities and Fiscal Competition
private goods, of payments for the purchase, at world price g^, of quotas if (^ ei e^i0 ), the amount purchased, is negative, or of receipts from the sale of quotas if (^ ei e^i0 ), the amount sold, is positive. Clearly the vector (^ ei0 , e^n0 ) defined by (6.8)–(6.9) is also the one that solves (6.7), hence (6.5)–(6.6) and thereby maximizes world consumption, since: n X
gi (^ ei0 ) þ g^
i¼1
n X
(^ ei e^i0 ) ¼ max
i¼1
n X
gi (ei ) subject to
i¼1
n X i¼1
ei ¼
n X
e^i :
i¼1
As a confirmation, it can be seen from the first-order condition for (6.8) that at the price g^ the equality gi (^ ei0 ) ¼ g^ is satisfied for all i’s, 0 0 ei ) ¼ gj (^ ej ) for all i, j ¼ 1, . . . , n, a necessary condition implying that gi (^ for a solution of (6.5)–(6.6). Trading thus allows countries to achieve the aggregate emissions reduction e^ with the highest level of world consumption compatible with this reduction or, in other words, at the lowest opportunity cost for the world. This holds even if for some countries (^ ei e^i0 ) is negative: the point is indeed that world consumption be maximized and not that all countries necessarily emit e^i0 lower than e^i . 5.3
Emissions trading: coalitional stability of equilibrium
If trade in emissions is allowed another question arises: will there be blocs of countries forming in emissions trading? We answer the question in this section by means of a simple argument based on the theory of market games. Let S W be a bloc of countries whose members would decide, given the vector (^ ei )i2S of their individual Kyoto quotas, toP adopt some joint policy of their own for meeting their aggregate quota, i2S e^i , such as, for example, trading only among themselves, or engaging in other bilateral or multilateral agreements that fulfill the same condition. To characterize the economic effect of the formation of such a bloc, define: X X X y(S) ¼ max gi (ei ) subject to ei ¼ (6:10) e^i , i2S
i2S
i2S
that is, is the maximum total gross19 output that the countries in bloc S can hope to achieve jointly, given their aggregate emissions constraint. Consider now again (^ e10 , . . . , e^n0 ), the world competitive emissions trading equilibrium with respect to (^ e1 , . . . , e^n ). If we can show that the 19
We need not subtract damages as they are already fixed in the aggregate by the aggregate emissions constraint.
Ch.10 The Kyoto Protocol: An Economic and Game Theoretic Interpretation
209
members of S are better off at that worldwide competitive equilibirum than at their best actions as a separate bloc (as identified in (6.10)), we shall have established that bloc S has no interest in forming, thus answering in the negative the question raised in this section. This is in fact straightforward. Indeed, with our notation it is equivalent to showing that: X [gi (^ ei0 ) þ g^ þ (^ ei e^i0 ] $ y(S), (6:11) i2S
that is: X X [gi (^ ei0 ) þ g^(^ ei e^i0 )] $ gi (~ei ), i2S
i2S
where (~ei )i2S is the solution to (6.10). Clearly, we have Hence we must show that: X X X X 0 ~ei gi (^ ei ) þ g^ gi (~ei ): e^i0 $ i2S
i2S
i2S
P
^i i2S e
¼
P
ei : i2S ~
i2S
But this inequality is true since from concavity of gi we have for each i in S: gi (^ ei0 ) þ g^(~ei e^i0 ) $ gi (~ei ),
(6:12)
using the fact that g^ ¼ g0i (^ ei0 ) at the world competitive emissions trading 20 equilibrium. Repeating this argument for any conceivable bloc of countries leads to the conclusion that no bloc has an interest in forming,21 once a competitive emissions trading equilibrium prevails at the world level. We have thus shown that the outcome of competitive trade in emissions among the countries cannot be improved upon by the formation of coalitions of countries, such as, for example, trading blocs. We are thereby rediscovering – in fact, just applying – a general property of market equilibria known as their ‘core’ property, which says that such equilibria belong to the core of some appropriately formulated cooperative game.22 This is irrespective of whether (~ei e^i0 ) is positive or negative. Not only no bloc S taken in the aggregate, as formulated in (6.11), but also each member of the bloc, as (6.12) shows. 22 In technical game-theoretical terms, the expression y(S) defined above is the payoff that S canPachieve for its members and any vector (ei )i2S that meets the condition P ^i is an emissions strategy for S. Then the pair [W, y] satisfies the definition i2S ei ¼ i2S e of an n-person game in characteristic function form where y is the function y: 2W ! R defined by (6.10). A strategy for Pthe grand coalition W , (ei )i2W , is said to be in the core of this game if for each S W , i2S gi (ei )$y(S). That there exists such a strategy, that is, that the core of our game is non-empty can be asserted in general terms by showing that the game is balanced (in the sense of Shapley (1967)). But we provide above the same 20 21
210
5.4
Public Goods, Environmental Externalities and Fiscal Competition
Desirability of worldwide and competitive trading
While the Kyoto Protocol can be seen as allowing for trading in emissions among the Annex I or more parties, it leaves open the questions of the extent and nature of such trading.23 Economic and game-theoretic considerations can be further called upon to resolve these questions. As to the extent of trading, that is, the number of participants in the trade, market equilibrium theory makes a case in favor of emissions trading with the largest number of traders possible. Thus, worldwide emissions trading is desirable. This is implied by the previous argument on subgroups, be they trading blocs or any other form of ‘coalitions’. Indeed, if it is not to the benefit of any such subgroup of countries to form and act independently of the other countries, the outcome is also not more beneficial for these other countries, if a subgroup were to form. This is because their only best actions would be to act also as a subgroup, and for this subgroup the inequality (6.11) also applies. We claim on that basis that it is in the world’s overall economic interest that non-Annex I countries, whose emissions are not subject to quotas, nevertheless be allowed to participate in the trading process. The CDM contains provisions to that effect. A policy implication of our claim is that this mechanism be designed so as to make it as open as possible to the largest number of countries. The fact that no quota was assigned to many countries is irrelevant to the beneficial property, both for the world in general and for those countries in particular, of a worldwide emissions trading equilibrium. As to the nature of trading, the same body of theory advocates that the institutions governing the trades be designed so as to ensure that they are as competitive as possible – competitiveness meaning here that all participants behave as price takers. It is indeed only for markets with that property that efficiency, coalitional stabiltiy and worldwide maximal benefits are established. Regulatory provisions that would result in restricting competitiveness in the emissions trading process are thus to be avoided, and absence of regulations designed to prevent restrictions to competition should be positive answer in an economically more interesting way by exhibiting an actual strategy for W – namely the equilibrium outcome of worldwide competitive emissions trading – that we show to belong to the core. Note that the cooperative game defined here (and hence its core) is not the same as the one proposed in Chander and Tulkens (1995, 1997). The present game is a pure market game in the sense of Shapley and Shubik (1969), where externalities play no role since, once the quotas are fixed, the public good aspect of the problem disappears. One is left with only the private goods-type problem of allocating the emissions between the countries. It is worth pointing out, finally, that the game is one for an economy with production, and not of the usual pure exchange type. 23 These were addressed at the subsequent Conferences of Parties.
Ch.10 The Kyoto Protocol: An Economic and Game Theoretic Interpretation
211
avoided as well. For example, provisions allowing for market power to be exerted by some traders so as to influence price formation to their advantage, as well as regulatory controls that would impede sufficient price flexibility; or, as proposed by some, the capping of the quantities that participants are allowed to trade on. As is well known, the larger the number of participants, the more competitive the market is likely to be: our argument favoring a large extension of the market is thus also one that favors competition.24 Large numbers are admittedly neither the only way nor a sufficient condition to ensure the competitive character of a market, but they are a powerful factor. 5.5
A numerical illustration
Using the carbon emissions reduction commitments made by the Annex I parties to the Kyoto Protocol, as well as the marginal cost abatement curves generated by MIT’s EPPA model (which is a multiregional, multisectoral computable general equilibrium model of economic activity, energy use and carbon emissions), Ellerman and Decaux (1998) develop a method for estimating quantitatively the outcome in 2010 of various trading regimes, including the world competitive emissions trading equilibrium. They highlight the substantial differences in the outcome of the various trading regimes – confirming our theoretical claim of maximal efficiency of worldwide trading, but they leave open the question of which one might be agreed upon by the parties to the Protocol. Our analysis above brings an answer to this question, again in favor of world competitive emissions trading, based on showing that strategic behavior of coalitions of countries cannot be more beneficial to them than world-wide emissions trading. For illustrative purposes we reproduce here (see Table 6.2) Ellerman and Decaux’s estimate of the world competitive ^ at that time 24.75 emissions trading equilibrium in 2010 and of its price g US$/ton.
6.
Beyond the Kyoto quotas: the possibility of a scheme achieving a world coalitionally stable optimum
The outcome of the competitive emissions trading equilibrium with en ) is described in the last row of respect to the Kyoto quotas (^ e1 , . . . ,^ 24 With large numbers, our previous argument on the role of markets to achieve coalitional stability is also reinforced by a central result in economic theory (due to Debreu and Scarf 1963, elaborating on Edgeworth 1881) according to which the only coalitionally stable outcome (in our case, the only emissions allocation with that property) is the competitive one.
Table 6.2 The Ellerman and Decaux characterization of the world competitive emissions trading equilibrium with respect to the Kyoto quotas
Reference non-cooperative emissions in 2010 (Mton) ei Kyoto quotas of permitted emissions (Mton) e_ i Post-trading emissions reductions (Mton) 0 ei ^ei Emission permits (Mton) imported (þ) / exported () 0 ^ei ^ei Marginal cost of abatement ($/ton) 0 0 ^ ¼ gi (^ei ) g Total cost of own abatement ($ bn) 0 gi (ei ) gi (^ei ) Cost (þ) / receipt () of emission permits exports / imports ($ bn) 0 ^ ¼ (^ei ^ei ) g
USA
JPN
Annex I countries EEC OOE EET
EEX
CHN
Non-Annex I countries IND DAE BRA
1838.25
424.24
1063.27
472.04
394.76
ROW
World
873.32*
927.39
1791.96
485.76
308.32
97.27
531.61
9208.63
1266.67
280.05
756.51
300.66
247.45
873.32
927.39
1791.96
485.76
308.32
97.27
531.61
7866.95
186.22
12.33
74.96
60.07
52.98
213.36
52.54
447.93
104.87
42.78
2.50
91.07
1341.61
385.36
131.86
232.25
111.31
94.33
213.36
52.54
447.93
104.87
42.78
2.50
91.07
0.07
24.75
24.75
24.75
24.75
24.75
24.75
24.75
24.75
24.75
24.75
24.75
24.75
24.75
1.77
0.15
0.76
0.44
0.46
0.86
0.57
4.49
1.01
0.47
0.03
0.86
11.86
9.54
3.26
5.75
2.75
2.33
5.28
1.30
11.09
2.60
1.06
0.06
2.25
0.00
FSU
Notes: Annex I countries: USA; Japan (JPN); European Union, 12 countries (EEC); other OECD countries (OOE); Eastern Economies in Transition (EET); Former Soviet Union (FSU). Non-Annex-I: Energy Exporting Countries (EEX); China (CHN); India (IND); Dynamic Asian Economies (DAE); Brazil (BRA); Rest of the World (ROW). For non-Annex-I countries: Kyoto quotas of permitted emissions ^ei have been taken to be equal to their estimated noncooperative emissions in 2010, that is, ei , since it was agreed that their emissions need not be capped in this round of negotiations. For FSU, we have taken the reference emissions, ei , to be equal to the Kyoto commitment (873.32), although the actual emissions have been estimated to be only (762.79). This is equivalent to giving credit for emission reductions that would happen in any case. Source: Ellerman and Decaux (1998, Table G; August version).
Ch.10 The Kyoto Protocol: An Economic and Game Theoretic Interpretation
213
Table 6.2. It is seen that it results in monetary transfers among the countries and equalizes their marginal costs of abatement. This equilibrium thus very much looks like the worldwide treaty described above (Chander and Tulkens (1997)) which also requires transfers among the countries and equalizes their marginal costs of abatement (see (6.2) and (6.4)), except for the fact that that treaty leads to the worldwide optimal emissions (e1 , . . . , en ) while the Kyoto quotas do not. This prompts our final question: could an appropriate emission quotas and trading scheme of the Kyoto type nevertheless be used to reach a world optimum with the same coalitional stability property as ensured by the Chander–Tulkens transfers? The answer is yes, because that optimum can be shown to be equivalent to an emission quotas and trading scheme. To that effect define quotas (^ e1 , . . . , e^n ) from the optimal emissions (e1 , . . . , en ) and the reference emissions (e1 , . . . en ) such that for each country i, " # X X X d i ð^ ei ei Þ dj ¼ gi (ei ) gi (ei ) P gj (ej ) gj (ej ) (6:13) d j j2W j2W j2W j2W The left-hand side of this expression is what country i pays (or receives) ifPit buys (sells) emission rights amounting to (^ eP i ei ) at a ^ ¼ j2W dj are ^ ¼ j2W dj . This suggests that (e1 , . . . , en ) and g price g nothing more than a competitive emissions trading equilibrium with respect to the quotas (^ e1 , . . . , e^n ), in the sense of (6.8)–(6.9). And the right-hand side is precisely the Chander and Tulkens transfer Ti advocated in Section 3 (see (6.4)) above to achieve optimality in a coalitionally stable way. Note that while the world optimum emissions (e1 , . . . , en ) as defined in (6.2) are independent of the reference emission levels (e1 , . . . , en ) as defined in (6.3a) and (6.3b), the emission quotas (^ e1 , . . . , e^n ) as defined in (6.13) are not. In fact, since the optimal emissions are independent of reference emissions, there is a one-to-one correspondence between (^ e1 , . . . , e^n ) and (e1 , . . . , en ). This means that if the reference emission levels (e1 , . . . , en ) are not in dispute, then the emission quotas (^ e1 , . . . , e^n ) along with competitive emissions trading would also be acceptable to all countries since by definition these would not only lead to the optimum emissions (e1 , . . . , en ) but also to transfers that would make each country or group of countries better off compared to (e1 , . . . , en ). The significance of this shift in perspective lies in the fact that, as noted earlier, the currently considered reference emission levels (e1 , . . . , en ) are felt to be unfair, typically by the countries that are in the
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Public Goods, Environmental Externalities and Fiscal Competition
early stages of economic development with comparatively low emission levels. Therefore, the emissions of such countries may not be subjected to quotas, as agreed upon at Kyoto, at least until the time when their emission levels become comparable to those of Annex I countries. With time their emissions will rise as a result of economic development and those of the Annex I countries will fall as a result of abatement. While the Kyoto Protocol is a step in the right direction in terms of the actual emissions, we are suggesting here that the effective ultimate aim should in fact be to reach an agreement on appropriate reference emission levels (or pollution rights) (e1 , . . . , en ) at some future round of negotiations. The discussion in the preceding paragraph clarifies that once an agreement is reached regarding reference emissions (e1 , . . . , en ) then an agreement would also be reached regarding the target emission quotas (^ e1 , . . . , e^n ) and competitive emissions trading which by definition lead to optimum emissions (e1 , . . . , en ) and transfers that ensure coalitional stability. Such an agreement requires the countries first to agree on equity principles to be adopted, as for instance per capita or per unit of GDP emissions. The currently considered baselines of business-as-usual Nash equilibrium or historically grandfathered emissions are known to be problematic. Something else seems to be required, making explicit room, for instance, for principles such as ‘common but differentiated responsibilities’. If such new reference levels can be agreed upon, our analysis suggests that a quotas and trading scheme of the kind pioneered in Kyoto is an appropriate tool to reach stable world optimality in the future.
References Chander, P. and Tulkens, H., 1992. Theoretical foundations of negotiations and cost sharing in transfrontier pollution, European Economic Review, 36 (2/3), 288–99. (Chapter 6 in this volume). Chander, P. and Tulkens, H., 1995. A core-theoretic solution for the design of cooperative agreements on transfrontier pollution, International Tax and Public Finance, 2 (2), 279–94. (Chapter 9 in this volume). Chander, P. and Tulkens, H., 1997. The core of an economy with multilateral environmental externalities, International Journal of Game Theory, 26, 379–401. (Chapter 8 in this volume). Debreu, G. and Scarf, H., 1963. A limit theorem on the core of an economy, International Economic Review, 4, 235–46. Edgeworth, F.Y., 1881. Mathematical Psychics. London: Paul Kegan. Ellerman, A.D. and Decaux, A., 1998. Analysis of post-Kyoto CO2 emissions trading using marginal abatement curves. MIT Joint Program on the Science and Policy of Global Change, report no. 40, Massachusetts Institute of Technology (October).
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Energy Prices and Taxes, Paris: International Energy Agency. Fankhauser, S., 1995. Valuing Climate Change: The economics of the greenhouse. London: Earthscan. Germain, M., Toint, Ph., Tulkens, H. and de Zeeuw, A., 2003. Transfers to sustain cooperation in international stock pollutant control, Journal of Economic Dynamics and Control, 28, 79–99. (Chapter 12 in this volume). Intergovernmental Panel on Climate Change (IPCC)., 1995. Climate Change 1995: The economic and social dimensions of climate change, contribution of Working Group III to the Second Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge: Cambridge University Press. Nordhaus, W.D. and Yang, Z., 1996. A regional dynamic general equilibrium model of alternative climate-change strategies, American Economic Review, 86(4), 741–65. Shapley, L., 1967. On balanced sets and cores, Naval Research Logistics Quarterly, 14, 453–60. Shapley, L. and Shubik, M., 1969. On market games, Journal of Economic Theory, 1, 9–25.
Chapter 11 SIMULATING COALITIONALLY STABLE BURDEN SHARING AGREEMENTS FOR THE CLIMATE CHANGE PROBLEM
Johan Eyckmans, Henry Tulkens Resource and Energy Economics 25, 299–327 (2003).
The CLIMNEG world simulation (CWS) model is introduced here for simulating cooperative game theoretic aspects of global climate negotiations. The CWS model is derived from the seminal RICE model by Nordhaus and Yang [Am. Econ. Rev. 86 (1996) 741]. We first state the necessary conditions that determine Pareto efficient investment and emission abatement paths under alternative regimes of cooperation between the regions. We then show with a numerical version of the CWS model that the transfer scheme advocated by Germain, Toint and Tulkens (1997) induces an allocation in the (‘‘gamma’’) core of the world carbon emission abatement cooperative game.
1.
Introduction and summary
International environmental agreements involving substantial emission reduction efforts are unlikely to be reached without provisions for international transfers. The reason is that, although there generally is a substantial surplus to be gained from cooperation, there are most often countries for which the abatement effort required by the world optimum is so large that they end up worse off under this world optimum compared This research is part of the CLIMNEG project, funded by the Belgian Federal Office for Scientific, Technical and Cultural Affairs (OSTC), contracts CG/DD/241 and CG/DD/ 243. The authors wish to express their thanks to Professor William Nordhaus for making his program code available to them, to Marc Germain for careful reading as well as to their colleagues from UCL, K.U. Leuven and Bureau Fe´de´ral du Plan/Federaal Planbureau in the CLIMNEG team for stimulating discussions in numerous working sessions. Useful comments and suggestions by two anonymous referees, by Carlo Carraro, by Juan C. Cı`scar and by Jacques Dre`ze are gratefully acknowledged. All remaining errors remain, of course, ours.
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Public Goods, Environmental Externalities and Fiscal Competition
to the noncooperative laissez-faire situation. The world optimum thereby appears to be unstable. That international transfers can compensate for such an undesirable outcome was argued quite long ago (e.g. Tulkens (1979)) and taken up again more recently, albeit with some skepticism, by Carraro and Siniscalco (1993) or Barrett (1994). This paper investigates how such international transfers might look like in the case of the global climate change problem. In particular, we employ a variant of the transfer scheme initially proposed by Chander and Tulkens (1995, 1997) (CT hereafter) in a static context and extended by Germain, Toint and Tulkens (1998) (GTT hereafter) to a dynamic context. Essentially, the scheme consists in redistributing the surplus of cooperation over noncooperation in proportion1 to the (marginal) climate change damage costs that countries experience. In these papers, it is shown that the proportional transfer scheme results in an allocation in the core of a cooperative emission abatement game. This core property is a necessary (but not sufficient) condition for full, voluntary cooperation among the countries involved as explained in Tulkens (1998). If it is not satisfied, coalitions of countries can obtain a better outcome by coordinating their emission strategies among themselves and such coalitions have no incentive to join a worldwide environmental treaty. The core results just mentioned have been obtained only under a linearity assumption of the damage cost functions, which is hard to maintain for the climate change problem. Do these results still hold in the nonlinear case? Given the absence of an analytical answer, we may turn to an empirical model and test the core property of the transfer scheme numerically. For this purpose we introduce in Section 2 the CLIMNEG world simulation (CWS) model which we derived from the seminal multi-region economy–climate RICE model of Nordhaus and Yang (1996). Like RICE, the CWS model is an optimal growth model for the world economy, coupled with a basic representation of the carbon cycle and climate system. The main differences between RICE and CWS are that we work with a model in which utilities are linear in consumption and that we do not consider international trade flows. Both of these modifications are motivated below; they are made essentially to focus on the game theoretic aspects of the cooperation problem. In Section 3, we describe various scenarios w.r.t. cooperation in the CWS model. First, absence of cooperation is modeled as an open-loop Nash equilibrium of carbon emissions. Also a Ramsey–Keynes rule is 1 For international environmental agreements, proportionality w.r.t. damages has been advocated not only for its strategic stability properties, but also for incentive compatibility reasons, see Eyckmans (1997).
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
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formulated, determining an inter-temporal allocation of investment and consumption that disregards spillovers to neighboring countries. Next, Pareto efficient allocations are characterized. In such allocations, the rule determining emission abatement efforts is shown to be a dynamic version of the Samuelson (1954) rule for the optimal provision of public goods. Parallel to that, capital accumulation is determined by a generalization of the Ramsey–Keynes rule that internalizes the carbon emission externalities. Contrasting Nash equilibria and Pareto efficient allocations of abatement effort is not new to the literature: the point has been made before by, among others, Ma¨ler (1989), Hoel (1992), van der Ploeg and De Zeeuw (1992) and Kaitala et al. (1992). In this paper, however, we go one step further by characterizing intermediate scenarios of incomplete cooperation by subgroups (coalitions) of countries. To that effect, we define partial agreement Nash equilibria with respect to a coalition which is the counterpart for dynamic models of a concept introduced in CT. Optimality rules driving investment and emission abatement decisions at such equilibria turn out to be a combination of the optimality rules for the Pareto efficient allocations and the open-loop Nash equilibrium. In Section 4, the partial agreement Nash equilibrium w.r.t. a coalition concept is used to define the core of a carbon emission abatement cooperative game. Core allocations satisfy both individual and coalitional participation constraints, i.e. these allocations are such that no individual country, nor any coalition of countries, can gain by returning to its partial agreement Nash equilibrium. In Section 5, simulations with the numerical CWS model are reported. We first compute three reference scenarios (business-as-usual, Nash equilibrium and Pareto efficient allocation without transfers) and we compare them in terms of carbon emissions, carbon concentrations, temperature change, emission abatement effort and consumption levels over the period of analysis. The value of cooperation over noncooperation is estimated on that basis. We next check for the core property in the numerical CWS model. We therefore compute all possible partial agreement Nash equilibria w.r.t. any coalition and compare each coalition’s payoff to its joint allocation of consumption under the Pareto efficient solution without transfers. We observe in the simulations that the core property is violated, both for China as an individual region, and for several intermediate coalitions containing both China and rest of the world. We then consider whether the participation problem raised thereby can be overcome using the GTT international transfer scheme appropriately modified to allow for nonlinear damage cost functions. It turns out that all coalitions are better off under the transfer scheme than under their respective partial agreement Nash equilibrium. Hence, the Pareto efficient scenario with transfers
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belongs to the core of the game; in this sense, it can be sustained as a voluntary agreement. We also exploit our study of the 64 conceivable coalitions to evaluate some post-Kyoto partial cooperation policies: cooperation within Annex B, the US noncooperation, and cooperation within Annex B without the US. Finally, we point to the necessity of extending this analysis to closed-loop Nash equilibria so as to allow for transfers defined at every point in time rather than as single number discounted aggregates. Our conclusion in Section 6 emphasizes the specific merits of the cooperative game theoretic approach in the framework of international environmental agreements.
2.
CWS, an integrated economy–climate world model
2.1
Statement of the model
In the CWS model, each national economy is represented by a discrete time optimal growth model with a long but finite horizon. Growth is driven by exogenous population growth and technological change as well as by endogenous capital accumulation. Letting N denote the set of countries/regions2 indexed i ¼ 1, 2, . . . , n, the following equations describe the economy of region i at time t: Yi,t ¼ Zi,t þ Ii,t þ Ci (mi,t ) þ Di (DTt )
(1)
Yi,t ¼ Ai,t Fi (Ki,t )
(2)
Ki,tþ1 ¼ (1 dK )Ki,t þ Ii,t
(Ki,0 given)
(3)
Eq. (1) defines the claims of consumption Zi,t , investment Ii,t , cost of abatement Ci (mi,t ) and climate change damage Di (DTt ) upon production Yi,t . The costs of abatement and of climate change damages are both assumed to be strictly increasing and strictly convex in abatement mi,t and temperature change DTt , respectively. Eq. (2) defines production as a strictly increasing and strictly concave function of capital input Ki,t . Ai,t measures overall productivity. It is assumed that productivity increases exogenously as time goes by and technological progress is Hicks neutral. Population, identified with labor supply, is assumed to be an exogenous input in production and is subsumed in the productivity factor Ai,t . Finally, expression (3) is a capital accumulation equation, where dK stands for the rate of capital depreciation.
2
In the sequel we will indifferently speak of regions or countries, even if a region contains only one country.
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This economic model is coupled with a simple climatic model of global mean temperature change. The carbon emissions, the carbon cycle and their effect on climate are, respectively, modeled by the following equations: Ei,t ¼ si,t (1 mi,t )Yi,t Mtþ1 ¼ (1 dM )Mt þ b
X
(4) Ei , t
(M0 given)
(5)
i2N
DTt ¼ G(Mt )
(6)
According to expression (4), carbon emissions Ei,t are proportional to production. The emissions to output ratio si,t declines exogenously over time due to an assumed autonomous energy efficiency increase. Emissions can be reduced at a rate mi,t 2 [0, 1] in every period though this is costly according to Eq. (1). Eq. (5) describes the accumulation of carbon in the atmosphere. This process is modeled similarly to a standard capital accumulation process, where dM denotes the natural decay rate of atmospheric carbon concentrations and b is the airborne fraction of carbon emissions. Expression (6) translates atmospheric carbon concentration levels into global mean temperature change.3 We assume that G is a continuously differentiable and increasing function. For the purpose of this section, there is no need to make the function G more explicit. In the numerical simulation of CWS, we will adopt exactly the same formulation as in RICE for the carbon cycle and temperature change equations; this formulation satisfies the general properties we assume for G presently. Finally, the welfare Wi of each country i is measured by its aggregate lifetime discounted consumption: Wi ¼
T X t¼0
Zi,t þ wi (Ki,Tþ1 ) (1 þ ri)t
(7)
where ri stands for the discount rate of region i and the strictly increasing and strictly concave function wi stands for the scrap value of the terminal capital stock Ki, Tþ1 . 2.2
Differences with the RICE model
Conceptually, the model outlined above is very similar to the RICE model by Nordhaus and Yang (1996). In this section, we clarify and 3
This formulation is an extreme simplification of the physical processes behind climate change. In Eyckmans and Bertrand (2000), we provide a more realistic model of the carbon cycle and climate change process which takes into account regional differences in temperature change and cooling from sulfate aerosols.
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motivate the differences that our formulation introduces. First, we do not allow for international trade. Trade complicates the analysis considerably because it creates, besides the climate change externality, additional interdependencies between the regions that we want to avoid in order to better concentrate on the cooperation issues raised by the environmental externality. In addition, we feel that the way Nordhaus and Yang (1996) introduce trade in RICE is not fully satisfactory. In their Negishi solution, what they call exports are in fact some kind of (restricted) normative transfers among the regions whose justification is not clear. Finally, since the magnitude of actual net exports (exports minus imports) is relatively small, we feel that not much is lost from leaving out international trade. Secondly, we use an additive instead of a multiplicative formulation of the effect of emission abatement costs and climate change damages on consumption possibilities. Translated into our notation, Nordhaus and Yang’s formulation of the budget Eq. (1) is given by: Vi,t Yi,t
1 Ci (mi,t )=Yi,t Yi,t ¼ Zi,t þ Ii,t 1 þ Di (DTt )=Yi,t
Conceptually, the two formulations reflect the same fact that costs of emission abatement and of damages from climate change reduce the amount of production that can be devoted to consumption or investment. The difference lies in the fact that Nordhaus and Yang (1996) allow for cross effects between emission abatement costs and climate change damages: this type of cross effects are excluded by our formulation. Thirdly, in contrast to Nordhaus and Yang (1996), we assume that utilities are linear in consumption. We make this simplification in order to represent the global carbon emission game as a transferable utility (TU) game. For our purpose of game theoretic stability analysis by means of numerical simulations, a TU framework is better suited than a non-TU game. In particular, for the cooperative solution concept of the core in a non-TU context, one cannot use the additive form of the worth of a coalition on which our present computations rest. Fourthly, the CWS model allows for different regional discount rates. We feel that the huge differences between world regions in terms of economic development and openness to financial markets do not justify that a uniform discount rate be applied in all regions, as is the case in RICE. In our simulations, we have chosen systematically higher discount rates for developing regions than for industrialized countries.
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3.
Alternative scenarios as to cooperation in the CWS model
3.1
Nash equilibrium
We first describe what would happen if the regions do not sign a voluntary international environmental agreement. We characterize such a situation by means of the concept of open-loop Nash equilibrium. An open-loop Nash equilibrium (Nash equilibrium hereafter) is a family of strategies, one for each player, that maximize every region i’s welfare, given the strategies of all other players j 6¼ i. In such an equilibrium, no individual region has an incentive to deviate as long as the other regions stick to their equilibrium strategies. In our CWS model, a Nash equilibrium is obtained by maximizing every region’s welfare (7) with respect to its control variables Ii,t , Zi,t and mi,t while respecting its individual resource and capital constraints and the j,t of all other regions j 6¼ i and 8t. climate modules for given emissions E Formally, for each region i: max
{Zi,t , Ii,t , mi,t }t¼0,...,T
Wi ¼
T X t¼0
Zi,t þ wi (Ki, Tþ1 ) (1 þ ri)t
(8)
subject to (for all 0 # t # T): Ai,t Fi (Ki,t ) $ Zi,t þ Ii,t þ Ci (mi,t ) þ Di (G(Mt )), Ki,tþ1 ¼ [1 dK ]Ki,t þ Ii,t ,
8t
8t
[ci,t ]
Mtþ1 ¼ [1 dM ]Mt þ bsi,t [1 mi,t ]Ai,t Fi (Ki,t ) þ b
[zi,t ] X j6¼i
j ,t , 8t E
[fi,t ]
In addition, initial values of the state variables Ki,0 and M0 are given, all variables are constrained to be non-negative and mi,t # 1. We associate Lagrange multipliers zi,t to the resource constraint,4 ci,t to the capital accumulation constraint and fi,t to the carbon accumulation process. Differentiating with respect to all variables we get the following first-order conditions for an interior optimum (the superscript ‘‘NE’’ refers to the equilibrium values of the variables at the Nash equilibrium): zNE i ,t ¼ 4
1 ¼ cNE i,t , 8t (1 þ ri )t
(9)
Since the objective is monotonically increasing in consumption Zi,t , the inequality in the resource constraint can be replaced by an equality and a Lagrange multiplier can be used.
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0 NE NE NE zNE i,t Ci (mi,t ) ¼ bsi,t Ai,t Fi (Ki,t )fi,t ,
8t
(10)
NE 0 NE NE 0 NE NE cNE i,t1 ¼ ci,t [Ai,t Fi (Ki,t ) þ (1 dK )] bsi,t (1 mi,t )Ai,t Fi (Ki,t )fi,t , 8t
(11) 0 NE cNE i,T ¼ wi (Ki,Tþ1 ),
(12)
NE 0 NE NE 0 NE fNE i,t1 ¼ G (Mt )zi,t Di (G(Mt )) þ (1 dM )fi,t ,
fNE i,T ¼ 0:
8t
(13) (14)
A Nash equilibrium of the CWS model is a solution to this system of first-order conditions, holding simultaneously for all i 2 N. Conditions (9) say that for every region i and in every period t the shadow cost of capital equals the shadow cost of the resource constraint and that both are equal to the discount factor of region i. Eq. (10) determines the optimal amount of carbon emissions control for country i. The evolution of the capital stock is described by condition (11) for all but the last period and condition (12) for the last period. Expression (13) describes the evolution of the Lagrange multiplier associated with the atmospheric carbon concentration equation. In economic terms, this multiplier can be interpreted as the shadow price of carbon concentration. Finally, condition (14) says that last period’s shadow price of carbon equals zero. By using the terminal condition fNE i,T ¼ 0 for solving iteratively the difference Eq. (13), it appears that the shadow price of carbon for region i in period t is equal to the sum of all its future marginal damages caused by an additional unit of carbon emissions at time t: fNE i,t
¼
T X t¼tþ1
(1 dM )tt1 G0 (MtNE )
D0i (G(MtNE )) (1 þ ri )t
(15)
Notice that, since Eq. (15) holds for each region i separately, the shadow price of atmospheric carbon concentration at a Nash equilibrium only takes into account the climate change damage occurring within a region’s territory. Spillover effects to neighboring regions are not taken into account in the region’s individual decision process. Substituting Eq. (15) for the shadow price in Eq. (10), we derive for each country i a rule determining its optimal amount of carbon emission control at a Nash equilibrium, that reads as follows: T 0 NE X Ci0 (mNE 1 i ,t ) tt1 0 NE NE Di (G(Mt )) ¼ bf ¼ b (1 d ) G (M ) M i ,t t (1 þ ri )t (1 þ ri )t si,t Ai,t Fi (KiNE ,t ) t¼tþ1
(16)
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The left-hand side (LHS) stands for the discounted marginal cost for region i of reducing its carbon emissions by an additional tonne in period t. The denominator denotes gross emissions without abatement and is used to convert the units of the marginal abatement costs into US$ per tonne of carbon.5 Eq. (16) is the traditional optimality condition for a noncooperative Nash equilibrium, saying that in each i marginal abatement costs should be equal to individual marginal damage of climate change. We now turn to the Ramsey–Keynes condition that drives capital accumulation for country i. Substituting Eq. (9) into Eq. (12), the latter condition can be written as follows: t NE NE ri þ dK ¼ Ai,t Fi0 (KiNE ,t )[1 bsi,t (1 mi,t )(1 þ ri ) fi,t ]
(17)
This condition says that along an optimal investment path, region i should be indifferent between consuming an additional dollar at time t 1 and postponing consumption for investing in next period’s capital stock. The Ramsey–Keynes rule for the Nash equilibrium only internalizes climate change damage occurring domestically since negative climate change externalities to neighboring countries are not taken into account in the shadow price of carbon fNE i,t . It is interesting to notice that if a region does not value climate damages (D0i (x) ¼ 0, 8x $ 0), the Ramsey–Keynes rule boils down to simply: Ai,t Fi0 (Ki,t ) dK ¼ ri . This is the golden rule of capital accumulation saying that along an optimal investment path, the net marginal product of capital should be equal to the pure rate of time preference. The fact that there is a production externality causing detrimental climate change in the economy–climate model deflates the marginal return of capital by a factor that depends upon the shadow price of carbon. Therefore, even in a noncooperative Nash equilibrium, the rate of capital accumulation and output growth will be lower in the presence of climate change externalities compared to a BAU scenario without explicit climate change policy. 3.2
Pareto efficiency
Pareto efficient consumption, investment and emission abatement paths are obtained as the solution to the following problem: we maximize the utility of one particular country (without loss of generality, we will maximize country 1’s utility W1 ) subject to the restriction that every other 5 Recall that mi,t 2 [0, 1] has no dimension since it is the fraction of emissions that are abated. si,t Ai,t Fi (KiNE ,t ) stands for gross carbon emissions without emission abatement (mi,t ¼ 0) and is measured in tonnes of carbon.
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j and subject to the global country achieves at least some given6 utility level W economy–climate relationships. We also introduce transfers Ci (expressed in units of period 0 consumption good) of a once-only nature, with Ci > 0 ( < 0) denoting a transfer received (paid), respectively. Formally: max W1 ¼ {Zi,t ,Ii,t ,mi,t ,Ci }i2N ,t¼0,...,T
T X t¼0
Z1,t þ w1 (K1,Tþ1 ) þ C1 (1 þ r1)t
(18)
subject to (for all i 2 N and for all 0 # t # T): Wj ¼ X
T X t¼0
Zj ,t j þ wj (Kj,Tþ1 ) þ Cj $ W (1 þ rj)t
Cj ¼ 0
8j 6¼ 1
[lj ]
[n]
j2N
Ai,t Fi (Ki,t ) $ Zi,t þ Ii,t þ Ci (mi,t ) þ Di (G(Mt )), 8i, t Ki,tþ1 ¼ (1 dK )Ki,t þ Ii,t , 8i, t [ci,t ] X sj,t [1 mj,t ]Aj,t Fj (Kj,t ), Mtþ1 ¼ (1 dM )Mt þ b
[zi,t ] 8t
[ft ]
j2N
with Ki,0 and M0 given, Ci unrestricted in sign, non-negativity constraints on all other variables and mi,t # 1. The Greek symbols between square brackets denote Lagrange multipliers for the corresponding constraints. Without loss of generality, we normalize the li ’s such that l1 ¼ 1. The first set of constraints requires that all players other than player 1 enjoy a j . These constraints combined with the utility level at least as high as W utility maximization for player 1 exhaust all possibilities for Pareto improvements and do therefore lead to a Pareto efficient allocation of resources. The second constraint refers to the transfers that are expressed in period 0 consumption units; the constraint requires that these transfers balance. The other constraints repeat the basic economy and climate relationships, i.e. the individual countries’ budget conditions, capital accumulation and carbon concentration accumulation. Notice that the third constraint (on the countries’ budgets) are specified for each i separately: each country is supposed to survive out of its own productive resources only, at each time period t, with neither these resources nor the generated consumption good being transferable to other countries.7 This is a more restrictive setting than if such transfers 6 j for regions j ¼ 2, 3, . . . , n should correspond to a feasible The given utility levels W j one can allocation in the underlying economy–climate model. By parameterizing these W trace out the entire set of Pareto efficient allocations. 7 There should be no confusion between this restriction and the transfers Ci introduced earlier: transfers were independent of time (no index t), in spite of being expressed in units
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were allowed at each t but it is consistent with our deletion of international trade in the RICE model.8 Thus, the solution we compute is actually one of constrained Pareto efficiency. In all what follows, we mean efficient in that sense. The first-order conditions for all i 2 N and 0 # t # T for an interior optimum of (18) resemble strongly the Nash equilibrium first-order conditions stated above. Only condition (13) is fundamentally modified implying that the Pareto efficient stock of carbon in the atmosphere Mt should satisfy instead (the superscript asterisk refers to the values of the variables at the Pareto efficient solution): X ft1 ¼ G0 (Mt ) zj ,t D0j (G(Mt )) þ (1 dM )ft , fT ¼ 0 (19) j2N
Through iterative substitution from (19) the Pareto efficient shadow price of carbon ft is calculated for each t < T in the same way as before. Using this Pareto efficient shadow price of carbon, we obtain the following condition determining the Pareto efficient amount of carbon emission control for country i in period t: zi,t
Ci0 (mi,t ) si,t Ai,t Fi (Ki,t )
¼
bft
¼b
T X
(1 dM )tt1 G0 (Mt )
t¼tþ1
X
zj ,t D0j (G(Mt ))
j2N
(20) This expression can be simplified further using the first-order conditions for the variables Zi,t and Ci . These read: zi,t ¼ li
li , 8i, t (1 þ ri )t
(21)
¼ n , 8i
and together they imply for all i 2 N: zi,t ¼
n , 8i, t (1 þ ri )t
(22)
Using the latter expression to substitute for the multipliers zi,t in expression (20) we obtain the Samuelson rule for the Pareto efficient provision of emission reduction:
of the consumption good at time zero, and the moment(s) of their implementation was left unspecified. A model with time dependent transfers is proposed in Germain et al. (2003). 8 While allowing for international trade in capital, Nordhaus and Yang (1996) have to impose severe quantity constraints on the trade flows in order to get realistic results (see their Eqs. (A.16)–(A.18)), so severe that the role of trade is virtually annihilated.
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Public Goods, Environmental Externalities and Fiscal Competition 0
Ci (mi,t ) 1 ¼ bft t (1 þ ri ) si,t Ai,t Fi (Ki,t ) ¼b
T X t¼tþ1
(1 dM )
tt1
G
0
(Mt )
X D0j (G(Mt )) (1 þ rj )t j2N
(23)
The latter condition is a dynamic extension of the traditional optimality rule for static public good models that was first stated by Samuelson (1954). The LHS of the expression stands for the discounted marginal cost for region i of reducing its carbon emissions by an additional tonne in period t. The RHS consists of the sum from period t þ 1 until the final period T of all regions’ discounted future marginal damages from climate change, taking into account the airborne fraction of emissions and the natural decay of atmospheric carbon. In contrast to the Nash equilibrium, the Pareto efficient allocation of emission abatement takes into account total climate change damages affecting all regions in the world (sum of marginal damages over all j 2 N). Thus, the climate externality is fully internalized in Pareto efficient allocations. Rule (23) does not say that all regions should reduce their emissions in such a way that their discounted marginal abatement costs in each period t be equalized. This would be the case only if all countries had the same discount rate ri ¼ r. Our Samuelson rule is instead a weighted extension of the traditional optimality rule for the provision of public goods. Countries characterized by a high discount rate are required to perform relatively more emission abatement since their opportunity cost of an additional dollar of consumption is lower. For a more precise treatment of this argument and the trade off between equity and efficiency, see Eyckmans et al. (1993). It is also interesting to note that the allocation of abatement efforts j . Changing this does not depend on the vector of given utility levels W vector of utilities for regions 2 to n will modify the values of the optimal transfers Ci and the shadow price n , but it does not alter the Samuelson condition (23). The allocation of abatement is therefore independent of the allocation of transfers.9 This is an important property for the rest of the paper since we will introduce in Section 4 a particular transfer scheme to deal with participation constraints. Whatever these transfers will look like, they will not alter the allocation of abatement according to condition (23). 9
Since we have assumed linear utility functions for all players, the Pareto efficient allocation of abatement efforts (mi,t )i2N ;t¼0, 1,...,T can be Pcharacterized also as the solution of an unweighted sum of all the players welfare: max j2N Wj . We will use this formulation in Section 3.3 to solve the abatement choice problem of any coalition S N.
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Finally, we derive the condition for the Pareto efficient accumulation of capital in the presence of an environmental externality: ri þ dK ¼ Ai,t Fi0 (Ki,t )[1 bsi,t (1 mi,t )(1 þ ri )t ft ]
(24)
Though this condition looks exactly the same as condition (17), it is fundamentally different since in the Pareto efficient case, the shadow price of carbon ft internalizes all climate change externalities whereas it only internalizes domestic damages in the Nash equilibrium. 3.3
Partial agreement Nash equilibria w.r.t. a coalition
Sections 3.1 and 3.2 described two extreme cases as to cooperation. In a Pareto efficient scenario, all regions take action jointly to reduce their emissions of carbon dioxide and they do so by internalizing completely the external effects of their carbon emissions. In the Nash equilibrium, every region reduces its carbon emissions also but to a lesser extent because they only internalize the external effects of their emissions that affects their own territory. Intermediate cases are conceivable, when only some subgroup of regions agrees to coordinate its emission reduction policies.10 In order to characterize this situation of partial cooperation, we use the concept of partial agreement Nash equilibrium w.r.t. a coalition (PANE), introduced by Chander and Tulkens (1995, 1997). Suppose a coalition S N forms with s members. In a partial agreement Nash equilibrium w.r.t. coalition S, this coalition chooses actions that are most beneficial from the group’s point of view while the outsiders to the coalition choose actions that maximize their individual utility. The PANE w.r.t. S can be interpreted as a special type of Nash equilibrium in which a coalition S coordinates its policies taking as given the emission strategies of the outsiders who, in turn, are playing a noncooperative Nash strategy against S. Formally, a partial agreement Nash equilibrium w.r.t. coalition S is a combination of strategies that solves simultaneously the following n s þ 1 maximization problems: def
for the insiders (i 2 S): max WS ¼
X i2S
Wi ¼
T XX i2S t¼0
Zi,t (1 þ ri )t
(25)
10 The 1997 Kyoto Protocol on greenhouse gases emission reduction may be seen as an example of such partial cooperation. However, this view was not shared by Chander et al. (2002), who proposed instead an interpretation in terms of an all players coalition. Indeed, although many developing countries have not assumed quantified emission ceilings in the 1997 Kyoto Protocol, they are official signatories of it. Now, the ratification stage that followed the withdrawal of the US and a few other countries while all other countries ratify, brings one back to a situation of PANE w.r.t. the ratifying coalition.
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for each outsider (i 2 N nS): max Wi ¼
T X t¼0
Zi,t (1 þ ri )t
(26)
subject to the individual resource constraints (1), the production function (2), the capital accumulation conditions (3), the definition of carbon emissions (4), carbon concentration (5) and temperature change (6) equations being satisfied. This concept encompasses both those of Pareto efficiency (for S ¼ N) and of a Nash equilibrium (for S ¼ {i} for some i 2 N). First-order conditions for a PANE w.r.t. a coalition S can be derived in the same way as before. These first-order conditions appear as a mixture of the first-order conditions for a Pareto efficient allocation and a Nash equilibrium. For the outsiders, first-order conditions are exactly similar to the conditions (16) and (17). Indeed, outsiders take into account only domestic climate change damages. As a limiting case, for S ¼ {i} for some i 2 N, all the outsiders’ conditions coincide exactly with conditions (16) and (17), respectively. For the insiders of S, first-order conditions look very similar like the conditions (23) and (24) except for the fact that the summation of marginal damages bears only upon the members of coalition S. Insiders internalize the negative externalities from climate change only among themselves: T X D0j (G(MtS )) X Ci0 (mSi,t ) 1 tt1 0 S (1 d ) G (M ) ¼ b M t (1 þ rj )t (1 þ ri )t si,t Ai,t Fi (KiS,t ) j2S t¼tþ1
(27) ri þ dk ¼ Ai,t Fi0 (KiS,t )[1 bsi,t (1 mSi,t )(1 þ ri)t fSS,t ]
(28)
where fSS,t stands for the shadow price of the atmospheric carbon concentration for coalition S. It is given by the RHS of expression (27). For S ¼ N, the insiders’ conditions reduce to (23) and (24), respectively, and the PANE w.r.t. N coincides with the Pareto efficient allocation.
4.
Transfers ensuring individual and coalitional rationality
4.1
Individual and coalitional rationality
As is well known, the fact that a particular allocation is Pareto efficient does not imply that all regions are better off compared to a Nash equilibrium. While many regions are net winners, some other regions may be net losers. And if a region is worse off, it will not accept an agreement that proposes to implement such an allocation. In this case we will say that the proposed agreement does not satisfy individual rationality.
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Not only individual regions may be worse off under the Pareto efficient solution: also coalitions of two or more regions may find out that they can do better if the joint payoff of their members in the partial agreement Nash equilibrium is higher than at the efficient allocation. If this is the case, we say that the efficient allocation does not satisfy coalitional rationality. Allocations that satisfy both individual rationality for all regions, and coalitional rationality for all possible coalitions of the regions are said to belong to the ‘‘core’’ of a cooperative game associated with the economic model under consideration. In the present case of the CWS model, the players of the game are the regions, and the players’ strategies are the emission abatement policies chosen by the regions. In this setting, the core property of an allocation thus ensures that no coalition S could be better off by proposing a partial agreement Nash equilibrium w.r.t. itself. This property can be interpreted as a necessary (though not sufficient) condition for a voluntary international agreement to be sustained, as argued in Tulkens (1998). 4.2
The transfers formula
These considerations imply that a climate treaty implementing the Pareto efficient allocation prescribed by the Samuelson rule (23) may not emerge as a voluntary agreement among the emitters of carbon dioxide. However, transfers of consumption offer a way to induce such voluntary cooperation. In particular, we consider in this paper the transfer scheme proposed by GTT for stock pollution problems. In this section, we present a reinterpretation of this transfer scheme for the CWS model. We start from a Pareto efficient allocation of emission abatement efforts that solves the Samuelson conditions (20) for all i and 0 # t # T, and we then modify this allocation by introducing transfers of the consumption good defined as follows. Let WiNE be the discounted consumption stream of region i under the Nash equilibrium and Wi as the discounted consumption stream of region i in the Pareto efficient outcome: WiNE ¼
T X
ZiNE ,t
t¼0
(1 þ ri )t
and Wi ¼
T X
Zi,t
t¼0
(1 þ ri )t
GTT suggested the following transfer of the consumption good P (with shares pi $ 0 such that i pi ¼ 1): ! X X (29) Wj WjNE Ci ¼ (Wi WiNE ) þ pi j2N
j2N
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Public Goods, Environmental Externalities and Fiscal Competition
This transfer formula takes away, from every region, the difference between its Pareto efficient consumption allocation Wi and its consumption level WiNE at the Nash equilibrium; moreover, it divides the surplus of cooperation over noncooperation in proportion P to the weights pi . These weights pi are equal to the ratios D0i = j D0j . Regions with a relatively high share pi get relatively more of the surplus. The transfer scheme then yields the following consumption level for each i 2 N: ! X X Wj WjNE $ WiNE Wi þ Ci ¼ WiNE þ pi j2N
j2N
Clearly, the resulting consumption allocation is preferred over the Nash equilibrium allocation WiNE by all i 2 N as long as there is a positive surplus to cooperation. Hence, the allocation with transfers is always individually rational. Moreover, GTT have shown that the transfer scheme gives rise to an allocation of consumption which belongs to the core of the cooperative emission abatement game, provided that damage cost functions are linear. However, damage functions in the CWS model are nonlinear, implying that it is not sure whether the core property still holds. Nevertheless we can experiment numerically with the transfer formula (29) and check by means of simulations whether or not, with the transfers (29), coalitions have an interest in forming. With nonlinear damage cost functions Di (DTt ) the pi ’s in the transfer formula (29) are no longer constant over time. In order to take that into account we generalize the ratios pi by introducing p ~ i defined in the following way: P t 0 # t # T Di (DTt )=(1 þ ri ) p ~ i ¼ P 0P (30) t 0 j2N 0 # t # T Dj (DTt )=(1 þ r j ) that is, the share of region i in the aggregate discounted world marginal climate change damage costs. With these generalized ratios p ~ i introduced ~ i that we will call generalized in expression (29), we obtain new transfers C GTT transfers: ! X X NE NE ~ (31) ~i W W Ci ¼ (W W ) þ p i
i
j
j2N
4.3
j
j2N
Checking for the core property
To check whether the Pareto efficient allocation, supplemented with generalized GTT transfers is a core allocation, it is sufficient to check that the following inequalities hold for all coalitions of S N:
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements def
WSS ¼
X i2S
WiS #
X
~ i ] def ~S, [Wi þ C ¼ WS þ C
233
8S N
i2S
~ S $ W S , the coalition S obtains a higher Indeed, when WS þ C S payoff under the transfer scheme than it would obtain at the partial agreement Nash equilibrium w.r.t. itself; thus, the corresponding coalitional rationality constraint is satisfied.
5.
Simulations
5.1
Numerical specification of the CWS model
From the general specifications (1)–(6) of the CWS model we now move to a numerical specification, suitable for simulations to test the ‘‘core’’ property just discussed. The temperature change equation (function G in (6)) is taken from the economy–climate model RICE by Nordhaus and Yang (1996) as well as most of the parameter values and all basic data on GDP, population, capital stock, carbon emissions and concentration and global mean temperature (see Tables 1–3). A complete list of the equations and of the parameter values that we use, is provided in Appendix A (see (A.1)–(A.11)). The simulation model partitions the world into six regions: USA, Japan (JPN), European Union (EU), China (CHN), former Soviet Union (FSU) and rest of the world (ROW). However, contrary to Nordhaus and Yang (1996), we select different parameter values on two crucial points. First, we choose a discount rate of 1.5% for the industrialized regions (USA, JPN, EU and FSU) and a higher discount rate of 3% for the developing regions CHN and ROW.
Table 1 List of variables Yi,t Ai,t Zi,t Ii,t Ki,t Li,t Ci,t Di,t Ei,t s i ,t mi,t Mt Ft Ftx DTt Tto
Production (billion 1990 US$) Productivity Consumption (billion 1990 US$) Investment (billion 1990 US$) Capital stock (billion 1990 US$) Population (million people) Cost of abatement (billion 1990 US$) Damage from climate change (billion 1990 US$) Carbon emissions (billion tonnes of C) Emission–output rate Emission abatement Atmospheric carbon concentration (billion tonnes of C) Radiative forcing (W=m2 ) Exogenous radiative forcing (W=m2 ) Temperature increase atmosphere (8C) Temperature increase deep ocean (8C)
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Table 2 Global parameter values Capital depreciation rate Capital productivity parameter Airborne fraction of carbon emissions Atmospheric carbon removal rate Parameter temperature relationship Parameter temperature relationship Parameter temperature relationship Parameter temperature relationship Initial carbon concentration Initial temperature change atmosphere Initial temperature change deep ocean
dK g b dM t1 t2 t3 l M0 DT0 T0o
0.10 0.25 0.64 0.0833 0.226 0.44 0.02 1.41 590 0.50 0.10
This difference reflects taste differences across regions concerning the priority of economic development over environmental concerns. Secondly, we increase the value of the exponent in the climate change damage function (A.5) from 1.5 to 2.0. This choice was made after learning from our colleagues climatologists that temperature changes of 78C or more are to be considered as catastrophic ones: indeed, this difference is larger than the change in temperature between the last Ice Age and current temperature. Finally, we also revised downward the projected exogenous technology growth for FSU in order to match more closely current predictions on economic production for this region. 5.2
Computing equilibria
In order to calculate partial agreement Nash equilibria w.r.t. coalitions (including plain Nash equilibria), we use a standard numerical algorithm to compute noncooperative Nash equilibria where the coalition S is treated as one player in the emission game. In every iteration, a strategy is determined for each player, consisting of an investment and emission abatement path that maximizes its lifetime utility given the strategies of the other players. This iteration process is continued until the Euclidean distance between the strategy vectors in two consecutive iterations is smaller than a given threshold value.
Table 3 Regional parameter values USA JPN EU CHN FSU ROW
ui,1
ui,2
b i ,1
bi,2
ri
0.01102 0.01174 0.01174 0.01523 0.00857 0.02093
2.0 2.0 2.0 2.0 2.0 2.0
0.07 0.05 0.05 0.15 0.15 0.10
2.887 2.887 2.887 2.887 2.887 2.887
0.015 0.015 0.015 0.030 0.015 0.030
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
235
As perfect information is assumed, the resulting equilibrium is an open-loop equilibrium. The algorithm is equivalent to the one used by Yang (1998) to calculate numerically so-called ‘‘hybrid’’ coalition solutions. It was implemented using the optimization software GAMS. With 32 periods (decades), solving for one partial agreement Nash equilibrium only takes a couple of minutes on a Pentium 4, 1 GHz PC.11 Theoretically, a sufficient condition for convergence of this kind of algorithm is that the absolute value of the slope of the reaction functions of all players be smaller than one. In that case, the reaction mapping is a contraction and convergence is assured. In the CWS model (and in RICE) this condition on the slope of the reaction functions is not easy to check because of the dynamic specification of the carbon cycle and climate change model. In practice, however, we never encountered any convergence problem during the numerous simulations. We never found multiple equilibria by changing the set of starting values. 5.3
Reference simulations: BAU, NASH, EFF
5.3.1 Carbon emissions Fig. 1 shows annual world carbon emissions in three scenarios: business-as-usual (BAU), Nash equilibrium (NASH) and Pareto efficiency without transfers (EFF).12 We only consider carbon emissions 70 BAU 60
NASH
50
EFF
40 30 20 10
2230
2200
2170
2140
2110
2080
2050
2020
1990
0
Figure 1. World carbon emissions (GtC). 11
All data and simulation programs are available from the authors upon request. All figures report data for a time horizon of 2000–2250. However, all calculations were made until the year 3200 so as to avoid distortions from end period effects on the period we are interested in. 12
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Public Goods, Environmental Externalities and Fiscal Competition
originating from fossil fuel use. World carbon emissions in 1990 amount to approximately 6 gigatonnes of carbon.13 In the BAU scenario, we assume that countries do not value climate change and do nothing to restrict their carbon emissions, i.e. mi, t ¼ 0 for all i and t. BAU emissions grow continuously to reach nearly 40 GtC by the year 2100 and more than 62 GtC in 2200. BAU emissions continue to grow throughout the entire time horizon although the pace of growth gradually slows down. In the NASH equilibrium scenario, emissions grow at a slightly slower rate to reach about 38 GtC by the year 2100 and 59 GtC by 2200. Also in NASH, emissions continue to grow though growth decelerates. Pareto efficient carbon emissions (EFF) are substantially lower than BAU and NASH emissions: by the year 2100 they amount to some 24 GtC, and only 21 GtC by 2200. This is about half the BAU emission level in 2100 and almost one-third of BAU emissions in 2200. In contrast to BAU and NASH emissions, the EFF emission path rises until 2150, levels off at about 26 GtC and decreases afterwards. 5.3.2 Atmospheric carbon concentrations Fig. 2 shows the atmospheric carbon concentration in the BAU, NASH equilibrium and EFF scenarios, respectively. 1990 Atmospheric carbon concentration amounted approximately 750 GtC. Under BAU, the atmospheric carbon concentration rises steadily and reaches about 1718 GtC in 2100 and 3443 GtC in 2200. Doubling of the concentration 4500 BAU
4000
NASH
3500
EFF
3000 2500 2000 1500 1000 500
2250
2230
2210
2190
2170
2150
2130
2110
2090
2070
2050
2030
2010
1990
0
Figure 2. World carbon concentration (GtC). 13 All carbon emission and concentration data will be expressed in gigatonnes of carbon (GtC) which is the same as billion tonnes of carbon (btC), i.e. 109 tonnes of carbon.
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
237
w.r.t. 1990 takes place between 2080 and 2090. The NASH carbon concentration path follows closely the BAU path and continues to grow steadily all over the time horizon. By contrast, in the EFF scenario, atmospheric carbon concentrations grow at a much slower rate and reach 1368 GtC in 2100 and 2017 GtC in 2200. Doubling of the atmospheric carbon concentration w.r.t. 1990 is postponed until somewhere between 2110 and 2120. The carbon concentration levels off at about 2000 GtC by the year 2200. In that respect, the Pareto efficient outcome can be considered more sustainable than the BAU and NASH scenarios. 5.3.3 Temperature changes Fig. 3 shows the temperature increase compared to preindustrial times for the three reference scenarios. By the year 2100 temperature rises with 2.77, 2.70 and 2.248C in the BAU, NASH and EFF scenarios, respectively. By the year 2200, differences are more pronounced: 5.42, 5.30 and 3.938C. Whereas BAU and NASH temperatures continue to rise steadily, the Pareto efficient temperature change levels off at about 48C by the end of the time horizon shown in Fig. 3. This occurs only about 50 years after the atmospheric carbon concentration has leveled off because of the long time inertia of the climate system. 5.3.4 Emission control rates Fig. 4 shows the time path of emissions control (mNE i, t ) for the NASH equilibrium. There are substantial differences across regions. Taking averages of abatement effort over time, we see that CHN produces the 7 BAU 6
NASH
5
EFF
4 3 2 1
Figure 3. Global mean temperature change (8C).
2250
2230
2210
2190
2170
2150
2130
2110
2090
2070
2050
2030
2010
1990
0
238
Public Goods, Environmental Externalities and Fiscal Competition 10 9 8 7 6 5 4 3 2 1 0
USA JPN EU CHN FSU ROW
2230
2200
2170
2140
2110
2080
2050
2020
1990
WORLD
Figure 4. NASH emissions abatement (%).
highest abatement level (about 7.70%), followed by EU with 7.24% and USA with 6.44%. The lowest abatement effort is by ROW with only 1.45%. World average abatement amounts to 3.74%. The time path of emissions control rate of ROW lies far below the paths of the other regions due to strong free riding incentives within this heterogeneous region.14 CHN and EU are situated at the other end of the spectrum. For CHN, this is due to the combination of low emission abatement costs and substantial climate change damages. For EU, this is due to their relatively high climate change damage valuation. Finally, Fig. 5 shows the time path of emission control rates (mi, t ) for the Pareto efficient scenario EFF. Average world emission abatement 100 90 80 70 60 50 40 30 20 10 0
USA JPN EU CHN FSU ROW
2230
2200
2170
2140
2110
2080
2050
2020
1990
WORLD
Figure 5. EFF emissions abatement (%).
14
As in Nordhaus and Yang (1996) we model free riding behavior in the ROW region by revising downward their climate change damage parameter in all noncooperative scenario in which ROW is standing on its own. Without this modification, noncooperative emission control rate by ROW is unrealistic because of its size.
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements 600
Y=Z+I+D
500
239
D I
400 300 200
Z
100 2250
2170 2190 2210 2230
2150
2110 2130
2090
2070
2030 2050
1990 2010
0
Figure 6. Composition of world GDP in BAU (trillions 1990 US$ undiscounted).
w.r.t. BAU emissions rises from 3.74% in BAU to about 37.14% in EFF. In EFF, both CHN and ROW should reduce their emissions substantially more than the other regions (68.13 and 55.50%, respectively) and, more strikingly, their abatement effort rises over time. This last fact is due to the higher discount rates of CHN and ROW. Since they value the future less than the other regions, they are asked to perform ever more effort as time goes by. For them, the opportunity cost of forgoing an additional $ of consumption is valued less than for industrialized regions, cf. formula (20). 5.3.5 Macroeconomic magnitudes Figs. 6 and 7 exhibit the time profiles of aggregate output (Y), consumption (Z), investment (I), abatement costs (C) and damage costs (D) of the BAU and EFF scenarios (in the NASH scenario, these magnitudes are indistinguishably close to the BAU values). One first notices that output growth is fairly similar, as well as the consumption profile (but see further). The small differences between both scenarios stem from the way in which the difference Y – Z is allocated: the EFF scenario entails much less damage costs D, but this is only obtained from devoting part of the
600
D C I
Y=Z+I+C+D
500 400 300 200
Z
100 2250
2230
2210
2190
2170
2150
2130
2110
2090
2070
2050
2030
2010
1990
0
Figure 7. Composition of world GDP in EFF (trillions 1990 US$ undiscounted).
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Public Goods, Environmental Externalities and Fiscal Competition
production to emission abatement (measured by C), with investment (I) not observably affected by this substitution. 5.4
On the value of cooperation over noncooperation
Fig. 8 offers a closer look at the evolution of abatement damage costs for the extreme scenarios BAU and EFF. In the former scenario, where no emission abatement occurs, climate change damages rise to more than 10% of GDP by 2250. On the other hand, in the EFF scenario, where the abatement effort goes to 4% of GDP by 2250, damage costs nevertheless rise to about 5% by the end of the period considered. Taking together abatement costs and damage costs, the EFF scenario yields a better result than the BAU scenario, the difference amounts to approximately 1% of world GDP by 2250. In this sense, we can conclude that the EFF scenario is more sustainable than the BAU scenario. The latter conclusion is confirmed by Table 4 which summarizes total discounted consumption (over the entire horizon of the computation, i.e. 320 years) for each region under the alternative cooperation scenarios. The last row World reveals the overall magnitudes at stake. Discounted consumption amounts to US$ 337,692 billion (BAU), US$ 338,065 billion (NASH) and US$ 339,837 billion (EFF), respectively. The gain at the world level between the BAU and NASH is rather small (þ0.11%), but the additional gain obtained by moving from NASH to EFF is more important (þ0.52%). Overall, the welfare gain of moving from the BAU to the EFF scenario is about 0.64%. How do these figures relate to the original RICE model as reported in Table 4, p. 757 in Nordhaus and Yang (1996)? The overall difference is much larger in our calculations than in RICE. The difference between BAU and EFF amounts to US$ 1772 billion in Table 4 against only US$ 344 billion in Nordhaus and Yang (1996). However, this comparison is
12 10
C+DBAU
8 C+DEFF
6 4 DEFF
2
CEFF
Figure 8. Abatement and damage costs (percent of GDP).
2230
2200
2170
2140
2110
2080
2050
2020
1990
0
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
241
Table 4 World discounted consumption over 320 years (billion 1990 US$) Scenario
BAU
NASH
NASH/BAU (%)
EFF
EFF/BAU (%)
EFF/NASH (%)
USA Japan EU China FSU ROW World
78270 42867 102612 9131 23763 81049 337692
78353 42909 102731 9141 23794 81137 338065
0.11 0.10 0.12 0.11 0.13 0.11 0.11
78986 43222 103650 8862 24025 81093 339837
0.91 0.83 1.01 2.95 1.10 0.05 0.64
0.81 0.73 0.89 3.06 0.97 0.05 0.52
not quite correct because we are using different discount rates and a different time horizon (320 years). In particular, we use 1.5% for industrialized regions and 3.0% for developing regions whereas Nordhaus and Yang (1996) use ‘‘region-specific discount rates on consumption [ . . . that] average about 4.5% per year’’ (p. 755). Clearly, through our choice of discount rates we value climate change damages avoided in the far future much more heavily than Nordhaus and Yang (1996). Moreover, we use a more convex damage function. Overall, our value of cooperation must be considerably higher than the one calculated by RICE. The most striking difference is the pattern of winners and losers in the EFF scenario compared to NASH. Whereas Nordhaus and Yang (1996) find that USA is the only loser, we find that both CHN and ROW would lose from joining an efficient agreement without transfers. Again, it is difficult to compare this pattern with RICE since we have no trade in our cooperative solution. Unfortunately, Nordhaus and Yang (1996) did not report the trade flows in their efficient (Negishi) solution so we cannot judge whether the difference in the pattern of winners and losers is due to the absence of trade flows in our model. Though there is only an increase of total world consumption of about 0.5% between NASH and EFF, emission abatement policies do differ substantially between both scenarios as it appears from Figs. 4 and 5. Similarly, Fig. 3 shows importantly smaller temperature changes under EFF than under both BAU and NASH. Thus, while internationally coordinated policies appear to be importantly different from uncoordinated ones as far as environmental variables are concerned, the present model reveals that there may be some economic indifference between them.15
15
This last feature leads some authors, e.g. Hackl and Pruckner (2002), to minimize the importance of cooperative policies. Our findings concerning abatement and temperature change suggest that from an environmental point of view, optimality does matter a lot. Our point in Section 5.5, on how to achieve it cooperatively becomes all the more important.
242
5.5
Public Goods, Environmental Externalities and Fiscal Competition
Checking the core property of the Pareto efficient scenario without transfers
Concerning individual rationality, we observe from Table 4 that both China and ROW are experiencing a loss in lifetime consumption from moving from the Nash equilibrium to the Pareto efficient solution. This is not surprising since both regions are characterized by relatively low marginal emission reduction costs and/or relatively low marginal willingness to pay for environmental quality. From a global efficiency point of view, it might be desirable to require a substantial contribution from low cost regions in the global emission abatement effort. However, the value of the avoided climate change damage might be insufficient for these regions to compensate for their high abatement effort. In order to check for coalitional rationality, i.e. whether there exist groups of regions that would benefit more from forming a coalition of their own than from the Pareto efficient allocation, we calculated all possible partial agreement Nash equilibria for the CWS model. The number of all possible coalitions is given by 2N ¼ 64 for six regions. A complete list of all coalition values and their payoffs is given in Table 5. The entries in Table 5 have been sorted in increasing order of coalition size. The first six rows refer to the payoff of the individual countries in the Nash equilibrium. The next 15 rows refer to all pairs, the next 20 rows to coalitions of size three and so on. The last row refers the Pareto efficient allocation where S ¼ N. Payoff figures are reported in billion 1990 US$. The first column contains a six digit key from which the structure of the coalition can be deducted. If a region is a member of the coalition, it obtains a ‘‘1’’ at the appropriate position in the key. For instance, the key ‘‘111111’’ refers to S ¼ N ¼ {USA, JPN, EC, CHN, FSU, ROW}, ‘‘10000’’ refers to S ¼ {USA}, ‘‘111000’’ refers to S ¼ {USA, JPN, EC} and so on. P P Column 2 (WSS ¼ j2S t WjS, t =(1 þ rj )t ) contains the value of a coalition under itsPcorresponding partial agreement Nash equilibrium. P Column 3 (WS ¼ S t Wj,t =(1 þ rj )t ) contains the total of what the members of each coalition get in the Pareto efficient allocation. Columns 4 and 5 show the difference between WS and WSS in absolute amounts and in percentages. The differences WS WSS measure the gain, for each coalition S, from accepting the Pareto efficient allocation rather than sticking to a partial agreement Nash equilibrium w.r.t. itself. If this difference is negative, it means that S is worse off at the EFF allocation and that the voluntary participation constraint for coalition S is not satisfied. There are 12 coalitions with #S $ 2 for which the voluntary participation
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
243
Table 5 Coalitions payoffs at all PANE w.r.t. a coalition (WSS ) and at EFF (WS ); ~ S ) (billion 1990 US$) generalized GTT transfers (C Key
WSS
WS
WS
WSS
Percentage
~S C
WS
~S þC
WS þ ~S W S C S
Percentage
Coalitions of one country 100000 78353 78986 010000 42909 43222 001000 102731 103650 000100 9141 8862 000010 23794 24025 000001 81137 81093
633 313 919 279 231 44
0.808 0.729 0.895 3.057 0.969 0.054
282 121 423 333 123 616
78704 43102 103226 9195 23902 81709
351 192 496 54 108 572
0.448 0.448 0.482 0.591 0.452 0.705
Coalitions of two countries 110000 121264 122208 101000 181090 182636 100100 87535 87848 100010 102151 103011 100001 159829 160079 011000 145642 146872 010100 52062 52084 010010 66705 67247 010001 124262 124315 001100 111946 112511 001010 126531 127674 001001 184315 184743 000110 32944 32886 000101 90467 89955 000011 105134 105118
945 1546 312 860 250 1230 22 542 53 566 1143 427 58 512 17
0.779 0.854 0.357 0.842 0.156 0.845 0.043 0.813 0.043 0.505 0.903 0.232 0.175 0.566 0.016
403 706 51 405 334 544 213 244 495 90 546 192 210 949 493
121806 181930 87899 102605 160413 146328 52297 67003 124811 112421 127128 184935 33097 90904 105610
542 841 364 455 584 686 235 299 548 476 597 620 153 437 476
0.447 0.464 0.416 0.445 0.365 0.471 0.451 0.448 0.441 0.425 0.471 0.336 0.463 0.483 0.453
Coalitions of three countries 111000 224007 225858 110100 130486 131070 110010 145067 146233 110001 202879 203301 101100 190415 191497 101010 204903 206660 101001 263009 263729 100110 111367 111872 100101 169139 168941 100011 183752 184103 011100 154905 155734 011010 169448 170897 011001 227376 227965 010110 75880 76109 010101 133513 133177 010011 148160 148340 001110 135788 136536 001101 193681 193604 001011 208255 208767 000111 114376 113979
1851 584 1166 422 1083 1757 719 505 199 352 829 1448 589 229 336 180 748 76 512 397
0.826 0.448 0.804 0.208 0.569 0.857 0.274 0.453 0.117 0.191 0.535 0.855 0.259 0.301 0.252 0.121 0.551 0.039 0.246 0.347
826 69 526 213 372 829 90 72 667 211 211 667 72 90 829 372 213 526 69 826
225032 131001 145707 203514 191125 205832 263639 111800 169608 184314 155523 170230 228037 76198 134006 148712 136323 194130 208837 114805
1024 515 641 635 711 928 630 433 468 562 618 781 661 318 492 552 535 450 582 429
0.457 0.394 0.442 0.313 0.373 0.453 0.239 0.389 0.277 0.306 0.399 0.461 0.291 0.420 0.369 0.372 0.394 0.232 0.279 0.375
Coalitions of four countries 111100 233398 234720 111010 247830 249883 111001 306113 306951 110110 154332 155095
1322 2053 838 763
0.566 0.828 0.274 0.494
493 949 210 192
234227 248933 306741 154902
829 1104 628 571
0.355 0.445 0.205 0.370 (Continues)
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Public Goods, Environmental Externalities and Fiscal Competition
Table 5 (Continued ) Key
WSS
WS
110101 110011 101110 101101 101011 100111 011110 011101 011011 010111 001111
212255 226825 214285 272543 286996 193119 178761 236817 251338 157457 217685
212163 227326 215522 272590 287753 192965 179758 236827 251990 157202 217629
Coalitions of five countries 111110 257284 258744 111101 315738 315813 111011 330123 330976 110111 236267 236188 101111 296612 296615 011111 260851 260851 Coalitions of six countries 111111 339837 339837
~S WS þ C
WS þ ~S W S C S
Percentage
546 90 495 244 213 544 334 405 51 706 403
212710 227416 215027 272834 287540 193509 179425 237232 251938 157907 218032
454 591 741 292 544 390 664 415 600 451 346
0.214 0.261 0.346 0.107 0.190 0.202 0.372 0.175 0.239 0.286 0.159
0.568 0.024 0.258 0.033 0.001 0.000
616 123 333 423 121 282
258129 315936 330642 236611 296736 261134
845 198 519 344 124 283
0.328 0.063 0.157 0.146 0.042 0.108
0.000
0
339837
0
0.000
WS WSS
Percentage
92 501 1237 48 757 154 998 10 652 255 57
0.043 0.221 0.577 0.018 0.264 0.080 0.558 0.004 0.259 0.162 0.026
1461 75 853 79 3 1 0
~S C
constraint is violated. In particular, many coalitions containing CHN and ROW are in this situation. 5.6
Checking the core property of the generalized GTT transfers
~ S are reported In column 6 of Table 5, generalized GTT transfers C as computed from formula (29) with shares adjusted as in (30). Column 7 ~ S ) contains the value of total consumption available to the (WS þ C coalitions after these transfers have taken place. The last two columns ~ S W S in absolute amounts and in percentshow the differences WS þ C S ~ i should balance, we verify that P C ~ i ¼ 0 in the ages. As the transfers C N last row of Table 5. The shares of the regions in the surplus of cooperation p ~ i are as follows: USA: 19.8%, JPN: 10.8%, EU: 28.0%, CHN: 3.0%, FSU: 6.1% and ROW: 32.3%. Hence, EU and ROW seize each about 30% of the surplus of cooperation. Recall that these shares reflect the different regions’ share in total world discounted marginal climate change damages. These weights resemble closely the distribution of GDP in the reference year 1990 (see Table 6). Table 5 shows that the industrialized regions (USA, JPN, EU and FSU) pay transfers to the developing regions CHN and ROW. In particular, the transfer scheme (29) compensates CHN and ROW such that they
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
245
Table 6 1990 Reference year variables Yi0 (billion 1990 US$) (%)
Ki0 (billion 1990 US$) (%)
L0i (million people) (%)
USA JPN EU CHN FSU ROW
5464.796 (25.9) 2932.055 (13.9) 6828.042 (32.4) 370.024 (1.8) 855.207 (4.1) 4628.621 (22.0)
14262.510 (26.3) 8442.250 (15.6) 18435.710 (34.0) 1025.790 (1.9) 2281.900 (4.2) 9842.220 (18.1)
250.372 (4.8) 123.537 (2.4) 366.497 (7.0) 1133.683 (21.5) 289.324 (5.5) 3102.689 (58.9)
1.360 (20.5) 0.292 (10.9) 0.872 (28.9) 0.669 (3.0) 1.066 (6.8) 1.700 (29.9)
World
21078.750 (100.0)
54290.380 (100.0)
5266.100 (100.0)
5.959 (100.0)
Ei0 (GtC)
are better off under the Pareto efficient allocation with transfers than if there were no cooperation at all. Hence, both regions have no incentive to deviate individually anymore. Eventually, all regions are individually better off under the transfer scheme compared to the noncooperative open~ S W S is positive for loop Nash equilibrium since the difference WS þ C S all regions. Hence, the transfer solution satisfies individual rationality. It can also be seen from Table 5 that the coalitional rationality constraints are met for all possible subcoalitions S N with #S $ 2. In particular, the 12 coalitions for which coalitional rationality was violated in the Pareto efficient allocation without transfers receive sufficient compensation. Hence, the allocation with GTT transfers is a core allocation for the emission abatement game associated to the CWS model.16 5.7
Coalitional aspects of some post-Kyoto policies
In addition to providing a numerical proof of the core property of the efficient scenario with GTT transfers, Table 5 is also an instrument for evaluating policies that single countries, or coalitions of countries, might consider adopting. As examples, let us single out the following three such policies, which happen to be of particular interest in the current context: 1. The ‘‘Kyoto Annex B countries’’ policy, where coalition S corresponds to the row keyed as {111010} (see coalitions of four countries, row 2). The table reveals that for this coalition: X {i} WSS ¼ 247:830 > W{i} ¼ 247:787 i2S
16
We have run numerous simulations for different sets of parameters like discount rates, climate change damage parameters, free riding behavior in region ROW, etc. Details on this sensitivity analysis can be obtained from the authors. For none of these sensitivity analysis we found a violation of the core property of the transfer scheme. Of course, this is not a general proof of the core property for non-linear damage functions but still it indicates that the result is robust.
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that is, the aggregate payoff for the Annex B coalition in terms of discounted aggregate consumption is higher than if each of its members chooses the NASH scenario. Thus, although this coalition does not achieve optimality at the world level, its formation is preferable for all its members to isolated domestic actions (NASH). It is thus collectively rational for the Annex B countries to act on their own, as specified by the Kyoto Protocol, in spite of the fact that the rest of the world sticks to an individualistic, noncooperative policy. 2. The ‘‘George W. Bush’’ policy, where the coalition S corresponds to the row keyed as {100000}. Table 5 reveals that for this coalition: USA WUSA ¼ 78:353 < WUSA þ CUSA ¼ 78:704 < WUSA ¼ 78:986
that is, sticking to a domestic optimization policy only is for the USA less beneficial than the world optimal policy, both with and without transfers. 3. The ‘‘Annex B without USA’’ policy, where the coalition S corresponds to the row keyed as {011010} (coalitions of three countries, row 12). For this coalition: X {i} WSS ¼ 169:448 > W{i} ¼ 169:434 i2S
meaning that, in spite of the Bush policy of staying out of the Protocol, the Annex B group would benefit from the partial cooperation achieved among the non-USA countries. This comes as a support of the policy initiated by the EU in the Spring of 2001. Our simulations show also that the USA gains only very little from leaving the Kyoto coalition. By renouncing the Kyoto coalition (and free riding on the effort of the remaining Kyoto members), the USA’s payoff would be S WUSA ¼ 78:367 for S ¼ {011010}, which is only slightly more than what it would achieve if it would stick to its Kyoto commitment S (WUSA ¼ 78:366 for S ¼ {111010}).17 5.8
The transfers and time
Fig. 9 shows the differences in world consumption levels, at each point in time t, between the P noncooperative Nash tequilibrium and the Pareto efficient scenarios: j2N (Zj,t ZjNE ,t )(1 þ rj) . All values are discounted back to 1990. Hence, Fig. 9 depicts the evolution over time of the 17 Moreover, in Eyckmans (2001), it is shown that the deviation of USA from the Kyoto Protocol is not credible in the farsighted sense as defined by Chwe (1994). The initial deviation of the USA would trigger off subsequent deviations by, among others, the FSU, making the USA in the end worse off than under the original Kyoto Protocol. Further coalitional analysis of the CWS model is offered in Eyckmans and Finus (2003).
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
247
120 100 80 60 40 20
2310
2270
2230
2190
2150
2110
2070
−40
2030
−20
1990
0
Figure 9. EFF–NASH difference in world consumption (billion 1990 US$).
value of cooperation over noncooperation. Typically, up to the year 2080, the NASH scenario dominates the efficient scenario EFF. After 2090, however, the order of dominance is reversed. In total, the sum over time of all differences (the surface under the curve) is positive as the gains from cooperation in the far future more than compensate for the initial losses. This means that we are in a situation as in Assumption 3 in GTT. Obviously, the countries cannot borrow against future gains in order to compensate for early losses. We should therefore design a transfer scheme in such a way that the regions most affected initially be compensated partially by the less affected regions. An attempt to design such a transfer scheme with transfers evolving over time is reported, in theoretical terms, in Germain, Toint, Tulkens and de Zeeuw (2003) where the appropriate tool for that purpose is found to be closed-loop Nash equilibria replacing open-loop ones. However, computational complexity of this scheme requires further research before it can be applied to the CWS model.
6.
Conclusion
As briefly alluded to in the introduction, there are game theoretic approaches to the climate change problem that conclude with skepticism about the chance of ever reaching worldwide cooperation in this area. The present paper concludes with the opposite view. This difference in results stems from using different branches of game theory. One of these branches dealing with stability of coalitions, inspires the former approach; another branch—cooperative (or coalitional) game theory, provides the foundation for the present paper. Evaluating the respective degrees of validity of either approaches must be left to a pure game theoretical paper, still to be written.
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In the meantime, this paper contributes to that debate in a way whose generality is wider than most papers defending the skeptic thesis. Here, conditions for world cooperation are obtained in the framework of a dynamic, multi-regional integrated assessment model including stock externalities and numerical verification. The progress brought about by these features over the one-equation models used in the other approach does arguably give more credibility to the cooperative approach advocated here.
Appendix A Equation listing of the CWS model: Ui (Zi,t ) ¼ Zi,t
(A:1)
Yi,t ¼ Zi,t þ Ii,t þ Ci (mi,t ) þ Di (DTt )
(A:2)
Yi,t ¼ Ai,t Kig,t L1g i,t
(A:3)
bi,2 Ci (mi,t ) ¼ Yi,t bi,1 mi,t
(A:4)
Di (DTt ) ¼ Yi,t ui,1 DTt i,2
(A:5)
u
Ki,tþ1 ¼ (1 dK )Ki,t þ Ii,t Ei,t ¼ si,t (1 mi,t )Yi,t Mtþ1 ¼ (1 dM )Mt þ b
X
(Ki,0 given)
(A:6) (A:7)
Ei,t
(M0 given)
(A:8)
i2N
Ft ¼
4:1 ln (Mt =M0 ) þ Ftx ln (2)
(A:9)
o o Tto ¼ Tt1 þ t 3 [DTt1 Tt1 ]
DTt ¼ DTt1 þ t1 [Ft lDTt1 ] t2 [DTt1
(A:10) o Tt1 ]
(A:11)
References Barrett, S., 1994. Self-enforcing international environmental agreements, Oxford Economic Papers, 46, 878–894. Carraro, C. and Siniscalco, D., 1993. Strategies for the international protection of the environment, Journal of Public Economics, 52, 309–328. Chander, P. and Tulkens, H., 1995. A core-theoretic solution for the design of cooperative agreements on transfrontier pollution, International Tax and Public Finance, 2, 279– 293. (Chapter 9 in this volume). Chander, P. and Tulkens, H., 1997. The core of an economy with multilateral environmental externalities, International Journal of Game Theory, 26, 379–401. (Chapter 8 in this volume).
Ch.11 Simulating Coalitionally Stable Burden Sharing Agreements
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Chander, P., Tulkens, H., van Ypersele, J.-P. and Willems, S., 2002. The Kyoto Protocol, an economic and game theoretic interpretation, in P. Dasgupta, B. Kristro¨m, and K.G. Lo¨fgren, (eds.). Environmental economics: theoretical and empirical inquiries, Festschrift in honour of Karl-Go¨ran Ma¨ler. Cheltenham: Edward Elgar, chapter 6, 98–117. (Chapter 10 in this volume). Chwe, M.S.-Y., 1994. Farsighted coalitional stability, Journal of Economic Theory, 63, 299–325. Eyckmans, J., 1997. Nash implementation of a proportional solution to international pollution control problems, Journal of Environmental Economics and Management, 33, 314–330. Eyckmans, J., 2001. On the farsighted stability of the Kyoto Protocol. ETE Working Paper 2001–03, K.U. Leuven-CES-ETE and CLIMNEG Working Paper 40. Louvain-laNeuve: Universite´ catholique de Louvain. Eyckmans, J. and Bertrand, C., 2000. Integrated assessment of carbon and sulphur emissions, simulations with the CLIMNEG model. CLIMNEG Working Paper 32. Louvainla-Neuve: Universite´ catholique de Louvain. Eyckmans, J. and Finus, M., (forthcoming 2006). Coalition formation in a global warming game: how the design of protocols affects the success of environmental treaty-making, Natural Resource Modeling. Eyckmans, J., Proost, S. and Schokkaert, E., 1993. Equity and efficiency in greenhouse negotiations, Kyklos, 46, 363–397. Germain, M., Toint, Ph. and Tulkens, H., 1998. Financial transfers to sustain cooperative international optimality in stock pollutant abatement, in F. Duchin, S. Faucheux, J. Gowdy and I. Nicolaı¨ (eds.). Sustainability and firms: technological change and the changing regulatory environment. Cheltenham: Edward Elgar, 205–219. Germain, M., Toint, Ph.L., Tulkens, H. and De Zeeuw, A., 2003. Transfers to sustain dynamic core-theoretic cooperation in international stock pollutant control, Journal of Economic Dynamics and Control, 28, 79–99. (Chapter 12 in this volume). Hackl, F. and Pruckner, G.J., 2002. How global is the solution to global warming? Economic Modelling, 20, 93–117. Hoel, M., 1992. International environmental conventions: the case of uniform reductions of emissions, Environmental and Resource Economics, 2, 141–159. Kaitala, V., Pohjola, M. and Tahvonen, O., 1992. Transboundary air pollution and soil acidification: a dynamic analysis of an acid rain game between Finland and the USSR, Environmental and Resource Economics, 2, 161–181. Ma¨ler, K.-G., 1989. The acid rain game, in H. Folmer and E. Van Ierland (eds.), Valuation methods and policy making in environmental economics. Amsterdam: Elsevier, 231–252. Nordhaus, W.D. and Yang, Z., 1996. A regional dynamic general-equilibrium model of alternative climate change strategies, American Economic Review, 86, 741–765. Samuelson, P.A., 1954. The pure theory of public expenditure, Review of Economics and Statistics, 36, 387–389. Tulkens, H., 1979. An economic model of international negotiations relating to transfrontier pollution, in K. Krippendorff (ed.). Communication and control in society. New York: Gordon and Breach, 199–212. (Chapter 5 in this volume). Tulkens, H., 1998. Cooperation versus free-riding in international environmental affairs: two approaches, in N. Hanley and H. Folmer (eds.). Game theory and the environment. Cheltenham: Edward Elgar, 30–44. van der Ploeg, F. and de Zeeuw, A.J., 1992. International aspects of pollution control, Environmental and Resource Economics, 2, 117–139. Yang, Z., 1998. Incentive compatible coalitions in international agreements on CO2 emission controls, mimeo, MIT.
Chapter 12 TRANSFERS TO SUSTAIN DYNAMIC CORE-THEORETIC COOPERATION IN INTERNATIONAL STOCK POLLUTANT CONTROL
Marc Germain, Philippe Toint, Henry Tulkens, Aart de Zeeuw Journal of Economic Dynamics and Control 28, 79–99 (2003).
For international environmental agreements aiming at world efficiency in the presence of transboundary flow pollution, it is known that, in a static context, efficiency and stability in the sense of the core of a cooperative game can be achieved using appropriately defined transfers between the countries involved. However, for accumulating pollutants, such as CO2 in the atmosphere, a dynamic analysis is required. This paper provides a transfer scheme for which a core property is proved analytically in a dynamic (closed-loop) game theoretic context. The characteristic function of the cooperative dynamic game yielding this result is discussed and an algorithm to compute the transfers numerically is presented and tested on an example. The transfers are also compared with an open-loop formulation of the model.
1.
Introduction
This paper presents a cooperative game theoretic analysis of the economics of international agreements on transnational pollution control, when the environmental damage arises from stock pollutants This paper was first circulated as CLIMNEG Working Paper No. 6 and CORE Discussion Paper No. 9832. It was initiated as part of a research project (project 729039) financed by the Fonds de De´veloppement Scientifique of the Universite´ catholique de Louvain. It was completed as part of the CLIMNEG research project (project CG/DD/ 241) financed at UCL by the Belgian Government. The fourth author’s contribution was made possible thanks to a Human Capital Mobility programme (CHRX CT93) of the European Union. The authors gratefully acknowledge comments and suggestions from C. d’Aspremont, J.-P. van Ypersele, P.-A. Jouvet, Ph. Michel, A. Ulph, Y. Nesterov and especially R. Amir, as well as from two referees of the journal.
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that accumulate (and possibly decay). The issues raised by the necessity of cooperation amongst the countries involved, if a social optimum is to be achieved, have already been addressed in the literature in terms of game theory concepts; see e.g. Ma¨ler (1989). However, most of these contributions have only dealt with static (one shot) games, which are only suitable for flow pollution models. With stock pollutants the problem acquires an intertemporal dimension. In this case, dynamic game theory is a more appropriate tool of analysis, as is done in, e.g. van der Ploeg and de Zeeuw (1992), Kaitala, Pohjola and Tahvonen (1992), Hoel (1992), Tahvonen (1994), Petrosjan and Zaccour (1995). With exception of the last paper, these contributions leave aside the issue of the voluntary implementation of the international optimum. This is an important drawback since no supranational authority can be called upon to impose the optimum in a context where the countries’ interest in cooperation diverges strongly between one another, and especially if some countries lose when the social optimum is implemented. In view of ensuring such implementation, it has often been suggested that financial transfers between the countries involved would provide incentives towards cooperation.1 This property, understood in the sense of the core of a cooperative game, has in effect been demonstrated by Chander and Tulkens (1995, 1997), who propose a particular transfer scheme based on parameters reflecting the relative intensities of the countries’ environmental preferences. This result, established for flow pollutants only (that is, in a static game model), has been extended by Germain, Toint and Tulkens (1998) to the larger context of open-loop dynamic games. But this extension suffers from the fact that it implies the restrictive assumption that negotiations take place once and for all and bear on emissions trajectories that extend until the end of the planning horizon. Agreements are thus binding all along. The aim of the present paper is precisely to relax this assumption: here, cooperation is negotiated at each period, i.e. players reevaluate at each time the interest of cooperation, taking account of the current stock of pollutant. This approach is one of closed-loop (or feedback) dynamic games. In this context, financial transfers are considered again. They are now formulated in such a way that they induce cooperation in the coretheoretic sense at each period. The model thus yields a sequence of 1 In a two-countries framework, Petrosjan and Zaccour (1995) use financial transfers to obtain a cooperative solution as a Nash equilibrium. However, being limited to two countries, they cannot deal with the issue of coalitions, which is one of our aims here. Kverndokk (1994) deals both with coalitions and financial transfers for the greenhouse gas problem, but he resorts neither to dynamic games, nor to cooperative ones.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
253
cooperative international agreements. This result is obtained under more general convexity conditions than in the earlier attempt by Germain, Toint, Tulkens and de Zeeuw (1998) that was restricted to linear damage cost functions. Other approaches to the cooperation issue concentrate on the stable number of signatories by considering the trade-off for an individual country between joining some coalition and staying out of that coalition (see Carraro and Siniscalco (1993); Barrett (1994)). A critical comparison between these approaches and the one of this paper is given in Tulkens (1998). On the other hand, one must emphasize that this paper does not intend to describe the way actual international negotiations are conducted (e.g. the Kyoto Protocol). It chooses instead a normative approach, identifying transfers that make agreements less likely to be broken. For an alternative approach in this sense, see Bahn et al. (1998). The structure of the paper is as follows. Section 2 presents the international stock pollutant model and characterizes the international optimum as well as the closed-loop Nash noncooperative equilibrium. In Section 3, financial transfers are formulated so as to ensure that each country is not worse off when it participates, either in the finite or in the infinite horizon case. In Section 4, transfers are further specified so as to achieve stability in the core-theoretic sense, followed by a discussion of the characteristic function of dynamic cooperative games yielding that result. Section 5 describes an algorithm that solves the model developed in Sections 3 and 4, and presents numerical results obtained by application of the algorithm to a simple climatic change model with three regions, including a comparison with the open-loop case. Section 6 is a conclusion.
2.
International optimality and noncooperative equilibrium
Our economic model is written in discrete time. Consider n countries indexed by i 2 N ¼ {1, 2, . . . , n} and some planning period t ¼ {1, 2, . . . , T} (T, the planning horizon, is a positive integer, possibly infinite). In each country, pollution is entailed by economic activity: let Et ¼ (E1t , . . . , Ent )0 denote the vector of the different countries’ emissions of a certain pollutant at time t. These emissions spread uniformly in the atmosphere and contribute to a stock of pollutant S according to the equation St ¼ [1---d]St1 þ
n X
Eit ,
(1)
i¼1
where the initial stock of pollutant S0 is given and where d is the pollutant’s natural rate of degradation (0 < d < 1). This model describes, for
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Public Goods, Environmental Externalities and Fiscal Competition
example, the basics of the climate change problem where the flows of emissions of greenhouse gases accumulate into a stock, which only gradually assimilates and which is the cause of the climate change. This stock of pollutant causes damages to each country’s environment. For country i(i 2 N ), these damages during period t are measured in monetary terms by the function Di (St ), where Di is supposed to be a differentiable, increasing and convex function of the current stock St (D0i > 0, D00i $ 0). As described in (1), this current stock is a function of the inherited stock St1 and of the current emissions Et . The only way to control the stock of pollutant is through the control of emissions.2 More precisely, each country i can reduce its own emissions, and the cost of doing this is described by a differentiable, decreasing and strictly convex function Ci (Ei )(Ci0 < 0, Ci00 > 0). Ci (Ei ) measures the total costs incurred by country i from limiting its emissions to Ei . The convexity of the function reflects the intuitive idea that the marginal cost of reducing emissions is higher for lower levels of emissions. From Eq. (1), it is clear that the damages to the environment of country i will depend on the emissions of all countries. In what follows, we will consider two different modes of behavior of the countries. A first one assumes that they behave in an internationally optimal way, i.e. that each of them takes account of the impact of its pollution not only on itself but on all other countries as well. In this case, countries jointly choose at each period their emission levels in order to minimize total discounted costs, i.e. for each t 2 t, they solve the following problem: ( ) T X n X min bs [Ci (Eis ) þ Di (Ss )] {Es }s2{t,...,T} s¼t i¼1 subject to the constraints (1) and Eit $ 0 ( 8t 2 t, 8i 2 N ), b is the discount factor (0 < b # 1). According to Bellman’s principle of optimality, the solution can be found by solving the dynamic programming equations ( ) n X [Ci (EiT ) þ Di (ST )] , (2) W (T, ST1 ) ¼ min ET
W (t, St1 ) ¼ min Et
i¼1
( n X
)
[Ci (Eit ) þ Di (St )] þ bW (t þ 1, St ) ,
i¼1
(3)
t ¼ 1, 2, . . . , T 1
2
That is reducing pollution is done through the reduction of emissions, and not through the cleaning of the environment.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
255
subject to the constraints (1) and Eit $ 0 (8t 2 t, 8i 2 N ). In (2) and (3), W is called the value function. As the total costs of all countries are minimized, the resulting trajectories of emissions and stock constitute the international optimum. The convexity of the functions Ci and Di ( 8i 2 N ) suffices to guarantee that the minimum exists and is unique. In an alternative mode of behavior, one may assume that countries behave noncooperatively in the sense of a Nash equilibrium, where each of them minimizes at each period only its own discounted costs, taking as given the emissions of the other countries. Formally, at each t 2 t, each country i 2 N solves the following problem: ( ) T X s min b [Ci (Eis ) þ Di (Ss )] , {Eis }s2{t,...,T} s¼t subject to the constraints (1), Eit $ 0 and Ejt j 6¼ i given. Within the framework of dynamic programming this leads to the value functions Ni (T, ST1 ) ¼ min {[Ci (EiT ) þ Di (ST )]}, EiT
(4)
Ni (t, St1 ) ¼ min {[Ci (Eit ) þ Di (St )] þ bNi (t þ 1, St )}, Eit
t ¼ 1, 2, . . . , T 1
(5)
under the constraints (1) and Eit $ 0 8t 2 t. The convexity of the functions Ci and Di suffices to guarantee that the Nash equilibrium exists and is unique. The family of trajectories of emissions thus determined for each i in N, together with the resulting stock, constitute a noncooperative feedback Nash equilibrium (Bas¸ar and Olsder (1995)).3 The value functions Ni characterize the costs-to-go at this equilibrium for each country i. The emissions are functions of the current stock of pollutant and the equilibrium has the property of Markov perfectness (Maskin and Tirole (1988)), in the sense that it remains an equilibrium if the game is restarted at any intermediate year t from any value of the stock of pollutant. At the international optimum, and contrary to what happens at the Nash equilibrium, each country takes account of the impact of its pollution on the environment of all other countries. Therefore, from a collective point of view, the international optimum is better than the feedback Nash equilibrium. Nothing ensures, however, that this is also true at the individual level. Indeed, countries being different, it is possible that some country at sometime t is better off at the noncooperative equilibrium than 3
See Ma¨ler and de Zeeuw (1998) and van der Ploeg and de Zeeuw (1992) for applications of such an equilibrium to acid rains and climate change, respectively.
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at the optimum, so that cooperation is not profitable for this country (at least at time t). The same can occur for subsets of countries—i.e. coalitions—in the sense that, by limiting cooperation to such coalitions, the members of the latter could be better off than at the international optimum. The aim of this paper is to show that there exist financial transfers between countries that can make each one of them interested in achieving the international optimum at all periods t (individual rationality), and are such that, in addition, no sub-group of countries has ever an incentive to form a coalition and enact an optimum for itself only (coalitional rationality). We are thus aiming at a cooperative international optimum. To make the above argument precise, it is important to state explicitly what the fallback position is for each country when no transfers occurs. At each time t one could take the noncooperative feedback Nash equilibrium from t onwards as such point of reference, and determine the transfers accordingly. However, one should not neglect the fact that countries know that later on, thanks to the cooperative transfers to which they will have access, they will be better off than at the noncooperative Nash equilibrium. Hence, a better point of reference at time t is: noncooperation at time t, followed by cooperation afterwards.4 This is a rational expectations argument, that we introduce formally in Section 3 to deal with individual rationality, and that we use again in Section 4 to deal with coalitional rationality in the framework of a dynamic cooperative game.
3.
Transfers to sustain individual rationality at the international optimum
3.1
Transfers at final time T
Let us start by determining which transfers yield gains for all countries when they cooperate in the last period, for any level of the stock of pollutant S inherited from the past. In the noncooperative equilibrium the countries are supposed to solve problem (4). Country i’s total cost is then N ) þ Di (STN ), Ni (T, S) ¼ Ci (EiT
i 2 N,
(6)
N where EiT denotes the emission equilibrium level and STN denotes the resulting stock of pollutant given by
4
This idea was already put forward in Houba and de Zeeuw (1995) in the framework of a dynamic bargaining problem.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
STN ¼ [1 d]S þ
n X
N EiT :
257
(7)
i¼1
If countries cooperate, they jointly solve problem (2). Country i’s total cost is Wi (T, S) ¼ Ci (EiT ) þ Di (ST ),
(8)
where EiT is the optimal emission level and ST is the optimal stock of pollutant, given by
ST ¼ [1 d]S þ
n X
EiT :
(9)
i¼1
By definition of the optimum, one verifies that D
W (T, S) ¼
n X i¼1
Wi (T,S) #
n X
D
Ni (T,S) ¼ N(T, S):
(10)
i¼1
The difference between the two sides of this inequality measures the ecological surplus resulting from international cooperation. However, (10) is not sufficient to ensure cooperation. Indeed, if 9i 2 N such that Wi (T, S) > Ni (T, S), then country i will not cooperate without financial compensation for the higher cost it incurs. Since dynamic programming reduces the choice of emissions to one period at the time, one can use the transfers formula proposed by Chander and Tulkens (1997) in a static framework. Let ui (T, S) ¼ [Wi (T, S) Ni (T, S)] þ mi,T [W (T, S) N(T, S)]
(11)
be the transfer (< 0 if received, > 0 if paid) to country i at time T, where P mi,T 2 ]0, 1[,8i 2 N and ni¼1 mi,T ¼ 1. Then country i’s total cost including transfers becomes ~ i (T, S) ¼ Wi (T, S) þ ui (T, S): W
(12)
By construction, the budget of the transfers defined by (11) is P balanced (i.e. ni¼1 ui (T, S) ¼ 0). Since ~ i (T, S) Ni (T, S) ¼ mi,T [W (T, S) N(T, S)] # 0, 8i 2 N W
(13)
cooperation with transfers is individually rational at time T, in the sense that each country has interest to participate whatever the inherited stock of pollutant S.5 5 The fact that mi,T cannot be equal to 0 ensures that country i will benefit from cooperation if W (T, S) < N(T, S). The fact that mi,T cannot be equal to 1 excludes that country i monopolizes all the gains of cooperation.
258
3.2
Public Goods, Environmental Externalities and Fiscal Competition
Transfers at earlier periods
Countries know that, whatever they do at T 1, financial transfers exist (defined by (11)) that make the international optimum at T preferable for each of them with respect to the noncooperative equilibrium. Let us assume that these transfers induce cooperation,6 and that countries therefore expect, at T 1, that they will cooperate in period T. This is the rational expectations assumption announced at the end of Section 2. The problem we wish to consider now is whether under this assumption transfers can be designed that make the countries interested to cooperate at T 1 as well. In the absence of cooperation at T 1, each country i minimizes its own discounted total costs over the two periods T 1 and T, expecting cooperation and transfers in T. Thus, given the emissions of the other countries, country i solves problem (5) for t ¼ T 1 with Ni under the ~ i . This leads to min operator replaced by W ~ i (T, ST1 )}, i 2 N Vi (T 1, S) ¼ min {[Ci (Ei,T1 ) þ Di (ST1 )] þ bW Ei,T1
(14)
under the constraints (1) and Ei, T1 $ 0, 8i 2 N .7 This yields an equilibrium characterized at time T 1 by emission levels EV T1 as functions of the initial stock S at time T 1. The value function V ~ i (T, S V ), Vi (T 1, S) ¼ Ci (EiV,T1 ) þ Di (ST1 ) þ bW T1
i 2 N,
(15)
where V ¼ [1 d]S þ ST1
n X
EiV, T1
(16)
i¼1
denotes country i’s discounted equilibrium costs. We will call this equilibrium the fallback noncooperative equilibrium at time T 1. In the case where all countries cooperate, they solve problem (3) for t ¼ T 1. Optimal levels of emissions and of the resulting stock of pollutant are denoted by ET1 and ST1 , respectively. Both are functions of S. This yields ~ i (T, S ) Wi (T 1, S) ¼ Ci (Ei, T1 ) þ Di (ST1 ) þ bW T1
6
(17)
Note that, following Chander and Tulkens (1997, Section 5), one could indeed obtain the cooperative optimum with transfers as an equilibrium, called ratio-equilibrium. 7 ~ i (T, ST1 ) contain transfers that sum up to zero, convexity Since the value functions W of the cost functions Ci and Di does not ensure that the objectives in (14) are convex, unlike what happens for T. In Section 5, we present a numerical algorithm that allows to check the second-order conditions at each period.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
259
which is country i’s part in the optimal total discounted costs, taking into account the transfers and the resulting cooperation expected in T. As in period T (see (10)), one verifies that D
W (T 1, S) ¼
n X i¼1
Wi (T 1, S) #
n X
D
Vi (T 1, S) ¼ V (T 1, S):
(18)
i¼1
V (T 1, S) W (T 1, S) measures the ecological surplus induced by extending cooperation to period T 1, with respect to the alternative scenario where cooperation is limited to T. However, (18) is again not sufficient to induce cooperation at time T 1. If 9i 2 N such that Wi (T 1, S) > Vi (T 1, S), then country i will not want to extend cooperation to period T 1 without financial compensation. To induce country i to participate in T 1, we proceed as in period T. Let ui (T 1, S) ¼ [Wi (T 1, S) Vi (T 1, S)] þ mi,T1 [W (T 1, S) V (T 1, S)]
(19)
be the transfer paid or received by country i at time T 1, where P mi,T1 2 ]0, 1[, 8i 2 N and ni¼1 mi, T1 ¼ 1. Then country i’s total cost including transfers becomes ~ i (T 1, S) ¼ Wi (T 1, S) þ ui (T 1, S): W
(20)
By construction, the budget of the transfers defined by (19) is P balanced (i.e. ni¼1 ui (T 1, S) ¼ 0). Furthermore, it is easy to see that these transfers make cooperation individually rational at time T 1, whatever the inherited stock of pollutant S. The preceding analysis can be repeated for all earlier periods. The final result will be that the countries cooperate in each period. This determines the emission levels in each period and also the trajectory of the stock of pollutant, given its initial value S0 . In turn this trajectory ~ i , and therefore also determines the values of the functions Vi , Wi and W the values of the transfers ui . In Section 4 it will be shown that these transfers yield an interesting property from the perspective of cooperative game theory for specific values of mi . In the infinite horizon case, the backward reasoning considered above applies no more. However, we can consider the stationary solution by taking advantage of the fact that the cost functions Ci and Di as well as the sharing parameters mi do not depend directly on time. The functional forms of the solutions thus only vary in time through the varying stock of pollutant S. The structure of the problem is then the same as in the finite horizon case.
260
4.
Public Goods, Environmental Externalities and Fiscal Competition
Transfers to sustain coalitional rationality at the international optimum
The only condition so far on the mi ’s appearing in (11) or (19) is to take values between 0 and 1 and to sum up to 1. The aim of this section is to utilize the degrees of freedom left to obtain the property of ‘‘coalitional rationality’’ suggested by the core concept in cooperative game theory. We achieve this goal by extending to dynamic games the results obtained by Chander and Tulkens (1995, 1997) in a static framework. 4.1
Associating a dynamic cooperative game with the economic model
In general, a cooperative8 game (with transferable utility) is defined as a pair [N , w( )], where N ¼ {1, . . . , n} is the set of players (i.e. the n countries), and w( ) is the characteristic function of the game, that is, a function that associates to every subset (i.e. coalition) of N a real number called the worth of the coalition. In the present dynamic context, we consider such a game at each point in time of our economic model and we therefore specify a characteristic function wt ( , St ), at each t, where St is the inherited stock of pollutant at that time. We then consider the sequence over time of such cooperative games [N , wt ( , St )], t ¼ 1, . . . ,T; this sequence defines the dynamic cooperative game that we9 associate with our economic model. 4.2
Specifying the components of the dynamic game at each t
For each of the games [N , wt ( , St )] to be fully specified, the characteristic functions wt ( , St ) should further state: (i) what the strategies are that the players (countries) have access to, individually and as coalitions; (ii) which strategy they will actually choose; and (iii) the worth each coalition may expect from using its chosen strategy. As all this will be defined for a given time t, we drop from now on the time index until further notice, letting all emissions and stock variables be those of time t, which denotes the stock inherited from the past, St1 . except for S As to (i), we assume that the strategy space for each country i is the interval of possible emission levels Ei at time t, i.e. [0, 1[, and for each coalition U N the product of jUj of these intervals, where jUj denotes the number of countries in the coalition U. As to (ii), when some coalition 8 See, e.g. Osborne and Rubinstein (1994, p. 857). These authors use the more recent— and actually more appropriate—alternative terminology of ‘‘coalitional game’’. 9 Other concepts of dynamic cooperative games are discussed in Section 4.5.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
261
U N forms (including singletons), we consider that the vector EU of the strategies adopted by all players is as follows: (a) The emissions of the coalition members are the vector {EiU : i 2 U} which is the solution of the optimization problem n X X ) þ ~ i (S)] s:t: S ¼ [1 d](S [Ci (Ei ) þ Di (S) þ bW Ei , (21) min {Ei }i2U
i2U
i¼1
~ i (S) is defined as where 8j 2 N nU, Ej ¼ EjU as defined by (b), and W in (20). (b) The emissions EjU , j 2 N nU, of the countries outside of the coalition, are the solutions that simultaneously solve the optimization problems ~ j (S), min Cj (Ej ) þ Dj (S) þ bW Ej n X þ Ei , s:t: S ¼ [1 d]S
j 2 N nU, (22)
i¼1
~ j (S) is also as in (20). where 8i 2 U, Ei ¼ EiU as defined by (a) and W There are two assumptions present in this specification. The first one is the ‘‘g-assumption’’,10 according to which if a coalition forms, its members together minimize the sum of their discounted total costs (item (a)) while each country outside of it reacts by minimizing its own individual discounted total cost, that is, by turning to a natural fallback position (item (b)). The vector EU is thus of the nature of a Nash equilibrium of a noncooperative game where the players are the coalition U and the individual countries outside the coalition taken as singletons.11 Our second assumption is the rational expectations assumption, introduced in the preceding section; it is expressed here by the inclusion of ~ i (S) and W ~ j (S) in the objective functions of (21) and (22), respectively, W to be minimized. Finally, and on the basis of the above, the worth of any coalition U reads as X ~ i (S U )], w(U; S) ¼ [Ci (EiU ) þ Di (SU ) þ bW (23) i2U
þ Pn E U . Note that w(N ; S) is equal to W(S) as where S U ¼ [1 d]S i¼1 i defined in (17), with t ¼ T 1, i.e. to the optimal total cost for all countries together. The explicit form of the characteristic function of the games [N , wt ( , St )] is thus (23). 10 Borrowed from the concept of ‘‘g-characteristic function’’ introduced by Chander and Tulkens (1995, 1997). 11 This equilibrium is called by Chander and Tulkens a partial agreement Nash equilibrium with respect to the coalition U. The authors show that such an equilibrium exists and is unique under assumptions similar to those of our economic model.
262
4.3
Public Goods, Environmental Externalities and Fiscal Competition
Imputations at each t
For any game [N , w( ; S)] so specified, any n-dimensional vector of which the components sum up to w(N ; S) is called an imputation of the game. An imputation can thus be seen as a way of sharing the optimal total cost between the players. The vector (W1 (S), . . . , Wn (S)), where Wi (S) is defined as in (17), is an example of such an imputation, in which each country bears its own abatement and damage costs as induced by the optimal strategy {E }. However, the possibility of financial transfers between the countries implies that (an infinite number of) other imputations exist, associated ~ ~ with the same P strategy. Indeed, all vectors (W1 (S), . . . , Wn (S)) defined like (20), with i ui (S) ¼ 0, are imputations as well. A solution of the game is an imputation that satisfies certain properties. Among the imputations defined as (20), the ones that satisfy X ~ i (S)#w(U; S) 8U N (24) W i2U
are said to belong to the g-core of the game. In words, the core is the set of imputations with the property that each possible coalition bears a fraction of the total costs W(S) less than or equal to w(U;S), i.e. the least cost that this coalition can guarantee for itself. Imputations belonging to the core of the game defined above are therefore called ‘‘rational in the sense of coalitions’’ since the members of any coalition would suffer a total cost higher than the one at the optimum with transfers. 4.4
Transfers inducing g-core imputations at each t
Recall from Section 3 that E was defined as the vector of optimal emission levels and S as the optimal stock level, that EV was the vector of emission levels and SV the stock characterizing the fallback noncooperative equilibrium, and that (W1 (S), . . . ,Wn (S)) and (V1 (S), . . . ,Vn (S)) were, respectively, the vectors of discounted total costs at the optimum and at the fallback noncooperative equilibrium (cf. (17) and (15)), with W(S) and V(S) the sum of these vectors’ components. Recall also that all variables mentioned here are defined at time t. For the game [N , w( ; S)], consider now the imputation ~ 1 (S), . . . , W ~ n (S)) defined by (W ~ i (S) ¼ Wi (S) þ u~i (S) W
8i 2 N ,
where u~i (S) is a transfer of the form
(25)
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
~ i (S ) u~i (S) ¼ [Ci (Ei ) Ci (EiV )] þ m with S ¼ (1 d)S þ
P i
n X
[Cj (Ej ) Cj (EjV )]
263
(26)
j¼1
Ei and where
F 0 (S ) m ~ i (S ) ¼ Pn i 0 j¼1 Fj (S )
(27)
with ~ i (S ): Fi (S ) ¼ Di (S ) þ bW
(28)
We can now state the following. Theorem. The imputation (25) belongs to the g-core of the game [N , w( ; S)], if one of the following two conditions is satisfied: (i) 8i 2 N , Di is linear, or ~ i is monotonically increasing and convex and, 8U N , (ii) 8i 2 N , W X [Ci (EiU ) Ci (EiV )] $ 0, (29) i2U
where EU is the vector of emission levels at the partial agreement Nash equilibrium with respect to coalition U. Proof. Case (i) linear damage costs—is proven in Germain, Tulkens and de Zeeuw (1998). The proof of case (ii) is given in Germain, Toint, Tulkens & and de Zeeuw (1998) available from the authors upon request.12 4.5
Cooperative international optimum and alternative cooperative dynamic games
Turning back now to the cooperative dynamic game defined in Section 4.1, i.e. the sequence of games [N , wt ( , St )], t ¼ 1, . . . , T associated with the economic model of Section 2, consider the internationally optimal trajectories of emissions and stock defined there together with, at each time t, the transfers defined by (26)–(28). These optimal trajectories, combined with g-core-theoretic transfers are the solution we propose for the cooperative dynamic game. We call this combination the cooperative international optimum. As we have 12 Notice that the transfers (26) are formulated differently than in (11) or (19). The first part of the proof of case (ii) shows that the former with (27) and (28) are equivalent to the latter.
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Public Goods, Environmental Externalities and Fiscal Competition
argued above, this optimum is cooperative in a rational expectations sense. Recently, Kranich, Perea, and Peters (2001) have proposed three other core concepts for cooperative dynamic games, called, respectively, the ‘‘classical’’, the ‘‘weak sequential’’ and the ‘‘strong sequential’’ cores. Each one of these core concepts derives from a different form of the characteristic function, the differences residing in the assumptions made on how ‘‘deviations’’ by coalitions may (or may not) occur over time. Our characteristic function (and hence our core concept) is actually a fourth one, in its dynamic aspects (independently of its g nature). It is based on the assumption that if deviations occur at time t, they do not continue afterwards because from time t þ 1 on, cooperation with our transfers is more beneficial for all than noncooperation. This rational expectations attitude assumed to be adopted by the players at time t, was expressed summarily above as ‘‘noncooperation (i.e. deviation) at time t, followed by cooperation afterwards’’. Our characteristic function is thus a ‘‘rational expectations characteristic function’’, and hence our core concept a ‘‘rational expectations core’’. Referring to the formulation proposed in another paper by Filar and Petrosjan (2000), it seems to us that the economic model to which our dynamic games are associated implies (through the role played by the stock externality) that, in the notation of these authors (their Eq. (2.1)), the function v at time t þ 1 is determined by the solution retained at time t, that is, along the cooperative international optimum we have defined, by the Pareto optimal stock prevailing at t. We thus present an application of the model of dynamic games of these authors. 4.6
Two further comments
The transfers (26) used in the theorem are the sum of two components. The first term in the right-hand side of (26) is either (if > 0) an amount received by country i equal to the increase of its abatement costs due to its cooperating behavior during this period, or (if < 0) a payment made by country i equal to its savings in abatement costs, in the case where cooperation allows for an increase in this country’s emissions. The second term (always < 0) is country i’s contribution to cover the total cost of the aggregate abatement effort of the cooperating countries. Other remarks concern condition (ii). Notice that it is a sufficient, not a necessary one. It requires, with (29), that if a coalition forms, the total of the abatement costs of its members are larger than what this total would be at the fallback noncooperative equilibrium. How likely are such coalitions? In order for a coalition U to form—irrespective to its power to oppose (25)—it should at least satisfy
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
X
[Ci (EiU ) Ci (EiV )] þ
i2U
X
[Fi (SU ) Fi (S V )] # 0
265
(30)
i2U
since otherwise its members are all better off at the fallback noncooperative equilibrium. In (30) the magnitudes in the bracketed differences under the second summation sign are always negative, because S U is smaller than SV . Condition (ii) thus states that a coalition U that forms can only oppose (25) if in addition the first sum in (30) is negative, i.e. if the aggregate abatement costs of its members are lower than what they are at the fallback noncooperative equilibrium.13 However, one may think that the existence of such coalitions is unlikely because, as it is shown in the proof of the theorem, the sum of the emissions of the members of any coalition is always lower than what it is at the fallback noncooperative equilibrium.
5.
Computability issues
The aim of this section is to describe an original numerical algorithm that calculates the value functions and the transfers when the cost functions Ci and Di are convex. An original algorithm is indeed needed because in the backward induction with the complex transfer parameters, the problem is not analytically tractable, contrary to number of linear-quadratic problems in the literature (such as in Ma¨ler and de Zeeuw (1998), for example). It is written for finite horizon problems, which is appropriate for most practical applications. The emissions and stock trajectories associated with the international optimum can be calculated by traditional nonlinear programming techniques. The transfers are more difficult to calculate, because they make use of values calculated at the fallback noncooperative equilibrium, so that one must proceed by backward induction. The basic idea is to ~ i (t, St1 ), i 2 N , as construct explicit approximations for the surfaces W polynomial function of St1 , by using classical regression. 5.1
Statement of the algorithm
The algorithm consists of several steps: Step 1: The algorithm starts by solving the optimization problem (2) associated with the international optimum. The first-order conditions are Ci0 (Eit ) þ
T X t¼t
13
btt [1 d]tt
n X
D0j (St ) ¼ 0 8i 2 N , 8t 2 t
(31)
j¼1
In a static framework, Chander and Tulkens (1997) propose a condition on the marginal damages that ensures that all members of a coalition decrease their emissions w.r.t. the noncooperative equilibrium. Adapted to our framework, this condition is that P 0 0 V 8U N , U 6¼ N , jUj $ 2, i2U Fi (S ) $ Fj (S ), j 2 U. This condition guarantees (29).
266
Public Goods, Environmental Externalities and Fiscal Competition
and St ¼ [1 d]St1 þ
n X
Eit
8t 2 t,
(32)
i¼1
where Ci0 and D0j are the derivatives of functions Ci and Dj , respectively. Solving this T(n þ 1) equations yields the optimal values of the emissions and of the stock of pollutant for all periods. Step 2(a): To calculate the transfers, one must first solve the dynamic programming problems associated with the fallback noncooperative equilibrium. The algorithm proceeds backwards, starting from the last period. At the final time T, given the inherited stock of pollutant S, the fallback noncooperative equilibrium (which coincides with the Nash equilibrium at the last period) has to satisfy the conditions ! n X 0 N 0 N Ci (EiT ) þ Di [1 d]S þ (33) EjT ¼ 0, i 2 N : j¼1 N Once this system of equations has been solved for the variables EjT , and knowing by step 1 the optimal emissions at time T, it is possible P to calculate the transfers ui (S,T) using (26) (with m ~ i (ST ) ¼ D0i (ST )=( nj¼1 ~ i (S, T) using (25). D0j (ST ))) as well as the value functions W Step 2(a) is done for a set VT of given values of the inherited stock of pollutant S. This set is chosen to be representative of the interval of possible values of S. This interval is bounded below by the value of S that would be obtained with zero emissions during periods 1, 2, . . . , T 1, given S0 , and bounded above by the value of S that would be obtained with maximum emissions during the same period. Step 2(b): We now assume that total costs with transfers at time T have the following polynomial form:
~ i (T, S) ¼ kiT , m S m þ kiT , m1 S m1 þ . . . þ kiT , 0 , W
i 2 N,
(34)
where m is the order of the polynomial chosen so that the fit is good enough. To identify the parameters kiT , m , kiT , m1 , . . . , kiT , 0 , we use an ordinary least square regression method implemented in the software ~ i (T, S) are now approximated with Matlab, so that the functions W explicit analytical functions of S. Steps 2(a) and (b) are then repeated for period T 1. Given (14) and the inherited stock of pollutant S, first-order conditions associated to the fallback noncooperative equilibrium are
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
Ci0 (EitV )
þ
D0i
[1 d]S þ
" þ
~0 bW i
[1 d]S þ
n X
j¼1 n X
267
! EjV,T1
EjV,T1
#
! ,T
(35) ¼ 0, i 2 N :
j¼1
Once this system of equations has been solved for the EjV,T1 , knowing the optimal emissions at time T 1 (by step 1) and the value ~ i (S, T) (by application of steps 2(a) and (b) for time T), it is functions W possible to calculate the transfers ui (S,T 1) using (26)–(28) and the ~ i (S,T 1) using (25). This is done for a set VT1 of value functions W given values of the inherited stock of pollutant S, so that by regression of ~ i (S, T 1) on VT1 , the the calculated values of the value functions W ~ value functions Wi (S, T 1) can be approximated with explicit analytical functions of S. The algorithm continues backwards by repeating steps 2(a) and (b) until period 1. ~ i are known as functions of S for Step 3: Once the value functions W all times of the planning period t ¼ {1, . . . , T}, the algorithm calculates the actual values of these value functions and of the transfers for all countries all along the optimal trajectory calculated at step 1. 5.2
A numerical example
In the following, we show some numerical results obtained by application of the algorithm to a simple climatic change model with three regions and 30 periods (originally considered by Germain and van Ypersele (1999) in a six regions framework). The three regions considered are Industrialized Countries (IC),14 China and Rest of the World (RW). Periods are decades, starting with decade 1990–2000. Details of the equations and values of the parameters are given in Appendix. The polynomials used are of degree three. ~ i on a large set of possible First, by regressing the value functions W values of the inherited stock of pollutant S, we were able to verify that the objective function of each country is well behaved and yields existence and uniqueness of the fallback noncooperative equilibrium at each period. We show the results for a planning horizon extending over 11 periods (i.e. until the end of the 21st century), but to avoid boundary 14
The region IC groups Europe, US, Japan and Former Soviet Union.
268
Public Goods, Environmental Externalities and Fiscal Competition
problems, computations were made until T ¼ 30.15 Table 1 shows the evolution of the optimal abatement rates of emissions hi (i.e. the reduction of emissions induced by the optimal solution w.r.t. the BAU scenario, see Appendix) and atmospheric temperature (which is itself a function of the stock of CO2 in the atmosphere, see Appendix). The evolutions of the abatement rates are contrasting: increasing slowly for IC (indexed as 1 in the table) and RW (indexed as 3), decreasing slowly for China (indexed as 2). Despite the control of emissions, atmospheric temperature continues to increase monotonically during the whole century. The increase of temperature amounts to around 2.58C at the end of the 21st century compared to its preindustrial level. The first three columns of Table 2 give the differences between the payoffs of the three regions along the international optimum (without current transfers) and at the fallback noncooperative equilibrium, (Vi Wi ). Positive (negative) values mean that the region is better (worse) off at the optimum. While IC and RW are better off in each period, China is worse off in every period. Thus, the international optimum does not satisfy individual rationality. The following three columns of Table 2 give the transfers ui received or paid by each region to make the international optimum trajectory a cooperative one. A negative (positive) value means that the transfer is received (paid). China receives at each period a transfer that induces this country to cooperate. The principal donor is IC, but RW is also a donor until the middle of the century. After 2060 RW also becomes a receiver, but to a far less extent than China. Table 1 Optimal abatement rates (%) and atmospheric temperature changes (8C) t
h1
h2
h3
DT
1995 2005 2015 2025 2035 2045 2055 2065 2075 2085 2095 2105
14.29 14.47 14.67 14.94 15.19 15.37 15.51 15.63 15.74 15.83 15.91 15.98
30.45 28.74 27.46 26.50 25.81 25.31 24.95 24.70 24.53 24.42 24.34 24.29
22.49 23.08 23.61 24.09 24.54 24.96 25.34 25.68 25.99 26.28 26.53 26.76
0.87 1.00 1.15 1.29 1.44 1.59 1.74 1.89 2.03 2.18 2.32 2.47
15
As t approaches T, results become meaningless because one ignores environmental damages after T.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
269
Table 2 Individual payoff differences between fallback equilibrium (Vi ) and optimal ~ i ) transfers (ui ) between the regions i ¼ 1---3 (in trajectories without (Wi ) and with (W billion 1990 US$) t
V1 W1
V2 W2
V3 W3
u1
u2
u3
~1 V1 W
~2 V2 W
~3 V3 W
1995 2005 2015 2025 2035 2045 2055 2065 2075 2085 2095 2105
41.63 41.26 40.42 39.48 38.44 37.15 35.62 33.90 32.03 30.04 27.97 25.85
12.76 11.44 10.30 9.34 8.54 7.86 7.27 6.75 6.28 5.86 5.45 5.06
51.65 51.17 50.58 49.67 48.50 47.20 45.72 44.04 42.16 40.09 37.83 35.43
10.34 11.09 11.56 11.97 12.28 12.35 12.22 11.93 11.51 10.98 10.38 9.72
17.25 16.13 15.16 14.34 13.64 13.01 12.42 11.86 11.31 10.75 10.18 9.57
6.91 5.04 3.60 2.37 1.36 0.66 0.20 0.07 0.20 0.23 0.20 0.14
31.30 30.17 28.86 27.51 26.17 24.81 23.41 21.97 20.52 19.05 17.59 16.14
4.48 4.69 4.86 5.00 5.09 5.15 5.15 5.11 5.03 4.90 4.72 4.51
44.73 46.13 46.98 47.30 47.14 46.54 45.52 44.11 42.36 40.32 38.04 35.57
That these transfers are sufficient to induce cooperation from all regions at all periods is shown by the last three columns of Table 2, which give for each region the difference between its payoff at the cooperative optimum (i.e. with current transfers) and at the fallback noncooperative ~ i ). All values are positive which means that all regions equilibrium (Vi W (in particular China) win from cooperation at all periods. In other words, taking into account that cooperation prevails after t, all regions are induced to extend cooperation to current period t thanks to current transfers at period t. Table 3 reports on differences between the payoffs obtained by coalitions along the international optimum (with and without transfers) and at the partial agreement Nash equilibria w.r.t. these coalitions (as defined in Section 4.1). The purpose is to check coalitional rationality. It turns Table 3 Coalition payoff differences between partial agreement Nash equilibria w.r.t. ~ () ) transfers (In billion coalitions (V() ) and optimal trajectories without (W() ) and with (W 1990 US$. Coalitions: I, R: IC and RW; C, R: China and RW; I,C: IC and China.) t
VC , R WC , R
~ C, R VC , R W
VI , R WI , R
~ I, R VI , R W
VI , C WI , C
~ I, C VI , C W
1995 2005 2015 2025 2035 2045 2055 2065 2075 2085 2095 2105
27.75 28.27 28.56 28.39 27.89 27.24 26.407 25.39 24.21 22.90 21.47 19.97
38.08 39.36 40.12 40.36 40.17 39.58 38.62 37.31 35.72 33.88 31.85 29.69
68.69 66.88 65.26 63.80 62.42 60.93 59.26 57.35 55.17 52.70 49.97 46.99
51.45 50.75 50.10 49.47 48.78 47.93 46.84 45.49 43.85 41.95 39.79 37.42
21.32 23.18 24.24 24.85 25.11 24.91 24.34 23.47 22.36 21.08 19.67 18.20
28.23 28.22 27.84 27.23 26.47 25.57 24.55 23.40 22.16 20.85 19.47 18.06
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Public Goods, Environmental Externalities and Fiscal Competition
out that, according to Columns 1, 3 and 5, no coalition can do better by itself than what it obtains at the international optimum. Coalitional rationality is thus satisfied without transfers. But this property is particular to this example and parameter values: indeed, it is not satisfied in other similar cases, as for example in Eyckmans and Tulkens (1999), where coalitional rationality only obtains with financial transfers. 5.3
Comparing open-loop and closed-loop models
It is interesting to compare the negotiation process described in this paper with the open-loop case, where the countries negotiate once and for all at the beginning of the planning period. Germain Toint and Tulkens. (1998) have defined the following transfers that ensure that all countries are interested to cooperate in the open-loop framework: Qi ¼
T X
at [Cit (Eit ) Cit (Eito )]
t¼1
þ
T X t¼1
n X D0i, tþ1 (Stþ1 ) at Pn [Cjt (Ejt ) Cjt (Ejto )], 0 ) D (S j¼1 j , tþ1 tþ1 j¼1
(36)
where Eito (i 2 N , t 2 t) is the emission of country i at time t characterizing the noncooperative open-loop Nash equilibrium.16 Table 4 summarizes the results. Columns 1 and 2 give the total discounted costs for each region at the optimum (Wi (S0 )) and at the noncooperative open-loop Nash equilibrium (Oi (S0 )). It is clear that China has no interest to cooperate without financial compensation. Column 3 gives the transfers Qi (S0 ) calculated by formula (36). With these transfers, it appears from the comparison of columns 2 and 4 that all regions gain from cooperation. Finally, column 5 gives the sum of transfers given or received by the three regions all along the planning period in the closed-loop case. Comparison of columns 3 and 5 shows that although of the same magnitude, the transfers given by IC and RW to compensate China are different in the closed-loop framework than what they are in the open-loop case. Table 4 Comparison with the open-loop case i
Wi (S0 )
Oi (S0 )
Qi (S0 )
Wi (S0 ) þ Qi (S0 )
t ui (t,St1 )
IC CHINA RW
10352.0 2321.5 20078.9
10839.4 2159.0 20627.6
284.4 240.1 44.3
10636.4 2081.5 20034.6
202.3 222.1 19.9
16
Not to be confused with the fallback Nash equilibrium defined in Section 3.
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
271
The closed-loop case in general leads to different results than the open-loop case. The reason is that the closed-loop case allows for more strategic behavior because the countries observe the stock of pollution at each point in time and condition their emissions on these observations. When a country considers to increase emissions, this country will argue that it leads to a higher stock and thus to less emissions by the other countries, which then partly offsets the increase. Therefore, in equilibrium, emissions will be higher in the closed-loop case than in the openloop case. In the model of this paper two effects influence the transfers when compared to the open-loop case. On the one hand, the closed-loop case leads to more emissions so that more transfers are needed. On the other hand, since the construct with a fallback equilibrium assumes cooperation in the future, it brings the outcome closer to the full-cooperative one, so that less transfers are needed. One cannot say a priori whether the closed-loop case leads to higher or lower transfers than the open-loop case. It is clear that when the solutions coincide, the transfers are the same. This occurs when the damage function is linear, as shown in Germain, Tulkens and de Zeeuw (1998), because in that case emissions do not depend on the stock.
6.
Conclusion
This paper develops transfer schemes that yield both individual and coalitional rationality for the design of international agreements that seek to achieve global optimality in stock pollutant control problems, in case these agreements can in principle be negotiated at each point in time. The paper also provides an algorithm to calculate the transfers, whose functioning is illustrated by a numerical example applied to the climate change problem in the context of a partition in three regions of the world. Our approach presents nevertheless some limits, that invite further research. First, the transfer scheme is supposed to guarantee the cooperative outcome simply because it makes cooperation beneficial for all countries. It would be interesting to drop this hypothesis and try to sustain the cooperative outcome as a noncooperative equilibrium. Another track of research would be to study what happens when, for one reason or another, a country deviates from the international agreement at a certain period. Technical work could also be done on the algorithm. Its main strengths are its relative simplicity and its ability to solve problems that are not linear-quadratic, but it is limited to one state variable problems and several mathematical questions could be deepened (for example the question of the propagation of numerical errors due to the successive regressions).
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Public Goods, Environmental Externalities and Fiscal Competition
Appendix. Equations and parameters used in the numerical example The model: hit ¼ 1
Eit , sit Yit
(A:1)
Stþ1 S0 ¼ [1 d][St S0 ] þ b
n X
Eit ,
(A:2)
i¼1
St , DTt ¼ g ln S0
(A:3)
Eit ai2 Cit (Eit ) ¼ ai1 haiti2 Yit ¼ ai1 1 Yit , sit Yit bi2 Mt bi2 Yit , Dit (St ) ¼ bi1 DTt Yit ¼ bi1 g ln M0
(A:4) (A:5)
where Eit is the CO2 emissions of region i at time t, sit the CO2 emissions/ output ratio (exogenous), Yit the output (exogenous), hit the control rate of the emissions, St the stock of CO2 in the atmosphere, DTt the variation of the atmospheric temperature (w.r.t. to its preindustrial level), Cit the abatement costs, Dit the damage costs. The value of the parameters S0 ¼ 590 billion tons of carbon (preindustrial level), d ¼ 0:0833 (rate of decay of CO2 in the atmosphere between two periods), b ¼ 0:64 (marginal atmospheric retention ratio of CO2 ), g ¼ 2:5= ln (2), annual discount rate ¼ 0:02. Parameters of the cost and damage functions are as follows: i
IC
CHI
RW
ai1 ai2 bi1 bi2
0.07 2.887 0.01102 1.5
0.15 2.887 0.015523 1.5
0.1 2.887 0.02093 1.5
Output (Yit ) and CO2 emission/output ratio time series are borrowed from Germain and van Ypersele (1999).17
17 Data of these authors are mainly issued from different versions of the well-known RICE model, developed by Nordhaus and Yang (1996) and Nordhaus and Boyer (2000).
Ch.12 Transfers to Sustain Dynamic Core-Theoretic Cooperation
273
References Bahn, O., Haurie, A., Kypreos, S. and Vial, J.-P., 1998. Advanced mathematical programming modeling to assess the benefits from international CO2 abatement cooperation, Environmental Modeling and Assessment, 3, 107–115. Barrett, S., 1994. Self-enforcing international environmental agreements, Oxford Economic Papers, 46, 878–894. Basar, T. and Olsder, G.-J., 1995. Dynamic noncooperative game theory. London: Academic Press. Carraro, C. and Siniscalco, D., 1993. Strategies for the international protection of the environment, Journal of Public Economics, 52, 309–328. Chander, P. and Tulkens, H., 1995. A core-theoretic solution for the design of cooperative agreements on transfrontier pollution, International Tax and Public Finance, 2, 279– 293. (Chapter 9 in this volume). Chander, P., and Tulkens, H., 1997. The core of an economy with multilateral environmental externalities, International Journal of Game Theory, 26, 379–401. (Chapter 8 in this volume). Eyckmans, J. and Tulkens, H., 2003. Simulating with RICE coalitionally stable burden sharing agreements for the climate change problem, Resource and Energy Economics 25, 299–327. (Chapter 11 in this volume). Filar, J.A. and Petrosjan, L.A., 2000. Dynamic cooperative games, International Game Theory Review, 2(1), 47–65. Germain, M. and van Ypersele, J.-P., 1999. Financial transfers to sustain international cooperation in the climate change framework. CLIMNEG Working Paper No. 19. Louvain-la-Neuve: Universite´ catholique de Louvain. Germain, M., Toint, Ph. and Tulkens, H., 1998. Financial transfers to ensure cooperative international optimality in stock pollutant abatement, pp. 205–219 in S. Fauchaux, J. Gowdy, and I. Nicolaı¨ (eds.). Sustainability and firms: technological change and the changing regulatory environment. Cheltenham: Edward Elgar, Germain, M., Tulkens, H. and de Zeeuw, A., 1998. Stabilite´ strate´gique et matie`re de pollution internationale avec effet de stock: le cas line´aire, Revue Economique, 49 (6), 1435–1454. Germain, M., Toint, P., Tulkens, H. and de Zeeuw, A., 1998. Transfers to sustain coretheoretic cooperation in international stock pollutant control. CORE Discussion Paper No. 9832, Universite´ Catholique de Louvain, Louvain-la-Neuve. Pethig (ed.). Conflicts and cooperation in managing environmental resources, Microeconomic Studies. Berlin: Springer. Houba, H. and de Zeeuw, A., 1995. Strategic bargaining for the control of a dynamic system in state-space form, Group Decision and Negotiation, 4, 71–97. Kaitala, V., Pohjola, M. and Tahvonen, O., 1992. Transboundary air pollution and soil acidification: a dynamic analysis of an acid rain game between Finland and the USSR, Environmental and Resource Economics, 2, 161–181. Kranich, L., Perea, A. and Peters, H., 2005. Core concepts for dynamic TU games, International Game Theory Review, 7, 43–61. Kverndokk, S., 1994. Coalitions and side payments in international CO2 treaties, in E. Van Ierland (ed.). International environmental economics, theories, models and application to climate change, international trade and acidification, developments in environmental economics, vol. 4. Amsterdam: Elsevier. Ma¨ler, K.-G., 1989. The acid rain game, in H. Folmer and E. van Ierland (eds.). Valuation methods and policy making in environmental economics. Amsterdam: Elsevier. Ma¨ler, K.-G. and de Zeeuw, A., 1998. The acid rain differential game, Environmental and Resource Economics, 12 (2), 167–184.
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Maskin, E. and Tirole, J., 1988. A theory of dynamic oligopoly. I: overview and quantity competition with fixed wage costs, Econometrica, 56(3), 549–569. Nordhaus, W. and Boyer, J., 2000. Warming the world: economic models of global warming. Cambridge, MA: MIT Press. Nordhaus, W. and Yang, Z., 1996. A regional dynamic general-equilibrium model of alternative climate-change strategies, American Economic Review, 86, 741–765. Osborne, M. and Rubinstein, A., 1994. A course in game theory. Cambridge, MA: MIT Press. Petrosjan, L. and Zaccour, G., 1995. A multistage supergame of downstream pollution. G-95–14, GERAD, Montreal: Ecole des Hautes Etudes Commerciales. Universite´ de Montre´al. Tahvonen, O., 1994. Carbon dioxide abatement as a differential game, European Journal of Political Economy, 19, 685–705. Tulkens, H., 1998. Cooperation vs. free riding in international environmental affairs: two approaches, chapter 2 (pp. 330–44) in N. Hanley and H. Folmer (eds.). Game theory and the environment. Cheltenham: Edward Elgar. van der Ploeg, F. and de Zeeuw, A., 1992. International aspects of pollution control, Environmental and Resource Economics, 2, 117–139.
PART III EFFICIENCY ANALYSIS
Introduction C. Knox Lovell, University of Georgia
279
IIIa FDH and its early applications Chapter 13 Measuring labor-efficiency in post offices Dominique Deprins, Le´opold Simar, Henry Tulkens 1 Introduction and summary 2 Defining and measuring input-efficiency when the production set is fully known 3 Measuring input-efficiency from observed data 4 Measuring labor-efficiency in post offices 5 Conclusions References
285
Chapter 14 On FDH efficiency analysis: some methodological issues and applications to retail banking, courts and urban transit 311 Henry Tulkens 1 Introduction 2 Efficiency measurement from an FDH frontier: methodological issues 3 Retail banking: assessing relative productive efficiency performance for the branches of a public and of a private bank 4 Courts: using efficiency results to evaluate the backlog 5 Urban transit: efficiency over time and shifts in the production frontier 6 Conclusion References
Chapter 15 Assessing and explaining the performance of public enterprises: some recent evidence from the productive efficiency viewpoint 343 Pierre Pestieau, Henry Tulkens 1 Introduction 2 Defining economic performance for public enterprises 3 The productive efficiency approach 4 The expected impact of ownership, competition and regulation on productive efficiency 5 Recent evidence 6. Conclusions References
IIIb Dynamics Chapter 16 Non-frontier measures of efficiency, progress and regress for time series data 373 Henry Tulkens, Philippe Vanden Eeckaut 1 Introduction 2 Two alternative views of efficiency 3 Measuring efficiency, progress and regress from time-related data: limitations of nonparametric frontiers 4 Benchmark observations, progress and regress 5 Defining the benchmark production correspondence and computing progress and regress indexes from it 6 An empirical illustration References Chapter 17 Nonparametric efficiency, progress and regress measures for panel data: methodological aspects 395 Henry Tulkens, Philippe Vanden Eeckaut 1 Introduction: production sets and the efficiency concept 2 Constructing alternative reference production sets from observed data: the postulates behind FDH, DEA and other methodologies 3 Constructing reference production sets from alternative time-related observations subsets 4 Mathematical programming formulations of efficiency measurement from contemporaneous, sequential or intertemporal frontiers 5 Measuring progress and regress from frontier shifts
6 Measuring progress and regress from a ‘benchmark production correspondence’ 7 Conclusion Appendix References
IIIc Dominance Chapter 18 Efficiency Dominance Analysis (EDA): basic methodology Henry Tulkens 1 Motivation 2 Defining efficiency dominance 3 Measuring efficiency dominance 4 Concluding remarks References
431
INTRODUCTION
C. Knox Lovell
In much of economic analysis, agents optimize. In much of the real world, optimization is imperfect because it is really difficult, and because its pursuit requires the commitment of resources, which is costly. That is why we have management consulting gurus with their buzzwords. However despite the early work of Simon (who emphasized the public sector), Leibenstein (who emphasized the private sector) and others, mainstream economics has largely ignored the possibility of inefficient behavior, of a failure to optimize. Henry Tulkens has been an exception. Beginning in 1984, and continuing for over a decade, he explored the existence, the magnitude, the cost and the possible sources of inefficiency in production, more precisely in public sector service provision. His work has been important in several respects. First, the analytical framework within which he has modeled production possibilities is nonparametric, and creatively unconventional. Second, he has borrowed the notion of dominance from other fields and introduced it as a complement to efficiency in the evaluation of producer performance. Third, he has brought considerable insight to the intertemporal modeling of production possibilities. Finally, he has continually confronted his analytical framework with microeconomic data, with particular emphasis on the public sector in which Darwinian selection is much weaker than in the private sector. The contributions of his that have been selected for inclusion in this volume represent a small fraction of his work in the field, but they illustrate each of these contributions. ‘‘Measuring labor efficiency in post offices’’ (1984, with Dominique Deprins and Le´opold Simar), chapter 13 in this volume, seeks to measure the technical efficiency of 972 Belgian post offices. Technical efficiency is defined as the ability to minimize the number of labor hours required
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to provide a vector of exogenously determined postal service indicators. Not surprisingly, not all offices are equally efficient. But this finding is by no means the major contribution of the study. The major contribution is the introduction of nonparametric free disposal hull (FDH) analysis to economics. It is useful to put FDH in perspective. Let the feasible set of production activities be T ¼ {( y,x): x can produce y}, where y is an output vector and x is an input vector. A natural assumption to impose on the structure of T is that outputs and inputs be strongly, or freely, disposable, in the sense that ( y,x) 2 T ) (y0 ,x0 ) 2 T8 y0 # y, x0 $ x. A less compelling, but extremely popular, assumption is that T is a convex set, so that ( y0 , x0 ) 2 T and (y00 , x00 ) 2 T ) [l(y0 , x0 ) þ (1 l)(y00 , x00 )] 2 T, 0 # l # 1. Although neither assumption is necessary for duality theory, at least one seems indispensable for primal efficiency measurement. The first contribution of this study is to retain strong disposability and dispense with convexity, and to defend its deletion. Imposing only strong disposability creates an FDH technology, which constructs a free disposal hull of the data as an empirical approximation to T. The corresponding production frontier and its level curves have ‘‘staircase’’ shape, with vertices defined by non-dominated (and therefore efficient) producers. Efficient producer A is said to dominate all other producers for which ( y, x) # (yA , xA ), with at least one element of the vector inequality being strict. Conversely, inefficient producer B is dominated by all other producers for which ( yB , xB ) # (y, x), with at least one element of the vector inequality being strict. Dominance relationships provide performance evaluations that complement efficiency scores. The introduction of dominance to production analysis is the second contribution of the study, although it was not utilized to full effect until it was fleshed out in subsequent studies discussed below. Adding convexity to strong disposability connects the vertices and creates a technology that goes by the name data envelopment analysis. DEA constructs a polyhedral free disposal hull of the data as an empirical approximation to T, and so TFDH TDEA T. Neither FDH nor DEA imposes a parametric functional form on technology, and both are constructed using mathematical programming techniques. Adding a functional form assumption would create a ‘‘smooth’’ technology suitable for estimation using econometric techniques. The use of FDH to evaluate producer performance has four virtues. First, it is nonparametric and thus immune to specification error due to inappropriate functional form assumptions. Second, dispensing with the convexity assumption means that best practice is determined exclusively by actual producers, and not also by fictitious convex combinations of
Introduction
281
them. Third, and consequently, the productive efficiency of a dominated producer is based on a comparison of its activity with that of exactly one non-dominated best practice producer, and this makes the technique appealing to managers, who are unused to dealing with fictitious combinations of producers. Finally, since FDH envelops the data tightly, it is relatively generous in its performance evaluation. Since the best practice standard requires only that a producer not be dominated, efficiency scores tend to be relatively high. This in turn encourages a focus on the bottom of the performance distribution, where efficiency scores are relatively low and dominance counts are relatively high. These are precisely the producers most in need of managerial attention. ‘‘On FDH efficiency analysis: some methodological issues and applications to retail banking, courts, and urban transit’’ (1993), chapter 14 in this volume, reveals the benefits of nearly a decade of thought about how best to formulate the reference technology against which efficiency is measured. The analytical framework of FDH is rigorously presented, and computational issues are fully discussed. Here we see the FDH optimization program presented as a DEA linear program (which constructs a polyhedral free disposal hull) with a single integrality constraint appended, which converts it to a mixed integer program that constructs a free disposal hull. This program seems daunting, until we see how it can be solved by means of a simple minimax routine. The beauty of the minimax routine is that the ‘‘max’’ part identifies all dominating producers, while the ‘‘min’’ part identifies the single most dominant producer, for the producer under evaluation. The set of all dominating producers provides a list of role models from which the producer can learn, while the most dominant producer establishes the best practice standard against which the producer is evaluated. The applications to banking, courts and urban transit are summaries of studies conducted during the previous decade, and each is revealing in very different respects. . The application to 773 branches of a publicly owned Belgian bank highlights the differences between FDH and DEA performance assessments. The FDH frontier is supported by 75% of the observations, whereas the DEA frontier, even allowing for variable returns to scale, is supported by only 5% of the observations. On a goodness-of-fit criterion, FDH dominates DEA. . The focus shifts in the application to 187 Belgian courts, where the emphasis is on the relationship between FDH efficiency scores and the backlogs of unsettled cases. For each court, backlog is decomposed into that reducible by management through improved efficiency, and that reducible only through an increase in resources. The distinction between endogenous factors under managerial control and exogenous
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factors beyond the control of management, at least in the short run, is essential to the credibility of the analysis. . The application to urban transit is based on 153 monthly observations on the activities of the Brussels public transit authority. Since the data are a single time series, this study must confront the possibility of technical change. It does so by introducing the notion of ‘‘sequential free disposal hulls.’’ Best practice technology is built up from successive, or sequential, observations, the logic being that techniques once used are not forgotten. This procedure permits efficiency to be distinguished from technical progress, which is localized and not rigidly parameterized, although it does not allow for technical regress. Further developments of FDH in a time series context appeared in subsequent studies, most notably in a 1995 study published in International Journal of Production Economics and as (chapter 16 in this volume). ‘‘Assessing and explaining the performance of public enterprises’’ (1993, with Pierre Pestieau), chapter 15, represents a departure from the two previous studies, and from the final included study. Here FDH plays a subordinate role, the central issues being how to define performance in the public sector, and how to measure it. From a principal – agent perspective, the problem is that the principal (the ‘‘state’’) presumably has multiple objectives that are difficult to quantify, potentially incompatible, and difficult or impossible to weight with prices. In this context the agent’s performance cannot be evaluated on the basis of conventional criteria that would be applicable in the private sector. Moreover, the performance of a public enterprise depends on the degree of competition, perhaps from private providers, and on the nature of the regulatory environment. Against this complex background, Pestieau and Tulkens make a strong case for a primal measure of technical efficiency as being the only criterion consistent with any reasonable objectives of the principal. This argument applies within the public sector and also in public – private comparisons that have become so popular in this era of privatization. They illustrate their argument with a detailed survey of the literature on public sector performance in a variety of competitive and regulatory environments. Although many other such surveys of this important topic exist, this remains my favorite because ‘‘[I]t focuses on a single objective – productive efficiency – whose desirability is independent of any other considerations . . . ’’ (p. 318). Three of the applications discussed in the first two papers are based on cross-section data, and the fourth application is based on time-series data. ‘‘Nonparametric efficiency, progress and regress measures for panel data: methodological aspects’’ (1995, with Philippe Vanden Eeckaut), chapter 17, extends the use of FDH to panel data. This fully developed panel data model contains several innovative contributions.
Introduction
283
Three different notions of a production set are developed, complete with FDH and DEA optimization programs. One uses as many separate cross sections as there are time periods, and creates a series of independent ‘‘contemporaneous’’ production sets. A second constructs ‘‘sequential’’ production sets from a period and all previous periods that allow technical progress, but not regress because the sequential production sets are nested. The third constructs a single ‘‘intertemporal’’ production set from all the data in the panel, which does not allow for any sort of technical change. All three differ from the way an econometrician would treat panel data. This study concludes with an exploratory detour in which the frontier concept is abandoned entirely, in favor of a ‘‘benchmark correspondence.’’ The frontier concept being jettisoned, so is the efficiency concept. All interest centers on dominance and intertemporal progress/ regress relationships. The analysis is intricate, but the essence is that something like a ‘‘thick frontier’’ is created (although the authors would object to the phrase) from dominance and reverse dominance relationships. With a single cross section, this provides a variety of information for each observation, and this information is useful regardless of whether a producer is non-dominated. With panel data, information on technical progress or regress in subsequent periods is provided. The detour is a prelude to the final paper being discussed. In a series of studies the procedures used to construct the benchmark correspondence are renamed ‘‘efficiency dominance analysis,’’ with the clever acronym EDA. The final paper in this Section, ‘‘Efficiency dominance analysis: basic methodology’’, chapter 18, has been written especially for this volume. It summarizes the central themes appearing in ‘‘Mesurer l’efficacite´: avec ou sans frontie`res?’’ (1999, with Philippe Vanden Eeckaut), chapter 3 in P.-Y. Badillo and J. C. Paradi (eds.), La me´thode DEA: analyse des performances. Paris: Hermes Science Publications, and in detailed notes prepared for presentation at the third (Louvain-la-Neuve) and fifth (Copenhagen) European Workshops on Efficiency and Productivity Analysis. The published paper also provides an illuminating application to public service provision in a previously studied cross section of 235 Belgian municipalities. Although the acronym is clever, my interpretation of EDA is that ‘‘E’’ is superfluous. Since there is no explicit frontier, there can be no measure of efficiency, and the essence of the acronym boils down to DA. Three types of dominance are developed; one compares a producer with another producer, a second compares a producer with all other producers, and a third characterizes dominance in terms of the entire set of producers. Two directions are developed, active (dominante) and passive (domine´e). Three dimensions are mentioned: input, output and overall.
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Two measurement formulae are proposed, radial and average. Three counting mechanisms are proposed, an integer measure, an extreme measure and an average measure. I believe this taxonomy creates 108 ways of characterizing dominance. Each in its own way has the potential to provide useful information to management without recourse to the concepts of best practice, frontier or efficiency. Henry Tulkens has created an innovative and coherent body of work in the field of efficiency analysis. It is useful to conclude this introduction by emphasizing a few recurring themes. First, performance is evaluated on a rather minimalist technical efficiency criterion, for reasons clearly explained in Pestieau and Tulkens (1993). Second, throughout his work efficiency is most usefully measured using FDH, which dispenses with the convexity postulate, for reasons originating in Deprins, Simar and Tulkens (1984). Third, intertemporal analysis plays an increasingly prominent role, beginning with Tulkens (1993). Fourth, dominance relationships also play an increasingly prominent role, eventually eclipsing the efficiency concept in Tulkens and Vanden Eeckaut (1995) and subsequent EDA studies. Finally, necessity being the mother of invention, many of his innovations have been driven by challenges arising in projects with the public sector. This has two important consequences. It has forced him, from the very beginning, to structure his analysis in such a way that it provides maximum information to management, the ultimate consumer. It has also forced him to think creatively about what performance means in the public sector. Since this sector is large and growing in most countries, and because the optimization problems confronting public sector managers are so complex, the evaluation of public sector performance is not merely an interesting problem, it is an important problem. Henry’s work in efficiency and dominance analysis constitutes a significant contribution to the field of public economics. The field would benefit if more of us followed some of these themes.
Chapter 13 MEASURING LABOR-EFFICIENCY IN POST OFFICES
Dominique Deprins, Le´opold Simar, Henry Tulkens In The Performance of Public Enterprises: Concepts and Measurement, M. Marchand, P. Pestieau and H. Tulkens (eds.), 1984. Amsterdam: North-Holland, chapter 10, 243–268.
Three methods of measuring technical efficiency are defined, and their results compared in an application to Belgian postal data. The first one is that of adjusting a Cobb–Douglas production frontier; the second is that of computing the convex hull of the data, a` la Farrell; the third one is developed on the basis of the sole assumptions of input and output disposability. We conclude with an estimated average labor-efficiency of about 90 %.
1.
Introduction and summary
In this paper, we attempt to evaluate numerically the performance of a public enterprise from the point of view of technical efficiency – i.e. its ability to operate close to, or on the boundary of its production set. The problem to be analyzed is thus set in terms of physical input and output quantities, and no cost or price elements will be considered here. The concept of technical efficiency just mentioned was given its precise meaning by Koopmans (1951); it received its first empirical application The authors express their thanks to Mr. Bonnijns, Director General at the Belgian Re´gie des Postes, for giving them access to the data, and to Messrs. Herdewijn and Jadot for their technical help; to Etienne Loute, Christine Florival, Jean–Pierre Lemaıˆtre, Anne Campion, Jacques Van den Bol and Serge Wibaut for preparing the data for the computations; and to Jayasri Dutta, Maurice Marchand, Jack Mintz, Juan Muro Romero, Pierre Pestieau and Louis Phlips for their valuable suggestions. They also acknowledge the support of the Belgian Fonds de la Recherche Fondamentale Collective (Convention n8 2.4544.80) for the computation costs. Finally, Henry Tulkens’ work on this paper was done in part during a sabbatical visit at the Department of Economics, Princeton University, to whose members he expresses his thanks for their hospitality. He also gratefully acknowledges the sabbatical support secured by the Colle`ge Interuniversitaire pour les Sciences du Management, Brussels.
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in Farrell (1957) and, starting in 1968 with another key contribution made by Aigner and Chu, a fairly large amount of literature developed around this theme. For an interesting survey, see Førsund, Lovell and Schmidt (1980). From this literature we borrow here two methods of efficiency measurement that we apply to data from the Belgian Re´gie des Postes; we also introduce a third one, that we contrast with the other two, and claim to be probably more suitable for managerial purposes. These three methodologies, that will be exposed in detail below and then compared in our computations, can be easily described by means of the following simple example. Consider a one input (x)-one output (u) activity, about which statistical data are available, say in cross-Section form. The set Y0 of such observations is illustrated by the scatter diagram of Figure 1. Measuring efficiency for any data set of this kind requires first to determine what the boundary of the (unknown) production set Y can be; and then, to measure the distance between any observed point in Y0 and the boundary of Y. This we do here in three different ways. (1) Our first method, taken from Aigner and Chu (1968), postulates that the boundary of the unknown production set Y is representable by a production function of an a priori selected form, e.g. Cobb–Douglas. The parameters of this function are then determined in such a way that the graph of the estimated function lies on or ‘‘above’’ all the observations, and that the distance between the latter and this graph is minimized. In Figure 2, this distance is taken horizontally, i.e. parallel to the input axis, but other ways of measuring it are conceivable. In the present case, this distance also measures what is called the input-efficiency of each observation. u (output)
Y o
uk
0 xk
Figure 1
x (input)
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287
u (output)
Y 1
u
k
0 x
k
x (input)
Figure 2
(2) Our second method – in the spirit of Farrell (1957) – is induced by assuming only that Y be a convex and polyhedral set, with no particular functional form as to its boundary. It leads to constructing this set as Y2 (see Figure 3), which is the convex hull of the data set, to which have been added the origin, and some (fictitious) extreme points representing a very large input use. Input-efficiency of each observation is again computed as the horizontal distance between the observed point and the appropriate facet of the constructed polyhedron.
u (output)
uk
Y2
0 xk
Figure 3
x (input)
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(3) A third method, introduced in the present paper, characterizes Y with no other assumption than input and output disposability. This allows one to infer the set Y3 from Y0 as consisting of only these points that carry less output with the same amount of input as some observed point, and/or those that carry more input for the same amount of output. The resulting set appears in Figure 4. Here again, the horizontal distance between any point and the boundary measures the input-efficiency of the point. As to computational aspects, we explain in Section 3 that for any given data set, the use of the first method requires solving one single large linear program; use of the second method requires solving as many linear programs as there are observations; and the third method operates through simple comparisons of the input and output components of the vectors in the data set, so as to locate ‘‘dominating’’ ones. It was mentioned above that distances from any observed point to the efficiency boundary can be measured in other ways than in input terms, i.e. parallel to the input axis. In the case of the various plants of a monopolistic public enterprise such as the post office, however, there is a rationale for sticking to this ‘‘input’’ measure of efficiency : because of the ‘‘obligation of service’’ – i.e. the obligation of serving whatever demand arises at the prevailing prices – each of the individual plants has no control on its output; its only possible decisions are to adjust its input requirements to the traffic. A last qualification of general nature is that the input-efficiency so measured is of course a relative one. The true production set being essentially unknown, the one that is in fact constructed from the data can only derive from those observations that appear to be the ‘‘best’’ ones in the sense of some a priori posited assumptions. It is a major objective of
u (output)
uk
0
Figure 4
Y3
xk
x (input)
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this paper to state, as precisely as possible, which assumptions underlie each measure, and to exhibit with our data the differences their adoption can entail as to the numerical estimates of efficiency. Section 2 develops the arguments that lead to alternative definitions of input-efficiency measures in a general theoretical setting. Section 3 proceeds with describing the methodologies that these measures require when they bear upon actual data. Section 4 applies these methodologies to our Belgian postal data, and discusses the numerical results obtained. The concluding Section 5 evaluates our results from both the methodological viewpoint and with reference to the efficient management of a large public enterprise.
2.
Defining and measuring input-efficiency when the production set is fully known
2.1
The feasible set, its sections, and input-efficiency
Given a list of q > 0 inputs and p > 0 outputs, it is common practice nowadays in economic analysis (see e.g. Shephard (1970)) to describe the operations of any productive organization by means of a set of points, Y, defined as follows in the Euclidian space Rqþp þ : Y ¼ {(x,u)jx 2 Rqþ , u 2 Rpþ , (x,u)
is feasible}:
In this expression, x is called the input vector, u the output vector, and ‘‘feasibility’’ of the vector (x,u) designates the fact that, within the organization under study, it is physically possible to obtain the output quantities u1 , . . . , up when the input quantities x1 , . . . , xq are being used (all quantities being measured per unit of time). The set Y is called indifferently the ‘‘feasible set’’ or the ‘‘production set’’ of the organization in question. For our present purposes, it will prove useful to study the set Y in terms of some of its Sections, the latter being defined as the images of a relation between the input and the output vectors that are the elements of Y. More precisely, consider successively 18 8 u 2 Rpþ , the Section X(u) – a subset of Rqþ – defined by the correspondence u ! X(u) ¼ {x j (x,u) 2 Y}, and 28 8 x 2 Rqþ , the Section U(x) - a subset of Rpþ - defined by x ! U(x) ¼ {u j (x, u) 2 Y}: X() (respectively U()) is called the input (resp. output) correspondence; it associates to every output (input) vector u (x) the family of input (output) vectors x (u) such that (x, u) is a feasible point.
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The entire production set may then be described indifferently as Y ¼ {(x, u) j u 2 Rpþ , x 2 X(u)}, or ¼ {(x, u) j x 2 Rqþ , u 2 U(x)}: For each Section X(u) and U(x) of Y, let us now call its efficient boundary in the Farrell sense1 the subset denoted @ F X(u) (resp. @ F U(x)) and defined as follows: @ F X(u) ¼ {x j x 2 X(u),
lx 62 X(u) 8l 2 [0, 1[ }
@ U(x) ¼ {u j u 2 U(x),
uu 62 U(x) 8u 2 [1, þ 1]}:
F
(1)
Turning back to the production set itself, the above definitions allow us to characterize any points (x, u) 2 Y as: - ‘‘input [in-] efficient’’, if x 2 @ F X(u) (resp. x 62 @ F X(u)), and - ‘‘output [in-] efficient’’, if u 2 @ F U(x) (resp. u 62 @ F U(x)). Conditions can be determined under which a point (x, u) is both input- and output-efficient (see Deprins and Simar (1983), Theorem 1, p. 7). For the reasons explained in the introduction, we shall only deal henceforth with input-efficiency. 2.2
Measuring the degree of input-efficiency
The input-inefficiency of a point (x,u) has just been defined by its property of ‘‘not-belonging’’ to some a priori defined subset of a Section of Y, viz. @ F X(u). In geometric terms, one may view this property as the fact that the point in question is lying away from the efficient boundary set. This immediately suggests to define a distance between this point and 1
After Farrell’s 1957 proposal; this is actually a radial definition of the efficient boundary of a section. A more general definition of the same concept is given by Shephard (1970) as @ S X(u) ¼ {x j x 2 X(u), y # x ) y 62 X(u)} @ S X(x) ¼ {u j u 2 U(x),
v $ u ) v 62 U(x)}:
One has: @ S X (u) @ F X (u) and @ S U(x) @ F U(x): Note in addition that if x is a scalar (q ¼ 1), then @ S X (u) @ F X (u), and similarly, if u is scalar (p ¼ 1), @ S U(x) @ F U(x):
Ch.13 Measuring Labor-Efficiency in Post Offices
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the boundary set, and hence to measure - in the sense of this distance - the ‘‘degree’’ to which point (x,u) is inefficient. To this effect, given any point (x, u) 2 Y, the associated Section X(u) and its boundary @ F X(u), consider the real number determined by the function F(x, u) ¼ Min {l j lx 2 X(u)}:
(2)
Clearly, if (x,u) is input-efficient, this number must be equal to 1, according to the definition (1) above. We can thus restate this definition as @ F X(u) ¼ {x j F(x, u) ¼ 1}, and call ‘‘input-efficient’’ any point (x, u) 2 Y such that F(x, u) ¼ 1; similarly, for any point (x, u) 2 Y which is input-inefficient, the number F(x, u) is <1 by construction. We shall call this number the degree of inputefficiency of point (x, u). It also allows to compute (as [1 F(x, u)] x)) the number of units of input by which the productive activity represented by (x, u) should be reduced in order to achieve the same production u in an efficient way.
3.
Measuring input-efficiency from observed data
3.1
Alternative methods for determining the feasible set
For the observer of a productive activity, the production set is usually an unknown2. Of course, one may safely consider that the quantities of outputs and inputs he observes do correspond (when there are no measurement errors) to feasible points; they are thus elements of that set. But it is more delicate to answer the two questions whether (i) non observed points can also be considered as feasible ones; and (ii) if so, which ones? These questions are of utmost importance for our purposes because in measuring the efficiency of any observed point, a distance must be computed between this point and some other point in the set: should the latter only be an observed one, or can it be a ‘‘hypothetical’’ one? To our knowledge, most of the literature on efficiency measurement chooses the second alternative. In all cases, the unknown production set Y is in fact generated in some way or another from the set of observations; but then it is the method by which this is done that determines what the production set is going to be and, in particular, which non-observed points do belong to it. As suggested in the introduction, it is one of our purposes 2
Except, perhaps, in those rare circumstances where it can be determined from engineering data.
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in this paper to show to what extent alternative methods of generating Y from the observations induce different measures of efficiency, and yield, as it is the case with our data, strikingly different empirical results. In order to describe these methods precisely, let us denote by Y0 the set of points determined in the space Rpþq by n available observations, i.e. þ Y0 ¼ {(xk , uk ) j xk 2 Rqþ , uk 2 Rpþ ; k ¼ 1, 2, . . . , n}: Since in our postal application below we shall have data for one input only, we shall develop the rest of this exposition3 for the case where q ¼ 1; from now on, xk is thus a scalar, while uk remains a ( $ 1)-dimensional vector. Y0 is of course an arbitrary set; on the other hand, the fact that every observation is supposed to belong to the production set is equivalent to making on Y the minimal assumption that it contains Y0 . Given this preliminary convention, we describe now three of the methods announced above. 1. A first method (used by Aigner and Chu (1968), after a suggestion appearing in Farrell (1957); for a review of the development of its statistical version, see the survey by Førsund, Lovell and Schmidt (1980), p. 9) proceeds by postulating on the unknown set Y that for every output vector u 2 Rpþ , the input efficient boundary @ F X(u) of Y is determined as the image of a continuous function of an a priori chosen form, say x ¼ f(u; a), where a stands for the vector of the function’s parameters. This implies that the Section X(u) of Y is defined as X(u) ¼ {x j x $ f(u; a)};
(3)
and that the set Y itself is of the form Y ¼ {(x, u) j x $ f(u; a)}:
(4)
The form of function f(), including the numerical value of its parameters a, are then determined from the data set Y0 in such a way that the production set Y it induces is in some appropriate sense ‘‘closest’’ to Y0 . Specifically, suppose that the Cobb-Douglas form is selected; in our one input-p outputs case, it reads4 p
x ¼ a0 P uai i , a0 > 0, ai $ 0, i ¼ 1, . . . , p, i¼1
or, in logarithms (a on top of a symbol denotes its logarithm) ~ x¼a ~0 þ
p X
~i : ai u
(5)
i¼1 3
Most of our developments can fairly easily be extended to the multi-dimensional input case. 4 Contrary to the case of the usual form of a one output-several inputs Cobb–Douglas production function, the above expression is such that increasing returns to scale imply Si ai < 1.
Ch.13 Measuring Labor-Efficiency in Post Offices
293
The closest production set of the form (4) induced by the data in Y0 , and now denoted as Y1 , is obtained as ^ x$a ~0 þ Y1 ¼ {(x, u) j ~
p X
~i }, a ^i u
(6)
i¼1
^ 0, a where the values a ~ ^ 1, . . . , a ^ p are the optimal solution of the linear program (recall that k is the index referring to the observations in Y0 ): ! p n X X ~ ~ki (7) xk a ~0 ai u Min k¼1
i¼1
subject to 8 p P < k ~ki $ 0, k ¼ 1, . . . , n, x~ a ~ 0 ai u i¼1 : ai $ 0, i ¼ 1, . . . , p,
(8)
~ 0 , a1 , . . . , ai , . . . , ap . with the minimization being taken with respect to a The number of variables in this program is thus equal to the number of parameters in the production function, and the number of constraints is equal to the number of observations in Y0 . At the optimal solution of this program, the value of the objective function is equal to the sum of the distances between each observed point and the graph of the function (5). It is worth mentioning that, as shown by Schmidt (1976), the parameters a ^ estimated in this way have the statistical property of being maximum likelihood estimators5 under the assumption that the distances (interpreted now as non negative random disturbance terms) are exponentially distributed. 2. A second method (initiated by Farrell himself in his 1957 paper, and extended in Deprins and Simar (1983), p. 13, to allow Y to exhibit decreasing returns to scale) derives from postulating on Y that it is a convex polyhedron, containing the origin, that also satisfies the assumptions of strong disposability of outputs and free disposability of inputs6. A set Y2 that meets these assumptions is generated from Y0 by taking the set of all convex combinations of (18) the observed point (xk , uk ), (28) the origin 5
Without having, though, the usual asymptotic properties of such estimators. We remind the reader that these two assumptions, which bear upon the sections of Y, read formally as follows: x 2 X(u) (i) free disposability of inputs: 0 ) x0 2 X(u); x $x u 2 U(x) ) u0 2 U(x). (ii) strong disposability of outputs: 0 u #u 6
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(0, op ) - where op denotes the p-dimensional null vector -, and (38) fictitious extreme points that reflect the input and output disposability assumptions. Formally, a set so defined is given by: ( Y2 ¼
u x
! 2 Rpþ1 þ j
gk $ 0, k ¼ 1, . . . , n,
u x
n X
! ¼
n X k¼1
gk
uk xk
! þm
op 1
!
p X
ni
i¼1
)
gk # 1; m $ 0; ni $ 0, i ¼ 1, . . . , p ,
epi 0
! ;
(9)
k¼1
where epi is the ith-column of the p-dimensional identity matrix. In expression (9), the allowed values gk ¼ 08k, m ¼ 0 and ni ¼ 08i imply that the origin is included in Y2 ; on the other hand, the possibly very large values allowed for m and/or the ni ’s reflect the disposability assumptions, on inputs and outputs respectively; and finally, the constraint on the sum of the gk ’s expresses the decreasing returns assumption. If this constraint were dropped, the set Y2 would be a convex polyhedral cone implying constant returns to scale. 3. Our third method, presently introduced, rests on postulating on Y only strong disposability of outputs and free disposability of inputs for all output (respectively input) components of the points (x, u) that belong to Y0 , and of the origin. The set Y3 that meets these conditions is generated from Y0 as the union of all orthants, positive in x and negative in u, whose origin coincides with an observed point. Formally, ! ( p X p p o u u ei uk pþ1 Y3 ¼ þ m 2 Rþ j ¼ n , i x x 1 0 xk i¼1 (uk ,xk ) 2 Y0 U{(0, op )}; m $ 0; ni $ 0, i ¼ 1, . . . , pg: The production set so defined is also polyhedral, but not necessarily a convex one. 3.2
Induced measures of input-efficiency for the observed points
Given the alternative hypothetical production sets Y1 ,Y2 , and Y3 generated above from some data set Y0 , corresponding alternative In the single input case, that we consider more especially in this paper, free disposability of input is also equivalent to (i’) weak disposability of inputs: x 2 X(u) ) lx 2 X(u), l $ 1. On these concepts, see Shephard (1970).
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295
measures of input efficiency of any point in those sets can be defined and computed. We recall from 2.2 above that this requires identifying the Sections X(u) of the production sets, in order to derive the distance function F ( ) introduced there. We proceed with these two steps in what follows. 1. If method 1 is used, yielding Y1 , any Section X1 (u) of this set is, according to (3) and (6): ( ) p X ^ xa ~0 a ^ i u~i $ 0 : (10) X1 (u) ¼ x j ~ i¼1
With this expression for the Sections, the general form (2) of the measure of input efficiency for any point of Y1 , and in particular for an observed point (xk ,uk ), reads: F1 (xk ,uk ) ¼ Min{l j lxk 2 X1 (uk )}: But, from (10), this amounts to f(u ; a ^) F1 (xk ,uk ) ¼ ¼ k x k
a ^0
p Q i¼1
(uki )a^ i
xk
:
In this expression, let us denote the numerator by ^xk . We then get p X ^ xk k k k ^ ^ ~ki }, x ~ x } ¼ exp {~x a ~0 a ^iu F1 (x ,u ) ¼ exp ln k ¼ exp {~ x i¼1 k
k
where the expression within braces is the value of the kth-term of the objective function (7) (as well as of the left-hand side of the kth-constraint (8)) at the optimal solution of the linear program (7) – (8). The F1 efficiency measure is thus directly obtained from the construction of the set Y1 . 2. When the unknown set Y is generated by method 2, thus yielding the polyhedron Y2 , for any (observed or non observed) output vector u 2 Rpþ the Section of Y2 , denoted now as X2 (u), is given by ! ( p X p p n X o u e uh þm ¼ gh ni i ; X2 (u) ¼ x j h x 1 0 x h¼1 i¼1 ) n X gh # 1; m $ 0; ni $ 0, i ¼ 1, . . . , p : gh $ 0, h ¼ 1, . . . , n, h¼1
With the Section so defined, the general form (2) of the measure of input-efficiency for any point of Y2 , e.g. the observed point (xk ,uk ), reads
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F2 (xk ,uk ) ¼ Min{l j lxk 2 X2 (uk )}: But this may be written equivalently in mathematical programming terms as F2 (xk ,uk ) ¼ Min l, subject to 8 n p P P > h > g u ni epi ¼ uk > h > > h¼1 i¼1 > > n >
> n P > > > gh # 1 > > > : h¼1 l, m, gh , ni $ 0, h ¼ 1, . . . ,n; i ¼ 1, . . . ,p, where the minimization is taken with respect to l, m, g1 , . . . , gh , . . . , gn , n1 , . . . , ni , . . . , np . This is indeed a linear program in canonical form, with the 2 þ n þ p variables just mentioned and p þ 2 constraints. The coefficients of the constraint matrix are the components of all the vectors of observations that constitute Y0 and of the output- and input-disposability fictitious points, and the variables are the weights of their convex combination that is sought for. The input-efficiency of the vector (xk ,uk ) is obtained as the optimal solution of this program. 3. When Y3 is the production set generated from the data set Y0 , the Section of Y3 associated with any output vector u 2 Rpþ reads ! ( p X p p o u ei uh þ m X3 (u) ¼ x j ¼ n ; i x 1 0 xh i¼1 ! ) (11) uh 2 Y0 ; m $ 0; ni $ 0, i ¼ 1, 2, . . . , p xh Using this, for any point (xk ,uk ) in Y3 , the induced measure of input efficiency becomes, in the general terms of (2): F3 (xk ,uk ) ¼ Min{l j lxk 2 X3 (uk )}:
(12)
This number can be found by means of the following procedure. With the point (xk ,uk ), associate a set D(xk ,uk ) that includes the point (xk ,uk ) itself as well as those observed points (x, u) that ‘‘dominate’’ (xk ,uk ) in the sense that their input quantity x is strictly lower than xk , and their output vector u is weakly larger than uk . Formally, this reads:
Ch.13 Measuring Labor-Efficiency in Post Offices
( D(x ,u ) ¼ k
þ
p X i¼1
k
ni
epi 0
uk xk
!)
( u u [ 2 Y0 j ¼ x x )
297
uk xk
!
m
op
1 (13)
,m > 0, ni $ 0, i ¼ 1, 2, . . . , p :
Within D(xk ,uk ), select then the vector (x(k) , u(k) ) for which the component x is the smallest, and compute the ratio x(k) =xk . From the definition of D(xk ,uk ) this number is equal to or smaller than one according to whether (x(k) ,u(k) ) is the same point as (xk ,uk ), or a different, i.e. dominating, one. One can see from the definition (11) of the Section X3 (u) that the number x(k) =xk minimizes l in the sense of the definition (12) of F3 (xk ,uk ). We thus get the simple expression F3 (xk ,uk ) ¼ x(k) =xk : Note that the selected vector (x(k) ,u(k) ) is an observed one: the efficiency measure F3 for (xk ,uk ) thus rests on the explicit identification of an alternative observation that may reasonably serve as an example since it shows that the same or a larger amount of output can be achieved with a smaller amount of input. When several observations have that property (i.e. when there are more than two elements in D(xk ,uk )), the identified one (x(k) ,u(k) ) is, in addition, also the ‘‘best’’ example, in the sense of being the most input-saving one7. To conclude this methodological Section, note that the data set and the hypothetical sets generated by the methods described above are related to one another as follows: Y2 Y0 Y3 Y1 : Moreover, in the particular case where the function f(u, a ^ ) is convex, we have in addition Y2 Y1 . Thus, the stronger the assumption on Y, the ‘‘larger’’ the generated set. The above inclusions also imply that in general, for every (xk ,uk ) 2 Y0 . F3 (xk ,uk ) $ F2 (xk ,uk ) $ F1 (xk ,uk ), ^ ) is convex. with F2 (xk ,uk ) $ F1 (xk ,uk ) if f(u; a
If the vector (x(k) ,u(k) ) is not unique, any of them may serve as an example; but those with the largest u (in the vector sense) may perhaps be called the most convincing ones. 7
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4.
Measuring labor-efficiency in post offices
4.1
The setting
In the present application, we consider the operations performed in individual postal stations in Belgium. These are characterized by (i) an extreme diversity of activities, that range from collecting, sorting out, and distributing various kinds of mail, to quite a variety of so-called financial operations such as paying out postal checks and receiving deposits (‘‘giro’’ operations), registering postal subscriptions to newspapers and magazines, delivering fishing permits, selling tickets of the national lottery, etc.; and (ii) a heavily labor intensive technology – mechanization and automation being limited to very large scale stations and for only a small subset of the above mentioned activities (mostly mail sorting). At the level of the entire country, more than 80 % of the expenditures are salaries, and for the stations retained in our analysis this percentage is certainly still larger. Given these characteristics, we thought it justifiable, at least as a first approximation and for the sake of giving the above ideas a first test, to model the activities of a typical postal station in a one input-p outputs space, with labor as the sole input. Neglecting other inputs such as capital is a severe limitation, but in any event no other data could be made available in a form suitable for this study. 4.2
The data
Our original data consist of the monthly statistics collected in each one of the 972 postal stations of the country, concerning (i) the total number of hours effectively worked during the month in the station by the personnel of all categories; and (ii) the total number of items handled or of operations performed during the month, for each category of activity separately. There are 137 of these categories. These two rather huge data files being established separately and by different services of the central postal administration, a first technical problem was to merge them in such a way that the monthly labor time figure of each station be put in correspondence with the data describing its activities. Although data were provided for the twelve months of the calendar year 1980, the technical problems raised by merging the files allowed us only to work with the data for the single month of January 1980. We shall comment below on the managerial perspectives that would be opened by repeating the analysis for several successive months.
Ch.13 Measuring Labor-Efficiency in Post Offices
299
A detailed analysis8 of the postal network reveals that all stations do not make exactly the same operations, even though some of the latter are often described by the same term. This is especially true of mail handling, which in every individual station comprises one or several of the successive activities called collection, presorting, general sorting, sector sorting, delivery sorting, delivery classification, and delivery. Among those, general sorting and sector sorting are performed essentially in large scale offices, called sorting centers, where they are usually mechanized. On the other hand, there are postal stations where only financial (window) operations take place, and no mail handling at all. A complete study should of course take into account this variety of situations. The lack of available statistics (especially on capital equipment) led us to decide to work only with a fairly homogeneous subset of the stations, namely the 792 ones (from the total of 972) that share the common properties of (a) being all mail collection as well as delivery stations; (b) performing neither general nor sector sorting of their collected mail (the latter being shipped to sorting centers that we thus exclude from our present enquiry); and (c) offering all window financial services. This data set thus remains a census; it is not a sample. It is hardly conceivable, on the other hand, to work directly with the list of 137 different outputs mentioned above. Indeed, for computing the efficiency measure F2 for just one observation, one would need to solve a linear program with 2 þ 792 þ 137 variables and 137 þ 2 constraints; this is not an unfeasible task, but repeating it 792 times seems excessively costly at this exploratory stage. Being forced to aggregate, we settled for the six output categories listed in Table 1 below. In this table, besides the measurement units, the criterion is also mentioned that determines how
Table 1: Output Categories Aggregate Output Category 1. Financial operations 2. Registered mail (received and delivered) 3. Special delivery mail (received and distributed) 4. Unaddressed printed matter (bulk mail) (distributed only) 5. Outgoing mail (collection and presorting) 6. Delivery points
8
Unit of Measurement (all units are per month) # of operations performed # of items handled
Main Location of Input Requirements
# of items handled
window service window service and delivery activities window service and delivery activities delivery activities
# of items handled
collection activities
# of private mailboxes served
delivery activities
# of items handled
As given e.g. in Tulkens (1968), pp. 37–73.
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the category is constructed, namely the similarity in the location of input requirements for the items included in the category. A word of comment is in order concerning category 6, called ‘‘delivery points’’. In each one of the retained stations, the traffic statistics are taken only for the outgoing mail, right after its collection; this is recorded as category 5. The incoming mail however, to be delivered within the area served by the station, is not recorded as such. Since in the delivery offices we work with, delivery operations are important ones (they include delivery sorting – i.e. sorting among carrier routes –, delivery classification – i.e. classification of the mail by each carrier according to the succession of delivery mail boxes he meets along his route –, and finally delivery itself – i.e. travelling the delivery route), we can not miss some measure of this activity. As a proxy for it we use, for each station, the number of private mailboxes served by the carriers covering the area assigned to the station9. Measuring their size in terms of the number of hours worked in January 1980, our 792 post offices range from 592 hours (i.e. approximately 4 full time (i.e. 38 hours a week) working persons) to 157, 195 (i.e. 1,034 persons). However, as Table 2 below will show, only six offices exceed 50,000 hours, and the immense majority (more than 700, i.e. 90 % of them) recorded 10,000 hours (66 full time workers10) or less. Most elements of our data set are thus concentrated in the lower range of the input figures. 4.3
The computations and their results
For the 792 observations of our data set, the results of computing the labor-efficiency measures F1 , F2 and F3 are summarized in columns (3) to (5) of Table 2 below, and in Figures 5.1, 5.2 and 5.3 For the CobbDouglas production function (5) that underlies the F1 measure, the values of the estimated parameters are given in the first line of Table 3. C.P.U. computing times were of less than one minute for each one of F1 and F3 , but of 100 minutes for F2 . Before deriving from this the appropriate evaluative comments about efficiency in postal activities (for which we refer the reader to Section 5), we must first evaluate our methodologies. This we do in the present subsection.
9
In so doing, we are also missing the fact that in some large cities, there can be more than one delivery per day. 10 The total labor force of the Belgian post office in 1980 was of the order of 52,000 fulltime units.
Table 2 Original and adjusted data sets Classes of office size 792 observations 791 observations 782 observations (measured by the Obs. F Obs. F3 Obs. F 3 3 # of obs. # of obs. # of obs. # of obs. number of hours # of obs. # of obs. # of obs. efficient efficient efficient F2 F2 in the in the F1 worked in in the F1 efficient efficient number % of col. 2 class efficient efficient number % of col. 8 January 1980) number % of col. 6 class class 1 0 – 4,999 5,000–9,999 10,000–19,999 20,000–29,999 30,000–39,999 40,000–49,999 50,000–99,999 100,000–149,999 150,000–199,999 TOTAL
2 563 150 47 17 6 3 4 1 1 792
3 2 0 0 0 0 0 0 0 0 2
4 4 4 1 6 1 0 1 1 1 19
5 170 69 41 15 6 3 4 1 1 310
30 46 87 93 100 100 100 100 100 39
6 562 150 47 17 6 3 4 1 1 791
7 237 69 41 15 6 3 4 1 1 377
42 46 87 93 100 100 100 100 100 48
8 558 149 47 14 6 3 4 0 1 782
9 7 0 0 0 0 0 0 0 0 7
10 21 8 7 8 2 1 2 0 1 50
11 242 68 41 14 6 3 4 0 1 379
43 46 87 100 100 100 100 – 100 49
Table 3: Estimated Parameters of Function x ¼ f(u ; a ^) # of observations in data set 792 782
6 P
a ^0
a ^1
a ^2
a ^3
a ^4
a ^5
a ^6
1.0 1.239
.6521 .1689
.1430 .2364
.0 .0194
.0 .1430
.0 .0346
.0 .2600
a ^i
i¼1
.7951 .8623
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Public Goods, Environmental Externalities and Fiscal Competition
Note: In all following histograms, the numbers in the columns on the left denote respectively : – for col. 1, the number of post offices in the class; – for col. 2, the relative frequency within the class; – for col. 3, the efficiency classes (F values).
7 26 76 139 140 152 105 84 27 18 9 5 1 0 0 0 0 0 1 2
.009 .033 .096 .176 .177 .192 .133 .106 .034 .023 .011 .006 .001 .000 .000 .000 .000 .000 .001 .003
.09 .13 .18 .22 .27 .31 .36 .41 .45 .50 .54 .59 .63 .68 .73 .77 .82 .86 .91 .95 1.00
# of observations : 792 mean value of F1 : .322 standard deviation : .102
Figure 5.1
1 12 35 58 85 115 122 79 63 51
.001 .015 .044 .073 .107 .145 .154 .100 .080 .064
39 31 22 12 11 10 7 7 8 24
.049 .039 .028 .015 .014 .013 .009 .009 .010 .030
Figure 5.2
.08 .13 .18 .22 .27 .31 .36 .40 .45 .50 .54 .59 .63 .68 .73 .77 .82 .86 .91 .95 1.00
# of observations : 792 mean value of F2 : .437 standard deviation : .184
Ch.13 Measuring Labor-Efficiency in Post Offices
6 12 11 21 29 19
.008 .015 .014 .027 .037 .024
21 31 28 28
.027 .039 .035 .035
35 27 22 16 29 21 32 31 30 343
.044 .034 .028 .020 .037 .027 .040 .039 .038 .433
.20 .24 .28 .32 .36 .40 .44 .48 .52 .56 .60 .64 .68 .72 .76 .80 .84 .88 .92 .96 1.00
303
# of observations : 792 mean value of F3 : .787 standard deviation : .239
Figure 5.3
1. As only two observations are F1 efficient, and the histogram of Figure 5.1 reveals that an immense majority of all the other ones lie rather far away from the production frontier, the latter is clearly biased upwards by these two points. 2. Moreover, the four zeroes in the parameter values make the estimate rather suspicious: how could so many outputs have no influence on total input requirements? On the other hand, the increasing returns to scale property implied by the sum of the estimated parameters is somewhat in contradiction with the fact that all large offices are inefficient. This second point is confirmed by the F2 estimation: since several large observations become efficient with this method, the scatter of the observed points must be seen as bending inwards in that range, i.e. implying decreasing returns there. 3. But an equally immense majority of observations are also F2 inefficient in the small range of 0 to 10,000 hours. This suggests that the constant or decreasing returns implied by the convexification of Y0 that method 2 operates is perhaps an assumption inappropriate to our data. This is precisely what method 3 shows: dropping the convexification yields 30% of efficient F3 observations in the 0 –5,000 hours range, 46% in the 5,000 –10,000 hours range, and still larger percentages for higher ranges. Increasing returns are thus locally prevailing in the small
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Public Goods, Environmental Externalities and Fiscal Competition
observations range, and the weight of their very large number explains why they determined a shape of this nature for the estimated production function. 4. A last fact, unreported in Table 3, must be mentioned concerning the F3 measures. Recall that with each F3 inefficient observation, an alternative ‘‘model’’ observation, denoted above as (x(k) ,u(k) ), is identified and exhibited. This computation revealed that over the input range of 0–5,000 one single observation appears as the ‘‘model’’ one in a very large number of cases, and thus dominates systematically; probably half of the efficiency measures F3 shown in Figure 5.3 are computed with respect to this one. Although we cannot detect whether a measurement error is present in this case, we thought it of interest to repeat all of the F3 efficiency computations with this observation deleted. This led us to the results reported in columns (6)–(7) of Table 2 and on Figure 6. One can see that the effects of this deletion are striking ones: 1. over the whole set of 791 observations and for the 414 inefficient ones, no single ‘‘model’’ one appears systematically anymore (say, more than 10–15 times); 2. the number of efficient observations in the class of 0–5,000 hours increases from 30 to 42%;
2 0 0 2 5 6 9 8 14 14 21 15 18 29 31 39 55 48 50 425
.20 .003 .24 .000 .28 .000 .32 .003 .36 .006 .40 .008 .44 .011 .48 .010 .52 .018 .56 .018 .60 .027 .64 .019 .68 .023 .72 .037 .76 .039 .80 .049 .84 .070 .88 .061 .92 .063 .96 .537 1.00
Figure 6
# of observations : 791 mean value of F3 : .891 standard deviation : .156
Ch.13 Measuring Labor-Efficiency in Post Offices
305
3. this number remains unchanged in the higher classes, a fact that illustrates very well how, with this method, outlying observations have only a ‘‘local’’ influence on the overall efficiency estimate. Knowing, on the other hand, that the two other methods are more globally sensitive to outliers, and having noted above the possible role of the latter in our estimates of F1 and F2 , we attempted to locate them by examining the frequency distribution of the normalized outputs (uki =xk ), i ¼ 1, . . . , 6 for all the observations k ¼ 1, . . . , n. It turned out that seven of them had one normalized output far outlying an otherwise fairly regular distribution; one had two such outputs, and two (including the previously deleted one) had three. Ten observations were thus deleted, and a new computation made of the three efficiency measures F1 , F2 , F3 for the remaining 782 observations data set. The results appear in columns (8) to (11) of Table 2, in Figures 7.1, 7.2 and 7.3, and on the second line of Table 3 as far as the parameters of the Cobb–Douglas production function are concerned. The outcome of this last run of computations can be characterized as follows: 1. The overall efficiency measures have substantially increased for F1 and F2 , but very little for F3 (for the latter, at least in the case of 782 vs. 791 observations).
1 .001 3 .004 5 .006 17 .022 17 .022 33 .042 34 56 71 80
.044 .072 .091 .102
91 96 69 66 58 32 22 12 8 11
.116 .123 .088 .084 .074 .041 .028 .015 .010 .014
Figure 7.1
.15 .20 .24 .28 .32 .36 .41 .45 .49 .53 .58 .62 .66 .70 .75 .79 .83 .87 .92 .96 1.00
# of observations : 782 mean value of F1 : .609 standard deviation : .149
306
1 1 5 10 9 23 23 29 48 49 58 76 79 75 71 60 32 38 23 72
Public Goods, Environmental Externalities and Fiscal Competition .14 .001 .18 .001 .23 .006 .27 .013 .31 .012 .36 .029 .40 .029 .44 .037 .48 .061 .53 .063 .57 .074 .61 .097 .66 .101 .70 .096 .74 .091 .79 .077 .83 .041 .87 .049 .91 .029 .96 .092 1.00
# of observations
: 782
mean value of F2 : .687 standard deviation : .178
Figure 7.2
3 4 3 11 4 4 15 12 17 14 11 20 26 24 34 44 40 37 40 419
.34 .004 .37 .005 .40 .004 .43 .014 .47 .005 .50 .005 .53 .019 .57 .015 .60 .022 .63 .018 .67 .014 .70 .026 .73 .033 .77 .031 .80 .044 .83 .056 .87 .051 .90 .047 .93 .051 .97 .536 1.00
Figure 7.3
# of observations mean value of F3 standard deviation
: 782 : .894 : .152
Ch.13 Measuring Labor-Efficiency in Post Offices
307
2. A still very low number of only 7 observations determine the function underlying F1 . However, it is known to be a general property of the method, with the Cobb-Douglas specification, that only as many observations can be found efficient as there are parameters to be estimated (Førsund, Lovell and Schmidt (1980), p. 10). On the other hand, the estimated parameters are now more reasonable; but the weight of P small scale observations still determines, with 6i¼1 a ^ i < 1, increasing returns to scale for the entire function. 3. This last property is again disproved by the computations that follow. But the convexification imposed on Y0 by the Farrel non parametric method still leads to an excessively small number of F2 efficient observations in the lower ranges. This is indeed established by the fact that with the F3 measures, the number of efficient observations remains almost unchanged compared with the 791 data set. 4. The deletion of the ten outlying observations also leads to a substantial modification of the shape of the frequency distribution of the efficiency measures, at least, for F1 and F2 : average efficiencies move from .32 (respectively .43) to .61 (resp. .69), which are numbers somewhat more in line with plain common sense. As far as F3 is concerned, this average efficiency level remains almost unchanged between the 791 and 782 data sets; only three or four extreme cases of inefficiency disappear (see Figures 6 and 7.3).
5.
Conclusions
5.1
On the measurement of labor-efficiency
We have discussed above the relative merits of our three measurement methods, as they appear from their application to our particular data set. The main conclusions we draw from this discussion, as far as methodology is concerned, are as follows. The parametric method shows essentially that the Cobb–Douglas specification we have chosen for the production function is a little convincing one. In addition, Figures 5.1 and 7.1 do not support at all the implicit statistical assumption of an exponential distribution of the distance between the observed units and the theoretical production function (taken in logarithms). From our subsequent results, it seems clear that a translog function would do better; this is a further work that ranks high in our agenda of future work on this topic. On the other hand, the convexification of the data set is an assumption which in this case is unwarranted. One may consider, of course, the extension of the Farrell method to increasing returns (see Farrell and
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Fieldhouse (1962)). It has little appeal to us, however, because of its arbitrariness in determining the facets of the production set. Finally, the simplicity of our third method makes it not only a tool extremely easy to use: it also has the merit of resting on the weakest assumptions regarding the production set. And this is not devoid of managerial implications, in the following sense: while only few managers – whether private or public – would have any reason to impose that the operations of their firms be organized so as to fit the graph of an a priori selected function (or the boundary of a necessarily convex set), most of them surely would agree with the statement that using less input for the same or a larger output is better than using more. But this is nothing else than input disposability. In addition to having shown that with this only assumption an efficiency measure can be devised, we may point out that the identification of dominating observation(s) reveals information of direct use for managers thus motivated. Proving to a decentralized unit that it can do better is indeed more effectively done by showing it existing examples, than by exhibiting the image of a theoretical production function or some convex combination of other observed units. 5.2
Labor-efficiency within the Re´gie des Postes
Our comments here are based on the results of measure F3 (with 791 observations) only: we feel that the limitations inherent to the other two methods – at least as they have appeared in this case – prevent one from making evaluative statements on the Re´gie des Postes that are sufficiently grounded. From the synthetic figures of .89 for the average F3 efficiency (and a median above .96), of 48 % of the offices being fully efficient (and 66 % with F3 $ :88), and from the shape of the distribution of efficiency measures that appears on Figure 6, the over-all picture that we come up with is not an unfavorable one, given the size of the enterprise. The same distribution also suggests that extreme cases of inefficiency are very rare (for only 9 offices, i.e. about 1 % of them, F3 # :40), and heavily inefficient observations are not that many (there are 60 offices, i.e. 7.6 %, for which F3 # :60). This optimistic evaluation probably needs to be qualified, however, because of various limitations of our data set. For instance, our traffic output being that of January only, it is probably higher than that of the average month in the year, while the personnel does not vary much from month to month. On the other hand, labor is treated here as homogeneous, whereas distinguishing between categories would probably change the efficiency results (although it is not clear in which direction). Moreover, no account being taken of the fact that in some cities there are more
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than one mail distribution per day creates an obvious downward bias in the efficiency measure for these observations. And finally, the absence of distinction between rural and urban postal stations also can play a role. Correcting for all these limitations is obviously a task ahead. At this stage, however, we think that there is more ground for assuming that many of the recorded inefficiencies can be explained by elements left out of the computations, than for asserting that manpower waste prevails to a large extent in this enterprise.
References Aigner, D.J. and Chu, S.F., 1968. On estimating the industry production function, American Economic Review, 58, 826–839. Deprins, D. and Simar, L., 1983. On Farrell measures of technical efficiency. CORE Discussion Paper n8 8327, Louvain-la-Neuve: Universite´ catholique de Louvain. Farrell, M.J., 1957. The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, 120, 253–281. Farrell, M.J. and Fieldhouse, M. 1962. Estimating efficient production functions under increasing returns to scale, Journal of the Royal Statistical Society, Series A, 125, 252–267. Førsund, F.R., Lovell, C.A.K. and Schmidt, P., 1980. A survey of frontier production functions and of their relationships to efficiency measurement, Journal of Econometrics, 13, 5–25. Koopmans, T.C., 1951. Analysis of production as an efficient combination of activities, in T.C. Koopmans (ed.). Activity analysis of production and allocation. New York: Wiley, 33–97. Schmidt, P., 1976. On the statistical estimation of parametric frontier production functions, The Review of Economics and Statistics, 58, 238–289. Shephard, R.W., 1970. Theory of cost and production functions. Princeton, (New Jersey): Princeton University Press. Tulkens, H., 1968. Programming analysis of the Postal Service: a study in public enterprise economics. Louvain, Librairie Universitaire.
Chapter 14 ON FDH EFFICIENCY ANALYSIS: SOME METHODOLOGICAL ISSUES AND APPLICATIONS TO RETAIL BANKING, COURTS AND URBAN TRANSIT
Henry Tulkens Journal of Productivity Analysis 4 (1/2), 183–210 (1993).
The methodology of free disposal hull (FDH) measure of productive efficiency is defined and put in perspective vis-a`-vis other nonparametric techniques, in terms of the postulates on which they respectively rest. Computational issues are also considered, in relation to the linear programming techniques used in DEA. The first application bears on a comparison between a private and a public bank, in terms of the relative efficiency of their branches. Important characteristics of the data are revealed by FDH that are not by DEA, due to a better data fit. Next, efficiency estimates of judicial activities are used to evaluate what part of the existing backlog could be reduced by efficiency increases. Finally, with monthly data of an urban transit firm over 12 years, the FDH methodology is extended to a sequential treatment of time series, that supplements efficiency estimation with a measure of technical progress.
1.
Introduction
In all empirical studies of productive efficiency, a crucial role is played by the choice of the ‘‘reference technology,’’ that is, in the sense given to this expression by Grosskopf (1986), the production possibilities set whose frontier is used to evaluate the observed productive activities. In many cases, the results obtained are indeed quite sensitive to alternative specifications of This is a revised version of the CORE Discussion Paper No. 9050 presented, under a slightly different title, at the Third Franco-American Seminar on Productivity Issues in Services at the Micro Level held at the National Bureau of Economic Research, Cambridge, MA, July 1990. The author wishes to express his thanks to Bronwyn Hall and Charles Hulten, who discussed the paper in depth, as well as to Marie-Astrid Jamar and
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this set. While recognizing this fact, the efficiency literature does not provide much guidance on the issue of how to evaluate the appropriateness of choices in this respect. In Deprins, Simar, and Tulkens (1984), an effort was made in that direction. In that paper, results were presented regarding a single data set (on post office operations) that were obtained by using first two reference technologies that were standard at that time: one is the production set whose frontier is defined by the graph of a parametric Cobb-Douglas production function, and the other is a convex polyhedral production set (with nonincreasing returns to scale) of the type used in data envelopment analysis (DEA). In addition to noticing the differences just mentioned in the efficiency measures, the paper also stressed that both estimates of the technologies were characterized by an extremely bad data fit. This led the authors to use a third form of reference technology, not much noticed until then and labeled later on1 as the ‘‘FDH (free disposal hull) production set.’’ This yielded again quite different efficiency results, but also fared much better from the data fit point of view. Several other experiments having been made with this last method, many of which are reported in Tulkens (1986), Thiry and Tulkens (1988), and in Tulkens (1989), the present paper intends to take stock of the significance of this methodology, and to illustrate it further with three representative case studies.2 As far as ‘‘significance’’ is concerned, I consider in Section 2 successively (i) the economic production-theoretical assumptions on which rest the FDH methodology and its competing ones, (ii) the question of how efficiency computations with FDH compare with the linear programming techniques used in the literature on DEA, and finally (iii) some further issues regarding data fit and managerial relevance, replicability, and outliers. Philippe Vanden Eeckaut for excellent research assistance. He also benefitted from comments received at several earlier presentations of various parts of this material (Boston University, LOS Center Bergen, University of Odense, KU Leuven, GREQE Marseille, Journe´es de Microe´conomie Applique´e in Toulouse and Montre´al, the Productivity in Europe conference of the European Production Study Group held at ENSAE, Paris and the European Workshop on Productivity and Efficiency Measurement in the Service Industries held at CORE in October 1989). Gratefully acknowledged funding sources for the works surveyed here were for one part the Belgian Fonds de la Recherche Fondamentale Collective (under convention No. 2.4528.88 with the author), for another part the Belgian Ministry of Scientific Policy (under its Programme d’Actions de Recherche Concerte´es No. 87–92 with CORE), and for this revision, the Colle`ge interuniversitaire d’Etudes Doctorales dans les Sciences du Management (CIM), Brussels. 1 Actually on p. 4 of the 1988 discussion paper on which Thiry and Tulkens (1992) is based. 2 A fourth one, dealing with cost efficiency in municipalites and summarizing Vanden Eeckaut, Tulkens, and Jamar (1993), appeared as Section 4 of the first version of the present paper.
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The three illustrative applications are covered in Sections 3 to 5. Like the original post office one, they all bear on public sector activities. A strong justification for this choice may be found in the observation, made by Bronwyn Hall, that the public sector is one where frontier production analysis is the most useful since, unlike in the private sector, we cannot rely on Darwinian selection to eliminate inefficient units, or we may find that selection operates much more slowly. The method presented here for identifying inefficient units may thus serve as a way either to speed up that process or to pinpoint where to take corrective action.
2.
Efficiency measurement from an FDH frontier: methodological issues3
2.1
FDH and the postulates required to construct reference sets
The production set that economic theory associates with any productive activity is usually an unknown. Therefore, the efficiency analyst must construct the reference set he needs. When he does this from statistical data on outputs achieved and inputs used, as is usually the case, his method necessarily rests on a relationship between the statistical observations and the elements of the constructed set. As this relationship may be required to have various properties, our main objective here is to point out that differences in the properties this relationship is postulated to have are at the root of the differences between the alternative reference sets used, and hence between the results obtained in the efficiency measurement literature. Expressing these postulates in terms of the elements that are allowed to belong to the reference set, let us consider the following list (we call a ‘‘production plan’’ any vector of input (x) and output (y) quantities). The reference production set should contain as its elements: 1. determinist postulate: all the observed production plans. 2. free disposal postulate: any not observed production plan with output levels equal to or lower than those of some observed production plan and more of at least one input; or with input levels equal to or higher than those of some observed production plan and less of at least one output; or still with both of these properties. 3. convexity postulate: any not observed production plan that is a convex combination of some production plans induced by 1 and 2. 3 Beyond the Deprins, Simar, and Tulkens paper already mentioned, Tulkens (1986) and Thiry and Tulkens (1989) provide extended introductory presentations; in the last paper, the connection is also made with the usual productivity concepts. For a general exposition of efficiency theory in economic terms, the book by Fa¨re, Grosskopf, and Lovell (1985) is a basic reference.
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4. convexity and partial proportionality postulate: any not observed production plan that is a convex combination of some production plans induced by 1 and 2, or of some such plans and the origin of the inputoutput space. 5. full proportionality postulate: any not observed production plan that is proportional to some observed reproduction plan induced by 1 and 2. Together, postulates 1 and 2 are sufficient to induce a reference set that has all the properties economic theory requires of a production set. As these postulates make this set to be the ‘‘free disposal hull’’4 of the observed plans, we denote it henceforth as YFDH . Notice that postulate 1 rules out stochastic elements, as most of the nonparametric literature does5 and contrary to the parametric one.6 With the third postulate being added, i.e., convexity, one gets one of the convex polyhedral reference sets used in DEA, namely the one proposed in Banker, Charnes, and Cooper (1984). It is usually interpreted as exhibiting varying (i.e., first increasing and then decreasing) returns to scale, and is therefore denoted here as YDEA-V . Adding further postulate 4 yields a convex polyhedron with nonincreasing returns, denoted as YDEA-CD . Finally, with postulates 1 through 5, the obtained reference set is the one used in the original form of DEA proposed in Charnes, Cooper, and Rhodes (1978), which is the same as the one of Farrell (1957). It has the form of a cone issued from the origin of the space of input and output quantities, implying constant returns to scale; we denote it here as YDEA-F . That the above postulates are stated in order of increasing strength7 can be seen from the formal expressions of the corresponding reference sets, as they can be obtained from using the set-theoretical notation proposed on pp. 249–250 of the Deprins, Simar, and Tulkens paper. Here we shall dispense ourselves with that exercise, and notice instead directly from Figure 1, for the one input–one output case, that these sets appear as nested in one another: YFDH (whose frontier is the staircase line ABCDEF) is contained in YDEA-V (whose piecewise linear frontier is ABCEF), which in turn is contained in YDEA-CD (whose frontier is now 0CEF), and this one is further contained in YDEA-F (whose frontier is 4
A terminology introduced by MacFadden (1978, p. 8). This is of course quite a severe limitation that only recent works such as Land, Lovell, and Thore (1988) and Olesen and Petersen (1989) have attempted to overcome. 6 At least since Aigner, Lovell, and Schmidt (1977) and Meeusen and Van Den Broeck (1977). 7 We have restricted ourselves to postulates inducing reference sets with nonparameteric frontiers. When parametric methods are used, one central postulate is made, namely that the reference set’s frontier be the graph of a function with constant parameters, specified a priori. Whether any of the postulates 2–5 also apply entirely depends on the properties of this function. 5
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Figure 1. Alternative shapes of the boundary of the reference production set when constructed from observed data by means of the FDH, DEA-V, DEA-CD, and DEA-F methods.
0CG). Figures 2 and 3 further illustrate the many inputs and the many outputs cases (where the three DEA-type reference sets cannot be distinguished). The presentation made so far should make clear what the position is of the FDH methodology vis-a`-vis the other ones used in efficiency analysis. In particular, it shows that FDH makes the weakest postulates as to how the reference set is constructed from the statistical data.8
Figure 2. Input requirements reference sets constructed from observed data by means of the FDH vs. DEA-type methods. 8
FDH thus uses still more parsimonious axioms to characterize a production possibility set than those advocated by Charnes et al. (1985, p. 94).
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Figure 3. Output possibilities reference sets constructed from observed data by means of the FDH vs. DEA-type methods.
At this point, we also may offer already one comment on data fit: it is indeed clear that while the Farrell cone is the coarsest way to describe, or ‘‘envelop’’ the data set, the FDH frontier is, at the extreme opposite, the closest one among the four frontiers we have considered. This is due, essentially, to the absence of convexity assumption in the definition of the FDH set. 2.2
FDH computation of efficiency measures and linear programming
For computing efficiency measures when the assumed reference production sets are of the type discussed above, the influential paper by Charnes, Cooper, and Rhodes (1978) has rightly popularized the use of linear programming methods–except for the case of the FDH technology which was ignored in that paper. We therefore show here how the programming formulation of efficiency computations also applies (albeit with an important modification) with FDH technologies; this was not seen in the Deprins, Simar, and Tulkens paper, and was apparently not pointed out elsewhere.9 Given a set Y0 ¼ {(xk ,yk )jk ¼ 1, . . . ,n} of n observed production plans of a given organization, where xk is an I-dimensional nonnegative vector of the quantities of I inputs used and yk is J-dimensional nonnegative vector of the quantities of J outputs achieved, let us recall that radial efficiency measures with respect to the DEA-Farrell reference technology, YDEA-F , are obtained as follows. The efficiency degree in inputs of observation k is the value uk of the optimal solution of the linear programming problem (problem P1): 9
The development that follows originates in a private conversation with Sten Thore from IC2 Institute, to whom the author is grateful.
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
Min
{uk , g h , h¼1,..., n}
uk xki
n X
317
uk , subject to
g h xhi $ 0, i ¼ 1, . . . , I,
h¼1
n X
gh yhj $ ykj ,
j ¼ 1, . . . , J,
h¼1
uk , g h $ 0, h ¼ 1, . . . , n; and the efficiency degree in outputs of this observation is obtained as the value of 1=lk , where lk is the optimal solution of the programming problem (problem P2): {l
k
Max lk , subject to , , h¼1,..., n} gh
n X
gh xhi # xki ,
h¼1 n X
lk ykj
i ¼ 1, . . . , I,
g h yhj # 0, j ¼ 1, . . . , J,
h¼1
lk , g h $ 0, h ¼ 1, . . . , n: It is well known that adding to either one of these problems the constraint n X
gh # 1
(1)
h¼1
amounts to take YDEA-CD as the reference set; and changing further this constraint into n X
gh ¼ 1
(2)
h¼1
implies assuming instead that the reference set is of the form YDEA-V . It is now a simple matter to see that by substituting for (1) or (2) the 1 þ n constraints n X
gh ¼ 1
(3)
h¼1
and gh 2 {0,1}, h ¼ 1, . . . , n,
(4)
radial efficiency measures are obtained with respect to the reference set YFDH . Figure 4 illustrates the outcome of these alternative computations
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0R/0N = DEA-F measure; 0P/0N = DEA-CD and DEA-V measures; 0H/0N = FDH measure (all measures in input).
Figure 4. Alternative efficiency measures of observation M with FDH, DEA-V, DEA-CD, and DEA-F reference sets.
for an observation such as M, in the one input–one output setting (the multiple inputs and outputs cases are illustrated in Figures 5–6, which the following considerations regarding the computation technique will make clearer). For the first three pairs of problems just described, i.e., P1 & P2, P1 þ (1) and P2 þ (1), and P1 þ (2) and P2 þ (2), the actual computation of solutions requires the apparatus of the standard linear programming algorithms. In the fourth case however, i.e., P1 þ (3) þ (4) and P2 þ (3) þ (4) which is the FDH one, the problems are seen to be ones of mixed integer programming: hence, these standard algorithms do not apply. Solutions are easily obtained, though, by means of a simple vector comparison procedure that amounts to a complete enumeration algorithm. The procedure runs as follows.10 Given the observation (xk , yk ) to be evaluated, associate with it the set Di (k) containing the indices of the observation (xk , yk ) itself and of the subset of observations that weakly11 dominate it in inputs, that is, the subset of vectors (xh , yh ) 2 Y0 such that xhi # xki , i ¼ 1, . . . , I, with strict inequality for at least one i and yhj $ ykj , j ¼ 1, . . . , J. The solution uk of the linear programming problem P1 stated above augmented with constraints (3) and (4) then reads 10
We deal here with the general case, i.e., with several inputs and several outputs. On pp. 252–253 of the Deprins, Simar, and Tulkens paper, just the one input-several outputs case was considered, and for input efficiency measures only. 11 An alternative form of strong dominance may be defined by letting the inequality that follows be a strict one for all inputs. It allows for less observations to belong to Di (k).
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
319
x1
k⬙
k a
b
k⬘ 0
A
x2
K
DEA: q k* = 0b/0k FDH: q k* = 0a/0k = 0A/0K D i (k ) = {k ⬘, k ⬙}
Figure 5. Input efficiency measures with FDH and DEA reference sets.
uk ¼ Min i
d2D (k)
Max i¼1,...,I
d xi : xki
(5)
That (5) is indeed the solution of P1 with (3) and (4) is easily proved by first observing that all vectors that correspond to the elements of the set Di (k) are vectors that satisfy the first two sets of constraints of P1. Next, rewriting the first set of constraints as uk $
Snh¼1 g h xhi , i ¼ 1, . . . , I, xki
y1
d c
B K
k⬘ k⬙
k
0
y2 1/λ k*
DEA: = 0k/0d FDH: 1/λ k* = 0k/0c = 0K/0B D o (k ) = {k ⬘, k ⬙}
Figure 6. Output efficiency measures with FDH and DEA reference sets.
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shows that the role of the Max operator in (5) is to satisfy each one of these inequalities. Finally, the Min operator corresponds to what is sought for in the objective function of P1. The observation d in (5), with respect to which the efficiency of k is computed, thus appears to be the one which ‘‘most dominates’’ observation k in inputs. Let us consider next the measurement in output. To this effect, associate now with (xk , yk ) the set D0 (k) containing the indices of the observation (xk , yk ) itself and of the subset of observations that weakly dominate it in outputs, that is, the subset of vectors (xh , yh ) 2 Y0 such that xhi # xki ,i ¼ 1, . . . , I, and yhj $ ykj , j ¼ 1, . . . , J, with strict inequality for at least one j. The value 1=lk where lk solves the linear programming problem P2 augmented with constraints (3) and (4) is then given by ( ) ykj 1 , (6) ¼ Min Max lk d2D0 (k) j¼1,...,J ydj the proof of this assertion being the same as in the previous case. Here again, the observation d retained in (6), with respect to which the efficiency of k is computed, thus appears to be the one which ‘‘most dominates’’ in outputs observation k. Notice that it need not be the same as the one that most dominates in inputs. Figures 5 and 6 illustrate that radial computations almost always leave a positive slack for all inputs (or outputs) except for those retained in (5) or (6). In that sense, these efficiency measures may be seen as underestimates. As no radial measure can circumvent this, and in view of the easy interpretability of radial measures, it seems preferable to use them in numerical applications, with slacks recorded separately. 2.3
Dominance, proportionality versus replicability and outliers
Returning to the procedure just described, let us further observe that beyond its computational virtues, the procedure offers the additional interest of putting forward the notion of dominance between observations, which is of both theoretical and managerial relevance. From the theoretical point of view, let us notice that when the sets Di (k) and/or D0 (k) only contain the index k itself, the corresponding observation has the property of being undominated, in inputs and/or in outputs respectively, by any of the other observations. This is in fact how, with the FDH methodology, an observation is defined as being efficient (one may also check that the resulting efficiency measures uk and 1=lk are then each equal to 1). In this context, efficiency appears to be a property of an observation essentially vis-a`-vis other observations rather than vis-a`-vis a predefined frontier. Elaborating on this observation may have far reach-
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ing consequences, as explored recently in Tulkens and Vanden Eeckaut (1991). From a managerial point of view, I should like to point out that the identification of a set of dominating observations, by showing actually implemented production plans that are clearly more efficient, gives to the inefficiency scores a credibility that they usually lack when reference is only made to an abstract frontier. This argument is illustrated with the observation labeled D on Figure 1: its inefficiency alleged by all forms of DEA measurement is only supported by reference to some fictitious combination of other observations, and not by any real-life production plan. The gain in realism thus claimed for the FDH methodology may be seen as an implication of its already observed virtue of better data fit. Further conceptual developments are also suggested by the linear programming formulations P1 þ (3) þ (4) and P2 þ (3) þ (4). In Petersen (1990) for instance, other forms of nonconvexities of the reference set than those implied by FDH are obtained by putting an upper or lower bound equal to 1 (but no integrality constraint) on the g h variables of problems P1 and P2. In particular, he shows how such an upper (lower) bound induces nonincreasing (resp. nondecreasing) returns to scale of a somewhat special form. Although they are called ‘‘intensity’’ variables by the author, it should be clear that the role of the gh variables in the construction of the reference set is governed by our proportionality postulate 5. In fact, Petersen’s bounds amount to postulating restricted forms of proportionality: a restriction on expansion for the upper bound, and on contraction for the lower bound. Another variant of the proportionality postulate is suggested by replacing (4) by gh 2 {0, 1, 2, . . . }, h ¼ 1, . . . , n,
(7)
by introducing some integer larger than 1 in the right-hand side of (3) and by further substituting therein the weak inequality # for the equality. Alternatively, one may dispense with (3) and have an upper bound in the integers allowed in (7), possibly different for each observation h. The philosophy behind these new specifications is perhaps best expressed by formulating it as a replicability postulate, similar in spirit although weaker than proportionality, and running as follows: the reference set should contain any production plan which is a gh -fold replica of the hth observed one (thus not simply scaled up or down in a continuous way, as full proportionality allows). As to outliers, finally, we may derive the following general statement from Section 2.1: with nonparametric methods, sensitivity to outliers increases with the strength of the postulates made in constructing the reference production set—‘‘sensitivity’’ being understood here in terms of
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the size of the subset of the data whose efficiency measures are influenced by a given outlier. It is lowest with FDH, since this method rests on the weakest postulates. It is easy to see that the set of observations whose efficiency rating is affected by the presence of an outlier is at least as large (and most often much larger) when convexity assumptions are made, as in all forms of DEA, with sensitivity to outliers being the most extreme one with DEA-F.
3.
Retail Banking: assessing relative productive efficiency performance for the branches of a public and of a private bank12
3.1
The public bank
This application bears upon the activities of 773 branches of a large publicly owned bank operating in Belgium. Our data are a cross section of the output and input quantities of each branch for the month of January 1987. The outputs are measured in terms of the number of operations performed during the month. More than 60 different ones are reported, but we aggregate them into eight categories, the list of which is given in Table 1. Only the number of operations is considered, irrespective of their financial values.13 As to inputs, the statistics available for each branch
Table 1.
Description of inputs and of output aggregates for the bank branches.
Inputs
No. 1
Outputs
No. 2 No. 3 No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8
12
Labor (number of hours worked per month; in Tables 3 and 4 this data is converted into number of persons per month) Windows (number available) Automatic teller machine (0 or 1) Checking and savings accounts operations (deposits and withdrawals) Automatic teller machine operations International operations (transactions on foreign exchange and on travelers’ checks) Brokerage activities (stock and bonds) Credit operations Opening of new accounts Special services (e.g., issuing of credit and ATM cards) Miscellany (e.g., life insurance operations for a public insurance company closely related with the bank)
This section summarizes the work presented in Tulkens and Vanden Eeckaut (1989). Thus, we consider only the ‘‘production’’ aspect of the banking activity. Financial values are of course no less essential, but they do not lend themselves easily to a treatment in terms of production sets. In fact, the efficiency theory developed above should clearly be reformulated for that purpose, an extension that has not been made so far, to our knowledge. 13
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consist of data on manpower (in terms of the number of people employed in the branch), on the number of windows operated, as well as on whether or not the branch operates an automatic teller machine. As to methods, we concentrate here on the nonparametric FDH one, and we limit ourselves to input measures of efficiency. From a managerial point of view, the most interesting results of efficiency computations in such a case are those provided by the complete listing of the efficiency scores of each individual branch, ranked in increasing order. One can thus locate at the top of the list the most severe cases of inefficiency. The size of our data set clearly precludes providing this entire listing here, but an excerpt from its first lines is given in Table 2 as an example, where we present five of the worst cases as determined by the FDH method. On Table 2, various comments are in order. First, the efficiency degrees provided in column 3 are ‘‘radial’’ ones, i.e., equiproportionate input reductions as suggested on the theoretical Figure 5. Thus, with the 0.648 figure mentioned for the first quoted branch it is claimed that had this branch been operating on the assumed efficient frontier, it could have achieved the same output with only 64.8% of all its inputs (and possibly even less for some of them as will be made clear next). One further notes in column 4 that the method identifies, for each inefficient branch, one which dominates14 it, in the sense that it uses no more inputs and strictly less of some of them, while producing more of all of the outputs. As stressed in Section 2, this information is managerially very relevant as it provides from within the enterprise factual evidence of better achievement.15 Of course, it is not an absolute reference, in the sense that it may well be that the dominating branch operates under characteristics (other than output and input quantities) that are substantially different from those of the dominated unit. If so, what is measured as inefficiency is rather to be attributed to these characteristics. But if no such characteristics can be identified, the measure then reveals rather convincingly exactly what is sought for, namely situations where inputs are used in larger quantities than elsewhere, with less of all outputs achieved, i.e., pure technical inefficiency. For each branch found inefficient, such a careful case by case examination of characteristics is called for, before the inefficiency diagnosis can be confirmed. Finally, the table allows one to characterize the importance of the inefficiency in terms of each one of the inputs involved. Thus, for the first quoted branch, the reduction of all inputs to 64.8% of their use 14
Actually the most dominating one if there are several such. Among all the methods described in this paper, the FDH one is the only one to provide such information for each one of the activities it declares inefficient. 15
Table 2. Sample of worst FDH inefficiencies, with dominating unit and underlying data. Rank 1
Unit’s Code #
Efficiency Score
81832
.648
Dominating Unit’s Code # 81997
2
81619
.667
3
82894
.710
4
82753
.740
5
81667
.750
81561 81997 81591 81768
Input Quantities No. 1 No. 2 No. 3
No. 1
No. 2
No. 3
27,578 17,857 29,178 19,442 25,149 17,857 26,273 19,442 39,811 29,612
1109 1281 2030 2096 1159 1281 2060 2096 2673 3133
0 0 0 0 0 0 0 0 0 0
231 330 485 494 172 330 368 494 689 719
4 2 3 2 3 2 4 2 4 3
0 0 0 0 0 0 0 0 0 0
Output Quantities No. 4 No. 5 85 125 33 73 75 125 34 73 50 71
2 31 0 17 1 31 0 17 18 22
No. 6
No. 7
No. 8
17 18 38 67 24 18 18 67 29 51
9 18 35 108 10 18 46 108 20 37
3 21 5 18 0 21 4 18 7 14
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
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corresponds to reducing them to an amount such that at least one of them be equal to its usage in the dominating branch (0.648 is indeed the ratio of the usage of input 1 by the dominating branch, namely 17,857 hours of labor, to the 27,578 hours used in the dominated branch). If such a ratio had been computed for input 2, the reduction would even be to 50% of actual use. But this would mean that one would require the inefficient branch to do even better, in terms of input 1, than what the dominating branch does. As the dominating branch referred to happens to be an efficient one, this would imply requiring from the dominated branch to perform beyond the assumed frontier, a clearly excessive requirement. However, while a reduction to 64.8% is thus the maximum reduction one can require for all inputs, there remains ‘‘slack’’ for some of them since a reduction to 50% is possible for input 2. For the second branch quoted in Table 2, whose efficiency score is 0.667, notice that the slack is with input 1. When more than just two inputs are involved, this inputwise analysis clearly gains importance. However, I doubt that seeking for a single numerical index of efficiency including these ‘‘slacks’’ would be a useful extension. Additional columns exhibiting the values of all slacks would be more informative. Turning now to aggregate statistics of our results, Table 3 shows on the bottom line of columns 9 and 10 that, out of the total 773 branches, 136 (i.e., 18%) are found to be inefficient in the FDH sense, that is, dominated by at least one other well-identified branch. This leaves of course ample room for the case by case analyses mentioned above—a task surely beyond the grasp of the outside analyst and probably best tackled by the firm itself. A contrario, there are 637 branches found undominated, and thus declared FDH-efficient (column 3). Within this rather large subset, we further distinguish two classes: on the one hand (column 4), the class of efficient branches which do dominate some inefficient one(s): for those, superiority of behavior is hardly questionable. There are 141 such branches (i.e., 18% of the total). On the other hand (column 5), there is the class of branches that we call ‘‘efficient by default,’’ defined by the fact that there is no branch in the data set that they dominate. Essentially, the branches in this class are Table 3.
Summary statistics of FDH efficiency results.
1 Cat. by Size
2
3
# Obs.
# Eff.
0–2 2–3 3–4.5 4.5–7.5 > 7.5 Total
191 243 119 120 100 773
141 186 106 108 96 637
4 %
5 # Eff. & Domtg.
.74 .77 .89 .90 .96 .82
47 51 19 22 2 141
6
8
9
%
7 # Eff. by Default
10
%
# Ineff.
%
11 Aver. Degree
.25 .21 .16 .18 .02 .18
94 135 87 86 94 496
.49 .56 .73 .72 .94 .64
50 57 13 12 4 136
.26 .23 .11 .10 .04 .18
.976 .986 .995 .991 .997 .987
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Public Goods, Environmental Externalities and Fiscal Competition
compared to no other branch, and their ‘‘efficiency’’ label actually amounts to one of noncomparability. The number of branches in this class is 496 (i.e., 64%). To sum up, the 773 branches in our data set is partitioned into three groups: the group of worse-than-others, comprising 18%, the group of better-than-others, comprising another 18%, and finally the group of noncomparable ones (64%). On the first five lines of Table 3 these results are broken down according to branch size, size being measured by manpower input. Typically, inefficiency appears to be more frequent in small scale branches than in large ones. However, efficiency by default is also more frequent in large branches (especially in the largest class). Finally, the average efficiency scores are given in column 11. They appear to be quite high, due of course to the large percentage of the observations that are 100% efficient. More informative is the frequency distribution of the efficiency scores, given on Figure 7. It exhibits a typical decreasing shape, with the number of inefficiency cases declining gradually with severeness. 3.2
Comparing with a private bank
To put the above results in perspective, it is of some interest to attempt a comparison with similar computations made in the private sector. To this effect, we make use of a fairly similar data set pertaining to the 911 branches of a Belgian private bank–thus an institution of comparable size. The data were compiled and studied by Respaut 1.0 0.95 Detail for [0.50-1] domain 1.0 0.95
0.85
0.90
0.80
Efficiency degree
Efficiency degree
0.90
0.75 0.70 0.65
0.85 0.80 0.75 0.70 0.65 0.60 0.55
0.60
0.50
0.55
0
5
10 15 Number of branches
20
0.50 0
100
200
300 400 500 600 Number of branches
700
800
Figure 7. Frequency distribution of FDH efficiency scores of the public bank branches (3 inputs, 8 outputs).
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
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(1989). As the setting is somewhat different (outputs are aggregated into seven categories, and only one input—labor—is considered), we modified accordingly the aggregation of our public bank data, and recomputed efficiency scores. The computation methods used are this time not only FDH, but also DEA-F and DEA-V, applied to each bank separately. Table 4 and Figure 8 report the results and provide the elements for the comparison. First of all, the DEA methods, that impose convexity, yield in both cases strikingly less favorable results than FDH. A difference was to be expected, since we know that the former method less closely envelops the data than the latter; but the importance of that difference is considerable: with DEA-F the frontier is constructed with only about 2% of the observations in either case (since only that percentage is declared efficient; see the bottom line of column 4 in either table), and with DEA-V it is so with about 5% (see bottom of columns 6); but with FDH these percentages are 57.8 and 74.6 (columns 8) for the private and the public banks respectively. Without embarking here in a discussion of the virtues or defects of convexity, and of the merits or demerits of fitting a frontier
Table 4.
Efficiency results for branches of the private and of the public bank.
Input Categ. (#Pers) (1) Public Bank Branches 0–2 av.eff.deg. 2–3 av.eff.deg. 3–4.5 av.eff.deg. 4.5–7.5 av.eff.deg. >7.5 av.eff.deg. Total Private Bank Branches 0–2 av.eff.deg. 2.5–4.5 av.eff.deg. 4.5–6.5 av.eff.deg. 6.5–10.5 av.eff.deg. >10.5 av.eff.deg. Total
Total (2) 220 242 121 120 101 804 187 397 174 93 60 911
#Br.Dea-F Efficient # % (3) (4) 7 0.64 2 0.70 2 0.72 1 0.68 1 0.63 13 9 0.6 10 0.63 1 0.63 1 0.61 1 0.59 21
3.2 0.8 1.7 0.8 1.0 1.6 4.8 2.5 0.6 1.1 1.6 2.3
#Br.Dea-V Efficient # % (5) (6) 11 0.69 6 0.72 8 0.78 6 0.78 11 0.77 42 13 0.73 11 0.66 7 0.67 6 0.68 13 0.74 50
5.0 2.5 6.6 5.0 10.9 5.2 7 2.8 4 6.5 21.7 5.5
#Br.FDH Efficient # % (7) (8) 143 0.95 157 0.97 103 0.99 104 0.98 93 0.99 600 95 0.95 201 0.91 116 0.95 63 0.96 52 0.98 527
65.0 64.9 85.0 86.7 92.1 74.6 50.8 50.8 66.7 67.7 86.7 57.8
328
Public Goods, Environmental Externalities and Fiscal Competition 1.0 0.95 Detail for [0.50-1[ domain
0.90
1.0
0.90
Efficiency degree
Efficiency degree
0.95
0.85 0.80 0.75 0.70 0.65
0.85 0.80 0.75 0.70 0.65 0.60 0.55
0.60
0.50 0
0.55
10
20 30 Number of branches
40
50
0.50 0
100
200
300
400
500
600
700
Number of branches (a) Public bank 1.0 0.95 Detail for [0.50-1[ domain
0.90
1.0
0.90
0.80
Efficiency degree
Efficiency degree
0.95
0.85
0.75 0.70 0.65
0.85 0.80 0.75 0.70 0.65 0.60 0.55
0.60
0.50 0
0.55
10
20 30 Number of branches
40
50
0.50 0
100
200
300
400
500
600
700
Number of branches (b) Private bank
Figure 8. Frequency distribution of FDH efficiency scores for the branches of the private and the public bank (1 input, 7 outputs).
close to the data, let us just note with this example how important an impact these issues can have. The difference in the methodological assumptions also has a direct bearing on the public versus private comparison. Indeed, with the two DEA methods, the private bank appears to score somewhat better than the public bank, at least in terms of percentages of efficient units (columns
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
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4 and 6 in either table). With FDH, on the contrary, the public bank scores better, and even much better (compare columns 8). This is strikingly summarized in the total figures: only 57.8% of the private bank’s branches make up the FDH frontier, whereas 74.6% make it up for the public bank. To the extent that close data fit is considered to be a desirable property of the analysis, which is the case with FDH, one may conclude that these data reveal relative superiority of the public bank in terms of relative productive efficiency of its branches. This conclusion is reinforced by the information provided by the frequency distributions of efficiency scores appearing on Figure 8: the two diagrams clearly show that the tail of the distribution for the public bank is flatter and stretches less far in the low values than for the private bank. In the public bank, relative efficiency of the branches vis-a`-vis one another is stronger than in the private bank. There may be several reasons for the so identified evidence of superior productive performance in this public bank. However, the method used here is not designed to discover such reasons; they must be found in other statistical and factual data than inputs and outputs. But the method does seem to achieve what it is designed for, namely to reveal the specific cases and the magnitude of inefficient behavior. In our comparison of a public and a private bank, we can conclude in favor of the former. To further substantiate this finding, however, further data allowing one to repeat the experiment would be desirable.
4.
Courts: using efficiency results to evaluate the backlog16
In this application we attempt to evaluate productive efficiency in the provision of judicial services, i.e., essentially court decisions, using again the FDH nonparametric method explained earlier. From the results of this analysis, we then derive some implications concerning the phenomenon of the judicial backlog, which happens to have reached important proportions in Belgium. 4.1
The data
Our enquiry is limited to only one category of courts, the Justices of the Peace (‘‘Justices de Paix;’’ JPs henceforth), mainly for reasons of easiness of access to statistical data. In Belgium, these are courts with a single judge, dealing with cases of minor monetary value (up to BF 50 000, 16 The contents of this section is taken from Jamar and Tulkens (1990). Other efficiency studies in this area have been made by Lewin, Morey, and Cook (1982) and by Førsund and Kittelsen (1992).
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Public Goods, Environmental Externalities and Fiscal Competition
i.e., about U.S. $1,500) and of local character (i.e., arising within a welldefined geographical area called ‘‘canton’’). At the time our data were compiled (1983–85), the country was served by 187 such judges. As to further labor input, each judge is assisted by one or several clerks, the number of them ranging from 1 to 7, depending upon the importance of the canton. Most JPs (about 125 of them) have three or four such clerks. As no statistical material is available on other inputs, we conduct the analysis for the labor input only, actually clerical labor since the number of judges in each canton is a constant. As to outputs, the competence rationae materiae, (i.e., in terms of the nature of the cases they are entitled to deal with) of the Juges de Paix is fairly wide ranging. Available statistical data report their activities in yearly figures of three broad categories, namely (i) The number of settled civil and commercial cases (denoted CC) (ii) The number of family arbitration sessions held (‘‘conseils de famille’’) (denoted FA) (iii) The number of minor offense cases settled (denoted MO) Table 5 provides the orders of magnitude of this activity. Table 6 provides further information as to the backlog of unsettled cases (BL-1), as well as on the number of cases newly introduced each year (CI). Both bear on CC cases only, as these are the only ones for which a backlog phenomenon occurs. The importance of the backlog problem can be summarily appreciated from the last line of Table 6 where the total backlog is seen to be increasing over the three years under review; from
Table 5.
Summary statistics on outputs. 1983
C.C. 1984
1985
1983
F.A. 1984
1985
1983
M.O. 1984
1985
Min. Median Mean Max. Total
172 1,070 1,162 3,796 216,898
197 1,068 1,141 4,103 212,283
199 1,039 1,131 4,095 211,592
0 83 85 241 16,037
4 87 85 228 15,901
1 85 85 237 15,985
115 506 568 1,413 42,446
103 525 570 1,443 42,744
102 512 566 1,176 42,571
Table 6.
Summary statistics on backlog and new cases.
Min. Median Mean Max. Total
1983
BL-1 1984
1985
1983
CI 1984
1985
12 686 890 8,873 167,648
24 732 949 8,738 175,635
12 773 1,008 9,201 188,936
156 1,128 1,215 3,640 225,390
185 1,120 1,200 4,566 223,133
181 1,088 1,178 4,013 280,388
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues Table 7.
331
Summary of results of FDH efficiency measures for years 1983–1985.
Clerks (units) (1)
J.P. (units) (2)
1 2 3 4 5 6 7 Total
6 29 72 56 20 1 3 187
1983 Ineff. (units) (3)/(2) (%) (3) (4) 3 26 66 49 16 0 1 161
J.P. (units) (5)
50 90 92 87 80 0 33 86
6 29 72 56 20 1 3 187
1984 Ineff. (units) (6)/(5) (%) (6) (7) 3 26 65 50 16 0 2 162
50 90 90 89 80 0 66 87
J.P. (units) (8)
1985 Ineff. (units) (9)
(9)/(8) (units) (10)
6 25 78 56 19 1 2 187
2 18 67 50 17 0 0 153
66 72 86 89 85 0 0 82
the rest of the table, it also appears to represent more than half of the new cases introduced, and even more than double in the largest JPs. 4.2
FDH efficiency results
Number of JP’s
Table 7 presents summary results of applying FDH efficiency measurement, for each one of the three years separately, to the data just described. The results are presented in successive classes that correspond to increasing sizes of the clerical labor force in the various JPs. Figure 9 shows the frequency distribution of the efficiency degrees obtained for year 1985 only, but the distributions for 1983 and 1984 have approximately the same shape. Let us highlight two characteristics of the contents of the table. First, one reads from the bottom line that the percentage of inefficient units is in the range of 82% to 87%. This is a rather high proportion, especially in comparison with the FDH results obtained in the previous application, where that proportion was between 18% and 40%. Recalling the characteristically more ‘‘generous’’ nature of this methodology, in comparison with DEA, this high proportion is indeed surprising. Second,
40 35 30 25 20 15 10 5 0 .15 .20
.30
.40
.50 .60 .70 Efficiency degree
Figure 9. Frequency distribution of efficiency degrees.
.80
.90
1
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Public Goods, Environmental Externalities and Fiscal Competition
one observes in the table that the most frequent cases of inefficiency occur in the middle sized JPs (as measured by clerical staff). Considering now Figure 9, the frequency distribution of the efficiency degrees exhibits a strikingly different shape than the one such distributions had in the previous application. Indeed, while we found such distributions there to be typically decreasing, of the exponential type, in the present case the shape is bimodal, with a normal shape around the mode of low efficiency (.60–.65) and an exponential decreasing shape around the other mode. From these various observations we derive essentially two conclusions. On the one hand, the weakness of the criterion that inspires FDH calculations justifies that our results be a matter of concern for whoever cares about productivity and/or efficiency. A serious problem seems to be present here in this respect, in terms of the judges’ work organization, as well as motivation, given that almost no agency is monitoring their work. On the other hand, this is a case where even the free disposal hull lies away from a large proportion of the data. Why is that? Part of the answer might well be in a much stronger heterogeneity of the cases than what is assumed in the simple aggregation in three categories. And even among cases of the same nature, some can be much more difficult to handle than others. Heterogeneity might thus explain the wide dispersion of the output data for similarly staffed JPs. Unquestionably judges themselves would invoke this argument. However, let us mention as a reply that they could not use it all the time. In other words, if the same JP was found repeatedly dominated by one or several of its peers, it would come first in line for an examination of the characteristics of its activity, so as to find out whether the evaluation is due to lack of productivity or to a truly different kind of activity. 4.3
Inefficiency and the judicial backlog
The fact that figures are available on the number of unsettled cases pending before each JP at the end of each year allows one to connect them with the efficiency results in the following simple way. Consider a JP inefficient (i.e., dominated) at time t, denoted as k, whose inefficiency degree is Etk , and whose backlog is BLkt . Let d (k) denote the JP that most dominates it, with CC d (k) the number of CC cases handled by this dominating JP. If JP k had behaved as its peer d (k), it could have reduced its backlog– assuming homogeneity of the cases–by an amount at least equal to Xtk ¼ (1 Etk ) CC d (k):
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
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However, for this to be possible for the JP k, its backlog at time t must of course reach at least Xtk . Hence, we can define for the JP k the backlog reducible by productivity increase as RBLkt ¼ Min[Xtk , BLkt ], and the complement to the total backlog, NRBLkt ¼ BLkt RBLkt as the backlog reducible only by an increase in personnel, given the peers’ best practice. Table 8 provides aggregate estimates of the so defined reducible backlogs for all JPs, ranked according to the size of their current clerical staff. In total, about 35% of the prevailing backlog is claimed to be reducible by productivity increases that would match best peers’ practice. Naturally, given our previous findings, this percentage is largest with the middle sized JPs, where it reaches 40%. Still on the basis of best peer practice, about 70% of the backlog could not be so reduced. Personnel increases thus seem to be justified, all the more that this figure of course includes the backlog of the efficient JPs, which by construction is not reducible in the sense we have retained. Without going into further details, this brief account should prove sufficient to see how efficiency analysis can serve, in this case, as a way to evaluate to what extent the backlog figures do justify17 the repeated claims for increases in personnel in this part of Belgium’s judiciary.
Table 8.
Summary results of ‘‘reducible backlog’’ computations.
Pers
RBL (1) units
1983 BL (2) units
(1)/(2) (3) %
RBL (4) units
1984 BL (5) units
(4)/(5) (6) %
RBL (7) units
1985 BL (8) units
(7)/(8) (9) %
1 2 3 4 5 6 7 Total
476 8,669 26,851 23,821 7,095 0 380 67,292
2902 21,690 58,422 58,952 29,350 2,510 2,182 176,008
0.16 0.40 0.46 0.40 0.25 0.00 0.17 0.38
486 8,803 24,916 21,432 7,342 0.00 997 63,976
2,981 22,674 61,291 62,705 31,919 2,641 2,644 186,855
0.16 0.39 0.41 0.34 0.23 0.00 0.38 0.34
407 3,336 29,436 29,148 8,375 0 0 70,702
3,371 15,723 75,663 66,808 31,477 2,828 1,863 197,733
0.12 0.21 0.39 0.44 0.27 0.00 0.00 0.36
17 A referee has observed that the backlog may serve a substantive role, by reducing demand for the use of the courts since it increases the price of the service they provide. Alternatively, delays may be seen as a decrease in quality.
334
5.
Public Goods, Environmental Externalities and Fiscal Competition
Urban transit: efficiency over time and shifts in the production frontier18
The specific interest of this third application stems from the fact that it deals with time series data of a single firm, in contrast with the cross-sectional nature of the preceding studies. New and interesting problems do indeed arise with the nonparametric methodology19 when the time dimension is recognized. 5.1
Sequential FDH measurement
Letting Y0 ¼ {(xt , yt )jt ¼ 1, . . . ,T} denote an input (x)–output (y) data set extending over T periods, straightforward applications to such a set of any of the nonparametric techniques explained in Section 2 rest on the implicit assumption of an unchanging production set throughout the T periods—a severe assumption indeed when T is large. In Tulkens (1986) and Thiry and Tulkens (1988), this assumption is avoided by introducing the notion of ‘‘sequential free disposal hulls,’’ that is, a time sequence of reference production sets successively constructed at each time t from the observations made thus far. Formally, recalling the terminology of Section 2, at each time t: (i) A specific reference set Y 1,t is constructed from the following subset of the data: Y 1,t ¼ {(xt , yt )j1 # t # t}; and (ii) The efficiency degree of the observation (xt , yt ) is computed by referring to the boundary of this constructed set Y 1,t . Thus, the efficiency of each successive observation is determined by referring only to those who are preceding in time–hence the expression of ‘‘sequential’’ FDH measurement. 5.2
Measuring both efficiency and local shifts in the production frontier
For increasing values of the time index t, the successively constructed production sets necessarily contain each other, i.e., g01,t g01, t , 18
t $ t,
Much of the work reported on here was initiated in collaboration with Bernard Thiry during the preparation of Thiry and Tulkens (1988, 1992). Earlier (parametric) work on this firm was done by d’Aspremont (1984). 19 Parametric efficiency measures bearing on the same data set are reported, and compared with FDH results, in Chapter 3 (Section IV) of Thiry and Tulkens (1988). Chapter 2 of that book contains similar comparisons for two other transit companies.
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
335
and a ‘‘larger’’ production set also necessarily has a more (North-West) outward boundary—at least locally—than a smaller one. Such outward shifts are of the nature of technical progress. As they involve a distance between the new frontier and the previous one, one may think of measuring this distance in a way similar to the one used to measure inefficiency. Within the FDH conceptual framework—that is, in terms of the domination relation between the observations—this is easily done as follows. Consider an observation at some time t that is sequentially efficient in the sense just defined. Among the previous observations that this observation dominates, there may exist some which were themselves declared efficient, because of being undominated at the time they were made (see for instance, in a one input–one output setting, observation t ¼ 8 on Figure 10, which dominates observations t ¼ 3, and also t ¼ 6. When an observation has this double property of (1 8) being sequentially efficient and (2 8) of dominating observations previously declared efficient, one may logically conclude that the boundary of the production set has (locally) shifted outwards, in other words, that ‘‘local’’ technical progress—call it FDH technical progress—has occurred.20 This is illustrated by the cross-hatched area on Figure 10.
y
t=7
t=8
a
YFDH at time t = 3
t=3
b
t=6
t=5 t=4
t=2 t=1
0
c
d
x
Figure 10. Local technical progress with FDH sequential production sets. 20
Professor Ricottilli has pointed out to me that Atkinson and Stiglitz (1969) have proposed a similar notion in a framework of the DEA-CD type.
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Public Goods, Environmental Externalities and Fiscal Competition
A measure of the degree of FDH technical progress can be formulated in output terms and in input terms, just as it is the case for the degree of efficiency. In output terms, it can be defined as the ratio of the output achieved at time t to the output achieved by the previous observation with the smallest input level among those that are dominated and efficient (see the vertical distance indicated below observation t ¼ 8). In input terms a similar ratio is suggested by the horizontal distance indicated to the right of observation t ¼ 8. Both ways, the ratio is by construction larger than 1, thus indicating a distance away from the previous boundary.21 5.3
Urban transit data
Our data are monthly observations on quantities of output produced and quantities of inputs used by the Socie´te´ de Transports Intercommunaux de Bruxelles (STIB). The company operates buses, streetcars and a subway. The observation period covers almost 12 years (from January 1977 to September 1989). We thus have 153 observations. Output, denoted by Y, is measured by the number of seats-km monthly supplied. This is a standard measure of the output22 of mass transit firms. This output is considered to be produced by means of three inputs: labor, denoted by H, measured by the total number of hours worked; energy, denoted by E, measured in number of kWh; and vehicles, denoted by PV, measured by the number of seats-vehicles. Our data on output and inputs are graphically shown on the four charts of Figures 11a–d. 5.4
Results of sequential FDH measurement of efficiency and of local technical progress for STIB (1977–1989)
Results of FDH efficiency and technical progress computations in output terms are exhibited graphically on Figure 12. All efficiency and technical progress degrees are multiplied by 100. We note that: (i) In general, the sequential method implies by construction that at least the first observation is always efficient, and the immediately following ones have a high probability of being so too, because of the paucity of the observations set in early periods. A similar argument can be made also as to (FDH) technical progress when it is observed at early times: it may in fact be only efficiency. 21
In more than two-dimensional input or output cases, this ratio is to be defined in a radial way, as was mentioned in Section 2. 22 Passengers-km are often thought of as a more appropriate measure, but they are in fact an expression of the demand rather than the output, with the discrepancy between seats-kms and passengers-km being then to be interpreted as a measure of effectiveness rather than efficiency in the specific sense used here.
Ch.14 On FDH Efficiency Analysis: Some Methodological Issues
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600 500 400 300 200 100 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Figure 11a. STIB: output data (monthly seats-km 1,000,000). 1000 900 800 700 600 500 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Figure 11b. STIB: labor input (monthly hours ( 1000) worked). 500 400 300 200 100 0 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Figure 11c. STIB: energy input (monthly kWh ( 100,000) used). 150
125
100 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Figure 11d. STIB: vehicle input (number ( 1000) of available seats per month).
(ii) 61 observations are sequentially inefficient, 73 are efficient, and 19 exhibit technical progress. The inefficiencies occur during the first half of the observation period, reaching a bottom level of 81 in February 1981. On the other hand, technical progress appears occasionally in 1984–85–86, with no more inefficiency; progress occurs even systematically, and in increasing amounts, throughout 1988 and 1989 (the
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120
110
100
90
80 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
Figure 12. STIB: efficiency degrees and technical progress, 1977–1989 (output measure).
early case in 1977 is negligible in view of the argument made in (i)). The maximum level of technical progress achieved is 111.7 in the month of November 1988. Beyond a history of the productive performance of the firm that these results provide, and beyond the obvious question they raise as to which factors explain its ups and downs (which have not really been elucidated as yet), I should like to stress the potential interest of the sequential approach from a managerial point of view: the efficiency and progress degrees provide indeed each month a single index evaluation of the current activity, resulting from a comparison with the past one, and based on physical data. The domination criterion which is exclusively used also allows one to precisely identify which periods were better ones, in case of inefficiency, and which periods are improved upon in case of progress. However, the technique presented here characteristically cannot identify technical regress, as this phenomenon would be assimilated with inefficiency. This is a severe limitation of the domination criterion (at least in the way it is used here), which does not contain a way to distinguish between the two on the sole basis of input and output data. Recent research on this issue23 suggests that this difficulty may be circumvented, but at the cost of a rather radical reconsideration of the concepts in which efficiency measurement methods are rooted.
6.
Conclusion
To evaluate the FDH method of efficiency measurement, vis-a`-vis its nonparametric competitors DEA-V, DEA-CD and DEA-FARRELL, 23
Reported in Tulkens and Vanden Eeckaut (1991).
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let us consider four possible viewpoints: (i) theoretical; (ii) empirical; (iii) managerial; and (iv) computational. In this comparison we leave aside the parametric approaches (except for a brief albeit important remark below) because they have not been dealt with in this paper. They would indeed deserve a separate and systematic treatment. As far as theory is concerned, it seems natural to evaluate a methodology in terms of the strength (or for that matter, the weakness) of the postulates on which it rests. In this respect, the increasing strength of the postulates listed in Section 2.1 clearly shows that FDH rests on the most parsimonious ones, among the four methodologies here reviewed. Perhaps still more parsimonious postulates could be imagined: but I am not aware of any such, currently existing in the literature, and allowing one to construct frontiers appropriate for efficiency measurement. From the point of view of empirical analysis, goodness of fit (to the data) of the frontier model used to measure efficiency seems to be a desirable virtue. Here, the nested character (as illustrated on Figures 1– 3) of the four alternative frontiers whereby data are enveloped in each of the four methods, does imply that the FDH reference set always fares best: its frontier always lies at least as close (and often lies much closer) to the data than any of the other three frontiers. As a result, efficiency scores are of course always higher with FDH than with all other three reference technologies—a fact perfectly in line with the far reaching observation made recently by Fa¨re and Grosskopf (1991) according to which efficiency measures are actually nothing else than ‘‘goodness of fit’’ measures of the chosen reference technology. Concerning managerial relevance, i.e., relevance for decision making, our judgment must differ according to whether the results generated by the alternative methods are considered for predictive versus evaluative uses. By ‘‘predictive’’ uses I mean uses relating to decisions concerning the future—thus requiring forecasts, whereas ‘‘evaluative’’ uses are meant to relate instead to decisions that somehow involve the past, the typical example of which being decisions to reward performance. As to the former, FDH frontiers are of almost no use because of their extremely conservative character. Indeed, whether the needed predictions require extrapolations or interpolations of the productive activity in areas of the input-output space where no activity has ever been observed, FDH frontiers provide only trivial answers (precisely due to the weakness of the postulates they rest on). For interpolations, frontiers of the DEA type are somewhat more informative, by suggesting convex combinations of observed realizations; for extrapolations however, such frontiers cannot do better than FDH, unless the additional assumption of unlimited proportionality is made, as in DEA-FARRELL as well as in one of the methods suggested by Petersen (1990). Overall, nonparametric methods
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are, however, rather poor ways of making extrapolations in general, and parametric methods probably are of irreplacable help in this respect. As to evaluative uses of frontier efficiency estimates, the situation is reversed: because of the tight closeness of the FDH frontier to the data, and because of the fact that another observation is singled out when computing the radial measures of inefficiency as defined in (5) and (6), individual units are always rated in reference to another one, belonging to the frontier. Thus, the comparison is always a realistic one, and this makes it more convincing than the DEA evaluations, whenever the latter rate the unit by referring to frontier points that are not observed ones (i.e., not vertices of the constructed convex polyhedron). Turning finally to the computational aspect, there is hardly more to say than highlight the simplicity of the computational requirements of the FDH efficiency measures: they only involve making pairwise comparisons between all observation vectors and then taking appropriate ratios between some of their components. This may be a matter of surprise, if not a source of frustration for some. But the connection established in Section 2 with the linear programs of DEA shows that FDH measures do solve just a mixed integer variant of programs with the same structure: the role of mathematical programming in this area is thus reinforced, albeit in terms that hardly call for the development of new algorithms. Instead, we have here an example of a mixed integer problem where a technique of complete enumeration works easily even with large data sets. Beyond a confirmation of these general properties, the three empirical illustrations presented in Sections 3–5 do provide each some specific additional information: a more clearcut differentiation with FDH than with DEA between public and private efficiency performance in multibranch retail banking; the effects of the sparsity of observations, due to heterogeneity of the outputs, in the case of courts; and finally, in the case of urban transit time series data, a new possibility of distinguishing nonparametrically between efficiency gains and technical progress.
References Aigner, D., Lovell, C.K. and Schmidt, P., 1977. Formulation and estimation of stochastic frontier production function models, Journal of Econometrics, 6, 21–37. Atkinson, A.B. and Stiglitz, J.E., 1969. A new view of technical change, Economic Journal, 79, 573–578. Banker, R., Charnes, A. and Cooper, W., 1984. Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30, 1078–1092. Charnes, A., Cooper, W., Golany, B., Seiford, L. and Stutz, J., 1985. Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions, Journal of Econometrics, 30, 91–107.
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Charnes, A., Cooper, W.W. and Rhodes, E., 1978. Measuring the efficiency of decision making units, European Journal of Operations Research, 2(6), 429–444. d’Aspremont, A., 1984. Une mesure ge´ne´rale de l’efficacite´ au sens de Farrell: application au cas de la STIB, The`se de doctorat, Institut d’Administration et de Gestion. Louvainla-Neuve: Universite´ catholique de Louvain. Deprins, D., Simar, L. and Tulkens, H., 1984. Measuring labor efficiency in post offices, in M. Marchand, P. Pestieau, and H. Tulkens (eds.). The performance of public enterprises: concepts and measurement. Amsterdam: North-Holland, 243–267. (Chapter 13 in this volume). Farrell, M.J., 1957. The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, 120(3), 253–281. Fa¨re, R. and Grosskopf, S., 1991. Non-parametric tests of regularity, Farrell efficiency and goodness of fit, Journal of Econometrics, 69, 415–425. Fa¨re, R., Grosskopf, S. and Lovell, C.A.K., 1985. The measurement of efficiency of production. Boston-Dordrecht: Kluwer-Nijhoff. Førsund, F. and Kittelsen,S., 1990. Efficiency analysis of Norwegian District Courts, Journal of Productivity Analysis, 3(3), 277–306. Grosskopf, S., 1986. The role of the reference technology in measuring productive efficiency, Economic Journal, 96, 499–513. Jamar, M.A. and Tulkens, H., 1990. Mesure de l’efficacite´ de l’activite´ des tribunaux et evaluation de l’Arrie´re´ Judiciaire, paper presented at the European Workshop on productivity and efficiency measurement in the service industries held at CORE, October 19–20, 1989. Available as Document de travail No. 90/1, Fonds de documentation statistique et e´conomique sur les services publics belges. Louvain-la-Neuve: CORE, Universite´ catholique de Louvain. Land, K.C., Lovell, C.A.K. and Thore, S., 1988. Chance-constrained efficiency analysis, paper presented at the National Science Foundation conference on parametric and non-parametric approaches to frontier analysis held at the University of Carolina at Chapel Hill, September 30–October 1, 1988; revised April 1990. Lewin, A., Morey, R. and Cook, T.J., 1982. Evaluating the administrative efficiency of courts, OMEGA, 10 (4), 401–411. MacFadden, D., 1978. Cost, revenue and profit functions, in M. Fuss and D. MacFadden (eds.). Production economics: a dual approach to theory and applications. Amsterdam: North-Holland, 3–110. Marchand, M., Pestieau, P. and Tulkens, H., 1984. The performance of public enterprises: normative, positive and empirical issues, in M. Marchand, P. Pestieau, and H. Tulkens (eds.). The performance of public enterprises: concepts and measurement. Amsterdam: North-Holland, 3–42. Meeusen, W. and Van Den Broeck, J., 1977. Efficiency estimation from Cobb-Douglas production functions with composed error, International Economic Review, 18(2), 435– 444. Olesen, O. and Petersen, N.C., 1989. Chance constrained efficiency evaluation. Publication No. 9/1989, Odense: Department of Management, Odense University. Petersen, N.C., 1990. Data envelopment analysis on a relaxed set of assumptions, Management Science, 36, 305–314. Respaut, B., 1989. Mesures de l’efficacite´ productive des 911 agences d’une Banque Prive´e Belge, in H. Tulkens (ed.). Efficacite´ et management. Travaux et Actes de la Cinquie`me Commission du Huitie`me Congre`s des Economists Belges de Langue franc¸aise, Charleroi, Belgium: Centre Interuniversitaire de Formation Permanente (CIFOP), 53–71. Thiry, B. and Tulkens, H. (eds.), 1988. La performance economique des Socie´te´s Belges de Transports Urbains. Lie`ge, Belgium: Centre International de Recherche et d’Information sur l’E´conomie Collective (CIRIEC).
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Thiry, B. and Tulkens, H., 1989. Productivity, efficiency and technical progress: concepts and measurement, Annals of Public and Cooperative Economics/Annales de l’Economie Publique Sociale et Coope´rative, 60(1), 9–42. Thiry B. and Tulkens, H., 1992. Allowing for inefficiency in parametric estimates of production functions for urban transit firms, Journal of Productivity Analysis, 3(1/2), 45–66. Tulkens, H., 1986. La performance productive d’un service public. De´finitions, me´thodes de mesure et application a` la re´gie des postes en Belgique, L’Actualite´ Economique, Revue d’Analyse Economique, (Montre´al), 62(2), 306–335. Tulkens, H. (ed.), 1989. Efficacite´ et management. Travaux et Actes de la Cinquie`me Commission du Huitie`me Congre`s des Economistes Belges de Langue franc¸aise. Charleroi, Belgium: Centre Interuniversitaire de Formation Permanente (CIFOP). Tulkens, H. and Vanden Eeckaut, P., 1989. Productive efficiency measurement in retail banking in Belgium, paper presented at the European Workshop on Productivity and Efficiency Measurement in the Service Industries held at CORE, October 19–20. Louvain-la-Neuve: Universite´ catholique de Louvain. Tulkens, H. and Vanden Eeckaut, P., 1991. Non-frontier measures of efficiency, progress and regress, paper presented at the Second European Workshop on Efficiency and Productivity Measurement held at CORE, October 25–26, 1991, CORE Discussion Paper No. 9155 . Louvain-la-Neuve: Center for Operations Research and Econometrics, Universite´ catholique de Louvain. (Chapter 16 in this volume). Vanden Eeckaut, P., Tulkens, H. and Jamar, M.A., 1993. Cost-efficiency in Belgian municipalities, in H. Fried, C.A.K. Lovell, and S. Schmidt (eds.). The measurement of productive efficiency: techniques and applications. New York: Oxford University Press, 300–334.
Chapter 15 ASSESSING AND EXPLAINING THE PERFORMANCE OF PUBLIC ENTERPRISES: SOME RECENT EVIDENCE FROM THE PRODUCTIVE EFFICIENCY VIEWPOINT
Pierre Pestieau, Henry Tulkens Finanzarchiv Neue Folge 50 (3), 293–323 (1993).
From a general discussion of the notion of economic performance as applied to public enterprises, we first observe that productive efficiency is the least ambiguous and most generally desirable form of performance for such firms. We then discuss the type of relation expected to hold between a firm’s productive efficiency on the one hand and on the other hand characteristics pertaining to ownership, objectives, regulation, and competition. Next we review a number of empirical efficiency studies which are aimed at testing the effect of those characteristics on firms’ productive efficiency. In conclusion, we offer some evaluative remarks on the public policy implications of efficiency measurement in public sector economics.
1.
Introduction
In February 1990, the European edition of the Wall Street Journal devoted some of its first page columns to analyzing the case of ‘‘inefficient Belgium’’. The paper’s harsh statements on ‘‘public services creak(ing) with inefficiency’’ or on ‘‘a state-owned telecommunications service that The authors wish to express their special thanks to C. Knox Lovell, Maurice Marchand, Sergio Perelman and Philippe Vanden Eeckaut for valuable discussions and/or research assistance. This paper is part of a research program supported in part by the Belgian Fonds de la Recherche Fondamentale Collective (under conventions n8 2.4528.88 and 2.4505.91 with the authors) and for another part by the Belgian Ministry of Scientific Policy (under its Programmes d’Actions de Recherche Concerte´es n8 87–92/106 with CORE, Universite´ Catholique de Louvain and n8 90–94/141 with Universite´ de Lie`ge). Both authors also acknowledge the sabbatical support of CIM, the Colle`ge Interuniversitaire d’e´tudes doctorales dans les sciences du Management, Bruxelles.
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sometimes sinks to near Third World standards’’ were vividly commented on locally. However, as we observed the reactions to this incident (which after all could have occurred in a number of other European countries) we noted that once more, as is so often the case when public sector performance is questioned, what was exactly meant by performance and how it was measured did not seem to matter in any way. The purpose of this paper is precisely to provide a definition and a way to measure the performance of a public enterprise or service, explicitly rooted in the principles of production economics. Why do we need such a measure? Is it just to keep the record of the public sector straight? In fact, performance indicators are widely used by economists in both the public and the private sector, essentially, we think, because of a big concern for the factors that are contributing to performance. Among those factors, one finds for instance the nature of ownership, the firms’ objectives, the regulatory setting, or the market structure. Witness of that concern is the critical way in which the U.K. privatisation program is currently being assessed by some economists, focusing on the role of competition rather than ownership per se1; there also exists a large number of studies on the effect of organizational status, ownership, competition and deregulation on the performance of firms and services. Generally, performance is assessed in terms of factor productivity growth, employment functions, prices, and standard accounting ratios. Typical of this approach is the widely quoted paper by Yarrow (1986) who discusses the relative contributions of ownership and competition on the basis of indicators such as real turnover, profit margins, price increases, and share prices. Those indicators have the merit to be readily available but what they exactly reflect remains ambiguous, as is increasingly acknowledged2, because they say little about the objectives of the firms being evaluated or compared, and because they don’t allow for controlling for the effects of the market structure environment in which production is taking place. In this paper we precisely want to advocate the use of an indicator of performance, developed over the course of the last two decades or so, that we see to be free of many of the drawbacks just alluded to. This indicator essentially reflects the firms’ capacity to achieve productive efficiency, a concept originally due to Koopmans (1951) and Debreu (1951), and close to those of internal efficiency (Vickers and Yarrow (1990)) and X-efficiency (Leibenstein (1966), Leibenstein and Maital (1992)). In the next section, we present what we call the ‘‘performance approach’’ to public enterprises, focusing on the objectives assigned to 1 2
Kay and Thomson (1986), Hartley, Parker and Martin (1991). See on this Barrow and Wagstaff (1989) as well as Birch and Maynard (1986).
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firms. In Section 3, we discuss the pluses and minuses of the productive efficiency approach, and contrast it with other indicators developed to date. Section 4 discusses the theoretical links between production efficiency on the one hand and various characteristics of the firm on the other hand: ownership, objectives, competition and regulation. Section 5 reviews the evidence existing nowadays on these issues. The final Section 6 presents some conclusions regarding the merits of the productive efficiency approach and its expected contribution to public policy.
2.
Defining economic performance for public enterprises3
One of the main forms of performance studies is by comparison: comparisons over time, across space, international or interindustry comparison, comparison of firms having different ownership regimes or legal status. The gist of this paper is that, when based on the usual measures of performance, the comparisons are often not conceptually valid, for a series of reasons that we shall develop presently; by contrast, the productive efficiency criterion appears to be immune of these critiques, and therefore probably is a more reliable one for appreciating and comparing performance, at least in the case of public enterprises. To assess the performance of a firm, or more generally of any production unit, one first has to recognize the ‘‘principal-agent’’ relation that links the production manager with the authority ultimately concerned by his/her achievements. Performance is then defined by the extent to which the agent fulfills the objectives assigned to him/her by the principal. The simplest performance measure in this sense is undoubtedly the profit level of the neoclassical textbook private firm. Here, as the sole objective of the owners – the principal – is assumed to be the maximization of their revenue, performance in the sense just defined is easily and effectively measured by the profit level achieved by the firm’s manager, i.e. the agent. In the case of public sector activities, if the welfare economics view is followed, suggesting the State (taken as representing Society as a whole) to be the principal, the objectives appear to be multifold because of the many aspects of social welfare4. As a result the missions assigned to public managers are also multifold, and the performance assessing issue becomes a more complex one. Performance in nonprofit firms raises the same difficulty. 3
This approach was developed in Marchand, Pestieau and Tulkens (1984); see also Tulkens (1986). 4 As they were couched in a by now well-established taxonomy proposed by Rees (1976, chapter 1, and 1984, chapter 2).
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As a matter of fact, some have argued that even private firms are to be evaluated from multidimensional viewpoints, involving other objectives than those of their owners only. Thus Scherer (1980) takes the viewpoint of social welfare to define what he calls ‘‘good performance’’ of firms. He notes that the concept embodies at least the four following components: 1. ‘‘Decisions as to what, how much, and how to produce should be efficient in two respects: scarce resources should not be wasted outright, and production decisions should be responsive qualitatively and quantitatively to consumer demands. 2. The operations of producers should be progressive, taking advantage of opportunities opened up by science and technology to increase output per unit of input and to provide consumers with superior new products, in both ways contributing to the long-run growth of real income per capita. 3. The operations of producers should facilitate stable full employment of resources, especially human resources. Or at minimum, they should not make maintenance of full employment through the use of macroeconomic policy instruments excessively difficult. 4. The distribution of income should be equitable. Equity is a notoriously slippery concept, but it implies at least that producers do not secure rewards far in excess of what is needed to call forth the amount of services supplied. A subfacet of this goal is the desire to achieve reasonable price stability, for rampant inflation distorts the distribution of income in ways widely disapproved’’ (Scherer, 1980, p. 3). The first objective is allocative and comprises both technical and price efficiency5. The second objective is one of growth and the third concerns macroeconomic considerations. The fourth objective is one of equity6. Multidimensional objectives thus appear to be the rule, in assessing firms’ performance in general, rather than an exception specific to public enterprises. Beyond the recognition of this fact, a number of difficulties arise that we now briefly deal with, each in turn. First, Scherer notwithstanding, the need for a performance measure based on the achievement of multiple objectives is more undisputable for public enterprises (or even nonprofit organizations) than it is in the case of for-profit enterprises. Typically, the measure must be one from the viewpoint of society as whole, whereas the sole viewpoint of the owners has its justifications for private firms. 5
In the rest of this paper, we use indifferently ‘‘technical’’ or ‘‘productive’’ efficiency; we also use as synonyms the terms ‘‘allocative’’ and ‘‘price’’ efficiency. 6 It is interesting to note that the above list rather closely parallels the taxonomy of objectives that Rees proposes for public enterprises.
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Second, these goals may not always be completely compatible with one another. One knows, for example, that peak-load pricing in services such as transport and communication is desirable from an allocative viewpoint but often distributively objectionable. Thus, when assessing the overall performance of a service, a delicate balance has to be struck between these two criteria. Third, measuring the degree to which those objectives are satisfied is quite a difficult task because it involves computing first an indicator of partial performance for each of them, and then proceeding to some weighing of those indicators. This cannot be resolved without involving basic value judgements. Fourth, the only objective the achievement of which does not impede that of the others is technical efficiency. Producing too little or employing too many factors as compared to what is technically feasible cannot be justified in terms of any of the other objectives listed above. By contrast, hiring labor in (too large) quantities that are allocatively inefficient can be legitimated by macroeconomic considerations of employment policy. Fifth, the trade-offs between allocative and non-allocative objectives can have effects on the controllability of firms. Indeed, a firm can be allocatively inefficient for two reasons, the first being that it has to fulfill non-allocative goals and the second being that the manager pursues objectives of his/her own (e.g., the three P’s, power, prestige and pay). The difficulty then arises in sorting out these two sources of inefficiency, and this has often been presented as an argument in favour of privatization and deregulation. In a competitive setting, a private firm is indeed supposed to be efficient both technically and allocatively. Given the above, we advocate in this paper that the performance of public enterprises be measured and compared on the basis of productive efficiency only. In summary, our reasons are twofold: on the one hand, the global performance evaluation problem, i.e. measuring how close a public firm comes to achieving all the objectives just listed is, in our opinion, too ambitious. Both data and techniques of analysis currently available make such an undertaking unachievable. On the other hand, as productive efficiency allows for evaluations that are consistent with the manifold objectives of the firms at stake, it does definitely constitute a step in the right direction.
3.
The productive efficiency approach
3.1.
The frontier method of measuring productive efficiency
Over the course of the last two decades, a number of methods for measuring productive efficiency have been proposed. They all have in
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common the frontier concept: efficient firms are those operating ‘‘on’’ the frontier of their respective production set, while inefficient firms operate ‘‘below’’ that frontier (i.e. in the interior of their production set). The methods also have in common that they all rely on inference. Indeed, statistical data that report on the outputs achieved and the inputs used by a productive organization do not provide per se the production set, no more than they yield the production frontier; both must be inferred, i.e. constructed from the data, prior that any efficiency computation can be made. Hence the happy expression of ‘‘best-practice frontier’’ whereby Farrell (1957) originally designated the production frontier that he was first to derive from statistical data. To estimate best practice frontiers two main alternative methodologies are available: parametric ones, and nonparametric. The difference lies in the technique used to formally describe the frontier. In the first case, a usual function with constant parameters – e.g. Cobb-Douglas, or translog – is specified a priori, and its parameters are estimated by statistical or other methods in such a way that the graph of the function best ‘‘envelops’’ the data ‘‘from above’’, that is, so that all observations appear to lie on or below7 this graph. Then, the efficiency of each observation is computed in terms of the distance that lies between the observation and the graph of the estimated function8, now considered to be the frontier of the production set. This distance is usually expressed in terms of the ratio between the achieved output and the output predicted by the function9. Notice that the efficiency measure so obtained may differ according to which functional form is specified a priori. Usage of this parametric method is mostly found in the econometric literature10. In the second case, the nonparametric methodology, what is specified a priori is not an explicit function but rather some formal properties that the points in the production set are assumed to satisfy: e.g. free disposal, convexity (implying either nonincreasing returns to scale for the frontier, or some form of variable returns) or proportionality (implying constant returns). Data are then ‘‘enveloped’’, too, however not by the graph of a function whose parameters are estimated, but instead by determining whether each observed point can be considered as being an element of the frontier or not, under the chosen assumption(s). This is 7
Instead of being scattered around the function, as in standard econometric estimation. As a result, residuals are all of the same sign, and the estimation technique is sometimes called that of ‘‘one sided’’ residuals. 8 That is, the residual associated with the observation. 9 Thus a strictly positive number less than or equal to 1, depending upon whether the observation is inefficient or efficient. 10 A classic survey on parametric production frontiers estimation is the one by Førsund, Lovell and Schmidt (1980). A more recent one is Bauer (1990).
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done by solving an appropriately defined system of linear equations – one such system for each observation. The same system can then serve to associate with each observation a numerical efficiency score, that measures again the distance lying between the observation and the frontier11. Here as above, the efficiency measure obtained for each observation may differ according to which formal assumptions are a priori specified on the production set. Usage of nonparametric methodologies mostly prevails in the management science and operational research literature12. For both categories of methods, parametric and nonparametric, the data set can indifferently be either a cross-section of several productive units, or a time series of observations of the same unit. In the first case, direct application of any one of the above methods implicitly assumes that for all units the production set is the same; in the second case, it assumes that the production set remains unchanged over time. Either one of these assumptions can be relaxed, provided information other than just input and output quantities can be included in the analysis, such as extraneous characteristics specific to some subsets of cross-sectional observations, or time-related such characteristics in the case of time-series observations. When time is involved, considerations relating to technical progress (or regress) are particularly relevant, because there arises the issue of sorting out between efficiency gains (movements towards the frontier) and progress (frontier shifts). Both parametric and nonparametric methods have recently offered extensions in this direction. As both classes of methodologies do operate on exactly the same data base, viz. input and output quantities, a given data set can always be subjected to both parametric and nonparametric efficiency measurements (and furthermore, within each class, to different functional specifications or set theoretic assumptions). The respective virtues of one or the other approach are therefore to be evaluated not so much on the basis of the nature of the data, but rather in terms of the answers each one provides the analyst given what he is seeking for. At the present stage of their respective development, an important difference between the parametric and the nonparametric methods must be mentioned, namely that allowance for random components in the data can be made with the former while it is ignored in the latter, making them 11
More precisely, it is measured (by means of linear programming) as the maximum possible equiproportionate reduction in all inputs, given the outputs, or the maximum possible equiproportionate expansion in all outputs, given the inputs, that still allow to be on the production frontier (these are thus radial measures). 12 A recent introductory survey on methods assuming convexity (called DEA for ‘‘data envelopment analysis’’) is given by Banker, Charnes, Cooper, Swarts and Thomas (1989); a variant that operates without convexity and labelled FDH is presented in Tulkens (1986, 1993).
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completely deterministic. Among other implications, this serious drawback makes nonparametric methods more sensitive to outliers. 3.2.
Pros and cons
We now turn to the merits of productive efficiency-based indicators of performance vis-a`-vis other ones. First, as was pointed out in Section 2, productive efficiency does not prejudge whether and how other objectives are fulfilled. Second, the production efficiency approach can be applied not only where public and private activities can be compared but also where public enterprises do operate in isolation, with no comparable or competing activity. Indeed, the only prerequisite for measuring productive efficiency is the possibility of constructing a production frontier. Note in this respect that most traditional (i.e. not productive efficiency oriented) empirical studies of performance of public enterprises concern only areas where there are both public and private firms engaged in similar activities. As these areas are not very representative of the bulk of the public sector, their findings can hardly be generalized. A third advantage of the productive efficiency viewpoint is that it relies on physical data which is readily available in many instances and basically more reliable than financial or accounting data. Fourth, unlike most partial indicators of performance, productive efficiency can encompass a large number of inputs and outputs including qualitative aspects without having to go through disputable aggregation. Finally, the concept of productive efficiency being both intuitive and unambiguous, its measure usually meets a wide consensus. Admittedly, production efficiency is only a partial indicator of performance. However, compared to traditional indicators such as price, profitability or productivity, it is by far more robust to application involving firms operating in changing market structure or other aspects of the production setting. Partial productivity can be misleading within a multi-output/multi-input process. Price differentials are quite imperfect indicators of slacks in allocative efficiency. Finally, one knows that efficiency can imply low profitability and in case of decreasing costs, it is even compatible with deficits. The idea that technical efficiency is achievable independently of the other objectives assigned to the firm and particularly of allocative efficiency has been challenged on various counts. We now turn to three of the most usual objections which pertain to the term of adjustment, to the institutional setting and to the idea of entrepreneurial motivations. The ‘‘short term’’ objection is quite intuitive and goes as follows: a firm can be constrained to be technically inefficient if it is forced to employ too
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much of a factor of production or to supply too little of a service without being able to quickly adjust its input-output vector so as to remain on the efficiency frontier. Take the example of a public railway company, the demand of which just fell by a lot, and assume that it is not allowed to lay off any of its employees. It is obvious that in the short run, such a firm is doomed to be technically inefficient; however, after some time, it should be able to reduce its idle labor force through attrition or to use the idle workers to bring up the quality or increase the variety of its services. The second objection has been forcefully raised in works on privatization. The point is made that privatizing natural monopoly amounts to prefer the unavoidable allocative inefficiency of the private firm to the often observed technical inefficiency of the public firm. Without questioning this line of argument, which clearly calls for empirical backing, let us just note that our own argument is developed for a given institutional setting. In other words, we state that a given firm, whatever its legal status or its ownership regime, can improve its technical efficiency without conflict with its other objectives13. Finally, there is Stigler’s classic argument on the inexistence of Xinefficiency, a concept closely connected to that of technical inefficiency. Stigler (1976) notes that there are only two sources of technical inefficiency defined ‘‘as the situation in which foregone products could be obtained for less than they cost’’. It can ‘‘arise ex post because ex ante plans rested upon erroneous predictions.’’ It can also arise because the producer is not really ‘‘engaged in a maximizing behavior’’. Stigler admits that the first type is unavoidable. As to the second, he does not find it a useful concept as it rests on a nonmaximizing assumption, at odds with standard economic theory. It remains that modern agency theory as well as empirical evidence show that managers maximizing their own objective function can imply slacks in production, particularly when both their position and their actions are sheltered by all sorts of regulation and lack of competition.
4.
The expected impact of ownership, competition and regulation on productive efficiency
4.1.
Efficiency and ownership
When comparing public and private firms, it is important to specify whether or not the institutional and market structure settings are kept constant. There are indeed two types of classification, one according to ownership and one according to the conditions of competition and 13
See, on this, Peters (1985).
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regulation. In Table 1, we present a taxonomy of firms according to these two dimensions; it yields six categories according to which firms may be ranked and several possible ways to make comparisons between the public and the private sector. There is little doubt that expert opinion would rank competitive private firms as the most efficient, both allocatively and productively, and noncompetitive firms as least efficient. Beyond that there is no clear consensus as to, e.g., whether private monopolies are more efficient than public ones. There are two views on this. The first predicts no efficiency differentials on the basis of type of ownership. This is the view of Leibenstein (1966) who argues that monopolies, public or private, are likely to be equally X-inefficient. This view is forcefully developed by Hicks (1935) when he talks of monopolies’ managers who ‘‘are likely to exploit their advantage much more by not bothering to get very near the position of maximum profit, than by storming themselves to get very close to it’’14. The only expected difference is in the way inefficiency is expressed: expensive plushy carpets in private corporations and bridge playing on the dole in public corporations. According to this view comparison between public and private enterprises becomes only relevant when considering the effect of regulation under private ownership or when contrasting public and private production in a competitive setting. The contrary view argues that all things being equal private firms tend to be more efficient than public firms on the grounds of both allocative and productive efficiency. This view is advocated by several strands of literature including the property rights literature, the public choice school and the theory of bureaucracy. The property rights view most commonly associated with the name of Alchian (1965) suggests that public ownership attenuates property rights. Public ownership is indeed diffused among all members of society and no member has the right to sell his/her shares15. There is thus little incentives to monitor the firm’s management and to ensure efficiency in the production of goods and services. Table 1 Ownership structure vs. alternative competitive and regulatory settings Ownership structure Public Private
14 15
No regulation Pure public monopoly Pure private monopoly
No competition Regulation Autonomous public monopoly Regulated private monopoly
Hicks (1935, p. 8) quoted by Lovell (1992). A view that was elaborated upon in Tulkens (1976).
Competition or contestability Public firm in competitive setting Private competitive firm
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A similar point is argued by public choice economists such as De Alesi (1974, p. 7) who writes that managers of public firms are less constrained by market considerations and ‘‘find it easier to obtain subsidy and to mask bad management under the guise of fulfilling other ‘social’ goals’’. Over the last decade, however, particularly following the experience of privatization in the U.K. and its controversial assessment16, it became clearer and clearer that it was impossible to sort out the ownership matter from the market structure and regulation matters. Some analysts even argued that privatization without enhancing competition and deregulation could be harmful: the greater profit orientation of private ownership may indeed reduce allocative efficiency whereas its effect on productive efficiency is ambiguous. Add the continuing need for regulation when public monopolies are sold off and it is scarcely surprising that privatization is not a panacea. We are now going to discuss the effects of competition and deregulation on productive efficiency. But before that, a few words on alternative organizational forms are in order. Private (for-profit) and public firms are not the only organizational forms of production even if they are the most usual. There are also organizations such as self-managed firms, cooperatives and nonprofit organizations which have different and specific objectives. What does economic theory predict concerning the capacity of each of them to achieve productive efficiency? The important characteristics of nonprofit organization is that if there are profits they must remain with the organization and used to further its purposes. One important line of theory stresses that those organizations have to be found in situations where consumers don’t have enough information to evaluate the quality of a product and therefore must place their trust in the enterprise that is producing it. Then they may be more willing to trust nonprofit organizations which don’t have any profit incentive to downgrade quality17. Another line of theory acknowledges that those organizations are indeed more trustworthy but at the same time they have less productive efficiency. Alchian and Kessel (1962) thus argue that if no one has a property right in the residual, no one has an incentive to keep the organization free from sloth and waste. The case of self-managed firms is a bit different; they have for objective function value added per worker, and thus have a strong incentive to avoid slacks. In a competitive setting, self-managed firms are expected to be fully efficient. In a noncompetitive setting, worker participation in management is presumed to produce a more democratic environment 16 17
Thomson, Weight and Robbie (1990). Easley and O’Hara (1983).
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as well as superior channels for information-processing and conflict resolution. In contrast, worker participation entails less power and discretionary authority for managers and this can affect productive efficiency negatively. Whether this negative argument can outweigh the positive ones is a question that can only be solved empirically18. 4.2.
Efficiency and competition
The proposition that competition enhances performance is deeply rooted in economics. Hayek (1945) was terribly effective in making the point that through price competition the correct information and the right incentives were conveyed to agents and induced them to act efficiently. Recent theory in the economics of information shows that within a setting of uncertainty and asymmetric information, competitive pressures are the most effective way to foster productive efficiency. Competition induces managers to operate as closely as possible to their production frontier and further it gives their principal, the owners or the State, the relevant information for better monitoring their activities. It also goes without saying that competition induces allocative efficiency. In trying to eliminate managerial slacks, shareholders, governments, or any other principals face difficulties arising from lack of information. Hence, they cannot directly sort out the sources of inefficiencies: managerial weaknesses or the firm’s true opportunities. Observations of the performance of competing firms in the same market or of firms in similar environments do provide relevant information for the development of more efficient incentive schemes. Competition makes interdependent the performances of different firms. In the theory of tournaments, it is shown that by rewarding a firm’s manager not only according to his/her own performance but also according to the performance of other managers, productive efficiency can be fostered and even fully obtained19. Even when other firms’ profits are not observable, the market system makes the actions and utilities of managers interdependent via prices. Thus the market system acts as a sort of incentive scheme20. In that respect, a possible utilization of the productive efficiency measures inside firms suggests itself, at least when the measures bear on cross-sectional data of several plants or units. A principal, when dealing with several agents operating within the same institutional and economic environment, can put them in the setting of a tournament, with rewards 18
Jones and Svejnar (1985). The idea of tournaments is presented in Nalebuff and Stiglitz (1987), albeit not in a context of productive efficiency. 20 Hart (1983). 19
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for those on the efficient frontier and fines for those with lower performance. One could thus conceive of incentive schemes designed so as to induce productive efficiency in all units. 4.3.
Efficiency and regulation
There are various ways of dealing with the relation between efficiency, particularly productive efficiency, and regulation. But before let us say that there are also various ways of defining regulation. In the U.S. where this concept is the most commonly used, it consists of measures to control price, sale, rate of returns, production decisions of firms in an avowed effort to prevent private decision making that would take inadequate account of the public interest. Without being more specific, one cannot say much about its effect on efficiency. The question is what are the terms of comparison: regulated versus unregulated private firm or regulated private monopoly versus public enterprise. In the first case, theory fairly consistently predicts that deregulation will enhance productive efficiency. The most cited example is rate of return regulation, to which many utilities have been subjected for years and which implies an inefficient mix of capital and labor. In this paper, we are rather interested in comparing a public and a private regulated firm. There exists a growing literature21 on this subject which has a double feature: (i) it focuses on informational differences between public and private enterprises; (ii) it tries to generate differences in efficiency rather than postulate them. For example, government officials are supposed to possess intimate knowledge of the workings of the public enterprise, while such information is controlled by private parties if the activity is private. Public enterprise is motivated to satisfy the goals of public management, while the private regulated enterprise is motivated to maximize profits, subject to whatever government regulation is imposed on it. Three results are worth being underlined. First, Shapiro and Willig (1990) show under quite general conditions that there is no difference between the performance of a public enterprise and that of a private company subject to an optimally designed regulatory and tax scheme. In a slightly different setting with asymmetric information, Pint (1991) argues that both public and private regulated firms are inefficient but in a different way: the privately-owned regulated firm will use relatively too much capital and the publicly-owned firm will use relatively too much labor. Finally, Laffont and Tirole (1990) contrast the cost of public ownership due to need of allocating resources away from profit enhancing uses, to the cost of private ownership arising from the duality of principals: 21
See Laffont and Tirole (1990), Pint (1991) and Shapiro and Willig (1990).
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the regulator and the shareholders. Their main result is that productive efficiency (measured by managerial effort) is lower in the regulated private firm than in the public firm. One last issue one should raise is that the link between regulation and efficiency, both allocative and technical, is not necessarily monotonic, in the following sense. One knows for sure that a firm subject to heavy regulation is going to be less efficient than the same firm operating in a setting of full deregulation. Yet, it is not clear that in the process of deregulating an economy, efficiency always increases.
5.
Recent evidence
In this section, we wish to review those empirical studies that are aimed at relating firms’ productive efficiency to features such as ownership, objectives, competition and deregulation. Unfortunately, most empirical studies of productive efficiency in public enterprises do not pursue that objective: they usually limit themselves to quantifying efficiency and productivity of specific activities and they often do not try enough (sometimes not at all) to identify the sources of slack. This is regrettable as a good understanding of these sources is of course essential for the appropriate design of policies to improve performance22. 5.1.
The effect of ownership
In their widely quoted survey paper, Borcherding, Pommerehne and Schneider (1982) present a table summarizing a large number of studies up to 1980 that compare the costs of private and public firms23. The table leads one to conclude that public firms have generally higher unit cost structures than private firms. We have attempted here to construct a similar table summarizing the literature specifically comparing productive efficiency of public and private firms. Contrasting the two exercises might be interesting. While productive efficiency measures are more recent, the activities studied in either surveys are nevertheless similar. They relate either to services provided by local governments or at a similarly local level, or to national public enterprises, in which case the efficiency comparison is international. This is not surprising as both costs comparisons and production frontier construction require the use of a large sample of activities. In general, the spirit of most productive efficiency studies is less
22 23
We have attempted a survey in Pestieau and Tulkens (1990). See also Yarrow’s survey (1986).
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polemical than that of previous studies. Their purpose is not to put down public production but rather to explain differences in performance and thus to supply information if not tools for betterment. The literature tersely presented in Table 2 indicates that in terms of productive efficiency public enterprises are not at all outperformed by private enterprises. Let us highlight three specific cases. First, there is Cubbin, Domberger and Meadowcroft’s study (1987) on the productive efficiency of refuse collection. It breaks a long tradition of earlier studies based on cost functions that had yielded mixed results. Employing the nonparametric DEA technique, they compare local authorities that use private contractors and those that have not tendered their service; the difference in productive efficiency is 17% in favor of the former. Such result allows for clarifying the debate on the benefits of privatisation. In another sector, water utilities, where there existed conflicting evidence arising mainly from the use of poor data on wages, Byrnes, Grosskopf and Hayes (1986) eschew the cost function approach and focus instead on technical (and scale) efficiency. They found no significant difference in efficiency across ownership types. One of us, finally (Tulkens, 1993, Section 3) compares the relative efficiency of the monthly operations24 of 804 branches of a publicly owned bank with a similar measure for the branches of a private bank of about equal size (911 branches) in Belgium. Here, the nonparametric FDH methodology reveals a substantially better performance of the public bank, both in terms of the number of branches lying on the frontier (74.6% of the branches of the public bank against 57.8% of those of the private bank) as in terms of the average efficiency scores (which are respectively 0.96 and 0.93). A look at the respective shapes of the frequency distributions of the efficiency scores in each case is particularly instructive (see Figures 1 a and 1 b): the two diagrams show that the tail of the distribution for the public bank is flatter and stretches less far to low values of the scores than for the private bank. Although such a higher relative efficiency of branches vis- a`-vis one another in the public bank than in the private bank may be due in part to a lesser variety in the range of services offered by the former – a fact that complicates the terms of the comparison, this is nevertheless another clear instance of no obviously inferior performance under public ownership25.
24 Only the number of physical operations in each branch is considered here, and not their financial values. Both lack of accessibility to data and methodological uncertainties have made it impossible to extend the analysis in that direction. 25 In Tulkens (1989, pp. 292–293), similar observations are derived from comparing the private bank with about 800 postal stations.
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Table 2 Productive efficiency comparative studies of public and private firms Sector & authors
Number of units
Type and period of data
Numb. of outputs & inputs
Airlines Good, Roeller & Sickles (1991 a)
9 airlines
Panel annual 1976–1986
1 output 4 inputs
Good, Roeller & Sickles (1991 b)
16 airlines
1976–1986
1 output 4 inputs
Barla & Perelman (1989)
26 airline companies
Panel annual 1976–1986
1 output 2 inputs
Manzini (1990)
50 airline carriers
Cross-section
1 output 1987
773 branches of a public bank and 911 branches of a private bank
Monthly data 1987
7 outputs 1 input
1986
7 outputs 1 input
64 public and 57 private universities in US
Panel annual 1971, 1974, 1981
5 outputs 5 inputs
Fa¨re, Grosskopf & Logan (1985)
30 public and 123 private utility plants in US
Cross-section 1970
1 output 3 inputs
Cote (1989)
62 utilities: 37 private 16 cooperatives 9 public
Panel annual 1965–1970
1 output 3 inputs
Hjalmarsson & Veiderpass (1991)
289 Swedish retail electricity distributors
Annual data 1970–1986
4 outputs 4 inputs
Banking Tulkens (1993)
Education Rhodes & Southwhick (1988) Electric Utilities
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Method
Mean efficiency degrees
Parametric
–
– U.S. airlines are 10–15% more efficient than French airlines.
Parametric and nonparametric
–
– U.S. carriers are 15–20% more efficient than European carriers. – Over the period, European technical efficiency declined.
Parametric
About 80%
– Main purpose is to compare efficiency performance of European and American deregulated companies. Public companies are performing better during crisis periods while deregulated ones are performing better in favorable economic conditions.
Parametric 4 outputs attributes 2 inputs
73.1%
– Main purpose is to estimate a production function in order to get technical information on the production process and to assess the influence of property rights and regulation on technical efficiency. American carriers are more efficient.
Nonparametric FDH
about 97%
– Proportion of branches on the frontier: 75%.
Nonparametric FDH
about 95%
– Proportion of branches on the frontier: 58%.
Nonparametric
About 88% a year
– Private universities have slightly higher efficiency scores, for every year considered.
Nonparametric
– 95%
– Public plants have better ratings in terms of technical efficiency measures than private ones. – Congestion is more a problem for public than for private utilities.
Parametric
91% (public) 86% (cooperative) 83% (private)
Nonparametric
–
Remarks and other findings
– Ownership or economic organization does not seem to be related to productivity change in any significant way. – The municipality and the state owned companies display the highest efficiency values during most years.
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Table 2 (Continued ) Type and period of data
Numb. of outputs & inputs
22 urban public and 60 private hospitals in California
Cross-section 1982
4 outputs 4 inputs
Fecher, Kessler, Perelman & Pestieau (1993)
243 French non-life and 84 life companies
Panel 1984–1989
3 outputs 2 inputs
Delhausse, Fecher, Perelman & Pestieau (1991)
243 French and 191 Belgian non-life companies
Panel 1984–1989
1 to 3 outputs 2 inputs
Oum & Yu (1991)
21 railway companies
Annual data
1 output 1978–1988
Filippini & Maggi (1991)
57 railways under mixed ownership
Annual data 1985–1988
1 output 3 inputs þ2 network characteristics
Cubbin, Domberger & Meadowcroft (1987)
Refuse collection in 317 local authorities
Cross-section 1984–1985
10 outputs 2 inputs
Burgat & Jeanrenaud (1990)
Refuse collection in 98 municipalities in Switzerland
Cross-section 1989
2 outputs 2 inputs
Distexhe (1993)
Refuse collection in 176 Belgian municipalities
Cross-section 1989
2 outputs 2 inputs
Sector & authors
Number of units
Hospitals Grosskopf & Vladamis (1987)
Insurance
Railways
Refuse collection
Ch.15 Assessing and Explaining the Performance of Public Enterprises
Method
Mean efficiency degrees
Nonparametric
About 97%
– Sample partitioned into public & private hospitals. – Comparison of results of each hospital with the whole sample frontier and the separate ownership frontier. – Best-practice frontiers differ, public hospitals using less resources. – Public hospitals are more efficient in both samples.
Nonparametric
at most 50%
– Public companies outperform private companies and mutuals.
Parametric and nonparametric
at most 50%
– Non-profit companies are more efficient than for-profit ones. French companies are more efficient than Belgian ones.
Nonparametric 5 inputs
–
– Managerial freedom (autonomy and government agency variables) has significant effect on efficiency. – The dummy variable for quasi-public firms (partial private ownership) is not statistically significant.
Parametric
1 each year
– Limited evidence has been found for a relationship between the share of state in capital and cost efficiency. – Positive correlation appears between cost efficiency and the importance of the cantons’s participation in the deficit of firms.
Nonparametric
81%
– Tendered services have higher efficiency scores than non-tendered ones.
Parametric and nonparametric
–
– Technical gains can be obtained by contracting out the service to a private collector.
Parametric
75% (public) 72% (private)
– Productive efficiency increases with biddings.
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Remarks and other findings
Moreover, account should be taken of the fact that these two banks are engaged, together with at least two more of comparable size, in a rather acute competition on the Belgian retail banking market. This case
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Table 2 (Continued ) Sector & authors
Number of units
Type and period of data
Numb. of outputs & inputs
239 Indian sugar factories
Annual data 1981–1985
1 output 4 inputs
84 public firms and 301 private firms in Tunisia
Cross-section 1985
–
Sugar industry Ferrantino & Ferrier (1991)
General Chaffai (1989)
study thus illustrates what we shall argue below, namely that competition appears to be a more determining factor than ownership in the achievement of productive efficiency. 5.2.
The effect of alternative objectives
So far, we have focused on the distinction between public-private ownership. One could also use another related distinction, that rests on the objectives of firms. Besides private and public forms, one has nonprofit, self-managed, cooperative and mutual forms of organization. There exists an increasing number of studies comparing the relative productive efficiency of firms having different objectives. Recent empirical work on the comparison between profit maximizing firms and sharing firms (self-managed or cooperatives) is summarized by Cote and Desrochers (1990). Using a parametric approach, Sterner (1990) compares the efficiency of multinational, domestic firms and workers cooperatives in the Mexican cement industry. He finds considerable efficiency differences among firms within each group but not significant differences between groups. Defourny (1988) compares productive efficiency of French workers’ cooperatives and for-profit firms over 10 years. He uses indicators based on a parametric approach. Medium size cooperatives tend to be more efficient than their for-profit counterparts whereas the opposite is observed for small cooperatives. Finally, in the insurance industry, besides the traditional public and private organization, there are the so-called mutuals. In a study of the
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Method
Mean efficiency degrees
Parametric
–
– Cooperative and private firms appear as more efficient than public firms. The authors note that in many cases public firms were ‘‘sick’’. – Private firms that were taken over by the government.
Parametric
–
– Example of findings, for the food industry: – Public: – average efficiency: 80% – range: 89–44% – Private: – average efficiency: 75% – range: 90–16%
Remarks and other findings
insurance market in Belgium and in France, Delhausse, Fecher, Perelman and Pestieau (1991) and Fecher, Kessler, Perelman and Pestieau (1993) reach the conclusion that insurance mutuals fare quite well in terms of productive efficiency relative to for-profit and to public companies. Table 3 presents a sample of such results for the French nonlife insurance companies. It provides for the 243 companies studied over years 1984–88 overall average efficiency as well as average efficiency for alternative institutional forms. Both parametric (stochastic) and nonparametric approaches have been used yielding rather low efficiency indicators. One clearly sees that public owned firms are the most efficient and that mutuals fare quite well. 5.3.
The effect of competition
There are a number of studies casting evidence on the positive relation between competition and productive efficiency. Caves and Barton (1990) or Fecher and Perelman (1991) have found a statistically significant inverse relation between trade barriers and productive efficiency (of firms, within each industry) for the U.S. and the OECD member countries, respectively. Another line of research concerns the comparison of public services which are more or less open to contestability. On this matter, the case of refuse collection is quite interesting. Three studies on the U.K. (Cubbin, Domberger and Meadowcroft (1987), Belgium (Distexhe (1993) and Switzerland (Burgat and Jeanrenaud (1990)) have shown how much can
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Figure 1 Frequency distributions of FDH efficiency scores for branches of a public and a private bank (1 input–7 outputs). The ‘‘detail’’ parts are a close-up view for efficiency degrees < 1. 1.0 0.95 Detail for [0.50-1] domain
0.90
1.0 0.95 0.90 Efficiency degree
Efficiency degree
0.85 0.80 0.75 0.70
0.85 0.80 0.75 0.70 0.65 0.60
0.65
0.55
0.60
0.50 0
10
20
0.55
30
40
50
Number of branches
0.50 0
100
200
300 400 Number of branches
500
600
700
Figure 1a Public bank branches 1.0 0.95 Detail for [0.50-1] domain
0.90
1.0 0.95 0.90
0.80
Efficiency degree
Efficiency degree
0.85
0.75 0.70 0.65
0.85 0.80 0.75 0.70 0.65 0.60 0.55
0.60
0.50 0
0.55
10
20
30
40
50
Number of branches
0.50 0
100
200
300
400
Number of branches
Figure 1b Private bank branches
500
600
700
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Table 3 Productive efficiency of French nonlife insurance companies for 1984–88 Institutional form Foreign Mutual Public Stock Total
Number
Non-parametric
Parametric
78 86 4 75 243
0.499 0.563 0.974 0.524 0.537
0.493 0.477 0.574 0.386 0.412
Source: Fecher, Kessler, Perelman and Pestieau (1993).
be gained from the tendering of the service of collection, particularly when the process of contracting out the service to a private collector goes through open bidding. Using an index of tendering out (in addition to information on autonomy, to be mentioned below), Perelman and Pestieau (1988, 1989, 1993) observe that the efficiency performance of 18 postal services is fostered by two variables reflecting the contestability of the market. 5.4.
The effect of deregulation
It is often asserted that deregulation has a positive effect on productive efficiency. This has indeed been observed in the case of U.S. airlines, as in a recent paper by Good, Nadiri and Sickles (1991) who find important gains in productive efficiency taking place during the 1977–83 period. This thesis is to be qualified, however, when the process of deregulation is only a partial one, and therefore leaves firms with an odd set of regulatory constraints. In a recent paper, Sickles and Streitweiser (1991) study the case of the U.S. national gas pipeline industry which experienced an unfinished process of deregulation. Using both parametric (stochastic) and nonparametric (DEA) techniques, their concordant findings suggest a perverse pattern of declining technical efficiency in the industry during the period in which the major deregulatory measures were implemented. The thesis is even challenged in a paper on the relative performance of 26 European and North-American airlines by Barla and Perelman (1989), who show that even though on average they have the same efficiency level, during crisis periods the European public companies outperform their North-American counterparts. Similarly, in their study on the productive performance of financial services in OECD countries, Fecher and Pestieau (1993) observe that the relation between productive efficiency – both in level and in growth over 12 years – and regulation is positive.
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As another form of deregulation, internal to the public sector, one may consider the granting of more or less autonomy in the management of the public firms (i.e. a move from the first to the second column in our Table 1 above). A series of studies bearing on railways and postal services show that efficiency is indeed enhanced by increased managerial autonomy. By this is meant the relaxation of government’s control on various aspects of management such as choice of inputs, or pricing and investment policies26. The gist of this line of research is to purge traditional efficiency measures from factors over which the firms’ management has no control and then obtain an indicator of truly managerial efficiency. As an example, one can cite a recent study by Gathon and Pestieau (1992) which purports to decompose efficiency into a managerial and a regulatory component. Their study is based on a survey conducted by Gathon (1991) among 21 European railway companies on three issues pertaining to managerial freedom: freedom in pricing, commercial, and hiring policy. This allows for constructing indicators of pricing, commercial and hiring autonomy as well as an indicator of overall autonomy. It appears that there is a significant positive correlation between the measure of what can be called gross efficiency and the indicator of overall autonomy or that of pricing autonomy. Managerial freedom in hiring labor, purchasing inputs and marketing products does not seem to be correlated to gross efficiency as measured through a parametric approach. One can then express gross autonomy as the product of managerial efficiency and regulatory efficiency (based on the indicators of autonomy). The interest of such decomposition is obvious: management is responsible for managerial inefficiency whereas governments are responsible for slacks in regulatory efficiency. Table 4 gives for 21 railway companies the indicators of gross, regulatory and managerial efficiency. One observes that the efficiency ranking among railways changes when going from the usual gross efficiency concept to that of managerial efficiency. One also sees what can be gained by national governments by adopting a more autonomous environment for their railways.
6.
Conclusions
The productive efficiency approach to public enterprises which has been presented and illustrated in this paper offers two attractive features which should characterize its contribution to recent public economics: (1) it focuses on a single objective – productive efficiency – whose desirability
26
Perelman and Pestieau (1988, 1989), Gathon (1991), Oum and Yu (1991). See also Lioukas, Bourantas and Papadakis (1991).
Table 4 Efficiency decomposition into managerial and regulatory efficiency, European railways 1986–1988
Railways
Country
BLS BR CFF CFL CH CIE CP DB DSB FS NS NSB OBB RENFE SJ SNCB SNCF TCDD VR
Switzerland U.K. Switzerland Luxembourg Greece Ireland Portugal Germany Denmark Italy Netherlands Norway Austria Spain Sweden Belgium France Turkey Finland
Source: Gathon and Pestieau (1992)
Autonomy %
Rlways
Managerial efficiency
100.0 76.3 66.0 63.5 47.3 58.3 64.0 61.0 41.5 67.0 70.3 45.3 41.8 52.3 80.0 64.5 69.8 60.0 40.0
VR TCDD NS CIE CFF SNCF RENFE OBB CP BR FS SNCB DB SJ BLS CH DSB NSB CFL
1.000 0.987 0.964 0.963 0.962 0.942 0.939 0.939 0.938 0.927 0.901 0.890 0.888 0.875 0.873 0.867 0.830 0.827 0.803
Efficiency measures 1986–1988 Regulatory Rlways efficiency BLS SJ BR NS SNCF FS CFF SNCB CP CFL DB TCDD CIE RENFE CH NSB OBB DSB VR
1.000 0.955 0.946 0.930 0.929 0.921 0.918 0.914 0.912 0.911 0.903 0.900 0.895 0.875 0.857 0.850 0.836 0.834 0.828
Rlways
Gross efficiency
NS TCDD CFF BR SNCF BLS CIE CP SJ FS VR RENFE SNCB DB OBB CH CFL NSB DSB
0.897 0.888 0.883 0.877 0.875 0.873 0.862 0.856 0.835 0.830 0.828 0.822 0.813 0.802 0.785 0.743 0.731 0.702 0.693
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is independent of any other considerations; (2) it is backed by a set of fairly robust techniques of measurement. There is an unavoidable price to pay for this. First, as mentioned above, productive efficiency is only one of the many goals which might be assigned to public firms. These cannot be ignored and the extent to which they are achieved does of course matter when assessing the overall performance of public production. Second, the approach requires appropriate statistical data (that most enterprises can provide), which can rarely be exhaustive. The main problem in that respect concerns the qualitative dimension of inputs and outputs, especially when they are of the nature of services. One of the big advantages of the productive efficiency approach is that it yields evidence on the impact of ownership, competition and regulation on firms’ performance which is conceptually robust. On the basis of the work existing to date it appears that firms’ performance is quite independent of ownership, for a given competitive and regulatory setting. In particular, there is no clear-cut performance differential between public enterprises and privately owned regulated firms. One also observes that introducing competition increases performance regardless of ownership. Furthermore, the effect of deregulation, especially when it is only partial, appears to be rather ambiguous. Finally, the contribution of the productive efficiency approach is not only to yield more robust comparisons among enterprises operating in different settings but is also to provide firms’ managers with guidance for improvement.
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Borcherding, T. E., Pommerehne, W.W. and Schneider, F., 1982. Comparing the efficiency of private and public production: the evidence from five countries, Zeitschrift fu¨r Nationalo¨konomie, supp. 2, 127–156. Burgat, P. and Jeanrenaud, C., 1990. Mesure de l’efficacite´ productive et de l’efficacite´couˆt: cas des de´chets me´nagers en Suisse, Working paper n8 9002. Switzerland: l’Institut de Recherches Economiques et Re´gionales, Universite´ de Neuchaˆtel. Byrnes, P., Grosskopf, S. and Hayes, K., 1986. Efficiency and ownership: further evidence, Review of Economics and Statistics, 68, 337–341. Caves, D. W. and Christensen, L.R., 1980. The relative efficiency of public and private firms in a competitive environment: the case of Canadian Railroads, Journal of Political Economy, 88, 958–976. Caves, R. E. and Barton, D.R., 1990. Efficiency in U.S. manufacturing industries, Cambridge/Mass: MIT Press. Chaffai, M., 1989. Efficacite´ technique compare´e des entreprises publiques et prive´es en Tunisie, Annals of Public and Cooperative Economics, 60, 451–462. Cote, D., 1989. Firm efficiency and ownership structure: the case of U.S. electric utilities using panel data, Annals of Public and Cooperative Economics, 60, 431–450. Cote, D. and Desrochers, S., 1990. L’efficacite´ des organisations participatives: propositions the´oriques et etudes empiriques, Coope´ration et De´veloppement, 22. Cubbin, J., Domberger, S. and Meadowcroft, S.,1987. Competitive tendering and refuse collection: identifying the sources of efficiency gains, Fiscal Studies, 8, 69–87. De Alesi, L., 1974. An economic analysis of government ownership and regulation: theory and the evidence from the electric power industry, Public Choice, 19, 1–42. Debreu, G., 1951. The coefficient of resource utilization, Econometrica, 19, 273–292. Defourny, J., 1988. Comparative measures of technical efficiency for 500 French workerscooperatives, in D. C. Jones and J. Svejnar (eds.). Advance in the economic analysis of participatory and self-managed firms, 4. Greenwich, CT: JAI Press. Delhausse, B., Fecher, F., Perelman, S. and Pestieau, P., 1991. Measuring productive performance in the non-life Insurance industry: the case of French and Belgian markets, mimeo. Distexhe, V., 1993. L’efficacite´ productive des services d’enle`vement des immondices en Wallonie, Cahiers Economiques de Bruxelles, 137, 119–138. Distexhe, V., Gathon, H-J. and Pestieau, P.,1992. L’efficacite´ des services communaux de collecte des de´chets me´nagers. Liege: Universite´ de Lie`ge, mimeo. Domberger S., Meadowcroft, S.A. and Thompson, D.J., 1986. Competitive tendering and efficiency: the case of refuse collection, Fiscal Studies, 7, 69–87. Easley, D., and O’Hara, M., 1983. The economic role of nonprofit firms, Bell Journal of Economics, 14, 531–538. Fa¨re, R., Grosskopf, S., Logan, J. and Lovell, C.A.K., 1985. Measuring efficiency in production with an application to electric utilities, in A. Dogramaci and N. Adam (eds.). Current issues in productivity. Boston:Kluwer, 185–214. Fa¨re, R., Grosskopf, S. and Pasurka, C., 1986. Effects on relative efficiency in electric power generation due to environmental controls, Resource and Energy, 8, 167–184. Farrell, M. J., 1957. The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A – General, 120(3), 253–281. Fecher, F., Kessler, D., Perelman, S. and Pestieau, P., 1993. Productive performance of the French insurance industry, Journal of Productivity Analysis, 4, 77–93. Fecher, F. and Perelman, S., 1990. Productivity growth, technological progress and R & D in OECD industrial activities, in G. Krause-Junk (ed.). Public Finance and steady economic growth. The Hague: Foundation Journal Public Finance, 231–249. Fecher, F., Perelman, S. and Pestieau, P., 1991. Scale economies and performances in the French insurance industry, Geneva Papers, n860. Fecher, F. and Pestieau, P., 1993. Efficiency and productivity growth in OECD financial services, in H. Fried, C.A.K. Lovell and S. Schmidt (eds.). The measurement of
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productive efficiency: techniques and applications. Oxford: Oxford University Press, 374–385. Ferrantino, M. J. and Ferrier, G.D., 1991. The technical efficiency of the vacuum-pan sugar industry in India: an application of a stochastic frontier production function using panel data. Dallas: Southern Methodist University, mimeo. Filippini, M. and Maggi, R., 1991. Efficiency and regulation in the case of the Swiss private railways, Journal of Regulatory Economics, 5, 199–216. Førsund, E. R., Lovell, C. A. K. and Schmidt, P., 1980. A survey of frontier production functions and of their relationship to efficiency measurement, Journal of Econometrics, 13, 5–25. Fried H., Lovell, C.A.K. and Schmidt, S. (eds.), 1993. The measurement of productive efficiency: techniques and applications. Oxford: Oxford University Press. Gathon, H.-J., 1986. La mesure des gains de productivite´ globale dans les chemins de fer: une comparaison internationale, Annales de l’Economie Publique, Sociale et Coope´rative, 459–476. Gathon, H-J., 1991. La performance des chemins de fer Europe´ens: gestion et autonomie, PH. D. thesis. Liege: Universite´ de Lie`ge. Gathon, H.-J. and Perelman, S., 1992. Measuring technical efficiency in European railways: a panel data approach, Journal of Productivity Analysis, 3, 135–151. Gathon, H.-J. and Pestieau, P., 1995. Decomposing efficiency into its managerial and regulatory components: the case of European railways, European Journal of Operations Research, 8(3), 500–507. Good, H. D., Nadiri, M.I. and Sickles, R.C., 1991. The structure of production, technical change and efficiency in a multiproduct industry: an application to U.S. Airlines. NBER Working Paper no. 3939. Good, H. D., Roeller, L-H. and Sickles, R., 1991a. EC integration and the structure of the Franco-American airline industries: implications for efficiency and welfare, in W. Eichhorn et al. (eds.). Models and measurement of welfare and inequity. Berlin: Springer . Good, H.D., 1991b. Airline efficiency differences between Europe, and the U.S; implications for the pace of E. C. integration and domestic regulation, mimeo. Grosskopf, S. and Vladamis, V., 1987. Measuring hospital performance: a non-parametric approach, Journal of Health Economics, 6, 89–107. Hartley, K., Parker, D. and Martin, S., 1991. Organizational status, ownership and productivity, Fiscal Studies, 12, 46–60. Hart, O. D., 1983. The market mechanism as an incentive scheme, Bell Journal of Economics, 14, 366–382. Hayek, F. A., 1945. The use of knowledge in society, American Economic Review, 35, 519– 530. Hicks, J. R., 1935. The theory of monopoly: a survey, Econometrica 3, 1–20. Hjalmarsson, L. and Veiderpass, A., 1991. Productivity in Swedish electrical retail distribution. Gothenburg: University of Gothenburg, mimeo. Jones, D. C. and Svejnar, J., 1985. Participation, profit sharing, worker ownership and efficiency in Italian producer cooperatives, Economica, 52, 449–465. Kay, J. A. and Thomson, D.J., 1986. Privatization: policy in search of a rationale, Economic Journal, 96, 18–32. Koopmans, T. C., 1951. Analysis of production as an efficient combination of activities, in T. C. Koopmans (ed.). Activity analysis of production and allocation. New York: Wiley, chapter III. Laffont, J. J. and Tirole, J., 1990. Privatization and incentives. Toulouse: Universite´ de Toulouse, mimeo. Leibenstein, H., 1966. Allocative efficiency vs. x-efficiency, American Economic Review, 56, 392–415.
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Leibenstein, H. and Maital, S., 1992. Empirical estimation and partitioning of x-inefficiency: a data envelopment approach, paper presented at the New Orleans AEA Meeting. Lioukas, S., Bourantas, D. and Papadakis, V., 1991. Managerial autonomy of stateowned enterprises: determining factors, Organizational Science, 4, 645–666. Lovell, C. A. K., 1993. Production frontiers and productive efficiency: an introduction and readers guide, in H. Fried, C.A.K. Lovell and S. Schmidt (eds.). The measurement of productive efficiency: techniques and applications. Oxford: Oxford University Press, 3–67. Manzini, A., 1990. Efficacite´ publique et prive´e: analyse the´orique et ve´rification empirique dans le cas du transport ae´rien. The`se de Doctorat, Geneva: Universite´ de Gene`ve. Marchand, M., Pestieau, P. and Tulkens, H., 1984. The performance of public enterprises: normative, positive and empirical issues, in M. Marchand, P. Pestieau and H. Tulkens (eds.). The performance of public entreprises: concepts and measurement, Amsterdam: North-Holland, chapter 1. Nalebuff, B. and Stiglitz, J., 1987. Prizes and incentives, towards a general theory of compensation and competition, Bell Journal of Economics, 14, 21–43. Oum, T. H. and Yu, C., 1991. Economic efficiency of passanger railway systems and implications on public policy. Vancouver: University of British Columbia, mimeo. Perelman, S., 1986. Frontie`res d’efficacite´ et performance technique des chemins de fer, Annals of Public and Cooperative Economics, 4, 449–459. Perelman, S. and Pestieau, P., 1988. Technical performance in public enterprises: a comparative study of railways and postal services, European Economic Review, 32, 432–441. Perelman, S., 1989. The performance of public enterprises: a comparative efficiency study of railways and postal services, in M. Neumann and K. W. Roskamp (eds.). Public finance and performance of enterprises, proceedings of the 43rd Congress of the International Institute of Public Finance, Paris 1987, Detroit, 365–381. Perelman, S., 1994. Comparative performance study of postal services: a productive efficiency approach, Annales d’Economie et de Statistique, 33,187–202. Pestieau, P. and Tulkens, H., 1990. Assessing the performance of public sector activities: some recent evidence from the productive efficiency viewpoint, CORE Discussion Paper n8 9060. Louvain-la-Neuve: Center for Operations Research and Econometrics, Universite´ catholique de Louvain. Peters, W., 1985. Can inefficient public production promote welfare? Zeitschrift fu¨r Nationalo¨konomie, 45, 395–407. Pint, E. M., 1991. Nationalization vs. regulation of monopolies, Journal of Public Economics, 44, 131–164. Rangan, N., Grabowski, R., Aly, H.Y. and Pasurka, C., 1988. The technical efficiency of US banks, Economic Letters, 28, 169–175. Rees, R., 1976 (1984). Public enterprise economics. London: Weidenfeld and Nicholson. Rhodes, E. L. and Southwick, L., 1988. Comparison of university performance differences over time, mimeo. Scherer, F. M., 1980. Industrial market structure and economic performance. Boston: MIT Press. Shapiro, C. and Willig, R.D., 1990. Economic rationale for the scope of privatization, in E. N. Suleiman and J. Waterburg (eds.). The political economy of public sector reform and privatization. Oxford: Oxford University Press, 55–87. Sickles, R. and Streitwieser, M., 1992. Technical inefficiency and productive decline in the U.S. interstate natural gas pipeline industry under the U.S. Interstate Natural Gas Policy Act, Journal of Productivity Analysis, 3, 115–130
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Chapter 16 NON-FRONTIER MEASURES OF EFFICIENCY, PROGRESS AND REGRESS FOR TIME SERIES DATA
Henry Tulkens, Philippe Vanden Eeckaut International Journal of Production Economics 39, 83–97 (1995).
We deal in this paper wih efficiency, progress and regress without using the frontier concept. For describing the relation between inputs and outputs in production, we introduce a notion that we call ‘‘benchmark production correspondence’’ with which to measure progress and regress without running into the problems of distinguishing between efficiency gains and progress, or between efficiency losses and regress. An illustration with a monthly time series covering almost 15 years of the activities of an urban transit company is presented and contrasted with DEA and FDH nonparametric frontier approaches.
1.
Introduction
In the classical literature on efficiency measurement, the notion of efficiency is intimately linked with the one of production frontier: specifically, a productive activity is called efficient if the vector of input–output quantities that describes it belongs to the (efficient) frontier of some Paper presented at the Second European Workshop on Efficiency and Productivity Measurement) held at CORE, 25–26, October 1991 and at GREQE, Marseille, the Stockholm school of Economics, Union College, and the University of Munich. Thanks are due to Professors C. Knox Lovell for helpful discussions and Le´opold Simar for a careful reading of a first draft, to Marie-Astrid Jamar for assistance in handling of the data and to Ms. Hanocq and Ms. D’Hondt from STIB for providing the data. We acknowledge the support of Apple-Belgium who made appropriate computational equipment available, the financial support of the second author successively by FRFC (convention No. 2.4528.88 with Henry Tulkens) and by the Belgian Ministry of Scientific Policy (PAI convention No. 26 with CORE), and sabbatical support of the first author by C.I.M (the Colle`ge Interuniversitaire d’e´tudes doctorales dans les sciences du Management), Brussels.
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appropriately defined production set; and the activity is called inefficient if this vector belongs to the interior of the set. Moreover, when time-related data are involved, a distinction is usually made between shifts of the frontier and gains in efficiency. Running against this tradition, we deal in this paper with efficiency, progress and regress – first conceptually and in terms of measurement, and then with an illustrative application – without using the frontier concept. Actually, we substitute an alternative benchmark production relation, with respect to which progress and regress are measured. In the process, the notions of efficiency and inefficiency are merged with those of progress and regress, in the sense that it is no longer necessary to distinguish between progress and efficiency gains, nor between regress and efficiency losses. We deal only with nonparametric methods, leaving for another occasion an examination of the implications for parametric methods of the ideas presently developed. The next Section is devoted to the conceptual basis of our approach. Section 3 explains why it is useful for the study of time-related data, be they time series or panels. Section 4 introduces the basic tools of ‘‘benchmark’’ observations and correspondence, and Section 5 presents a formal exposition of the method of computing progress and regress indexes that stems from it. Section 6 presents a numerical application to data from an urban transit company, where a rather striking illustration is provided of the relative power of the method proposed presently, compared to competing ones. The same methodology, adapted for panel data, is presented in Section 6 of Tulkens and Vanden Eeckaut (1995).
2.
Two alternative views of efficiency
2.1
Efficiency as a set-frontier property – a quick reminder
Given a firm or organization, let x ¼ (x1 , . . . , xi , . . . , xI ) 2 RIþ denote an input bundle, i.e. a vector whose components represent the quantities of I different inputs it uses, and u ¼ (u1 , . . . ,uj , . . . ,uJ ) 2 RJþ an output bundle, i.e. a vector of the quantities of J different outputs it produces – all quantities being defined over the same unit of time, and call any pair of such vectors, (x, u), a ‘‘production plan’’. Standard efficiency theory begins with positing as a primitive concept the notion of a ‘‘production set’’, Y, defined as Y ¼ {(x, u) 2 RIþJ þ j(x, u) is feasible for the organization} . By so recognizing a distinction between feasible vs. unfeasible production plans, this definition implicitly refers to the existence of a relation
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between input and output bundles. The literature makes this relation explicit in the following two alternative and complementary ways: (i) by defining for any output vector u, the input Section of Y relative to u, denoted L(u), as the set of input bundles that permit to produce u; formally,1 L(u) ¼ {xj(x, u) 2 Y }; (ii) and by defining for any input vector x, the output Section of Y relative to x, denoted P(x), as the set of output bundles that can be produced by means of x; i.e. the set2 P(x) ¼ {uj(x, u) 2 Y }: The efficiency notion is then introduced by referring to a subset of the boundaries of these two sets, called their efficient boundary. However, there are at least two different ways to define efficient boundaries. One is the ‘‘efficient boundary in the Farrell sense’’, denoted here as @eF L(u) for an input Section, and @eF P(x) for an output Section, and defined in each of these cases as the sets:3 @eF L(u) ¼ {xjx 2 L(u), ux 62 L(u) 8u 2 [0, 1[}, and @eF P(x) ¼ {uju 2 P(x), lu 62 P(x) 8l 2 ]1, þ 1[}: Another definition is the one of ‘‘efficient boundary in the Shephard sense’’, that reads as follows4 for the input and the output Sections:5 @eS L(u) ¼ {xjx 2 Lj (u), y4x ) y 62 L(u)}, and @eS P(x) ¼ {uju 2 P(x), v5u ) v 62 P(x)}, Then, a production plan (x, u) is called: – input-efficient in the Farrell (respectively Shephard) sense if its input bundle x belongs to the boundary @eF L(u) (respectively @eS L(u)); otherwise, the production plan is input-inefficient; and 1
Fa¨re et al. (1985) call L(u) the input requirement set relative to u. For P(x), the terminology of the authors just mentioned is the output possibility set relative to x. 3 The superscript F is for Farrell. 4 See Shephard 1970 (Shephard (1970), p. 13). The superscipt S is for Shephard. 5 For comparing vectors, our notation is as follows: y > x ) yi > xi for all components of y and x; y$x ) yi ^ xi for all components i and yi > xi for at least one i, and y ^ x ) yi ^ xi for all components. 2
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– output-efficient in the Farell (respectively Shephard) sense if its output bundle u belongs to the boundary @eF P(x) (respectively, @eS P(x)); otherwise, the production plan is output-inefficient. The characterization of feasible programs (x, u) as efficient vs. inefficient may thus not be the same depending upon which notion of efficient boundary one refers to. In what follows, we shall most often refer to the Shephard notion of efficiency. 2.2
Efficiency as a pairwise dominance relation between production plans
Consider now a pair of production plans (x, u) and (x0 , u0 ), irrespective of whether or not they are feasible – thus independently of the production set notion. For any such pair, a dominance relation may be defined between the two plans expressed both in input terms, in output terms or still in terms of both the inputs and the outputs: Definition. (1) (x0 , u0 ) dominates (x, u) in inputs if both u0 ^ u and x0 #x. (2) (x0 , u0 ) dominates (x, u) in outputs if both u0 $u and x0 % x. (3) (x0 , u0 ) dominates (x, u) if either (1) or (2) holds, or both. By replacing the term ‘‘dominates’’ by the expression ‘‘is more efficient than’’ one is led to see efficiency as a pairwise relation between production plans. In this perspective, the efficiency notion appears more as a property that may serve to characterize a production plan vis-a`-vis any other such plan, instead of a specific quality of belonging or not to some predefined subset of production plans (e.g. ‘‘frontier’’ ones). Notice that when it is considered in this way, efficiency is also independent of the feasibility notion. By this remark a connection is established however with the frontier definition of efficiency. Indeed, call undominated any feasible production plan such that there exists no other feasible production plan that dominates it in input and/or in output; then, any input- (output-) efficient production plan in the Shephard sense6 is also an undominated one. On the other hand, without the feasibility notion, undomination has no meaning. Notice also, finally, that all our developments so far have been made for production plans in general, irrespective of whether these plans are actually observed or not. No measurement issue has been raised either, but it is probably clear that while measuring efficiency in the frontier context amounts to computing distances between production plans and the frontier, similar measurement in the pairwise dominance relation perspective 6 Efficient production plans in the Farrell sense are also undominated ones, but with respect to the stronger definition of dominance where ‘‘<’’ and ‘‘>’’ are substituted for ‘‘#’’ and ‘‘$’’ in the definition stated above.
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should consist of taking distances between pairs of production plans. We now turn to these measurement issues in some more detail.
3.
Measuring efficiency, progress and regress from time-related data: limitations of nonparametric frontiers
When the activities of an actual productive organization are considered over time, and time is defined in discrete units t ¼ 1, 2, . . ., the observations take the form of a succession of dated production plans (xt , ut ), t ¼ 1, 2, . . . ,T, where T denotes the end of the observation (1, T) period. Let Y 0 denote the set of such observations. In this Section, we shall not need to distinguish whether at each t a single observation is (1, T) made or a cross Section (making Y 0 a panel). In order to make possible the efficiency analysis of such data in frontier terms, it is necessary that ‘‘reference’’7 production sets and boundaries be defined over time also, that is, that for each t, a welldefined reference set Y t be specified, with respect to which the efficiency of the plan (xt , ut ) can be evaluated. A series of alternative ways of defining such dated reference pro(1, T) duction sets, and of constructing them from the available data set Y 0 , was proposed and used in (Tulkens (1986), Section 6) and in (Tulkens and Vanden Eeckaut (1995)). They are briefly recalled as follows: (i) define a single reference set, called intertemporal, for the whole observation period [1, T], and construct it from all the observations in (1, T) Y0 ; (ii) define a distinct reference set, called contemporaneous, for each time t, and construct it from the observations made at that time only; (iii) define a distinct reference set, called sequential, for each time t, and construct it from all the observations made from the beginning of the observation period, t ¼ 1, until time t; thus from the subset (1, T) Y0 ¼ {(xt , ut )j1 # t # t} of the data. In such a temporal context, an inescapable question is whether the frontier of the so-constructed reference sets remains the same or shifts during the observation period – ‘‘upwards’’ shifts being associated with the classical notion of technical progress, and ‘‘downwards’’ shifts suggesting regress. On this point, it is clear that the first approach simply begs the question, since it postulates that only one frontier prevails altogether. Nevertheless, analyses have been made under that assumption (see e.g. d’Aspremont (1984)). The second approach somewhat lies at the other extreme, not only because it allows for different frontiers at each time t (with all possibilities 7
Useful terminology introduced by Grosskopf (1986).
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of upward and downward shifts), but also because it postulates that each frontier’s form can only be determined by the data observed at that time. This approach is typically the one used in the recent literature on Malmquist productivity indexes allowing for inefficiency (see Fa¨re, Grosskopf, Lindgren and Roos (1993); Fa¨re (1991); Førsund (1991)). Notice that these are all panel data studies; the contemporaneous approach is indeed ill-suited (i.e. somehow trivial, although not impossible) for time series, since only one observation is available at each point in time. The third approach, namely the sequential one, somehow lies in between the other two, by allowing different reference frontiers at each time, and postulating that these be determined by all the past history of the activity up until that time. This makes the approach suitable for both time series data sets (as in Thiry and Tulkens (1992)) and for panel data (as in Tulkens (1986), Tulkens and Vanden Eeckaut (1995)). The way in which the past plays a role is somehow asymmetric, however. Indeed, while progress, i.e. an upward shift, is clearly determined each time an observation appears to be more efficient than an earlier one that was found efficient when it was made, regress cannot be similarly defined. Whenever an observation is found to be less efficient than a previous one, this mere inefficiency property does not allow one to identify whether it corresponds to a move towards the interior of an unchanged production set, or to an inwards move of the sets’ frontier. Regress can therefore not be identified, nor measured, as a phenomenon distinct from inefficiency. Our present endeavour towards freeing the efficiency notion from the frontier concept may be seen precisely as an attempt to retrieve regress in time-related data sets, symmetrically to progress.
4.
Benchmark observations, progress and regress
When the dominance relation is substituted for the frontier concept in gauging the efficiency of a given production plan (xt , ut ), the issue of whether or not the frontier shifts at time t simply disappears, since there is no question of frontier anymore! However, one must determine what the plan dominates, or is dominated by; that is, one must determine, just as in frontier analysis, a reference point or set of points, with respect to which dominance will be characterized and subsequently measured. In this paper, we propose that this reference set of points be the graph of some relation between the inputs and outputs, that we shall call the ‘‘benchmark correspondence’’. In this expression the first term is self-explanatory, while the second one will be explained briefly. What could be an appropriate choice for serving as a benchmark in evaluating observed production plans over time? If some exogenous information were available, e.g. engineering blueprints, or a previously estimated
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production function for the organization (by ‘‘previously’’ we mean: estimated from data other than those being analyzed), it could surely serve our purpose: each observation could easily be assessed as showing progress, or regress with respect to it, or as being just ‘‘on’’ the benchmark. If such exogenous information is not available and only statistical data on input and output quantities can be used, as it is most often the case with efficiency analyses, we shall argue now that the benchmark can be constructed from the data themselves, in an endogenous way. There will even be some reasons, we believe, to consider such endogenous benchmark as preferable to an exogenous one – or at least to compare carefully the two. In defining this endogenous benchmark, two elements play a crucial role: nondominance on the one hand, and time on the other. Indeed, we proceed by identifying at each point in time t, from within the data set (1, t) (1, t) Yo then available, a subset of the observations YoB that we call ‘‘the set of benchmark observations at time t’’, whose elements are selected recursively as follows: (i) the first observation (x1 , u1 ) is always retained as a benchmark observation; (ii) the second observation (x2 , u2 ) is retained as a benchmark observation if and only if it neither dominates nor is dominated by the previous observation (that was retained as a benchmark observation at time 1); (iii) for the third and each one of the following observations, (xt , ut ), t ¼ 3, 4, . . . , t, it is retained as a benchmark observation if and only if it neither dominates nor is dominated by any of the previous observations that were retained as benchmark observations at the time they were made. (1, t) The set of benchmark observations at each time t, YoB , thus comprises only observations that neither dominate nor are dominated by one another. It is by means of this set that we shall formally define below what we have announced as the ‘‘benchmark correspondence’’, yielding the reference set to be used for evaluating all subsequent observations. Notice that any observation included in the benchmark subset at the time it is made belongs to all subsequent benchmark subsets. Thus, (1, tþ1) (1, t) successive benchmark subsets are nested, i.e. YoB
YoB for every t. (1, t) As the set YoB thus varies over time, the correspondence it induces will vary too; we shall therefore always speak henceforth of the benchmark correspondence at time t. Consider now all observations not included in the benchmark observations set at the time they are made. By construction, these observations either dominate, or are dominated by some observation in the benchmark set at that time. Now, the expression ‘‘to dominate’’ was assimilated above with ‘‘being more efficient’’ and ‘‘being dominated’’ with ‘‘being less efficient’’. Therefore, one could be tempted to call ‘‘more efficient than the
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benchmark’’ the benchmark-dominating observations, and ‘‘less efficient than the benchmark’’ the benchmark-dominated observations. However, this terminology might convey the impression that the benchmark observations themselves are efficient ones. This is inappropriate since in selecting them we have ignored the frontier notion altogether.8 To avoid this confusion, it is thus necessary to use another vocabulary, and to choose instead to call ‘‘in progress (vis-a`-vis the benchmark)’’ any observations that dominate one or several benchmark observations and ‘‘in regress (vis-a`-vis the benchmark)’’ any observation that is dominated by one or several benchmark observations. This terminology also naturally leads one to define ‘‘progress indexes’’ and ‘‘regress indexes’’, when it comes to measurement.
5.
Defining the benchmark production correspondence and computing progress and regress indexes from it
In this Section, we deal only with time series data, as the panel data case is treated in (Tulkens and Vanden Eeckaut (1995), Section 6). The computation of numerical indexes of progress and regress in the sense described above – let us say performance indexes for short – takes place by treating the data set in a sequential way, i.e. by computing an index first for the first observation from the observation set Yo1 ¼ {(x1 , u1 )}, then for the second observation from the observation (1, 2) ¼ {(xt , ut ), t ¼ 1, 2}, and in general for the tth observation set Yo (1, t) from the observations set Yo ¼ {(xt , ut ), t ¼ 1, 2, . . . , t}. 1. Consider thus first t ¼ 1 and the observation set Yo1 . Partition the whole space RIþJ þ of non-negative inputs and outputs into three subsets defined as follows:9
Dd (1) ¼
(" # u x
2 3 " # " 1 # " I# J I u 0I ei u X X IþJ þ , 2 Rþ mj 4 5 ni ¼ x eJ 0J x1 j¼1 i¼1 j
(" u1 #) mj $ 08j , ni $ 08i n x1
8 The benchmark correspondence to be defined below will be seen definitely not to be a frontier of the observations. 9 In what follows, the notations 0I and 0J denote I-dimensional and J-dimensional zero vectors, respectively, and eIi and eJj denote I-dimensional and J-dimensional vectors whose ith and jth component, respectively, is equal to 1. mj , 8j and ni , 8i , are scalars.
Ch.16 Non-Frontier Measures of Efficiency
381
i.e. the set of points in the space that are weakly dominated10 by observation (x1 , u1 ); 82 3 2 I3 2 3 2 3 2 1 3 0 u J I < u eIi u X X IþJ mj 4 5 þ n i 4 5, Dg (1) ¼ 4 5 2 Rþ 4 5 ¼ 4 5 : j¼1 i¼1 eJj x x 0J x1 ) 82 1 39 < u = mj $ 08j , ni $ 08i n 4 5 : 1 ; x i.e. the set of points in the space that are dominating observation (x1 , u1 ); and " # " 1 # " I# " I# (" # u J I 0 u ei u X X 2 RIþJ Di (1) ¼ þ þ , m n ¼ i j þ x eJ x 0J x1 j¼1 i¼1 j
) mj $ 08j , ni $ 08i [
(" # u x
I X i¼1
j
" ni
" # " 1 # " I# J u 0 u X 2 RIþJ m ¼ j þ x eJ x1 j¼1 eIi 0J
#
) , mj $ 08j , ni $ 08i
i.e. the set of points in the space that are dominance-indifferent with respect to observation (x1 , u1 ), i.e. neither dominating nor dominated by it. One can check that Dg (1) \ Dd (1) \ Di (1) ¼ Ø and Dg (1) [ Dd (1) \ Di (1) ¼ RIþJ , as requested for a partition. Fig. 1 illustrates such a partitioning of the space in the one inputone output case: Dd (1) is the set of points to the South–East of (x1 , u1 ) including the dotted boundary but excluding (x1 , u1 ); Dg (1) is the set of 10
In the sense of the definition stated in Section 2.2 above; the point (x1 , u1 ) is subtracted from the expression between braces as it cannot dominate itself.
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u
Dg(1)
Di(1)
u1 Dd(1)
x1
0
x
Figure 1. Subsets Dd (1), Dg (1) and Di (1) in the one input one output case.
points to the North–West of (x1 , u1 ) including the dotted boundary but excluding (x1 , u1 ) and Di (1) is the union of the points lying either to the North–East of (x1 , u1 ) or the South–West, excluding the dotted boundary but including (x1 , u1 ). Fig. 2 shows the same type of partitioning for the input subspace only in the two inputs case, and Fig. 3 does so for the output subspace in the two outputs case. Next, noting that the observation (x1 , u1 ) 2 Di (1): (i) Attach to it two ‘‘performance indexes’’ I i (1) ¼ I 0 (1) ¼ 1, one in input, the other in output. (ii) Define the ‘‘benchmark observations set at time 1’’ as
x2 Dd (1)
Di (1) x12 Dg (1)
x11
0
x1
Figure 2. Subsets Dd (1), Dg (1) and Di (1) in the inputs space. u1 Dg (1)
Di (1) u12 Dd (1)
0
u12
u2
Figure 3. Subsets Dd (1), Dg (1) and Di (1) in the outputs space.
Ch.16 Non-Frontier Measures of Efficiency
383
1 YoB ¼ {(x1 , u1 )}:
(iii) Define the ‘‘benchmark production correspondence at time 1’’ as the mapping b1 : RIþ ! RJþ ; x7!U(x) ¼ {uj(x, u) 2 Di (1)}: As the definition makes clear, the graph of this correspondence is simply the set Di (1) just defined. It is thus represented by the crosshatched area in Fig. 4 for the one input-one output case, while in the other two cases, a representation of this graph would require a third dimension as will appear in Figs. 7(a) and (b) as examples of the one output and two inputs case. This completes the treatment of the first observation. 2. Consider now t ¼ 2, the observation (x2 , u2 ) to be evaluated, and the observations set Yo(1, 2) then available for that purpose. (A) If (x2 , u2 ) 2 Dd (1), then call (x2 , u2 ) a regress observation and: (i) Attach to it a ‘‘performance index in inputs’’ computed as 1 xi i , R (2) ¼ max i¼1,..., I x2i and a ‘‘performance index in outputs’’ computed as ! 2 u j : Ro (2) ¼ max j¼1,..., J u1j Notice that in either case, this index is computed in reference to the graph of the benchmark correspondence (actually its frontier! . . . ) defined at time 1. Both indexes are, by construction, less than or equal to 1. They are radial measures, and derived from assumptions of free disposal of inputs and of outputs, respectively. An illustration of the performance
u
u1
0
x1
Figure 4. The benchmark correspondence at time t ¼ 1.
x
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Public Goods, Environmental Externalities and Fiscal Competition
indexes as well as of the graph of the benchmark production correspondence at t ¼ 2 is provided in Fig. 5(a) for the one input and one output case. (ii) Define the ‘‘benchmark observations set at time 2’’ as 2 1 YoB ¼ YoB :
(iii) Define the ‘‘benchmark production correspondence at time 2’’ as the same mapping as the previous one, i.e. b2 ¼ b 1 : (B) If (x2 , u2 ) 2 Dg (1), then call (x2 ,u2 ) a progress observation and: (i) Attach to it a ‘‘performance index in inputs’’ computed as 1 xi i , P (2) ¼ min i¼1,..., I x2i and a ‘‘performance index in outputs’’ computed as ! u2j o : P (2) ¼ min j¼1,..., J u1j Again, in either case, these indexes are computed in reference to the graph of the benchmark correspondence defined at time 1. By construction, they both are larger than or equal to 1. They are also radial measures, and derived from assumptions of free disposal of inputs and of outputs, respectively. An illustration of the performance indexes as well as of the graph of the benchmark production correspondence at t ¼ 2 is provided in Fig. 5(b) for the one input and one output case. (ii) Define also the ‘‘benchmark production correspondence at time 2’’ as 2 1 YoB ¼ YoB :
u Dg(1) d (1)
c (2)
0
a
b
Dd(1)
x
Figure 5(a). The benchmark correspondence at time t ¼ 2. Observation (2) is dominated by (1). Regress indexes are: Rj (2) ¼ oa=ob < 1 and Ro (2) ¼ oc=od < 1.
Ch.16 Non-Frontier Measures of Efficiency
385
u Dg(1)
c
(2)
d (1)
Dd(1)
0
b
a
x
Figure 5(b). The benchmark correspondence at time t ¼ 2. Observation (2) is dominating (1). Progress indexes are: Pi (2) ¼ oa=ob > 1 and Po (2) ¼ oc=od > 1.
(iii) Define also the ‘‘benchmark production corespondence at time 2’’ as the same mapping as the previous one, i.e. b2 ¼ b1 : (C) If (x2 , u2 ) 2 Di (1), then call (x2 , u2 ) a benchmark observation and: (i) Attach to it ‘‘performance indexes in inputs and in outputs’’ I i (2) ¼ I o (2) ¼ 1. (ii) Define the ‘‘benchmark observations set at time 2’’ as 2 1 YoB ¼ YoB [ {(x2 , u2 )}:
(iii) Partition again the whole space RIþJ into three subsets Dd (2), Dg (2) þ and Di (2) defined exactly as above for Dd (1), Dg (1) and Di (1) except for substituting (x2 , u2 ) for (x1 , u1 ). Define further Dd (1, 2) ¼ Dd (1) [ Dd (2), Dg (1, 2) ¼ Dg (1) [ Dg (2), Di (1, 2) ¼ Di (1) \ Di (2): Finally define the ‘‘benchmark production corespondence at time 2’’ as the mapping b2 : RIþ ! RJþ ;
x 7!U(x) ¼ {uj(x, u) 2 Di (1, 2)}:
This completes the treatment of the second observation. In Fig. 5(c), illustrations of the graph of this correspondence are provided for the one input and one output case, for the two possible cases that may occur, depending upon the values of observation (2) vis-a`-vis (1). The one input-one output case does not provide a sufficiently realistic visualization of the correspondence, however, as an example for the two inputs-one output case will show after presentation of the general case.
386
Public Goods, Environmental Externalities and Fiscal Competition Dg(1)∪Dg(2)
u
u (2)
Dg(1)∪Dg(2) (1)
(1)
Dd(1)∪Dd(2)
(2)
Dd(1)∪Dd(2) 0
x
0
x
Figure 5(c). Effects on the benchmark correspondance B2 if observation (2) becomes a benchmark one.
3. We now formulate the general case, by considering any point in time t$2, the observation (xt , ut ) made at that time and to be evaluated, and (1, t) the observations set then available for that purpose, Yo . (A) If (xt , ut ) 2 Dd (t 1), then call (xt , ut ) a regress observation and: (i) Attach to it a ‘‘performance index in inputs’’ computed as11 d xi i R (t) ¼ min max t g d2DoB (t) i¼1,..., I xi and a ‘‘performance index in outputs’’ computed as ( !) t u j , max Ro (t) ¼ min d d2DgoB (t) j¼1,..., J xj (1,t1) where DgoB (t) is the subset of YoB whose elements are dominating t t (x , u ). In either case, these indexes are computed in reference to the graph of the benchmark correspondence defined at time t 1. They both are less than or equal to 1. (ii) Define the ‘‘benchmark observations set at time t’’ as (1,t) (1,t1) YoB ¼ YoB :
(iii) Define the ‘‘benchmark production correspondence at time t’’ as the same mapping as the previous one, i.e. bt ¼ bt1 : (B) If (xt , ut ) 2 Dg (t 1), then call (xt , ut ) a progress observation and: 11
This computation, as well as the three following ones can be shown (see Tulkens and Vanden Eeckaut (1995), Section 6) to be the solution of some mixed integer linear programming problems, a property that makes explicit the link with the methodology used in the DEA literature.
Ch.16 Non-Frontier Measures of Efficiency
387
(i) Attach to it a ‘‘performance index in inputs’’ computed as d xi i , min p (t) ¼ max t d2DdoB (t) i¼1,...,I xi and a ‘‘performance index in outputs’’ computed as ( !) t u j po (t) ¼ max , min d d2DdoB (t) j¼1,...,J uj (1,t1) where DdoB (t) is the subset of YoB whose elements are weakly domint t ated by (x , u ). Again, in either case, these indexes are computed in reference to the graph of the benchmark correspondence defined at time t 1. By construction, they both are larger than or equal to 1. (ii) Define the ‘‘benchmark observations set at time t’’ as (1,t) (1,t1) YoB ¼ YoB :
(iii) define also the ‘‘benchmark production correspondence at time t’’ as the same mapping as the previous one, i.e. bt ¼ bt1 : (C) If (xt , ut ) 2 Di (t 1), then call (xt , ut ) a benchmark observation and: (i) Attach to it ‘‘performance indexes in inputs and in outputs’’ I i (t) ¼ I o (t) ¼ 1. (ii) Define the ‘‘benchmark observations set at time t’’ as (1, t) (1,t1) YoB ¼ YoB [ {(x0 , ut )}:
into three subsets Dd (t), Dg (t) (iii) Partition again the whole space RIþJ þ and Di (t) defined exactly as above for t ¼ 1 and t ¼ 2; define further Dd (1, t) ¼
t [ t¼1
Dd (t),
Dg (1, t) ¼
t [
Dg (t),
Di (1, t) ¼
t¼1
t \
Di (t):
t¼1
Finally define the ‘‘benchmark production correspondence at time t’’ as the mapping bt : RIþ ! RIþ ; x 7!U(x) ¼ {uj(x, u) 2 Di (1, t)}: This expression provides the general definition of the concept introduced in this paper. (1, t1) To sum up, any observation (xt , ut ), given the set Yo of observations made before it, is evaluated by the method proposed here as: – a regress observation if (xt , ut ) 2 Dd (1, t), i.e. is dominated by some benchmark observation; it is then attached to a performance index smaller than 1;
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Public Goods, Environmental Externalities and Fiscal Competition
– a progress observation if (xt , ut ) 2 Dg (1, t), i.e. is dominating some benchmark observation; it is then attached to a performance index larger than 1; – a benchmark observation if (xt , ut ) 2 Di (1, t), i.e. is neither dominating nor dominated by any benchmark observation; it is then attached to a performance index equal to 1. (D) A graphical representation. To illustrate, let us consider the two numerical examples shown in Tables 1 and 2. The graph of the benchmark correspondence at time t ¼ 6 of example 1 (Table 1) appears on Fig. 6. With two inputs and one output, however, as in example 2 (Table 2), this graph is more complex as shown in Fig. 7(a) and (b). The figures are cut at the values 10 on each axis (with the truncated parts of the benchmark represented by a white pattern). Fig. 7(a) shows the benchmark with all the Dg subsets removed (i.e. all points in the space dominating the four benchmark observations). Fig. 7(b) presents the other perspective with all the Dd subsets removed (i.e. all points in the space that are dominated by the four benchmark observations). Both Figs. 6 and 7 suggest that the larger the number of observations, the more the graph of the correspondence tends to become ‘‘thin’’, so as to approach, in the limit, a monotonically increasing relation between inputs and outputs, i.e. an increasing production function that can have any form. Table 1 Example 1 (one input-one output) Units (time)
Input
A(1) B(2) C(3) D(4) E(5) F(6)
Output
2 4 1 5 6 7
2 4 5 6 6 7
Table 2 Example 2 (two inputs-one output) Units (time) A(1) B(2) C(3) D(4) E(5) F(6)
Input 1
Input 2
2 4 1 5 6 7
2 3 0.5 3 7 5
Output 2 4 5 6 6 7
Ch.16 Non-Frontier Measures of Efficiency
389
u 6
7 D
6
4
5
5
6
F
E
C
3
5
2
4
B
3 A
1
2 1
0
1
2
3
4
7
x
Figure 6. The benchmark correspondance at time t ¼ 6 of Example 1 (Table 1) for the one input-one output case.
10 F
7.5
D
10 7.5 Input 2 5 2.5
5 output
B
C
2.5
A
0
0
0
2.5
7.5
5
10
(a)
Input 1
10 7.5
F E
Output
5 D
2.5 B 0 A
10 7.5 Input 1
5
2.5
00
2.5
5
7.5 Input 2
10
(b)
Figure 7. The benchmark correspondance at time t ¼ 6 for the two inputs-one output case: (a) forward presentation; (b) backward presentation.
390
6.
Public Goods, Environmental Externalities and Fiscal Competition
An empirical illustration
The methodology of nonfrontier progress and regress analysis expounded above is the best illustrated by the data which triggered its formulation. These are monthly statistical data from the Socie´te´ des Transports Intercommunaux de Bruxelles (STIB) on one output (seatskms) and three inputs (energy, labor and vehicles) over the period January 1977–August 1991, yielding a times series of 176 observations. This is a once more updated version of a data bank first used in d’Aspremont (1984), next in Thiry and Tulkens (1992) then in Tulkens, d’Aspremont and Nollet (1989) and most recently in Tulkens (1993), Section 6. These sources provide all necessary information as to the nature of the figures. The data are presented graphically in Figs 8–11. A downward trend is noticeable in output from 1977 to mid-1981, while an upward trend prevails during the same period for the vehicle and the labor inputs. In contrast, an upward trend is visible in output between mid-1981 and mid-1991, as well as a downward trend in vehicle and labor inputs. Figs 12 and 13 report on the numerical results of intertemporal and sequential efficiency analyses, using the traditional frontier concept (and allowing for ‘‘local’’ technical progress, i.e. frontier shifts, in the second case). The frontier used here is the one of so-called ‘‘free disposal hull’’ (FDH) production sets, amply documented in Thiry and Tulkens (1992), Tulkens (1993). For the intertemporal analysis, a comparison of the efficiency degrees (or scores) with those provided by ‘‘DEA’’ convex production sets is provided in Fig. 14. The Banker, Charnes and Cooper (1984) version with ‘‘variable’’ or more precisely increasing and then decreasing returns to scale of DEA is used (hence the acronym DEA-ID12). The comparison shows that the two methodologies do not yield, on the average, very different results, except for the fact that DEA does not capture the seasonal variation in efficiency that FDH makes apparent. This is however only a minor issue in the present context. The sequential analysis reported in Fig. 13 shows, in contrast with the intertemporal one, that the years 1984–1986 did in fact achieve a frontier-efficiency performance quite comparable to the situation that had prevailed in the first 15 months of the observation period, i.e. January 1977–April 1978. Thus, the severe inefficiencies suggested for the 10 years period 1978–1988 by the intertemporal measure appear from the sequential measure to be concentrated only in the period extending from mid-78–mid-83. This is a substantial difference, that shows how the results 12
As already defined in Tulkens and Vanden Eeckaut (1995).
5.50 105 5.00 105 4.50 105 4.00 105 3.50 105 3.00 105 2.50 105 2.00 105 77
78
79
80
81
82
83 84 Year
85
86
87
88
89
90
91
90
91
Figure 8. STIB: output data (monthly seats-km) January 1977–August 1991. 1.35 105 1.30 105 1.25 105 1.20 105 1.15 105 1.10 105 77
78
79
80
81
82
83 84 Year
85
86
87
88
89
Figure 9. STIB: vehicle input (number of available seats per month) January 1977–August 1991. 1.00 106 9.00 105 8.00 105 7.00 105 6.00 105 5.00 105 77
78
79
80
81
82
83 84 Year
85
86
87
88
89
90
91
Figure 10. STIB: labor input (monthly hours worked) January 1977–August 1991. 3.00 107 2.50 107 2.00 107 1.50 107 1.00 107 77
78
79
80
81
82
83
84
85
86
87
88
89
90
Year
Figure 11. STIB: energy input (monthly Kwh used) January 1977–August 1991.
91
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Public Goods, Environmental Externalities and Fiscal Competition
105 100 95 90 85 80
Inputs Output
75 70 77
78
79
80
81
82
83
84 85 Year
86
87
88
89
90
91
89
90
91
Figure 12. STIB: efficiency degrees with intertemporal measurement. 115 110 105 100 95 90 85 80
77
78
79
80
81
82
83
84 85 Year
86
87
88
Figure 13. STIB: efficiency degrees and technical progress with sequential measurement in input.
of the intertemporal analysis can be substantially affected by the end period chosen. As this is not the case with sequential analysis, we take it to be a major reason for superiority of the sequential method over the intertemporal one. The sequential frontier method has however another shortcoming, which lies on the progress measurement side. Not only, as explained earlier, is it unable to identify regress, but its identification of progress is in addition somewhat misleading in that it measures progress only with respect to the last recorded frontier. Thus, Fig. 13 shows that progress occurs several times at STIB from 1985 on, but it makes it not possible to know whether the progress made in the year say 1991, is progress vis-a`-vis the years 1984–1988 or vis-a`-vis the year 1990. In particular, while any kind of ‘‘progress’’ is recorded, cases of ‘‘progress over progress’’ are not identified.
Ch.16 Non-Frontier Measures of Efficiency
393
100 90 80
Measure DEA-ID Measure FDH
70
77
78
79
80
81
82
83
84 Year
85
86
87
88
89
90
91
Figure 14. STIB: comparison between DEA-ID and FDH intertemporal measurements in output.
Regress
Progress
This is what is revealed by progress–regress indexes computed from an appropriately defined benchmark, as illustrated13 in Fig. 15. The new information brought about by this diagram is that progress in the years 1985–mid 1988 is essentially occasional, vis-a`-vis the benchmark established by the years 1977 and 1983–1984. In sharp contrast to that, from October 1989 until August 1991, progress is recorded at a higher level than in the years 1985–mid 1988. This means that progress over progress has occurred, that is, that the productive performance in efficiency terms is definitely much better in end-of-1989–August 1991 than in the years 1985–1988. This was much less apparent with the frontier-related sequential measures of Fig. 13, and simply impossible to detect with the intertemporal methodology illustrated by Fig. 12. 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 −0.05 −0.10 −0.15 −0.20 −0.25 −0.30 −0.35
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
Year
Figure 15. STIB: progress and regress indexes with benchmark measurement in output. 13
The values of the indexes show on Fig. 14 are different from those used in the text: we use at every t $ 0 P0 (t) 1 for P0 (t) and R0 (t) 1 for R0 (t), so that for observations belonging to the the benchmark are always zero. We feel that this presentation is visually more natural.
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A final word of comment is in order concerning the sometimes noticeable differences between measures in output and in inputs. It should be remembered that we deal with one output only vs. three inputs. Now, logic suggests that finding dominant observations (or dominated ones) is always ‘‘easier’’, or more probable, in one dimension than in higher ones. This explains that in the case of regress (or inefficiency), the input measure is always more favorable than the output measure, and that the reverse holds for measures of progress.
References Banker, R., Charnes, A. and Cooper, W.W., 1984. Some models for estimating technical and returns to scale inefficiencies in data envelopment analysis, Management Science, 30, 1078–1092. D’Aspremont, A., 1984. Une mesure ge´ne´rale de l’efficacite´ au sens de Farrell: application au cas de la STIB, The`se de Doctorat, Louvain-la-Neuve: Institut d’Administration et de Gestion, Universite´ catholique de Louvain. Fa¨re, R., 1991. Malmquist productivity indexes and technical change, paper presented at the Second European Workhop on Efficiency and Productivity Measurement. CORE, Louvain-la-Neuve: Universite´ catholique de Louvain. Fa¨re, R., Grosskopf, S. and Lovell, C.A.K., 1985. The measurement of efficiency in production. Boston-Dordrecht: Kluwer-Nijhoff. Fa¨re, R., Grosskopf, S., Lindgren, B. and Roos, P., 1989. Productivity developments in Swedish hospitals: a Malmquist output index approach, in A. Charnes, W.W. Cooper, A.Y. Lewin, and L.M. Seiford (eds.), 1993. Data envelopment analysis: theory, method and process. Boston, MA: Kluwer. Førsund, F., 1990. The Malmquist productivity index, paper presented at the Second European Workshop on Efficiency and Productivity Measurement. CORE, Louvain-la-Neuve: Universite´ catholique de Louvain, and Memorandum no 28, Department of Economics, University of Oslo. Grosskopf, S., 1986. The role of the reference technology in measuring productive efficiency, Economic Journal, 96, 499–513. Shephard, R.W., 1970. Theory of cost and production functions. Princeton: Princeton University Press. Thiry, B, and Tulkens, H., 1992. Allowing for technical inefficiency in parametric estimates of production functions, Journal of Productivity Analysis, 3(1/2): 45–66. Tulkens, H., 1986. La performance productive d’un service public: de´finitions, me´thodes de mesure, et application a` la Re´gie des Postes de Belgique. L’Actualite´ Economique, Revue d’Analyse Economique (Montre´al), 62(2), 306–335. Tulkens, H., 1993. On FDH efficiency analysis: some methodological issues and applications to retail banking, courts, and urban transit, Journal of Productivity Analysis, 4(1/2), 183–210. (Chapter 14 in this volume). Tulkens, H., d’Aspremont, A. and Nollet, C., 1989. Mesures de la performance productive d’une entreprise publique, in CORE (ed.). Gestion de l’e´conomie et de l’entreprise: l’approche quantitative. Brussels: De Boeck-Universite´, chapter 4.1, 297–320. Tulkens, H. and Vanden Eeckaut, Ph., 1995. Non-parametric efficiency measurement for panel data: methodological aspects, European Journal of Operations Research, 80(3), 474–499. (Chapter 17 in this volume).
Chapter 17 NONPARAMETRIC EFFICIENCY, PROGRESS AND REGRESS MEASURES FOR PANEL DATA: METHODOLOGICAL ASPECTS
Henry Tulkens, Philippe Vanden Eeckaut European Journal of Operations Research 80, 474–499 (1995).
This purely methodological paper deals with the role of time in nonparametric efficiency analysis. Using both FDH and DEA technologies, it first shows how each observation in a panel can be characterized in efficiency terms vis-a`-vis three different kinds of frontiers: (i) ‘contemporaneous’, (ii) ‘sequential’, and (iii) ‘intertemporal’. These are then compared with window analysis. Next, frontier shifts ‘outward’ and ‘inward’, interpreted as progress or regress are considered for the two kinds of technologies, and computational methods are described in detail for evaluating such shifts in either case. These are also contrasted with what is measured by the ‘Malmquist’ productivity index. Finally, an alternative way of identifying progress and regress, independent of the frontier notion and referring instead to some ‘benchmark’ notion, is extended here to panel data.
1.
Introduction: production sets and the efficiency concept
The concept of efficiency in production has received a precise meaning in economics when Koopmans and Debreu introduced in 1951 the This is a considerably revised version of part I of the paper ‘‘Nonparametric efficiency measurement for panel data: methodologies and an FDH application to retail banking’’, presented and circulated at EURO XI, the 11th European Congress on Operational Research, RWTH Aachen, Germany, July 16–19, 1991. Part II of that paper will appear separately. This revision was written under convention No. 2.4505.91 between the Belgian Fonds de la Recherche Fondamentale Collective and the first author, who also benefitted from a sabbatical support received in 1991–92 from CIM, the Colle`ge Interuniversitaire d’e´tudes doctorales dans les sciences du Management, Brussels. The research of the second author was financed in part by the Belgian Fonds de la Recherche Fondamentale Collective (convention No. 2.4528.88) and for the other part by the Belgian Ministry of Scientific Policy under its Programme d’Actions Concerte´es No. 87/92-106 with CORE. The authors are grateful to Le´opold Simar, Shawna Grosskopf and anonymous referees for a careful reading of their Aachen paper, and to C.A. Knox Lovell for valuable conversations on this topic.
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production set notion in the theory of production. A production set, in their terminology,1 is a collection Y of pairs of vectors (x, u) – called hereafter production plans – where x ¼ (x1 , . . . , xi , . . . , xn ) is a vector of quantities of I inputs and u ¼ (u1 , . . . , ui , . . . , un ) a vector of quantities of J outputs, that have the property of being feasible ones. By ‘feasible’ it is meant that the quantities are such that the output bundle u can physically be produced by making use of the input bundle x. In formal terms, Y ¼ {(x, u)jx 2 RIþ , u 2 RJþ ; (x, u) is feasible}: The usefulness of the set notion for our purposes comes from the fact that it brings along two further notions, namely (i) that of the boundary (or frontier) of the set, and (ii) that of the interior of the set. This indeed permits one to distinguish between production plans that belong to the interior of the production set, which are called inefficient, and those that do belong to (a particular part of) the frontier, and are called efficient. Inefficiency so defined yields itself to measurement once the property of belonging to the interior is translated in terms of a distance – that is, a real number – between the production plan under consideration and the boundary of the set. In this setting, efficiency of the plan amounts to zero distance, and inefficiency to a strictly positive distance.
2.
Constructing alternative reference production sets from observed data: the postulates behind FDH, DEA and other methodologies
In most empirical analyses of efficiency, the analyst must construct a representation of the production set from the data he has available on input and output quantities, because the set itself, being an abstract concept, is not observable. He then uses the set’s frontier to measure the efficiency of the activities reported by the data. Grosskopf 1986 aptly calls ‘reference’ set2 the production set so constructed from empirical data. In so doing, any production plan that is included in the production set and is not observed necessarily results from some postulate, explicit or implicit, as to its feasibility. Hence, depending upon which postulates are retained, a different reference set is constructed. Formally, let 1 In Banker, Charnes and Cooper (1984) and Petersen (1990) the term used is that of ‘production possibilities set’. Fa¨re, Grosskopf and Lovell (1985) and Grosskopf (1986) prefer to call a production set a ‘technology’. In this paper, we take all these expressions as synonyms. 2 In her words, ‘reference technology’.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 397
Y0 ¼ {(xk , uk )jxk 2 RIþ , u k 2 RJþ , k ¼ 1, 2, . . . , n} [ {(OI , OJ )}
(1)
denote a set of n actually observed production plans, to which the origin of the input–output space is added by convention3 (OI and OJ are the I- and J-dimensional null vectors); for brevity we call Y0 the observations set or the data set. Let also Y (Y0 ) denote a reference production set constructed from Y0 . If the following two postulates are used:4 Postulate (i). Every observed production plan belongs to the constructed production set. Postulate (ii). Every nonobserved production plan that is weakly dominated in inputs and/or in outputs by some observed production plan also belongs to the constructed production set. Then a ‘free disposal hull (FDH) reference production set’, YFDH , constructed from Y0 can be written as follows: " # " # ( " h # I J X X eJj u OJ u IþJ u þ , 2 Rþ m n YFDH (Y0 ) ¼ ¼ j i x x eIi xh OI i¼1 j¼1 ) (xh , uh ) 2 Y0 ; mi $ 0; nj $ 0, i ¼ 1, . . . ,I; j ¼ 1, . . . ,J
(2)
where eIi denotes an I-dimensional zero vector with the i-th component equal to 1 and similarly eJj denotes a J-dimensional zero vector with the jth component equal to 1. The first postulate makes the ensuing methodology of efficiency measurement a deterministic one5. The methodology would indeed be stochastic if this belonging property were formulated in probabilistic terms. Postulate (ii) is the one of free disposal of inputs and of outputs, from which the name of the constructed reference set is derived6. We
3
The reason for including the origin in Y0 will appear below, in particular when we shall discuss the DEA case. 4 We expand here on a formulation originally used in Deprins, Simar and Tulkens (1984), and further developed in Tulkens (1993). 5 The actual computation of efficiency is dealt with in Section 4. 6 This terminology was introduced in efficiency analysis by Thiry and Tulkens (1992, p. 46). In private communication with one of the authors, Professor Cooper expresses a preference for ‘efficiency domination’, which had already been used in an unpublished note (Bowlin, Brennan, Charnes, Cooper and Sueyoshi (1984). Notice that free disposal is defined, here as in Debreu (1959, p. 42) as a property of the production set, rather than of market conditions – which we find irrelevant – as in the discussion by Banker, Charnes and Cooper (1984, p. 1082).
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express it here in terms of ‘dominance’, a term that for our present purposes may be defined as follows: Definition. In (2), observation (xh , uh ) weakly dominates (x, u) in inputs if 9i: mi > 0, and in outputs if 9j: nj > 0. The values taken by the variables mi (denoting disposal of the inputs) and nj (disposal of the outputs) thus express the free disposal assumption. Notice that an equivalent way to write (2), useful for what follows, is " # " # " # ( n I J J h X X X eJj u u O u h ¼ þ , YFDH (Y0 ) ¼ 2 RIþJ g m n j i þ x x eIi xh OI i¼1 j¼1 h¼1 n X g h ¼ 1, (xh , uh ) 2 Y0 ; gh 2 {0, 1}, h ¼ 1, . . . , n; h¼1
)
mi $ 0, i ¼ 1, . . . , I; nj $ 0, j ¼ 1, . . . , J :
(3)
In the DEA literature, efficiency is measured by referring to production sets similarly constructed from the data, and using a third postulate in addition to the two mentioned above. This third postulate differs, however, depending upon which DEA model is used: the one of Banker, Charnes and Cooper (1984), the one of Deprins and Simar (1983), or the original one of Farrell (1957) and Charnes, Cooper and Rhodes (1978). Let us consider each of these in turn, in that chronologically reverse order. Postulate (iii-1). Every nonobserved production plan that is a convex combination of production plans induced by postulate (i), excluding the origin, and by postulate (ii), also belongs to the constructed production set. Using the above notation, postulates (i), (ii) and (iii-1) induce a reference production set that reads YDEA-ID (Y0 ) ¼
2 3 " # " h# " J# n I J u ejJ O u X X X h 4 5, 2 RIþJ g m n þ ¼ j i þ x x eiI i¼1 j¼1 xh h¼1 OI
(" # u
(xh , uh ) 2 Y0 n{(OI , OJ )}, g h $0, h ¼ 1, . . . , n,
n X
g h ¼ 1,
h¼1
) mi $0, i ¼ 1, . . . , I, nj $0, j ¼ 1, . . . , J ,
(4)
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 399
to be called the ‘convex free disposal hull’ of the data. Because the origin is excluded from the data set, the constructed set does not contain the origin either. The piecewise linear frontier of this set is often interpreted as exhibiting ‘variable’ returns to scale (VRS). As Banker, Charnes and Cooper (1984) clearly show, this is only true however in the special sense that returns are increasing (I) only in the lower range of the inputs up to some point, and are decreasing (D) beyond. As there are other forms conceivable of variable returns7, we think it more accurate to use ‘ID’ rather than ‘VRS’8 in the subscripts used to identify the set constructed here. Postulate (iii-2). Every nonobserved production plan that is a convex combination of production plans induced by postulates (i) and (ii) also belongs to the constructed production set. With this postulate (iii-2), postulates (i) and (ii) induce a reference production set that reads " # " # " # ( n I J h X X X eJj u OJ IþJ u h u þ , YDEA-CD (Y0 ) ¼ g m n 2 Rþ ¼ j i x x eIi xh OI i¼1 j¼1 h¼1 (xh , uh ) 2 Y0 , g h $ 0, h ¼ 1, . . . , n;
n X h¼1
g h ¼ 1,
)
mi $ 0, i ¼ 1, . . . , I; nj $ 0, j ¼ 1, . . . , J :
(5)
Here too, the constructed set is the convex free disposal hull of the data, but the fact that it contains the origin implies that its (piecewise linear) frontier can only exhibit constant returns (in the lower range of the inputs, up to some point) and then decreasing returns to scale (which is what Deprins and Simar (1983, p. 136) aimed at introducing). Hence our ‘CD’ specification in the identifying subscripts. Postulate (iii-3). Every nonobserved production plan that is a linear combination of production plans induced by postulates (i) and (ii) also belongs to the constructed production set.
7 Typically, the FDH sets just considered may exhibit any form of returns to scale: returns are thus in some sense ‘more’ variable than in the ID case. This is analogous to the situation with parametric frontiers, where the flexibility of translog functions allows for more variability in the returns than ID type of production functions. 8 E.g. by Førsund (1992) or Lovell (1993, pp. 29–30).
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In our notation, the reference production set YDEA-C constructed from Y0 with postulates (i), (ii) and (iii-3) reads as follows: ( " # " # " # n I J h X X X eJj u OJ IþJ u h u YDEA-C (Y0 ) ¼ g m n ¼ þ , 2 Rþ j i x x eIi xh OI i¼1 j¼1 h¼1 (xh , uh ) 2 Y0 , g h $ 0, h ¼ 1, . . . , n;
)
mi $ 0, i ¼ 1, . . . , I; nj $ 0, j ¼ 1, . . . , J :
(6)
Because of the proportionality allowed by the weights of the linear combinations, this set is a cone, to be called the ‘free disposal cone’ constructed from the data. As well noted by both Farrell (1957) and Charnes, Cooper and Rhodes (1978), the linearity of its frontier implies that it exhibits constant returns to scale, hence our subscript ‘C’. It can easily be seen9 that for a given data set Y0 , YFDH YDEA-ID YDEA-CD YDEA-C :
(7)
This means that the postulates (i) through (iii-3) have been stated in order of increasing strength, this term being understood as referring to the importance of the parts of each reference set that are not made of observed points. Formally, the relative strength of the postulates is expressed by the alternative restrictions put on the weights gh whereby the observations are combined, on the sum of these weights, and on the convention to include or not the origin in the observations set. The convexity assumption, expressed in or implied by the postulates (iii) that induce some form of DEA, deserves a special further comment. The above exposition of the FDH methodology clearly shows that it can be dispensed with, without losing the economically essential element of efficiency analysis, namely a well-defined reference production set. Hence, the question of whether this assumption should be retained in this area becomes an essentially empirical one. The nonnecessary character of convexity in efficiency analysis has found a further confirmation in the recent contribution by Petersen (1990), who introduces two other kinds of nonconvex piecewise linear production sets, different from FDH. It can be shown that these reference production sets also derive from stronger postulates than (i) and (ii), among which some form of proportionality. They lie somehow in-between the FDH and the DEA-C reference sets.
9
A diagrammatic presentation of these inclusions is given in Tulkens (1993, pp. 185–186).
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 401
3.
Constructing reference production sets from alternative time-related observations subsets
3.1
Panels
The n observed production plans referred to in the above definition of the set Y0 are most naturally thought of as a cross-section, i.e. plans achieved by different firms10, each one indexed by k and all observed at the same point in time. Alternatively, the n production plans may be a time series of successive such plans realized by a single firm, the index k then standing for a time index. When these two situations are combined, Y0 is a panel. Two indexes are then better associated with each observation: k to denote the firm, and t to denote the point in time when the observation is made. An observation now reads (xkt , ukt ) and the data set is described as Y0KT ¼ {(x kt , u kt ) j x kt 2 RIþ , u kt 2 RþJ , k ¼ 1, 2, . . . , n; t ¼ 1, 2, . . . , m} [ {(O I , O J )},
(8)
where the superscripts K and T refer to the sets of firms and of observation times, respectively, and with the same convention as in (1) as to the origin. The presence of the time dimension in the data introduces a new variety of ways of evaluating the efficiency of observed productive behavior. Indeed, besides the necessity of choosing one among the several alternative reference production sets that we have seen in the previous section, there may arise reasons to construct it not from the full data set Y0KT , but instead from only alternative subsets of it, that we shall call ‘reference observations subsets’. In Tulkens (1986, Section 6) the following possibilities were described in this respect11: (a) A first one is to construct a reference production set at each point in time t, from the observations made at that time only. In this case, a different reference observations subset, denoted as Y0Kt ¼ {(x kt , u kt ) j k ¼ 1, 2, . . . , n} [ {(O I , O J )}
(9)
is used at each point in time t ¼ 1, 2, . . . , m. Over the whole observation period, a sequence of m reference production sets is constructed, one for each time t. We call ‘contemporaneous’ each of these production sets and denote them as
10
Or more generally ‘decision making units’, as nicely formulated by Charnes, Cooper and Rhodes (1978). 11 With an empirical application to monthly data, extending over 12 months, from the Belgian Post Office, with p ¼ FDH.
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Yp (Y0Kt ), t ¼ 1, 2, . . . , m, where the subscript p stands for the alternative sets of postulates that imply FDH, DEA-ID, -CD or -C reference production sets. Contrary to those that will be defined next, successive production sets so constructed, even if they are of the same type p, are essentially unrelated to one another; that is, they may or may not overlap in any possible way. (b) A second possibility is to construct a reference production set at each point in time t also, using the observations made from the point in time s ¼ 1 up until s ¼ t. The reference observations subsets at each t ¼ 1, 2, . . . , m being here denoted as K(1, t)
Y0
¼ {(x ks , u ks )jk ¼ 1, 2, . . . , n; s ¼ 1, 2, . . . , t} [ {(O I , O J )},
(10)
m successive reference production sets are thereby constructed, different of course from those under (a) above, that we call ‘sequential’ and denote as K(1, t)
Yp (Y0
), t ¼ 1, . . . , m:
Here too, the subscript p stands for FDH, DEA-ID, -CD or -C, according to which postulates of Section 2 are retained for their construction. However, whatever the shape thus obtained for these sets, since K(1, t) Y0
¼
t [
Y0Ks ,
s¼1
successive sequential reference production sets have the property of being nested into one another, that is: K(1, tþ1)
Yp (Y0
K(1, t)
) Yp (Y0
)
for every t ¼ 1, 2, . . . , m 1:
(11)
(c) A third possibility is to construct a single production set from the observations made throughout the whole observation period. The reference observations subset is here simply Y0KT , as defined at the beginning of this subsection. We call the resulting reference production set ‘intertemporal’ and denote it as Yp (Y0KT ). With p ¼ FDH, DEA-ID, -CD or -C, its shape depends once more upon the postulates of Section 2 that are retained. Notice that Yp (Y0KT ) Yp (Y0K(1, m) )
8p,
that is, intertemporal production sets are also the sequential production sets associated with the observation(s) made at the last point in time in the data set.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 403
3.2
Time series
Further reference observations subsets can be specified by treating each unit k separately, thereby singling out k-specific time series. This yields: (a) ‘k-specific sequential production sets’, denoted as Yp (Y0k(1,t) ), t ¼ 1, . . . , m and derived at each t from the observations set consisting of the time series of the activities of the sole unit k from s ¼ 1 up to s ¼ t; and (b) ‘k-specific intertemporal production sets’, denoted as Yp (Y0kT ). There is in this case a single time series of m observations of firm k. Notice that the indices k and T are here reversed with respect to K and t in the definition of Y0Kt given at the beginning of this section. Efficiency analysis of time series observations sets has been dealt with by Fa¨re, Grabowski and Grosskopf (1985) in terms of sequential production sets with p ¼ DEA-ID, and with an application to yearly data (one output, four inputs) extending over twenty years on Philippine agriculture. Both the sequential and the intertemporal approaches were used in Thiry and Tulkens (1992), and pursued in Tulkens and Vanden Eeckaut (1991) in the same spirit as that of the present paper, with applications to monthly data of an urban transit firm in Belgium extending over several years. We shall therefore limit ourselves in what follows to the panel situations covered by cases (a)–(c) above. 3.3
Window analysis
Finally, a connection should be made between the above and what is called ‘window analysis’, as initiated by Charnes, Clark, Cooper and Golany (1985). Window analysis is in fact a special case of (b). Indeed, a (time) window of size s 2 {1, 2, . . . , r; r < m} at time t is defined as a subset of adjacent points in time T st ¼ {tjt ¼ t, t þ 1, . . . , t þ s; t # m s} whose observations are used to construct an intertemporal reference production set, valid for the time period [t, t þ s] only. Successive such windows, defined for t ¼ 1, 2, . . . , m s, yield a sequence of reference production sets, that we might call ‘locally intertemporal’. These sets are not nested however, as it is the case with sequential analysis in (b) above. In contrast with this last methodology, whose spirit is to assume that what was feasible in the past remains feasible for ever, the treatment of time in window analysis is more in the nature of an averaging over the periods of time covered by the window. For the size of the window, as well as for the fact that part of the past is ignored, it seems to be hard to find more than an ad hoc justification. To summarize this section, for measuring the efficiency of observations (xkt , ukt ) in a panel, there is a double variety of ways to proceed:
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first, according to the postulated production set used as reference – FDH, DEA-ID, -CD or -C, and second according to the selected observations subset used to construct the production set – contemporaneous, sequential, intertemporal, or ‘window’. This makes for 16 possibilities! While the best choice to be made between them surely varies with the empirical application to be dealt with, their main interest lies perhaps in the fact that they also pave the way for the progress and regress analyses that are to follow in Sections 5 and 6.
4.
Mathematical programming formulations of efficiency measurement from contemporaneous, sequential or intertemporal frontiers
We now move to the techniques of numerical measurement of efficiency when the data are in panel form. The main purpose is to show that measurements can be made in all cases by means of mathematical programming techniques that are by now well established for computing efficiency from statistical data on inputs and outputs. This provides a unifying framework for the various approaches defined in the preceding two sections. We use a presentation similar to the one in Tulkens (1993) which is in turn close to those of Petersen (1990), Grosskopf (1986) and many others. To the usual distinction between input and output orientations we find it useful to add the graph orientation proposed in Fa¨re, Grosskopf and Lovell (1985, Chapter 5)12 and we call efficiency ‘degrees’ the resulting measures. These will be contrasted below (see Section 6) with what we shall call there ‘indexes’ 4.1
Alternative reference observations subsets
To allow for a flexible use of the various reference observations 0 , to denote subsets we have defined, we wish to define a further symbol, Y for any set Y0 the set of indexes whereby its observed elements are identified (thus, ignoring the origin); in the case of double indexes as 0 refers to index pairs: for instance, for Y0 ¼ Y KT with panel data sets, Y 0 0 ¼ {(kt)j(xkt , ukt ) 2 Y KT }. as defined in (7), Y 0 For any observation (xkt , ukt ), its input, output and graph efficiency degrees, relative to an FDH reference production set and to some refer ence observations subset Y0 , are determined from the values ukt , lkt and kt d , respectively, of the objective functions at the optimal solutions of the following mixed integer programs:
12
The input and output measures are radial ones whereas the graph orientation measure is a hyperbolic one.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 405
Problem 1.1 (input orientation): min
0 } {ukt , g hs ,(hs)2Y
subject to
ukt
X
0 (hs) 2Y
g hs ujhs $ ujkt , j ¼ 1, . . . , J,
u kt xikt X
X 0 (hs)2Y
g hs xihs $ 0, i ¼ 1, . . . , I,
0: g hs ¼ 1 and g hs 2 {0, 1}, (hs) 2 Y
0 (hs)2Y
Problem 1.2 (output orientation): max
0 } {lkt , g hs ,(hs)2Y
lkt
subject to lkt ujkt X 0 (hs)2Y
X
X 0 (hs) 2Y
g hs ujhs # 0, j ¼ 1, . . . , J,
kt ghs xhs i # xi ,
i ¼ 1, . . . , I,
0: ghs ¼ 1 and ghs 2 {0, 1}, (hs) 2 Y
0 (hs)2Y
Problem 1.3 (graph orientation): min
0 } {dkt , g hs , (hs)2Y
dkt
subject to uikt =d kt dkt xkt i X
X 0 (hs)2Y
X
(hs) 2 Y
g hs ujhs # 0, j ¼ 1, . . . , J,
g hs xihs $ 0, i ¼ 1, . . . , I,
g hs ¼ 1 and g hs 2 {0, 1},
0 (hs) 2 Y
where 0 ¼ {(hs)j(x hs , u hs ) 2 Y Kt }, or (a) Y 0 0 ¼ {(hs)j(x hs , u hs ) 2 Y K(1, t) }, or (b) Y 0 0 ¼ {(hs)j(x hs , u hs ) 2 Y KT }. (c) Y 0
0: (hs) 2 Y
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In these problems we have ukt # 1, lkt $ 1 and dkt # 1, respectively because in problem 1.1, ukt ¼ g kt ¼ 1, g hs ¼ 0, 8(hs) 6¼ (kt), is always a feasible solution; similarly in problem 1.2, lkt ¼ g kt ¼ 1, g hs ¼ 0, 8(hs) 6¼ (kt), is also a feasible solution; and in problem 1.3, dkt ¼ gkt ¼ 1, ghs ¼ 0, 8(hs) 6¼ (kt), is a feasible solution. Therefore, – If (a) applies, the numbers ExC (kt; FDH) ¼ ukt , EuC (kt; FDH) ¼ [lkt ]1 kt C and E(x , u) (kt; FDH) ¼ d , all three # 1, measure the contemporaneous FDH-efficiency degrees, respectively in input, in output and in the graph measure, of observation (xkt , ukt ). – If (b) applies, the three numbers ExS (kt; FDH) ¼ ukt , EuS (kt; 1 kt S FDH) ¼ [lkt ] and E(x , u) (kt; FDH) ¼ d , all # 1, measure the sequential FDH-efficiency degrees, respectively in input, in output and in the graph measure, of this observation. – If (c) applies, the numbers ExI (kt; FDH) ¼ ukt , EuI (kt; FDH) ¼ [lkt ]1 kt I and E(x , u) (kt; FDH) ¼ d , again all # 1, measure the intertemporal FDH-efficiency degrees, in input, in output and in the graph measure, of the same observation. To cover all the observations in the panel, n m problems, i.e. one for each pair (kt), are to be solved in cases (a) and (b), in each one of the input, the output and the graph orientations; in case (c), only n problems are involved in each orientation, i.e. one for each unit k. 4.2
Alternative reference production sets
Problems 1.1 to 1.3 are all formulated for gauging the efficiency of an observation relative to an FDH reference production set as defined in (3). However, if in either one of these problems, the last line is replaced by X 0, ghs ¼ 1 and g hs $ 0, (hs) 2 Y (12) 0 (hs) 2 Y
the efficiency degrees in input, output and graph are those relative to DEA-ID production sets as described by (4). We denote them as r Exr (kt; DEA ID), Eur (kt; DEA ID) and E(x , u) (kt; DEA ID), respectively, with the superscript r ¼ C, S, or I according to whether (a), (b) or (c) applies. If instead of line (12) the following is used: X 0, ghs # 1 and g hs $ 0, (hs) 2 Y (13) 0 (hs) 2 Y
the efficiency degrees obtained are those relative to the DEA-CD reference production sets defined in (5), that we denote as Exr (kt; DEA CD), r Eur (kt; DEA CD) and E(x ,u) (kt; DEA CD). Notice that the fact that the origin was included in (5) and excluded from (4) is reflected here in
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 407
the weak inequality vs. the equality with respect to 1 that is imposed in (13) and (12), respectively, on the sum of the weights ghs whereby the observation (kt) is expressed as a convex combination of other observations. Finally, replacing (13) by 0, ghs $ 0, (hs) 2 Y
(14)
implies assuming that the reference production set is of the DEA-C type as defined in (6), i.e. the smallest Farrell cone containing the data. The efficiency degrees are then naturally denoted by Exr (kt; DEA C), r Eur (kt; DEA C) and E(x , u) (kt; DEA C). Of course, the choice of the alternative observations sets referred to with (a), (b) and (c) is also offered when anyone of these three alternative forms of DEA is used. For computing the solutions of these mathematical programs there exist today appropriate algorithms and computer codes. In this respect, it should be mentioned that in spite of the integrality constraint that makes problems 1.1–1.3 mixed integer ones, they are in fact much simpler to solve than the linear programs obtained when any of the constraints (12), (13) or (14) are introduced. Indeed, an algorithm of complete enumeration, described in Tulkens (1993, p. 189) and consisting essentially in pairwise dominance comparisons of all input-output vectors in the data set, does the job13. We refer to this simple algorithm because it points to an important difference in the logic that lies behind FDH vs. DEA methodologies. The role of the integrality constraint is indeed essentially to identify a dominance relation between observed production plans. On the one hand, an observation is declared ‘efficient’ – and considered as lying on the boundary of the reference production set – if it is undominated: undomination, belonging to the frontier and efficiency are thus synonyms in FDH methodology. On the other hand, an observation is declared ‘inefficient’, i.e. it lies in the interior of the set, if it is dominated by one or several other ones; and in this case the mixed integer program identifies a ‘most dominating’ observation that serves as a reference to compute the efficiency degree, either in input or in output. By contrast, the linear programs used in DEA seek to compute a distance (in input, in output or graph) with respect to the frontier of a convex envelope of the data. While dominance plays also some role in identifying this envelope, the additional requirement of convexity induces 13 A computer code for large panel data sets, allowing for comparisons of FDH results with those obtained with each one of the DEA variants, and operating on Macintosh is currently being prepared by the second author.
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the possibility that undominated observations be declared inefficient because they do not lie on the convex envelope of the data.
5.
Measuring progress and regress from frontier shifts
5.1
Progress and regress with nonparametric frontiers
Over the time period covered by a panel, the question naturally arises whether the boundary of the reference production set can shift. The analyst’s answer to this question is contained in the reference observations subset he decides to select from within the panel data he has available: (i) at one extreme, if he uses the full data set to construct a single intertemporal production set, he makes the assumption of no shift at all, (ii) at the other extreme, if he uses at each t the contemporaneous observations only, to construct contemporaneous production sets, he is answering the question in the affirmative, by allowing the reference production sets at each point in time to be completely different from one another, without there being a priori any relation between them; and (iii) with the construction of sequential production sets, which are also different from one another over time, there is postulated some form of dependence between these sets, formalized by the property that they are nested. The dependence consists in assuming that ‘what was possible in the past remains always possible in the future’. This amounts to allowing only for outward shifts of the frontier over time, that is, enlargements of the constructed production possibilities set. In economics, beginning with Solow (1957), the notion of ‘progress’ – more specifically called ‘technical progress’ – has for a long time been associated with outward shifts of production frontiers. Symmetrically, inwards shifts refer to ‘regress’. While this literature deals essentially with parametric production functions, there is no reason why these notions of progress and regress could not be extended to shifts of nonparametric representations of the boundary of production sets. It is indeed not only a natural complement to nonparametric efficiency measures, but also an inescapable one when the data involve time. Our purpose in this section is therefore to provide suitable concepts and methods for measuring frontier shifts of nonparametric frontiers. Starting with the remark that observations inducing shifts of a frontier are by definition lying at some distance away from where this frontier was before these observations were made, it follows that measuring this distance is precisely what a measure of progress or regress should do. For this task, the techniques of efficiency measurement are of direct use, as we shall show presently. Frontier shifts can indeed be gauged by programming models that are somehow mirror images of those used to
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 409
gauge efficiency; and we shall show how to compute ‘progress–regress degrees’, just parallel to the ‘efficiency degrees’ considered so far. The methods vary, however, according to the postulated reference production set, i.e. FDH or DEA. 5.2
Defining and measuring progress and regress with FDH frontiers
a) Definitions As was suggested above, a logic of dominance – more than one of envelopment – governs the construction of FDH production sets as well as the formulation of the ensuing efficiency measurement. To be consistent with that, the description of shifts of the frontier of these sets should be done in the same spirit. We therefore propose the following concepts. Definition 1. An observation (xkt , ukt ) is said to induce ‘progress’ if it is (i) undominated at time t, and (ii) dominating one or several observations, made at some time s < t and found undominated at s. Since undomination is the same as belonging to the frontier, the distance between the observation made at time t and one of those (appropriately chosen if there are several of them) undominated made at the earlier time s provides a measure of the outward frontier shift locally achieved by the former. Time s as used here will henceforth be called ‘reference time’. In many recent applications, s is taken as t 1, but it need not be so in all cases. Definition 2. An observation (xkt , ukt ) is said to induce ‘regress’ if it is (i) undominated at time t, and (ii) dominated by one or several observations, made at times s < t and found undominated as s. As above, the distance between the observation at time t and one of those made earlier (appropriately chosen if there are several of them) provides a way to measure the (local) inwards frontier shift induced by the former. These definitions are incomplete however, to the extent that they do not specify the observations reference sets with respect to which the above domination-undomination properties are considered. In correcting for that, when this specification is introduced the following distinction needs to be made. On the one hand, if domination-undomination at times t and s < t are specified with respect to Y0Kt and Y0Ks , respectively, progress and regress are defined in terms of a relation between the two contemporaneous frontiers that these observations subsets induce; call them ‘contemporaneous progress and regress’. On the other hand, if domination-undomination are specified K(1, t) K(1, s) and Y0 , progress and regress should be with respect to Y0
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defined in terms of a relation between the two sequential frontiers that these observations subsets induce, and be called ‘sequential progress and regress’. Sequential regress is however a logical impossibility: indeed, going back to the definition, if an observation is sequentially undominated at time t, it cannot be dominated by any observation made at s < t. Thus with sequential reference sets, only progress can ever occur. What would be interpreted as regress with contemporaneous reference sets is in fact treated as inefficiency in a sequential context14. Bearing this distinction in mind, we now turn to how to compute degrees of progress or regress, with FDH reference sets. b) Measurement with contemporaneous FDH frontiers For any observation (xkt , ukt ) to be characterized vis-a`-vis the contemporaneous frontier at reference time s, proceed according to the following five steps (stated explicitly only for input measures; to alleviate this text, the parallel procedures for output and graph measures are given in the Appendix): Step 1. Compute first the FDH efficiency in input of (xkt , ukt ) according to problem 1.1 above, with Y0 ¼ Y0Kt as reference observations subset. Step 2. If (xkt , ukt ) is found inefficient, it is not a frontier observation and thus cannot be used for frontier shift measurement; the computation is therefore terminated as far as observation (xkt , ukt ) is concerned15. If (xkt , ukt ) is found efficient, go to Step 3. Ks Step 3. Consider Y0Ks (x; FDH), i.e. the subset of observations in Y0 that were found FDH input efficient at reference time s, and compute the optimal solution of the following Problem 2.1: max
0} {ukt , ghs , (hs) 2 Y
subject to
ukt ,
X
0 (hs) 2 Y
kt g hs uhs j # uj , j ¼ 1, . . . , J,
14 In the context of nonparametric efficiency analysis of time series, the above ‘progress’ notion was formulated first in Thiry and Tulkens (1992, Section 4, pp. 56–57), but only for sequential observations subsets. Indeed in that context, contemporaneous efficiency is trivially achieved by all observations (since there is only one at each t), making the progress as well as the regress measures from contemporaneous production sets not meaningful. 15 In spite of being inefficient, observation (xkt , ukt ) might well lie outside of the reference production set constructed from the observations subset Y0Ks (x; FDH) to be defined in the following step 3: the observation itself is then in progress vis-a`-vis the
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 411
ukt xkt i X
X 0 (hs) 2 Y
ghs xhs i # 0, i ¼ 1, . . . , I,
0, g hs ¼ 1 and ghs 2 {0, 1}, (hs) 2 Y
0 (hs) 2 Y
where 0 ¼ {(hs) j s < t; (xhs , uhs ) 2 Y Ks Y 0 (x; FDH)} [ {(kt)}:
(15)
For this problem again, the values ukt ¼ g kt ¼ 1, g hs ¼ 0, 8(hs) 6¼ (kt), of the variables are always a feasible solution. Therefore, one has ukt $ 1 for the value of the objective function at the optimum. Call this value PC x (kt, s; FDH), a short notation for denoting the ‘progress degree (P) of observation (kt) relative to an FDH frontier at time s, measured in input (x) from contemporaneous data (C)’. If PC x (kt, s; FDH) > 1, no further computation needs to be done as far as observation (xkt , ukt ) is concerned. Progress is indeed measured by this number in terms of the proportion of how much more of all16 inputs unit k at time t would have used if it had used the same input bundle as the unit it dominates most (in input) in Y0Ks . In Fig. 1a, such a value of PC x (kt, s; FDH) is illustrated by the ratio OB/OA. If PC (kt, s; FDH) ¼ 1, go to Step 4. x Step 4. With PC x (kt, s; FDH) ¼ 1, there is no progress, but there may be regress. To find that out, compute the optimal 0 defined as in (15). solution of Problem 1.1 above, with Y kt Here, one has u # 1 (since as stated there, ukt ¼ gkt ¼ 1, g hs ¼ 0, 8(hs) 6¼ (kt), is always feasible) and call RC x (kt, s; FDH) that value of the objective function at the optimum. RC x (kt, s; FDH) < 1 measures regress, in terms of the fraction of the actual input usage made by unit k at time t that was used at time s by the unit that dominates k most (in input). On Fig. 1b, RC x (kt, s; FDH) is illustrated by the ratio OB/OA. With this value the computation is also terminated. If RC x (kt, s; FDH) ¼ 1, go to Step 5. Step 5. Finally, if PC (kt, s; FDH) ¼ RC x x (kt, s; FDH) ¼ 1, neither progress nor regress prevail; the frontier observation (xkt , ukt ) is neither dominating nor dominated by any of
frontier at time s, and this progress can also be measured by the procedure that follows; this measure is however not one of a frontier shift. 16 This is indeed a radial measure.
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(b) u
u (kt)
YFDH(Y Ks ) o
YFDH(Y Ks ) o (kt)
A
O
B
X
O
B
A
X
Figure 1. Progress and regress with contemporaneous FDH frontiers.
the frontier observations at s and there is therefore no frontier shift. This completes the progress vs. regress characterization of observation (xkt , ukt ). Applying the above procedure to all observations in Y0Kt yields a complete description of the frontier shift achieved at time t. Figs. 2a and 2b show two typical cases: in the former, one of the two reference sets contains the other; in the latter, the two sets only partially overlap, exhibiting regress for low values of the output and progress for large ones. In comparison with efficiency measurement, we notice three differences in the evaluation of progress vs. regress: (i) Progress vs. regress may require up to three computations for each observation, namely those described at Steps 1, 3 and 4. (ii) The elements of the reference observations subsets used in the programming models of Steps 3 and 4 are characteristically the observations that make up the frontier at time t 1 plus (xkt , ukt ) itself which ensures that these problems always have a solution. In problems 1.1–1.3, the reference observations subsets used were containing only observations made at time t. (a) YFDH(Y Kt ) o
(b) YFDH(Y Kt ) o
u
u
YFDH(Y Ks ) o
YFDH(Y Ks ) o
O
X
O
Figure 2. Alternative shifts of contemporaneous FDH frontiers.
X
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 413
(iii) In Step 4 the problem whose solution measures regress actually has a structure identical to that of problem 1.1, designed to measure efficiency; only the reference observations subset is different. But this is only natural, since regress amounts to inefficiency vis-a`-vis the frontier at t 1. By contrast in Step 3 the problem 2.1 that measures progress is quite different from problem 1.1, but it is somehow a mirror image of it with the operator max substituted for min, and all inequalities reversed. Again, this is natural since the dominance relation that is sought for is in the reverse order compared with the one involved when efficiency is measured. c) Measurement with sequential FDH frontiers Similar computations can be done for measuring progress with sequential frontiers as defined in Section 3. In this case, however, the reference time s can only be t 1. For the rest, the steps to be applied to each observation (xkt , ukt ) are only slightly modified as follows: K(1, t) Step 1. Use Y0 ¼ Y0 as reference observations subset. Step 2. Unchanged. K(1, t1) Step 3. Consider Y0 (x; FDH), as the subset of observations found FDH sequentially input efficient at time t 1, and 0 defined compute the solution of problem 2.1 with Y K(1, t1) hs hs as Y0 ¼ {(hs)j(x , u ) 2 Y0 (x; FDH} [ {(kt)}. Call PSx (kt, t 1; FDH) (where the superscript S stands for ‘sequential’) the value of the objective function at the solution. If PSx (kt, t 1; FDH) > 1, it measures progress, in the same way that PC x (kt, t 1; FDH) would do, and the computation is terminated. If PSx (kt, t 1; FDH) ¼ 1, go to Step 4. Step 4. With PSx (kt, t 1; FDH) ¼ 1 there is no progress. Since there can be no regress with sequential reference production sets of this type, the observation (xkt , ukt ) is only sequentially efficient. Its characterization is thereby completed.
Repeating the procedure for all elements of Y0Kt provides a complete characterization of the sequential frontier shift between t 1 and t. A graphical illustration is given on Fig. 3. It shows that progress has a ‘local’ character, in the sense that the whole frontier does not necessarily shift. This is typical of the nonparametric approach in its sequential form17, quite in contrast with the description of progress with the parametric methods alluded to in Section 5.1. 17
Also encountered by Thiry and Tulkens (1992) in their efficiency analysis of time series.
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u
O
X
Figure 3. Local FDH progress.
5.3
Measuring progress and regress with DEA frontiers
With DEA, the logic of envelopment by means of the boundary of a convex set prevails. This implies that progress and regress are to be described by shifts of that boundary per se, more than by referring only to the observations themselves, as is the case with dominance on which FDH rests. As a formal definition of frontier shift is hardly necessary in this case, we directly proceed to the computational procedures. a) Measurement with contemporaneous DEA frontiers The procedure is basically similar to the one proposed above for FDH frontiers. However, specific problems arise at some points, that entail important differences. In this exposition we account for the fact that DEA can take on different forms according to the assumed returns to scale by using as before the index p ¼ {DEA-ID, DEA-CD, or DEA-C}. For an observation (xkt , ukt ), to be characterized in input terms: Step 1. Compute first its DEA efficiency according to problem 1.1, with the last line replaced by (12), (13) or (14) according to the value chosen for p (returns to scale option), and with Y0 ¼ Y0Kt as reference observations subset. Step 2. If (xkt , ukt ) is found inefficient, i.e. not a frontier observation at time t, the computation is terminated as far as this observation is concerned. If it is found efficient, go to Step 3. Ks Step 3. Consider Y0 (x; p), i.e. the subset of observations in Y0Ks that were found efficient in input at reference time s according to the same returns to scale option p, and compute the optimal solution of problem 1.1 with 0 ¼ {(hs)js < t, (xhs , uhs ) 2 Y Ks (x; p)}: Y 0
(16)
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 415
and the last line replaced by (12), (13) or (14) according to the value chosen for the returns to scale option p. If a solution exists18, one of three cases can occur: – If one has ukt > 1 for the value of the objective function at the optimum, call it PC x (kt, s; p). It measures the progress achieved by observation (xkt , ukt ) over the DEA-p frontier at time s (ratio OB/OA on Fig. 4a). – If one has ukt < 1, call it RC x (kt, s; p); it measures regress vis-a`-vis the DEA-p frontier at time s (ratio OB/OA on Fig. 4b). – Finally, ukt ¼ 1 denotes absence of frontier shift, i.e. the observation (xkt , ukt ) lies on the frontier at s. When one of these three cases occurs, the computation is terminated as far as observation (xkt , ukt ) is concerned. On the other hand if no solution exists, go to Step 4. 0 as Step 4. If problem 1.1 with (16) has no solution, redefine Y 0 ¼ {(hs)js < t, (xhs , uhs ) 2 Y Ks (x; p)} [ {(kt)}, Y 0
(17)
that is, the same observations subset except that observation (xkt , ukt ) is now included, and compute now the opti 0 , and mal solution of problem 1.1 with this expression of Y with the last line replaced by (12) or (13) in accordance with the constraint that was used in the previous problem. Here, the value ukt of the objective function at the optimum is $ 1, because ghs ¼ 1 for (hs) ¼ (kt), yielding ukt ¼ 1 is always a feasible solution for this form of problem 2.1.
(a)
(b)
u
u (kt)
YDEA(Y Ks ) o
YDEA(Y Ks ) o (kt)
O
A
B
x
O
B
A
x
Figure 4. Progress and regress with DEA contemporaneous frontiers. 18 It was spotted by Fa¨re, Grosskopf, Lindgren and Roos (1989) that there may not exist a solution to problem 1.1 with (16), when the constraints (12) or (13) are used. We return to this point in Step 4, in comment (ii) and in Section 5.4 below.
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Therefore, as in Step 3, if ukt > 1 for the value of the objective function at the optimum, call it PC x (kt, s; p), as it measures progress (ratio OB/OA on Fig. 5a). If ukt ¼ 1, no progress is revealed by the input measure (see Fig. 5b) and observation (xkt , ukt ) is only efficient. This terminates the progress vs. regress characterization of a single observation with DEA-type reference production sets. As it was the case with FDH, applying the above procedure to all observations in Y0Kt yields a complete description of the DEA frontier shift achieved at time t. Parallel to our comments above on contemporaneous progress vs. regress with FDH reference sets, we note here the following: (i) Progress vs. regress evaluation basically requires only two computations with DEA because the second one (Step 3) directly sorts out between progress and regress. A third computation is needed only in the case of no solution. This is however not an exceptional case as will be seen in the next paragraph. (ii) The way we propose to get around the nonexistence of a solution at step 4 deserves special attention. When does this phenomenon occur? Problem 1.1 with (12) or (13) has no solution when the observation hs (xkt , ukt ) to be characterized is such that 9j: ukt 0 {uj }, that j > max(hs)2Y is, when one of the output quantities of the observation is larger than the largest similar output reached by some observation in the reference observations set. This is clearly seen in Fig. 5a: to find a distance in input to the right of observation (xkt , ukt ) by means of problem 1.1 is indeed impossible since the frontier the problem specifies is abcde. Problem 2.1 with (17), by contrast, uses as reference the convex set whose frontier is labelled mbdn, of which the observation (xkt , ukt ) happens to be an element. This is why it always has a solution. This convex set hardly has an economic interpretation, however. Notice (b)
(a) n (kt)
u
d
e
u
d
c
(kt)
q e
c YDEA(Y Ks o)
YDEA(Y oKs )
b
b
m
m O
a
A
B
x
O
a
x
Figure 5. Progress and efficiency measurement in case of no solution to problem 1.1 with (16).
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 417
also that in the case depicted on Fig. 5b, the convex set used by problem 2.1 with (17) is mbqn, and efficiency but no progress in input is recorded. Symmetrically, no solution exists with measures in output when the observation (xkt , ukt ) to be characterized is such that 9i : xkt i < min(hs)2Y 0 {xhs }, that is, when one of the input quantities of the observai tion is smaller than the smallest similar input quantity reached by some observation in the reference observations set. This is illustrated in Fig. 6a, together with the solution OC/OD (revealing progress) provided by the solution of the equivalent of problem 2.1 in the output orientation (that is, problem 2.2 in the Appendix). Fig. 6b, finally, exhibits a case that combines the two preceding ones, where a solution to problem 1.1 exists neither in output nor in input, while progress can be measured by problem 2.1 with (17) and its equivalent in the output orientation. b) Measurement with sequential DEA frontiers Using sequential frontiers, the procedure just expounded for contemporaneous ones is to be modified as follows (recall that only s ¼ t 1 is valid here as reference time): Step 1. Use Y0 ¼ Y0K(1, t) as reference observations subset. Step 2. Unchanged. K(1, t1) Step 3. Consider Y0 (x; p), the subset of observations found DEA-p sequentially efficient in input at time t 1, and 0 defined as compute the solution of problem 1.1 with Y 0 ¼ {(hs)js < t, (xhs , uhs ) 2 Y K(1, t1) (x; p)} Y 0
(18)
instead of (16). If there is a solution, call PSx (kt; p) the value of the objective function at the solution. If PSx (kt; p) > 1, it (a)
(b) u
u
(kt)
C (kt)
C
YDEA(Y Kso )
YDEA(Y oKs )
D
D O
x
O
A
Figure 6. Measurement in two further ‘no solution’ cases.
B
x
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measures progress, in the same way that PC x (kt, s; p) with s ¼ t 1 would do, and the computation is terminated. If PSx (kt; p) ¼ 1, go to Step 4. If there is no solution, go to Step 5. Step 4. With PSx (kt; p) ¼ 1 there is no progress, the observation (xkt , ukt ) is only sequentially efficient, and the computation is terminated. 0 in (18) as Step 5. If there is no solution at Step 3, redefine Y 0 ¼ {(hs)js < t, (xhs , uhs ) 2 Y K(1, t1) (x; p)} [ {(kt)} Y 0
(19)
0 so defined and compute the solution of problem 1.1 with Y and the last line replaced by (12) or (13). The properties of the solution and its interpretation are as in Step 5 of the procedure with contemporaneous frontiers. The characterization of an observation (xkt , ukt ) in terms of sequential progress with DEA reference frontiers is thereby completed. Repeating the procedure for all elements of Y0Kt provides a complete characterization of the sequential frontier shift between t 1 and t. A graphical illustration is given on Fig. 7. As we observed with FDH sequential analysis, it shows the ‘local’ character of the frontier shift. This particular form of progress was already obtained, in a different context, by Atkinson and Stiglitz (1969, p. 573) who present an identical diagram. 5.4
On frontier shift measures used in Malmquist indexes
We want to conclude this section by comparing our approach to frontier shifts with the ‘Malmquist index’ recently developed by Fa¨re, Grosskopf, Lindgren and Roos (1989, 1992) (hereafter referred to as FGLR), also designed to characterize changes over time in productive activities described by panel data. A basic difference lies in the fact that
u
O
Figure 7. Local DEA progress.
x
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 419
the Malmquist index, denoted here as M(k; t 1, t; p) measures the productivity gain (if M(k; t 1, t; p) > 1) or loss (if M(k; t 1, t; p) < 1) achieved by a given unit k from time t 1 to time t19. As is well-known, productivity is a notion different from efficiency. However, recognizing the fact that the unit whose productivity change is to be assessed may be efficient or inefficient (relative to some reference frontier of type p, to be specified shortly) at either one of the two observation times, the authors show that the productivity index can be decomposed into two factors that we presently recall. The first factor does indeed measure the ‘efficiency gain or loss’ of the unit from t 1 to t, that is, in our notation, the extent to which (xkt , ukt ) is closer to or further away from the frontier of Yp (Y0Kt ) than (xkt1 , ukt1 ) was vis-a`-vis the frontier of Yp (Y0Kt1 ). Using in each case the computation prescribed by problem 1.1 above20, this is measured by the ratio ExC (kt; p)=ExC (kt 1; p): a value larger than 1 measures an efficiency gain, a value smaller than 1 an efficiency loss. The other factor measures ‘frontier shift’ in the sense that has been treated here, that is, the extent to which the frontier used to measure the efficiency of (xkt , ukt ) has shifted away from the one that has served to measure the efficiency of (xkt1 , ukt1 ). Now this can be measured in two ways: either from (xkt , ukt ) by a ratio SxC (kt, t 1; p)=ExC (kt; p), or from (xkt1 , ukt1 ) by a ratio ExC (kt 1; p)=SxC (kt 1, t; p), where the numerators and denominators are derived as follows from our previous developments: 8 C Px (kt, t 1; p) > > > > > > > > > > <
if a progress-regress evaluation of observation (xkt , ukt ) with respect to the contemporaneous p-frontier at time t 1 results in PC x (kt, t 1; p) > 1, SxC (kt, t 1; p) ¼ C (kt, t 1; p) if the same evaluation results in R > x > > C > > (kt, t 1; p) < 1, R > x > > > 1 if the same evaluation yields > : C PC x ( ) ¼ Rx ( ) ¼ 1:
Similarly for SxC (kt 1, t; p), where t 1 and t replace t and t 1, respectively, in the previous definition. ExC (kt; p) and ExC (kt 1; p) are the efficiency degrees already referred to, introduced here to correct for the possible inefficiency of the observation (xkt , ukt ) and (xkt1 , ukt1 ). 19 Berg, Førsund and Jansen (1991) use alternative points in time s in lieu of t 1. The issues raised by the choice of alternative reference times are of the nature of those arising in choosing a base for any index number. 20 This problem yields a Malmquist index in input; parallel measures in output or graph are easily defined.
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With this notation, the FGLR decomposition of the Malmquist index reads21: M(k; t 1, t; p) ¼
ExC (kt; p) ExC (kt 1; p) C 1=2 Sx (kt, t 1; p) ExC (kt 1; p) : C ExC (kt; p) Sx (kt 1, t; p)
(20)
Frontier shift as measured by the expression in the square brackets thus appears as being only a component of productivity as evaluated by the Malmquist index. The way this measure is done raises two problems, however. One is the issue of ‘where’ to measure the frontier shift: from the current observation (xkt , ukt ) to the frontier of the past, i.e. of Yp (Y0Kt1 ), i.e. the first factor in the square brackets; or from the past observation (xkt1 , ukt1 ) to the frontier of the current reference set Yp (Y0Kt ), i.e. the second factor? The authors’ compromising choice is to take a geometric mean of the two. How is that related with what we have done in the preceding sections? The other problem is that the authors confine themselves to DEA-C reference production sets when they need to use both input and output measures, because with DEA-CD and DEA-ID production sets output and/or input measures may not exist in certain cases (as recorded with problem 1.1 with (16) above, which is the one they use), and they offer no solution for that. On these two issues, we have the following points to make: (i) Our perspective in the preceding analysis was to measure frontier shift only, and this appeared not to need, per se, any kind of averaging. Of course, averaging in the productivity index is only due to the necessity to stick with the same unit k at times t 1 and t. Our approach just reminds one that imposing this constraint is not necessary for measuring technical progress in terms of a frontier shift, and shows how to do it. (ii) Because problem 1.1 with (16) may have no solution in some important cases, both FGLR (1989, 1992) restrict themselves to p ¼ DEA-C reference production sets. Our present claims in that respect are 1) that by using FDH instead of DEA reference production sets, the programming problems always have a solution, as was seen in Section 5.2; and 2) that with DEA models of all types, the no-solution cases can be solved by following what we propose in Step 4 of Section 5.3. Admittedly, our solution is in the spirit of substituting a domination– 21 For ease of comparison with the rest of this paper, the formula is presented here in terms of efficiency and progress–regress degrees instead of the distance functions actually used by the authors in their original presentation.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 421
undomination criterion for the one of convex envelopment, which may be seen as an unacceptable change of philosophy within a given evaluation methodology. However we feel that there lies a strong justification for our proposal in the fact that the no-solution cases always arise in parts of the input–output space where it is not the convexity assumption that creates the problem but well the free disposal one: remember indeed that both DEA-CD and ID construct convex free disposal hulls, and this is the source of the ‘horizontal’ and ‘vertical’ segments in the boundary of the corresponding production sets, as for instance in Figs. 6a and 6b.
6.
Measuring progress and regress from a ‘benchmark production correspondence’
It must be recognized that either one of the two ways of measuring progress-regress by frontier shifts, i.e. from contemporaneous or from sequential frontiers, has important drawbacks. When contemporaneous frontiers are used, it is hard to avoid arbitrariness in the choice of the reference time s; and if s is chosen to be always equal to t 122, it is characteristically of short memory: only the last frontier counts, and whatever progress (or regress) was achieved earlier is not accounted for in the numerical evaluation at time t. With sequential progress, on the other hand, there is no room for regress, neither conceptually, nor in terms of measurement. Regress, in this setting, is simply assimilated with inefficiency because the production frontier is postulated to only move outwards and not inwards. To overcome these drawbacks we call here upon a new concept23, namely the ‘benchmark correspondence’. The characteristic of this approach is to abandon the frontier concept, and to replace it by a special form of the input-output relation, derived from the dominance notion, that we call the ‘benchmark production correspondence’, and by reference to which both progress and regress will be defined. As a corollary, however, the distinction between efficiency and inefficiency, understood as belonging or not to a frontier, fades away since there is no frontier anymore! Hence our substitution of the word benchmark for those of frontier or hull. The essence of the approach is to associate with the data not a whole production set, but instead only a subset of it – technically called the graph of some correspondence – this subset being not postulated to be a frontier. The subset is induced from the data in an evolving way, that is, it is gradually modified over time, on the basis of the information revealed If s is chosen < t 1, all the intermediary information is ignored. What follows is actually the panel data version of a methodology expounded in greater detail for time series in our companion paper (Tulkens and Vanden Eeckaut (1991) that is entirely devoted to this topic. 22 23
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by the observations made at each time t ¼ 1, 2, . . . , m, in a typically sequential spirit. At every t, a subset of the observations in the panel is thus singled out, that we call the ‘benchmark observations subset at time t’. From these observations, the benchmark correspondence and its graph are defined. Finally, all observations made at t are characterized in terms of progress or regress vis-a`-vis this benchmark, by means of an index computed by a procedure very close to the one used before in sequential efficiency and progress analysis. We shall now describe these three steps more precisely. Since this is a progress-regress measure, we first specify what is to be done at time t ¼ 1, where there are no earlier observations to compare the observations (xk1 , uk1 ) with; then we move to any time t $ 2. Stage t ¼ 1 Step 1.1. Consider the set Y0K1 of the data observed at time 1, and K1 identify by means of problem 1 the subset Y0 of these observations that are contemporaneously FDH efficient. Denote henceforth this last subset K1 Y0B and call it the ‘benchmark observations set at time 1’. K1 , associate the set Step 1.2. With every observation (xk1 , uk1 ) 2 Y0B " # " # I J k1 X u u O u þ 2 RIþJ mi I Di (xk1 , uk1 ) ¼ ¼ þ x x ei xk1 i¼1 " # ) J X eJj þ , mi $ 0, nj $ 0, nj OI j¼1 " # " k1 # I J X [ u u O u mi I RIþJ ¼ x þ x ei xk1 i¼1 " # ) J X eJj , mi $ 0, nj $ 0, nj (21) OI j¼1 that is, the set of points in the whole input–output space (thus, nonobserved as well as observed points) that neither dominate nor are dominated strongly24 by (xkt , ukt ); in other words, points that are in ‘dominance indifference’ with respect to (xkt , ukt ) – whence the subscript K1 ‘i’ appended to D. With Y0B , itself, associate then the set \ K1 )¼ Di (xk1 , uk1 ): Di (Y0B K1 kj(xk1 , uk1 )2Y0B 24
Referring to weak domination in Section 2, an observation strongly dominates another one in inputs if mi > 0, 8i, and in output if nj > 0, 8j.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 423
This set may be seen as the graph (illustrated by the dashed area on Fig. 8) of the mapping K1 )} b1 : RIþ ! RJþ : x ! U(x) ¼ {uj(x, u) 2 Di (Y0B
which we call the ‘benchmark production correspondence at time 1’. Step 1.3. Finally, each one of the n observations in Y0K1 conventionally receives a numerical progress-regress evaluation – called ‘BPR index’ – equal to its FDH efficiency degree as computed at Step 1.1. General stage t $ 2 Step t.1. At any time t $ 2, consider the data Y0Kt and identify its subset Kt of contemporaneously FDH efficient observations Y0 . From defining K(1, t1) Kt Kt Y0B ¼ Y0 \ Di (Y0B ) we obtain K(1,t) Y0B
¼
t [
Ks Y0B ,
s¼1
which we call the ‘benchmark observations set for the period (1, t)’. K(1,t) Step t.2. With every observation (xkt , ukt ) 2 Y0B , associate a set K(1,t) itself associate Di (xkt , ukt ) defined as in (21), and with the set Y0B \ K(1,t) Di (xkt , ukt ) Di (Y0B ) ¼ K(1 , t) (ks)js # t; (xks ,uks )2Y0B This set is also the graph of the mapping b : RI ! RJ : x ! U(x) ¼ {uj(x, u) 2 D (Y K(1,t) )}, t
þ
þ
i
0B
i.e. the ‘benchmark production correspondence at time t’.
u
O
Figure 8. Graph of the BPR correspondence.
X
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Step t.3. Finally, for each one of the n observations (xkt , ukt ) in Y0Kt a numerical index of ‘benchmark progress-regress’ – the BPR index – is computed as follows: – Solve problem 2.1 above with (15) replaced by K(1 , t) K(1,t) ¼ {(h, s)j(xhs , uhs ) 2 Y0B } [ {(kt)}, (22) Y 0B i.e. the set of index pairs that denote the elements of the benchmark observations set at time t25 , and the observation (xkt , ukt ) itself. One has ukt $ 1 for the value of the objective function at the optimum. If ukt > 1, call it Px (kt; BPR) and the characterization of observation (xkt , ukt ) is terminated, as this measures the progress (in input) achieved by this observation relative to the ‘‘benchmark correspondence at time t.’’ – If ukt ¼ 1 in problem 2.1 then move to problem 1.1 and solve it 0 also defined as in (22). Here, ukt # 1. If ukt < 1, call it Rx (kt; with Y BPR) and the characterization of observation (xkt , ukt ) is also terminated, as this measures the regress (in input) achieved by the observation relative to the same benchmark. – If ukt ¼ 1 in both problems 1.1 and 2.1 ukt ¼ 1, there is neither regress nor progress achieved by the observation (xkt , ukt ) with respect to the benchmark correspondence at time t, and observation (xkt ,ukt ) belongs K(1,t) to the benchmark observation subset Y0B . A conventional BPR index equal to 1 is attached to it. With the corresponding computation in output terms given in the appendix this completes our description of progress and regress measurement by means of a ‘BPR index’.
7.
Conclusion
We have presented a wide variety of ways to measure nonparametrically efficiency, progress and regress from panel data. Yet our developments show that this variety reduces to two alternative basic formal structures: the one of problems 1.1–1.3 and the other of problems 2.1–2.3. While the former is designed to evaluate efficiency, the latter is designed to evaluate progress (revealing under way that evaluating regress is formally identical to evaluating inefficiency, and thus formally belongs to the category of problems 1.1–1.3). Nevertheless, this wealth of approaches may leave the impression that in empirical studies, the choice of the most appropriate method must be quite an intricate one. We would argue, however, that several methods are better 25 A superscript x or u should be added to these sets to allow for possible differences in their definition in input or in output terms, respectively. In the text, we stick to our convention of dealing only with input measures.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 425
than only a few: our experience is indeed that by analyzing the same data set in various ways, one gets a much better understanding of what these data contain, and one finally reaches conclusions that are more strongly founded.
Appendix A.1. Progress and regress measurement with contemporaneous FDH frontiers a) Output measures Step 1. Compute first the FDH efficiency in output of (xkt , ukt ) according to problem 1.2 above, with Y0 ¼ Y0Kt as reference observations subset. Step 2. If (xkt , ukt ) is found inefficient, it is not a frontier observation and the computation is therefore terminated. If (xkt , ukt ) is found FDH efficient, go to Step 3. Step 3. Consider Y0Kt1 (u; p), i.e. the subset of observations in Y0Kt1 that were found FDH output efficient at time t 1, and compute the optimal solution of the following Problem 2.2: {u
kt
lkt : min hs ,g ,(hs)2Y0 }
subject to lkt ukt j X 0 (hs)2Y
X
X 0 (hs)2Y
ghs uhs j $ 0, j ¼ 1, . . . ,J,
kt ghs xhs i $ xi , i ¼ 1, . . . ,I,
o, ghs ¼ 1 and ghs 2 {0, 1}, (hs) 2 Y
0 (hs)2Y
where 0 ¼ {(hs)j(xhs , uhs ) 2 Y Kt1 (u; FDH)} [ {(kt)}: Y 0
(A:1)
Write as PC u (kt) the inverse value of the objective function at the kt 1 C optimum, that is: PC u (kt) ¼ [l ] . Notice that Pu (kt) $ 1, because lkt ¼ gkt ¼ 1, ghs ¼ 0, 8(hs) 6¼ (kt), is always a feasible solution for problem 2.2. If the number PC u (kt) is strictly larger than 1, no further computation needs to be done as far as observation (xkt , ukt ) is concerned. PC u (kt) indeed measures progress in terms of the proportion of how much more of all outputs26 unit k at time t produces than if it had made no progress. 26
This is indeed a radial measure.
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Step 4. If PC u (kt) is found equal to 1, there is no progress, but there may be regress. To find that out, compute the optimal solution of problem 0 defined as in (A.1) above, and write as Rc (kt) the value of the 1.2, with Y u kt objective function at the optimum, that is: RC u (kt) ¼ l . Here, one has kt kt hs RC x (kt) # 1 since l ¼ g ¼ 1, g ¼ 0, 8(hs) 6¼ (kt), is also always a feasible solution for problem 2.2. A value strictly less than 1 for RC u (kt) measures regress. This value also terminates the computation. C Step 5. Finally, if both PC u (kt) and Ru (kt) are found to be equal to 1, neither progress nor regress prevail: the observation (xkt , ukt ) is also a frontier observation at t 1 and there is thus no frontier shift. This completes the progress vs. regress in output characterization of observation (xkt , ukt ). b) Graph measures Step 1. Compute first the FDH graph efficiency of (xkt , ukt ) according to problem 1.3 above, with Y0 ¼ Y0Kt as reference observations subset. Step 2. If (xkt , ukt ) is found inefficient, it is not a frontier observation and the computation is therefore terminated. If (xkt , ukt ) is found FDH efficient, go to Step 3. Step 3. Consider Y0Kt1 (x, u; p), i.e. the subset of observations in Kt1 Y0 that were found FDH graph efficient at time t 1, and compute the optimal solution of the following Problem 2.3: kt
{d
dkt max hs ,g ,(hs)2Y0 }
kt subject to ukt i =d
dkt xkt i X
X 0 (hs)2Y
X
(hs)2Y
g hs uhs j $ 0, j ¼ 1, . . . , J,
g hs xhs i # 0, i ¼ 1, . . . , I,
0, ghs ¼ 1 and g hs 2 {0, 1}, (hs) 2 Y
0 (hs)2Y
where 0 ¼ {(hs)j(xhs , uhs ) 2 Y Kt1 (x, u; FDH)} [ {(kt)}: Y 0
(A:2)
Write as PC x,u (kt) the value of the objective function at the optimum, kt C kt that is: Px,u (kt) ¼ dkt . Notice that PC x,u (kt) $ 1, because l ¼ g ¼ 1, g hs ¼ 0, 8(hs) 6¼ (kt), is always a feasible solution for problem 2.3.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 427
If the number PC x,u (kt) is strictly larger than 1, no further computation needs to be done as far as observation (xkt , ukt ) is concerned. PC x,u (kt) indeed measures progress in terms of the proportion of how much more of all outputs unit k at time t produces and, simultaneously the proportion of how much less of all inputs unit k at time t uses than if it had made no progress. Step 4. If PC x,u (kt) is found equal to 1, there is no progress, but there may be regress. To find that out, compute the optimal solution of problem 0 defined as in (A.2) above, and write as RC (kt) the value of 2.3, with Y x,u the objective function at the optimum, that is: RC (kt) ¼ dkt . Here, one x,u kt kt hs has RC x,u (kt) # 1 since l ¼ g ¼ 1, g ¼ 0, 8(hs) 6¼ (kt), is also always a feasible solution for problem 2.3. A value strictly less than 1 for RC x,u (kt) measures regress. This value also terminates the computation. C Step 5. Finally, if both PC x,u (kt) and Rx,u (kt) are found to be equal to 1, neither progress nor regress prevail: the observation (xkt , ukt ) is also a frontier observation at t 1 and there is thus no frontier shift. This completes the progress vs. regress in graph characterization of observation (xkt , ukt ). A.2. Progress and regress measurement with contemporaneous DEA frontiers Output and graph measures are obtained from our description in Section 5.3 of the computational procedure in input just by making the following substitutions: A.3. Progress and regress measurement from a ‘benchmark production correspondence’ Description of the procedure in the input orientation For . . .
Description of the procedure in the output orientation Substitute . . .
Description of the procedure in the graph orientation 27 Substitute . . .
Problem 1.1 Problem 2.1 Y0Ks (x; p) ukt PC x (kt, s; p) RC x (kt, s; p)
Problem 1.2 Problem 2.2 Y0Ks (u; p) [lkt ]1 PC (kt, s; p) u RC (kt, s; p) u
Problem 1.3 Problem 2.3 Y0Ks (x, u; p) dkt PC x,u (kt, s; p) PC x,u (kt, s; p)
Output and graph measures are likewise obtained from our description in Section 6 of the computational procedure in input with the following substitutions: 27
For the graph orientation, problem 2.3 is never used because problem 1.3 always has a solution.
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Description of the procedure in the input orientation For . . .
Description of the procedure in the output orientation Substitute . . .
Description of the procedure in the graph orientation Substitute . . .
Problem 1.1 Problem 2.1 ukt Px (kt; BPR) Rx (kt; BPR)
Problem 1.2 Problem 2.2 [lkt ]1 Pu (kt; BPR) Ru (kt; BPR)
Problem 1.3 Problem 2.3 dkt Px,u (kt; BPR) Rx,u (kt; BPR)
References Atkinson, A.B. and Stiglitz, J.E., 1969. A new view of technical change, Economic Journal LXXIX, 573–578. Banker, R.D., Charnes, A., and Cooper, W.W., 1984. Some models for estimating technical and scale inefficiency in data envelopment analysis, Management Science, 30/9, 1078–1092. Berg, S.A., Førsund, F.R., and Jansen, E.S., 1992. Malmquist indices of productivity growth during the deregulation of Norwegian banking 1980–89, Scandinavian Journal of Economics, 94 (Supplement), S211–S228 Bowlin, W.F., Brennan, J., Charnes, A., Cooper, W.W. and Sueyoshi, T., 1984. A model for measuring amounts of efficiency dominance. Unpublished Manuscript. Charnes, A., Cooper, W.W. and Rhodes, E., 1978. Measuring the efficiency of decision making units, European Journal of Operational Research, 2/6, 429–444. Charnes, A., Clark, C.T., Cooper, W.W. and Golany, B., 1985. A developmental study of data envelopment analysis in measuring the efficiency of maintenance units in the US air forces, in J.C. Balzer (ed.). Annals of Operations Research, 2, 95–112. Debreu, G., 1959. Theory of value. New York: Wiley. Deprins, D. and Simar, L., 1983. On Farrell measures of technical efficiency, Recherches e´conomiques de Louvain, 49/3, 123–138. Deprins, D., Simar, L. and Tulkens, H., 1984. Measuring labor efficiency in post offices, in M. Marchand, P. Pestieau and H. Tulkens (eds.). The performance of public enterprises: concepts and measurements. Amsterdam: North-Holland. (Chapter 13 in this volume). Fa¨re, R., Grabowski, R. and Grosskopf, S., 1985. Technical efficiency of Philippine agriculture, Applied Economics 17, 205–214. Fa¨re, R., Grosskopf, S. and Lovell, C.A.K., 1985. The measurement of efficiency in production. Dordrecht: Kluwer Nijhoff. Fa¨re, R., Grosskopf, S., Lindgren, B. and Roos, P., 1989. Productivity developments in Swedish hospital: a Malmquist output index approach, in A. Charnes, W.W. Cooper, A.Y. Lewin and L.M. Seiford (eds.), 1993. Data envelopment analysis: theory, method and process. Boston, MA: Kluwer. Fa¨re, R., Grosskopf, S., Lindgren, B. and Roos, P., 1992. Productivity changes in Swedish pharmacies 1980–1989: a nonparametric Malmquist approach, Journal of Productivity Analysis, 3(1–2), 85–101. Farrell, M.J., 1957. The measurement of productive efficiency, Journal of the Royal Statistical Society. Series A- General, 120(3), 253–281. Førsund, F., 1992. A comparison of parametric and nonparametric measures: the case of Norwegian ferries, Journal of Productivity Analysis, 3(1–2), 25–44. Grosskopf, S., 1986. The role of the reference technology in measuring productive efficiency, Economic Journal, 96, 499–513.
Ch.17 Nonparametric Efficiency, Progress and Regress Measures For Panel Data 429 Lovell, C.A.K., 1993. Production frontiers and productive efficiency, in H.O. Fried, C.A.K. Lovell and S.S. Schmidt (eds.). The measurement of productive efficiency: technique and applications. New York: Oxford University Press. Petersen, N.C., 1990. Data envelopment analysis on a relaxed set of assumptions, Management Science, 36(3), 305–314. Solow, R.M., 1956. A contribution to the theory of economic growth, Quarterly Journal of Economics, LXX, 65–94. Thiry, B. and Tulkens, H., 1992. Allowing for technical inefficiency in parametric estimates of production functions, Journal of Productivity Analysis, 3(1–2), 45–66. Tulkens, H., 1986. La performance productive d’un service public: de´finitions, me´thodes de mesure, et application a` la Re´gie des Postes de Belgique, L’Actualite´ Economique, Revue d’Analyse Economique, 62(2), 306–335. Tulkens, H., 1993. On FDH analysis: some methodological issues and applications to retail banking, courts and urban transit, Journal of Productivity Analysis, 4(1), 183– 210. (Chapter 14 in this volume). Tulkens, H. and Vanden Eeckaut, P., 1995. Non-frontier measures of efficiency, progress and regress, International Journal of Production Economics, 39, 83–97. (Chapter 16 in this volume).
Chapter 18 EFFICIENCY DOMINANCE ANALYSIS (EDA): BASIC METHODOLOGY
Henry Tulkens
This paper argues that for evaluating efficiency in the economic sense of an organization (any DMU, in the Charnes, Cooper and Rhodes (1978) terminology), the notion of frontier is less basic than the one of dominance. In Section 2, three different ways to define this concept are proposed : pairwise E-dominance, setwise E-dominance and global E-dominance. In Section 3, numerical measurement methods are proposed for each. Properties of EDA are highlighted and contrasted with the frontier methodology in the concluding Section 4.
1.
Motivation1
In the efficiency measurement literature, the notion of efficiency is typically linked with the one of frontier of a production possibilities set. Is that link a necessary one? This paper argues—after a few forerunners—that it is not, if one substitutes to the notion of frontier efficiency the alternative one of efficiency dominance (ED). ED is not intrinsically different: it will be shown to be more general, frontier efficiency being only a special case. Forerunners are those authors who abandoned the frontier notion because either they had to, or they wanted to get rid of the free disposal 1 This chapter, written for this book in November 2004, is a revised version of Tulkens (1993b). Occasions to put these ideas to empirical tests have been reported in Tulkens and Vanden Eeckaut (1995a) (urban transportation time series), Tulkens and Vanden Eeckaut (1995b) (bank branches), Tulkens and Malnero (1996) (bank branches), Tulkens (1997) (courts) as well as in Tulkens and Vanden Eeckaut (1999) (Municipalities). I take pleasure in acknowledging my intellectual debt for a fruitful collaboration with Philippe Vanden Eeckaut on this topic.
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assumption, which is common to all frontier approaches, be they the free disposal conical hull of Farrell (1957) and Charnes, Cooper and Rhodes (1978), the convex free disposal hull of Banker, Charnes and Cooper (1984), or the free disposal hull of Deprins, Simar and Tulkens (1984). Indeed, in the problem dealt with in Athanassopoulos and Storbeck (1992), which bears on the choice of alternative locations for some facility, the free disposal assumption makes no sense, whence there is no frontier anymore; but an efficiency computation remains possible. On the other hand, Hougaard and Tvede (1993) want to have a measure that rests on the data only and therefore ignore ‘‘envelopment’’ in any sense. Finally Fried, Lovell and Vanden Eeckaut 1993 also make a step in that direction when, in addition to their frontier efficiency measures, they use some form of dominance measure as a complementary criterion in evaluating what they call ‘‘performance’’. Another source of inspiration for the approach systematized here is the ‘‘goodness of fit’’ argument developed by Fa¨re and Grosskopf (1991): aren’t the usual efficiency scores not just a measure of the goodness of fit to the data of the postulated frontier of the reference technology? The lesser efficiency in the data, the lesser the hull of the reference set fits those data. Finally, some very simple but plausible structures of observed data, depicted in the two figures below, raise problems that frontier efficiency methodology does not answer satisfactorily. In the case of Fig. 1 for example, which represents successive observations of a DMU whose activity grows over time (u denotes quantities of one output, x, quantities of one input), the number of cases of inefficiency varies from zero to three, according to the reference technology which is chosen. Isn’t inefficiency
u
4
3 2
1
0
Figure 1
x
Ch.18 Efficiency Dominance Analysis (EDA): Basic Methodology
433
induced in the data by the assumptions on which the respective models rest? In the case of Fig. 2, where a large amount of inefficiency is obviously present, all frontiers take on either a trivial (DEA-V - also called BCC - and FDH) or an unrealistic (DEA-C, also called Farrell-CCR) shape. To provide an intuitive idea of what is developed formally below, i.e. efficiency analysis without using the frontier concept, consider Fig. 2 again. Under all usual criteria, the productive activity represented by a is efficient, whereas those represented by b, c, and d are not. However, b, while being ‘‘inefficient’’, is nevertheless more efficient than c, and much more so than d. And c is also more efficient than d. One can pursue this reasoning by considering actually all pairs of observations, and making an efficiency statement about each one of such pairs, irrespective of whether or not one of the two points involved is a frontier point. Defining such pairwise relations, and associating numbers with them so as to measure their importance is the beginning of what lies ahead. The argument is further developed by extending it to all elements of the observations set, so as to give it a structure in terms of these efficiency statements. We proceed in two stages: we first define ‘‘efficiency dominance’’ as an alternative to the notion of ‘‘frontier efficiency’’ (Section 2). Next, we propose computational formulae for obtaining numerical measures of efficiency dominance (Section 3). The concluding Section 4 emphasizes that the conceptual basis for the approach advocated here is to be found essentially in the economics of efficiency.
2.
Defining efficiency dominance
Let
Y0 ¼ yk (xk , uk )j k ¼ 1, . . . , n be a set of n observations yk , indexed k ¼ 1, . . . , n, where xk ¼ (xk1 , . . . , xki , . . . ,xk1 ) denotes a vector of observed input quantities u
DEA-C DEA-V
a
FDH
b c
d x
Figure 2
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(there are I inputs, indexed i) and uk ¼ (uk1 , . . . , ukj , . . . ,ukj ) denotes a vector of observed output quantities (there are J outputs, indexed j). In all figures below, observations are represented by the coordinates of dots in the input and/or outputs spaces. Efficiency dominance (or E-dominance, or ‘‘ED’’ for short) may be conceived of in three different ways, that correspond to the three following definitions2: Definition 1 – E-dominance as a property of one observation vis-a`-vis some other observation (pairwise E-dominance):
For any pair of observations yk , yd , (i) observation yd E-dominates observation yk : - in inputs if xd #xk and ud ^ uk ; - in outputs if xd % xk and ud $uk ; (ii) observation yd is E-dominated by yk (in inputs and/or in outputs) if the reverse inequalities hold; (iii) there is no E-dominance relation between yd and yk if neither (i) nor (ii) hold. In all three examples of Figure 3, observation yd E-dominates observation yk , in inputs as well as in outputs. Definition 2 – E-dominance as a property of one observation vis-a`-vis the set of observations (setwise E-dominance): a) Observation yk is dominated in Y0 if one has Da (k) ¼ y 2 Y0 jy is E-dominating yk 6¼ Ø, def
that is, if the subset of observations that dominate yk is nonempty.
u
x1
yd yk
x 1 input-1 output space
yk
yd
u1
yd yk
x2 2 inputs space (assuming equal outputs)
u2 2 outputs space (assuming equal inputs)
Figure 3 Three cases of E-dominance of yd vis-a`-vis yk 2 Notation for vector inequalities: ‘‘$’’ means ^ ’’ but not ‘‘¼’’ for each pair of components of two vectors to be compared; similarly, with ‘‘ % ’’, for the vectorial inequality ‘‘#’’.
Ch.18 Efficiency Dominance Analysis (EDA): Basic Methodology
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b) Observation yk is dominating in Y0 if one has Dp (k) ¼ y 2 Y0 jy is E-dominated by yk 6¼ Ø, def
that is, if the subset of observations that are dominated by yk is nonempty. An observation yk may be at the same time both dominating and dominated within Y0 . Alternatively, it is neither dominating nor dominated in Y0 when one has Da (k) ¼ Dp (k) ¼ Ø. Notice that the set of observations referred to may be a subset of Y0 . This definition is illustrated in the examples of Figure 4. In each case, the observation k is both dominating and dominated. Definition 3 – E-dominance as a global property of the set of observations (global E-dominance): The set of observations Yo exhibits inefficiency (that is, Edominance) if 9 yk 2 Yo such that Da (k) 6¼ ; otherwise, the set Yo exhibits no inefficiency. This definition is illustrated with Figure 4 for its first part and with Figure 5 for the second part. As neither the notion nor even the term of frontier are used in these definitions, the efficiency concept clearly appears to be, in this setting, independent of that notion.
3.
Measuring efficiency dominance
The purpose here is to associate an (index) number with each occurrence of dominance, in order to reflect the intensity of the phenomenon.
u
Dp (k)
Da (k)
x1
Da (k) u1
k
k
Dp (k)
Da (k)
Dp (k) x2
x 1 input-1 output space
k
2 inputs space (assuming equal outputs)
Figure 4 Three cases of E-dominance of y k in Yo .
u2 2 outputs space (assuming equal inputs)
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x1
u
u1
x2
x 1 input-1 output space
2 inputs space (assuming equal outputs)
u2 2 outputs space (assuming equal inputs)
Figure 5 Three cases of observations sets Yo exhibiting no inefficiency
The index varies according to the definition of dominance retained. Thus, the three definitions of E-dominance just stated induce corresponding measurement formulae. In addition, depending upon whether an observation k is considered as ‘‘dominated’’ or ‘‘dominating’’, it may be distinguished between passive and active dominance measures, respectively. And for each one of these two, three measures are proposed here below. 3.1
Measuring pairwise E-dominance
Given a pair of observations yk , yd where one observation dominates the other, two cases are to be considered: a) Passive dominance If yk is dominated by yd , two conceivable indexes of passive efficiency dominance of yk by yd , denoted as PD(k; d), can be formulated as follows3: – Radial 4indexes: " !# d k u x j i PDo2 (k;d) ¼ Max #1 PDi2 (k;d) ¼ Max #1 j2J udj i2I xki & index in inputs index in outputs
3 In the formulae below, the notation I and J is used under the operators Max and Min to denote the sets of inputs and of outputs, respectively. No confusion should arise with the cases where I and J denote the cardinals of these sets. 4 In Tulkens (1993b) this index and some of the following ones were given the mathematical name of ‘‘metric’’, as suggested in an early version of Bardhan, Bowlin, Cooper and Suyeoshi (1996). This terminology is inappropriate, however, because the above formulae do not satisfy the definition of a metric in topology.
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437
or: – Average indexes: " # X xd =xk i i i #1 PD1 (k;d) ¼ I i2I
"
PD01 (k;d)
&
index in inputs or:
2P
6i2I PDi&o 1 (k;d) ¼ 4
xdi =xki þ
# X ukj =udj #1 ¼ J j2J index in outputs
P j2J
ukj =udj
I þJ
3 7 5# 1:
index in both inputs and outputs
b) Active dominance If yk is dominating yd , two possible indexes of active efficiency dominance of yk over yd , denoted as AD(k; d), are as follows: – Radial indexes: d xi i AD2 (k;d) ¼ Min $1 i2I xki index in inputs – Average indexes: " # X xd =xk i i ADi1 (k;d) ¼ $1 I i2I
&
" &
ADo2 (k;d) ¼ Min
2P
6i2I ADi&o 1 (k;d) ¼ 4
xdi =xki þ
!#
udj
$1
index in outputs " # X ukj =udj AD01 (k;d) ¼ $1, J j2J
index in inputs or:
j2J
ukj
index in outputs P j2J
I þJ
ukj =udj
3 7 5$1:
index in both inputs and outputs 3.2
Measuring setwise E-dominance
To measure passive E-dominance of observation k vis-a`-vis the set Y0 one may use:
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(i) An integer index of passive E-dominance of k within Y0 . Denoted as IPD(k;Y0 ), it consists of just the number of elements (i.e. the cardinal, denoted #) of the subset Da (k) in definition 2 above. Formally: IPD(k;Y0 ) ¼ # Da (k), (0# #n 1): (ii) An extreme index of passive E-dominance of observation k within Y0 . Denoted as EPD(k;Y 0 ), it consists of the smallest pairwise dominance measure between k and each one of the observations that dominate it, that is, the elements of Da (k). Formally, using radial indexes in inputs: g xi i EPD (k;Y 0 ) ¼ Min #1: Max a i2I g2D (k) xki The operator Max makes it a radial measure; the operator Min makes it an extreme one. This formula is in fact identical to the radial one used in FDH efficiency analysis (see formula (5) p. 189 in Tulkens (1993a): it indeed measures radially the distance of an observation with respect to the FDH efficient boundary. (iii) A full index of passive E-dominance of observation k within Y0 . Denoted as FPD(k;Y 0 ), it consists of an aggregate of all pairwise dominance measures between k and all observations that dominate it (the elements of Da (k)): here too, the superscript i refers to a measure in inputs. Formally, using again radial indexes in inputs: g Y xi i FPD (k;Yo ) ¼ #1 Max i2I xki g2Da (k) This formula is made of the product of the radial E-dominance measures of all observations that dominate the observation k. These three passive measures in inputs are illustrated in Figure 6. To measure now active dominance of the same observation k, similar formulae can be presented, where the subset Dp (k) replaces Da (k), the replaces Max . operator Max replaces Min and the operator Min i i d d Finally, all the above measures can equally be made in terms of outputs, with appropriate changes in notation. Two remarks are in order. First, for every observation k, the second (extreme) and the third (full) measures of its E-dominance are related in the following way: FPDi (k)#EPDi (k):
Ch.18 Efficiency Dominance Analysis (EDA): Basic Methodology
u
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b Da (k)
a c
d k
h
e
j
f i
Dp (k)
0
xa
xc
xd xb x k
x
Integer index: IPD(k; Yo ) ¼ #{a, b, c, d} ¼ 4 0xa <1 Extreme index: EPDi (k; Yo ) ¼ 0x k
0xa 0xc 0xd 0xb Full index: FPDi (k; Yo ) ¼ 0x 0xk 0xk 0xk 1 k
Figure 6 Three measures of passive E-dominance of observation k vis-a`-vis the set of observations Yo (indexes in inputs for the extreme and full ones)
Second, the full measure is formulated above as a product of radial measures, instead of in terms of an average of these measures as proposed initially in Tulkens (1993b). That the product is preferable to the average appears from the fictitious numerical example that follows. Let an observation k be dominated by five other ones, namely a, b, c, d and e. The first five lines of the table below report the pairwise dominance measures of a with respect to k, of b with respect to k, a.s.o., until e with respect to k. On the basis of this, the extreme measure of k is 0,700 while the full measure of k is 0,810 or 0,339 depending upon whether the full measure is computed as the average or as the product, respectively, of the pairwise measures. Suppose now that a sixth observation, f, appears as an additional element of the set of observations that actively dominate k, and that its pairwise dominance measure with respect to k be 0.850. The presence of this observation obviously implies that k is more dominated; yet, the index of full dominance of k, if computed as the
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average, increases (to 0,817); if computed as the product, by contrast, it decreases to 0,288: this better reflects the stronger (passive) dominance of k after the addition of f. Averages observation a observation b observation c observation d observation e Full measure observation f Full measure
3.3
Products 0.950 0.850 0.800 0.750 0.700
0.810
0.339 0.850
0.817
0.288
Measuring global E-dominance
This measure corresponds to the third definition of E-dominance. It aims at characterizing by means of a single number the ‘‘amount’’ of ED present in an observations set considered globally. To this effect we again propose three indexes each of which corresponds to an average over all elements yk of Yo of: (i) the integer indexes (passive or active); or (ii) the extreme indexes (passive or active); or (iii) the full indexes (passive or active). Figures 7 show a few examples of observations sets whose structure implies a ‘‘full’’ measure typicaly high or low. In an empirical study, in the spirit of the first of these measures, Fried, Lovell and Vanden Eeckaut (1993) use the total number of (passive) dominance relations that they observe (150,000 cases) within a total set of about 9,000 units. In the spirit of the second measure, also with passive dominance, one often finds a figure of ‘‘average efficiency’’ in empirical studies. Thus, Deprins, Simar and Tulkens (1984) had estimated at 0,55 the average efficiency of 792 postal stations in Belgium. The third measue has not been used so far, to the best of my knowledge.
4.
Concluding remarks
Efficiency analysis in terms of E-dominance characteristically bears on each occurrence of a dominance relation within the observations set. It is not limited to the extreme measures. It therefore exploits more of the information contained in the data. Moreover, it is independent of any specification of a ‘‘technology’’, in particular of any form of ‘‘frontier’’. Its results are therefore immune of any bias due to frontier choice, and are thus robust with respect to frontier changes.
Ch.18 Efficiency Dominance Analysis (EDA): Basic Methodology
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u
x
x (b) Y0 with a high degree of ED
(a) Y0 with a low degree of ED
u
u
x (c) Y0 with a zero degree of ED
x (d) Y0 with a maximal degree of ED
Figures 7 Structures of observations sets exhibiting various degrees of E-dominance
Methodologically, in contrast with the well-established DEA method, EDA does not use mathematical programming (except for the case of extreme measures), simply because no maximization or minimization are involved. From another point of view, the search for explanations of inefficiencies (that is, of E-dominance), should now extend to all occurrences of it, and not only on extreme measures, as it is the case in so many studies in which efficiency scores are explained by regression analyses. Finally, let us turn to the essence of the matter and ask: ‘‘what are we actually after: enveloping data, or evaluating efficiency?’’ Isn’t the former only the means, while the latter is the end? My personal view as an economist is that with frontiers, the end is not served well enough by the means, because frontiers are only part of the reality to be understood, and what we can observe of them is not robust enough to alternative methodologies. With dominance relations, by contrast, the analysis of (in)efficiency is refocused on the central issue
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of resource usage, in all the forms and varying degrees of intensity it can take. This is possible only by directing the attention more to the inside of the data set.
References Athanassopoulos, A. and Storbeck, J., 1992. Convex vs. nonconvex models for spatial efficiency, Working paper n8 58, Warwick Business School Research Bureau. Banker, R.D., Charnes, A. and Cooper, W.W., 1984. Some models for estimating technical and scale inefficiency in data envelopment analysis, Management Science, 30(9), 1078–1092. Bardhan, I., Bowlin, W.F., Cooper, W.W. and Suyeoshi, T., 1996. Models for efficiency dominance in DEA, Journal of the Operational Research Society of Japan, 39 (3) (Part I : Additive models and MED measures, 333–344; Part II: Free disposal hull (FDH) and Russell measure (RM) approaches, 322–332). Charnes, A., Cooper, W.W. and Rhodes, E., 1978. Measuring the efficiency of decision making units, European Journal of Operations Research, 2(6), 429–444. Deprins, D., Simar, L. and Tulkens, H., 1984. Measuring labour efficiency in post offices, in M. Marchand, P. Pestieau and H. Tulkens (eds.). The performance of public enterprises: concepts and measurement. Amsterdam: North-Holland, chapter 10, 243– 267. (Chapter 13 in this volume). Fare, R. and Grosskopf, S., 1991. Nonparametric tests of regularity, Farrell efficiency and goodness of fit, Journal of Econometrics, (1995) 415–425. Farrell, M.J., 1957. The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A -General, 120(3), 253–281. Fried, H., Lovell, C.A.K. and Vanden Eeckaut, P., 1993. Evaluating the performance of US credit unions, Journal of Banking and Finance, 17, 251–265. Hougaard, J. and Tvede, M., 1993. Intertemporal dominance analysis, Working paper 5– 93. Copenhagen: Institute of Economics, Copenhagen Business School. Tulkens, H., 1993a. On FDH analysis: some methodological issues and applications to retail banking, courts and urban transit, Journal of Productivity Analysis, 4(1/2), 183– 210. (Chapter 14 in this volume). Tulkens, H., 1993b. Efficiency dominance analysis (EDA): a frontier free efficiency evaluation method, paper presented at the World congress of the International Federation of the Operations Research Societies (IFORS) Lisbon July 1993 and at the CORE Third European Workshop on Efficiency Analysis, October 1993. Louvain-la-Neuve: Universite´ catholique de Louvain, mimeo. Tulkens, H., 1997. Frontier or not frontier? that is the question . . . , Lecture at the Historical Overview Session of the Fifth European Workshop on Efficiency and Productivity Analysis held at The Royal Veterinary and Agricultural University, Department of Economics and Natural Resources, Copenhagen, Denmark, October 9. Tulkens, H. and Malnero, A., 1996. Nonparametric approaches to the assessment of the relative efficiency of bank branches, in D.G. Mayes (ed.). Sources of productivity growth. Cambridge U.K: Cambridge University Press, chapter 10, 223–244. Tulkens, H. and Vanden Eeckaut, P., 1995a. Non-frontier measures of efficiency, progress and regress for time series data, International Journal of Production Economics, 39, 83– 97. (Chapter 16 in this volume). Tulkens, H. and Vanden Eeckaut, P., 1995b. How to measure efficiency and productivity with special reference to banking. Introductory presentation at the Nordic Workshop in Productivity and Efficiency, held at the inauguration of Handelsho¨gskolan, Go¨teborg University, September 19–20, 1995, mimeo.
PART IV FISCAL COMPETITION AND OPTIMALITY
Introduction Jack Mintz, University of Toronto and C.D. Howe Institute
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Chapter 19 Commodity tax competition between member states of a federation: equilibrium and efficiency 449 Jack Mintz, Henry Tulkens 1 Introduction 2 Regional market equilibria and fiscal optima 3 Fiscal competition 4 Fiscal competition and efficiency 5 Conclusions References Chapter 20 On Pareto improving commodity tax changes under fiscal competition Alain de Crombrugghe, Henry Tulkens 1 Introduction 2 The main features of the model 3 The new results 4 Concluding comments References Chapter 21 Optimality properties of alternative systems of taxation of foreign capital income Jack Mintz, Henry Tulkens 1 Introduction 2 A simple economy with international real capital flows 3 Alternative systems of taxation of foreign capital income 4 Domestically optimal fiscal choices in one country
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5 Internationally optimal tax rates 6 Conclusion References Chapter 22 Tax interaction dynamics among Belgian municipalities 1984–1997 Jean-Franc¸ois Richard, Henry Tulkens, Magali Verdonck 1 Introduction and review of the literature 2 Specification of the economic model 3 Data and choice of variables 4 Econometric method 5 The results and their econometric interpretation 6 Summary and conclusions References
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Jack M. Mintz
Reductions in transportation and communication costs in the past several decades have led to deeper economic integration globally as well as on a regional basis in Europe, Asia and the Americas. No wonder that numerous books and articles on fiscal competition among governments in an international or a federal context have expanded the economics literature. Henry Tulkens’ papers have been significant contributions to the fiscal competition literature including the seminal paper on ‘‘Commodity Tax Competition between Members of a Federation’’, co-authored with myself, published in the Journal of Public Economics, 1986, which is cited in more than 100 articles since its appearance. In his work, Tulkens addresses issues related to both international and federal fiscal competition. Significant differences arise between these two cases that are critical to analysis. With international fiscal competition, economies are typically modeled with mobile capital and businesses that are attracted to where the highest after-tax income can be attained. Further, at the international level, a central government with sufficient authority does not exist that could promote fiscal coordination among governments. On the other hand, with fiscal competition in a federation, not only are capital and businesses mobile but also so are people who are able to migrate from one region to another without having to obtain approval from authorities. As well, in a federation, a central government plays a role in affecting fiscal competition outcomes – their spending and taxing powers can influence local government fiscal decisions. Similarly, local fiscal decisions could have an impact on central government fiscal decisions. The key insight derived from fiscal competition models is that fiscal decisions are dependent on interactions among governments. In
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particular, both beneficial and harmful fiscal externalities exist when independent governments set their spending and tax policies that affect economic welfare of other jurisdictions. Many examples of beneficial externalities can be provided. For example, tax rate increases can make neighbors better off since capital flees to other jurisdictions, providing opportunities for other governments to have more revenue to either spend or cut taxes. As another example, the provision of better highways, communication and energy infrastructure can improve national or international networks that benefit other jurisdictions. In these examples, fiscal competition leads to low tax rates or inadequate infrastructure expenditures since spillovers that benefit other jurisdictions are not included in a jurisdiction’s calculus in determining its optimal fiscal policies. Without taking into account these beneficial externalities, an independent government chooses to fund fewer positive externality-generating public services or cut tax rates. However, fiscal competition can also lead to harmful externalities. Local governments might impose few restrictions on pollution activities that harm the health of populations not only at home but also in neighboring jurisdictions (see Part II of this volume). Further, governments might choose to ‘‘export’’ taxes on non-residents by taxing tourism, natural resources and foreign-owned capital so long as the tax base is not entirely driven away. In these instances, expenditure and taxes are set too high since independent governments do not include the harmful effects of their fiscal policies on the economic welfare of non-residents. The above-referenced 1986 article on commodity taxation in a federation illustrates the importance of these different types of externalities to understanding whether taxes are too high or low in a game of tax competition. Mintz and Tulkens model two jurisdictions, each of which imposes origin-based taxes on production sold to both residents and nonresidents with another good (leisure) being non-taxable. The revenue funds local public goods so no other fiscal externalities exist in the model such as in the case of network infrastructure as described above. Only one-way trade exists for the taxable commodity depending on which jurisdiction has the lowest commodity price, inclusive of the commodity tax; as a result, a region either exports or imports the taxable commodity but not both. One fiscal externality – tax base flight – arises from a jurisdiction raising its commodity tax rate that either encourages residents to import cheaper products from the foreign jurisdiction or reduces exports sold to foreigners. The other fiscal externality, ‘‘tax exportation’’ only arises if the jurisdiction is an exporter – non-residents pay the commodity tax if importing the good. Mintz and Tulkens derive the properties of a fiscal competition game with pure strategies in commodity taxes – no ordinary problem since the multivalued reaction functions
Introduction
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(specifying the optimal tax rate chosen by a jurisdiction as a function of the other jurisdiction’s commodity tax rate) jumps from a high to low value when tax-inclusive prices become close to each other. Mintz and Tulkens suggest that the commodity tax rate chosen by the importing jurisdiction is too high (since only tax base flight is involved) while the tax rate chosen by the exporting jurisdiction may be too high or too low since the tax base flight externality is countered by a tax exporting externality that arises by taxing non-residents on their imported goods. A second paper by Alain de Crombrugghe and Henry Tulkens that also appears in this volume examines more carefully the Mintz-Tulkens result regarding whether fiscal competition leads to tax rates that are too high or low. They show that even in the exporting jurisdiction, fiscal competition leads to a lower commodity tax rate. The reason is that simply looking at the nature of externalities is not sufficient to understanding the properties of a fiscal competition game. In order for the reaction function to be decreasing for both countries (a property required for existence of an equilibrium in the Mintz-Tulkens game), fiscal competition must lead to both optimal tax rates being chosen at a level that is too low. In other words, for a fiscal competition game’s equilibrium to exist, the tax exportation harmful externality must be dominated by the tax base beneficial externality – therefore, tax rates are chosen at levels that are too low. The de Crombrugghe and Tulkens paper is a sharp reminder that the game-theoretic models can be quite tricky so that a careful elaboration of the properties of the game is critical to deriving results. A quite different model was developed by Henry Tulkens with Jack Mintz to analyze corporate tax competition. In the 1996 Journal of Public Economics on ‘‘Optimality Properties of Alternative Systems of Taxation of Foreign Capital Income’’, Mintz and Tulkens examine corporate taxes as a means of withholding rents earned on foreign capital as well as providing revenues for governments to fund public services. A key aspect of this paper is a typology of different foreign tax regimes depending on taxes imposed on domestic firms, foreign-owned firms and foreign-source income earned by domestic corporations. One approach, commonly found in many countries today, is for a country to exempt the foreignsource income of resident domestic companies (the exemption system). Another approach is to tax foreign-source income of resident multinationals but to provide a partial or full credit for foreign corporate taxes. Mintz-Tulkens show that in a Nash tax setting game, a multiplicity of equilibria are possible including both the pure exemption and foreign-tax cum credit regime. In a Nash equilibrium, however, each government would like to tax foreign-owned multinationals operating in their jurisdictions in order to withhold income from foreign investors, or in the case
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of tax crediting, foreign governments. However, to achieve Pareto efficiency, a residence-based tax regime (no tax on levied foreign-owned companies) would only qualify – a source-based corporate tax falling on domestic and foreign-owned capital is therefore inefficient. Yet, a residence-based tax regime is not sustainable in a Nash equilibrium. In recent years, researchers have been turning to empirical analysis of tax competition games. Work has concentrated on whether governments tend to raise or lower taxes in the presence of increased tax rates elsewhere in neighboring jurisdictions. Henry Tulkens, always with an eye to empirical applications, provides a fascinating empirical paper with Jean-Franc¸ois Richard and Magali Verdonck, entitled ‘‘Dynamique des interactions fiscales entre communes belges (1983–1996)’’, published in E´conomie et Pre´vision, 2003. Unlike some earlier efforts, the authors explicitly incorporate dynamics in fiscal decisions since it is unlikely that governments are likely to respond immediately to tax policies adopted by other governments. After all, tax adjustments not only have revenue implications but also impacts on other public policy decisions that would need to be considered. Thus, some lag might occur from the time a neighbor changes its tax rate to when a jurisdiction makes its own policy adjustment. Using a panel data set of 500 municipalities over 14 years, two reaction functions are developed for property and individual income tax rates. The analysis finds that reaction functions are upward sloping (meaning that tax increases in one jurisdiction increase tax rates in the other) but the lags are quite long before tax rates are adjusted to their desired values. In the case of the property tax, the adjustment takes 15 years while with the income tax 10 years. The conclusion of the authors is that tax competition is a slow process and therefore not a major obstacle to implementation of tax decentralization. Teasingly, the reader is left with a question – why would it take longer for the property tax rate to adjust compared to the income tax rate? All four papers on fiscal competition are significant contributions to the literature. One should expect that Henry Tulkens might hatch a few more soon so that all of us will benefit from his insights.
Chapter 19 COMMODITY TAX COMPETITION BETWEEN MEMBER STATES OF A FEDERATION: EQUILIBRIUM AND EFFICIENCY
Jack Mintz, Henry Tulkens Journal of Public Economics 29 (3), 133–172 (1986).
The purpose of this paper is to characterize the outcome of tax competition between autonomous fiscal authorities. It treats the case of a two-region economy, where an originbased commodity tax is levied by each region on some private good to finance a local public good. A second private good is untaxed. We first describe ‘regional market equilibria’, whereupon consumers of each region allocate their purchases of private goods between domestic and nondomestic ones according to the structure of relative prices, taxes, and transportation costs. Next, regional optimal tax levels and public good quantities are derived, the tax of the other region being held constant. Fiscal competition arises from the ability of one region in choosing its tax to alter the tax base of the other. A ‘noncooperative fiscal equilibrium’ (NCFE) is then defined as the pair of fiscal choices such that each region’s tax and public good supply are optimal for itself, given those of the other region. After examining the conditions for the existence of a NCFE, its efficiency properties are considered. Pareto efficient tax levels are computed and compared with the NCFE ones, showing the sources and nature of fiscal externalities. Finally, it is established that, in this model, fiscal choices that are Pareto improving with respect to a NCFE never reduce the taxes in both regions, and always increase the tax of a tax importing region.
1.
Introduction
In federal countries where local governments possess the power to levy taxes of their own, it is generally the case that fiscal decisions by one government affect the tax revenues of the others. Typically, by altering its This work was achieved for the most part during visits of the first author at Louvain-laNeuve (ISE and CORE) and of the second author at the Department of Economics, Princeton University. They express their gratitude to these institutions for their hospitality;
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tax rates relative to those of other jurisdictions, each government has the ability to modify the size of its tax base at the expense (or the benefit) of its neighbors. This phenomenon of fiscal competition – with commodity, income, and/or capital taxes – has received some attention from public finance specialists. Much of the analysis has been couched in terms of allocative efficiency [see for example Musgrave (1969), Oates (1972), Boskin (1973), Starrett (1980), Arnott and Grieson (1981), Boadway (1982), and Gordon (1983)]. Other analyses have concentrated on the incidence of differential taxes, in models in which the taxes are treated exogenously [Mieszkowski (1966) and McLure (1969)]. Other work has also been devoted to the definition and characterization of equilibria in models where migration induced by local fiscal decisions determines the size of the jurisdictions [such as Westhoff (1977) for Tiebout models]. In this paper we characterize the equilibrium arising from fiscal competition through a single commodity tax. The tax is assumed to be levied according to the ‘origin principle’, i.e. at the producer’s level, as opposed to the ‘destination principle’ (i.e. at the consumer’s level). One may remark that the latter, by removing tax competition from the model below, would make the problem disappear. But it also requires costly, and sometimes even impossible monitoring by governments of all individual purchases made, which is quite contrary to observation in federal states.1 Without such monitoring, a destination-based tax becomes de facto an origin-based tax. Moreover, some jurisdictions may wish to take advantage of the possibility of ‘exporting’ taxes, which the origin principle allows.2 The fiscal equilibrium concept we propose to use is the Nash equilibrium of a noncooperative game in which the players are the local governments, their strategies are the local taxes and public expenditure levels, while the payoffs are regional welfare functions. In this framework, to D. Bradford, R. Anderson, R. Willig, P. Kyle, J. Stiglitz, R. Harris, D. Usher, D. Wildasin, R. Gordon, A. Mas Colell, J. Thisse, S. Bucovetsky, and participants at seminar presentations at Princeton, Queen’s, Montre´al, York, Alberta, the NBER, CORE, and the Canadian Economic Theory Conference 1984, for their helpful comments; and to the Departments of Economics at Princeton and Queen’s, the Social Sciences and Humanities Research Council of Canada, and CIM, Brussels, for their support. Comments from two anonymous referees as well as an important correction in equations (17)–(19) suggested by Rosella Creatini are greatly appreciated. 1 Implementation of the destination principle does require establishing ‘fiscal borders’ as they currently exist within the European Economic Community. Removing these borders, which is a long-range objective of the fiscal harmonization policy of the EEC, implies switching to the origin principle. See, for example, European Communities, Commission (1980) on this issue. 2 For a useful discussion on origin and destination-based commodity taxes, see Musgrave (1969, pp. 274–276).
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there arise issues of positive and normative nature, such as: (i) Does a noncooperative fiscal equilibrium always exist? (ii) Is a noncooperative fiscal equilibrium Pareto efficient? And, if not, what are the sources of inefficiency? (iii) Are the taxes higher or lower at a noncooperative fiscal equilibrium, compared to Pareto efficient ones? These issues will be addressed by means of the following simple model described in detail in Section 2. We consider two local jurisdictions each levying an origin-based commodity tax on a private good, the tax revenue being used to finance a local public good (with no interregional spillovers). Residents of each region can purchase the taxed private good either in their own region, or in the other region after incurring transportation costs and paying there the local tax. Each region is assumed to choose optimally the levels of its domestic commodity tax and of its local public good, by maximizing an indirect utility function of the representative resident subject to balancing the local public budget. In making its fiscal decisions, the region is also assumed to take as constant the tax level of the other region. This leads, in Section 3, to the notions of a region’s ‘fiscal reaction function’, defined in the space of the region’s taxes, and of a ‘noncooperative fiscal equilibrium’ (NCFE) defined as a pair of fiscal choices made in each community that constitutes a Nash equilibrium for the game sketched out above. While apparently similar to Williams (1966), where reaction functions drawn in the public goods space illustrate interactions due to the technological externalities created by interregional spillovers of the local public good, our analysis bears on the essentially different interactions due to the tax levels only. Characteristics of a NCFE are examined, such as existence and possible nonuniqueness; some unexpected effect of its strategic nature is also brought to light. In Section 4 we examine efficiency properties of this fiscal equilibrium. We show, in agreement with Boskin (1973) and Gordon (1983), that Nash equilibrium taxes are not Pareto efficient in general. However, we also show that there may exist situations where a NCFE is efficient. We finally establish a proposition that answers the question whether Pareto superior tax levels would be higher or lower than equilibrium ones. In our concluding Section 5, some possible extensions of the approach and of the model are discussed.
2.
Regional market equilibria and fiscal optima
In this section we describe our two-region economy; we characterize alternative market equilibria and optimal fiscal choices for each region, and we examine the nature of interregional fiscal externalities.
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Public Goods, Environmental Externalities and Fiscal Competition
The model
Each regional economy is composed of a single consumer, or an aggregate of identical consumers, who consume a local public good (Ri ^ 0) and a private commodity (Qi ^ 0) which is taxed, and supply labor (X i % 0) which is untaxed. The index i ¼ a,b denotes the region. The consumer’s preferences are assumed to be representable by a continuously differentiable, strictly quasi-concave utility function U i (Qi , X i , Ri ), strictly increasing in each of its arguments. In order to consume Qi , the consumers of the ith region can purchase this private good either at home (in quantity Qii ^ 0), or in the other region (in quantity Qij ^ 0, j 2 {a, b}, j 6¼ i), a transportation cost being then incurred as the consumers travel from one region to the other. We thus have Qi % Qii þ Qij :
(1)
Similarly, letting Xii % and Xji % 0 denote the amounts of labor supplied at home and abroad respectively, we have X i % Xii þ Xji
(2)
as the aggregate amount of labor supplied in region i. When labor services are performed abroad (Xji < 0), one can view this labor as synonymous with the importation of leisure by the jth community.3 For trade to take place in the private sector, any positive consumption of the taxed commodity imported from the jth community (Qij > 0) leads to labor offered by the ith region’s consumers in the other community to pay for the imported taxed commodity. Production of the private goods Q in both regions is assumed to take place under conditions of perfect competition and constant marginal costs. Producer prices are thus fixed. If the untaxed commodity is taken as numeraire in each region, then the prices of X i are the same and equal to one in both.4 For the taxed commodity, the producer prices which are denoted as pi (i ¼ a, b) need not be the same across communities so long as one region has a production advantage. 3 One can interpret X i as any untaxed commodity under the appropriate specification of the budget constraint. We view X i as labor, keeping in mind an example such as residents of a suburb who work and purchase goods in the centre of a city. One could also make labor the taxed commodity and Q the untaxed commodity without affecting the analysis of the equilibrium examined later. 4 Alternatively, we could select the untaxed commodity in only one community to be the numeraire and prices differing across communities for the untaxed commodity. This extension to the model would add little to the basic characterization of the equilibrium which we develop later.
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Denoting by ti the per unit tax, the cost to consumers of region i of domestic purchases is ( pi þ ti )Qii , which is paid in Xii units of labor. The cost of purchases from abroad, on the other hand, is ( pj þ t j )Qij þ tij (Qij ), where t ij (Qij ) is the total transport cost incurred by the consumers of i in acquiring the amount Qij of produced goods from the jth region; this cost is paid in Xji units of labor.5 The balance of trade is thus expressed by the two equalities: ( pi þ ti )Qii þ Xii ¼ 0 and ( pj þ t j )Qij þ tij (Qij ) þ Xji ¼ 0: Summing side by side and using (2), we obtain the inequality: ( pi þ ti )Qii þ ( pj þ t j )Qij þ tij (Qij ) þ X i % 0,
(3)
which describes the private goods consumption opportunities for the members of region i. Expression (3) may be seen equivalently as a budget constraint or as a production possibility set. Transport costs are assumed to be an increasing and strictly convex 0 function of Qij , with tij (0) ¼ 0 and tij (0) ^ 0. To justify the convexity assumption, consider that individual consumers are responsible for their own transportation of commodities. On the other hand, no transportation costs are associated with trades in the untaxed good; alternatively, these costs can be viewed as embedded in the function tij .6 Note finally that the ith consumer in each community has no explicit lump-sum income. Differential taxation of the two private goods Q and X is thus necessary so that tax revenue can be raised to finance the local public good.7 Given taxation of only one private good, the total tax revenue raised in 5 Transport costs could alternatively be paid in Xii units of labor, since given our treatment of the numeraire, labor at home and abroad are treated as perfect substitutes. Inequality (3) is unaffected by this treatment of transport cost. 6 The way in which we have included transportation costs in the consumer’s budget constraint is equivalent to separating the consumption decision of the consumer from the transportation decision made by producers of transport services, under the assumption that the latter are wholly owned by the residents of the local community. Specifically, profits from transportation from i to j which can be written as pij ¼ ( pi þ ti )Qij ( pj þ t j )Qij t ij (Qij ), accrue to residents of i only. Note that the optimal choice of Qij for a profit maximizing transport firm only depends on consumer prices in each of the regions and transport costs. 7 As is well known in the optimal tax literature, a uniform tax levied on all private commodities when there is no lump-sum income would yield no tax revenue for the government [Sandmo (1974)]. Differential taxation is needed and, without loss of generality, one good can be made nontaxable.
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region i is equal to ti (Qii þ Qji ) – where Qji are purchases by the jth community residents of goods produced in the ith community – and the quantity of the public good, Ri , is constrained to be less than or equal to this amount [as stated in (9) below]; Ri is thus measured in units of numeraire (i.e. of X). In Sections 2.2 and 2.3 we shall separate market from fiscal decisions in each region by assuming that consumers choose Qi , Qii , Qij , and X i with Ri , ti , and t j fixed on the one hand, and that on the other hand fiscal authorities choose ti and Ri , taking into account the ensuing private choices of Q and X. The former assumption can be justified by considering that when there are a large number of consumers in each region, the impact of each consumer’s choice of Q and X on the fiscal variables ti , t j , and Ri is negligible. This also clearly sets our analysis in the framework of optimal tax theory. 2.2.
Regional market equilibria
2.2.1. Types of equilibria in one region In this model, a market equilibrium for one region consists of its consumers’ preferred choices of work effort, X i , and of consumption of the produced good, Qi , among those that are compatible with the ‘budget’ constraint (3), and for given levels of the taxes ti , t j , and of the public good Ri . More precisely: Definition 1. 8i, j 2 {a, b}, i 6¼ j, a regional market equilibrium (RME) relative to (ti , t j , Ri ) is a solution of max
{Qi , Qii , Qij , X i }
U i (Qi , X i , Ri )
(4)
s.t. (1), (2), and (3); X i % 0; Qi , Qii , Qij ^ 0 , Ri , ti , t j ¼ constants: According to the values taken on by the variables Qii and Qij at the solution, three different types of equilibria can be distinguished, which we index by ui ¼ I, II, or III, and refer to as follows: i (ui ¼ I); (a) if Qi i > 0 and Qj ¼ 0, ‘autarkic’ equilibrium i (b) if Qi (ui ¼ II); i > 0 and Qj > 0, ‘mixed’ equilibrium i i (c) if Qi ¼ 0 and Qj > 0, ‘no-production’ equilibrium (ui ¼ III). The assumed properties of the functions U i and t ij , and the firstorder conditions for a solution of the problem, yield a simple character-
Ch.19 Commodity Tax Competition between Member States of a Federation
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ization of each one of these three types of equilibria in terms of the relative values of the prices, taxes, and marginal transportation costs. Proposition 1. 8i, j,8 a RME (Qi , X i ) 2 R2 {0} is unique, and characterized by @U i @U i [ min {pi þ ti ; pj þ t j þ ti0j (Qi j )}] ¼ 0; @Qi @X i
(5)
in particular, if ui ¼ I (ui ¼ II, ui ¼ III), then pi þ ti % ( ¼ , ^ , respectively) pj þ t j þ ti0j (Qi j ):
(6)
Proof. See the appendix. Condition (6) suggests that at an autarkic equilibrium, where region i consumers buy goods produced only by their own region, the consumer price of their region’s taxed good (pi þ ti ) is less than that of the other 0 region (pj þ t j þ t ij (0)); this is of course why they have no incentive to travel to the other region. At a mixed equilibrium, where the consumers of region i buy goods produced by both regions, the consumers purchase goods from the other region until the consumer price of their own region is equal to the marginal consumer price inclusive of transportation costs in the other region: pi þ ti ¼ pj þ t j þ t i0j (Qi j ). For a mixed equilibrium to occur in the ith region, it is thus required that pi þ ti > pj þ t j þ t i0j (0). Finally, at a no-production equilibrium, where the consumers of region i purchase abroad only the taxed good, one has ( pi þ ti ) > ( pj þ t j þ t i0j (Qi j )); then, it is also the case that the ith region raises no tax revenue and hence supplies no public good (it will be shown later that this type of equilibrium may not occur when optimal taxes are chosen; see proposition 4). On the other hand, the uniqueness property – which rests in part on the strict convexity9 of tij – is important for deriving below the indirect utility function for each region, and thereby regional optimal taxes. Being analogous to a standard consumer equilibrium, each type of RME induces demands for commodity Q and a supply of X, all of which are continuous and differentiable functions of the parameters that have been held fixed, namely the two taxes and the quantity of the public good. More specifically, there are three demand functions for commodity Q: 8 For notational brevity we write, from now on, ‘8i, j’ always meaning ‘8i, j 2 {a, b}, i 6¼ j’. 9 If t ij were not strictly convex – e.g. linear – uniqueness would fail to hold for the variables Qii and Qij at a mixed equilibrium. If it were strictly concave, the budget set (3) would not be convex and (5) would no longer be a sufficient characterization of a RME.
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Public Goods, Environmental Externalities and Fiscal Competition
Qii (ti , t j , Ri ),
(7a)
Qij (ti , t j , Ri ),
(7b)
Qi (ti , t j , Ri ):
(7c)
They are related to one another through condition (1), which is always satisfied as an equality at any equilibrium. On the other hand, the labor supply function X i (ti , t j ,Ri ) is related to the demands for Q through the budget constraint (3). The form of these functions varies, however, according to the type of equilibrium from which they are derived. In particular, some of their partial derivatives are of a definite sign under some equilibria, and identically zero in other ones. Since these partials will play a key role in the tax competition analysis that follows, we spell them out in detail here. Proposition 2. If commodity Q is not an inferior good, the partial derivatives of the demand functions (7a)–(7c) have the following signs (the derivatives not mentioned are identically zero): (a) for ui ¼ I,
@Qi @Qii ¼ i < 0; @ti @t @Qi @Qii ¼ b 0; @Ri @Ri
@Qij @Qi @Qii < 0; < 0; > 0; @ti @ti @ti @Qij @Qi @Qii % 0; b 0; < 0; @ti @t j @ti @Qi @Qii ¼ b 0; @Ri @Ri @Qi @Qij ui ¼ III, j ¼ j < 0; @t @t i @Qij @Q ¼ b 0: @Ri @Ri
(b) for ui ¼ II,
(c) for
Proof. See the appendix. To motivate the results stated in proposition 2, note first that an increase in the tax ti of the consumers’ own region affects the demands for Q only in autarkic (ui ¼ I) and mixed (ui ¼ II) equilibria; and this occurs in terms of the usual substitution and income effects as expressed by the Slutsky equation. In a mixed equilibrium, however, there is an additional
Ch.19 Commodity Tax Competition between Member States of a Federation
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effect. Due to the strict convexity of the transport cost function, which creates a nonlinear budget constraint, pure profits arise from transportation activities; since an increase in ti causes the consumers to import more of the taxed good from abroad (@Qij =@ti > 0), this also increases the pure profits and thus reduces the income effect associated with the change in ti . On the other hand, when the other region raises its tax, t j , the demands for Q in region i are affected in equilibria of types II and III only. When a mixed equilibrium prevails (ui ¼ II), since the left-hand side of (6-II) remains constant, the slope of the budget constraint is unchanged at the equilibrium point and the entire effect of the change in t j translates in terms of a shift of the constraint towards the origin. There is thus only an income effect, which can be substantiated by the fact that the reduced imports (@Qij =@t j < 0) also reduce the pure profits. Under a no-production equilibrium (ui ¼ III), this income effect is also accompanied by a substitution effect. Finally, the impact of changes in the public good Ri on the final demand Qi cannot be made specific at this level of generality; as documented by Wildasin (1979), it depends upon whether the public and private goods are complements or substitutes, and upon whether the private good is normal or inferior (since Ri has a real income effect on its demand). In this paper we shall not need specific signs for the partial derivatives with respect to Ri , but we shall take advantage of the somewhat unexpected property that the demand function Qij does not vary with Ri in a mixed equilibrium. Intuitively, Qij is an intermediate good, the demand for which depends on the price at which consumers are willing to purchase it (pi þ ti ) and the marginal cost of acquiring it 0 (pj þ t j þ tij (Qij ) )) (see footnote 5). The public good has an effect only on final demands not on the demand for an intermediate good. As for Qii , it is simply the residual demand: Qii ¼ Qi Qij . Since Ri affects Qi (and not Qij ), then Qii is also affected. We shall therefore henceforth write the demand function for Qij simply as Qij (ti ,t j ). 2.2.2. Regimes of the economy We shall now move from considering a single region’s market choices to a characterization of the two-region economy taken as a whole, when each region is at a RME. To this effect we now introduce the notion of a ‘regime’, defined as follows. Definition 2. For the economy consisting of the two regions a and b, a regime is the pair of RMEs induced by the taxes prevailing in each region. In view of the uniqueness property of the RMEs noted earlier, the regime induced by any tax pair (ta ,tb ) is also uniquely determined. With two regions and three possible types of equilibria in each of them, the induced
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Public Goods, Environmental Externalities and Fiscal Competition
regime will be one out of the nine conceivable ones, namely (Ia ,Ib ), (Ia ,IIb ), (Ia ,IIIb ), (IIa ,Ib ), (IIa ,IIb ), etc. However, four out of the latter can be ruled out [namely (IIa ,IIb ), (IIa ,IIIb ), (IIIa ,IIb ) and (IIIa ,IIIb )], since they involve ‘cross hauling’ of the commodity Q (i.e. both Qa b > 0), Qb > 0), and the following proposition shows that this phenomenon is not a compatible with the simultaneous occurrence of a RME in each region. j Proposition 3. When (5) holds in both regions, Qi j > 0 ) Qi ¼ 0, 8i, j.
Proof. See the appendix. An exhaustive list and indexing of the regimes that can emerge in a two-region economy will prove useful (see table 1). Note that the regimes are indexed ‘from region i’s viewpoint’: indeed, what is denoted above as regimes 2 or 4, in both of which region i is exporting taxes, corresponds from region j’s viewpoint to the regimes labelled here as 3 and 5, respectively, where j is tax exporting. Table 1 Regime index (from region i’s viewpoint) r¼1 r¼2 r¼3 r¼4 r¼5
Type of equilibrium in region i ui ui ui ui ui
¼I ¼I ¼ II ¼I ¼ III
Type of equilibrium in region j uj uj uj uj uj
¼I ¼ II ¼I ¼ III ¼I
While every tax pair determines a single regime, a given regime may be induced by many different tax pairs. Therefore, a natural question is to ask which tax pairs induce which regime. More precisely, letting T ¼ {(ta ,tb )jta ^ 0, tb ^ 0} be the set of all possible tax pairs, the problem amounts to identifying five subsets Tr , r ¼ 1,2, . . . ,5, of T, whose elements determine each one of the five regimes listed above. This can be done by using the conditions (6) characterizing the equilibria. Indeed: (i) By virtue of (6) for (ui ¼ I) applied to both regions i and j: T1 ¼ {(ti ,t j ) 2 Tj pi þ ti % p j þ t j þ t i0j (0); p j þ t j % pi þ ti þ t ij0 (0)}: In fig. 1 the area T1 contains the set of pairs (ti ,t j ) that satisfy the two 0 inequalities appearing in this definition. Note that if 0 tij (0) ¼ tji (0) ¼ 0, the two lines AA and BB collapse into a single one, CC, with intercept pj pi ; if, in addition, pj ¼ pi , this line coincides with the first diagonal.
Ch.19 Commodity Tax Competition between Member States of a Federation
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ti T5 A
>0 pj
−
pi
+
τ i'
[Q i* j j
(t i
,0)]
C
T3 B
+1 T1 p j − p i + τ ji' (0)
pj − pi
A
+1 T2
>0 T4
C 0
p j − p i − τ j' (0) i
tj B
Figure 1. Subsets of the tax-space that induce the same regime.
(ii) By virtue of (6) (for ui ¼ II and III) applied to region j: T2 ¼ {(ti ,t j ) 2 Tj pj þ t j ¼ pi þ ti þ tij0 (Qij ) with Qij ^ 0 and Qj j > 0; pj þ t j % pi þ ti þ tij0 (Qij ) with Qij > 0 and Qjj ¼ 0}: This subset T2 is also represented in fig. 1. Since the equality appearing in its definition implies that j j i0 pi þ ti ¼ pj þ t j t ij0 (Qj i ) < p þ t þ t j (0),
(8)
it also implies that region i is indeed in equilibrium of type I. The same equality with Qj i ¼ 0 yields the boundary between T2 and T1 ; and the last expression, taken in equality form, yields the boundary between T2 and T4 (for a justification of the positive slope of the latter, see below). (iii) Permuting i and j we have in a symmetric way: i i T3 ¼ {(ti ,t j ) 2 Tj pi þ ti ¼ p j þ t j þ ti0j (Qi j ) with Qj ^ 0 and Qi > 0; i i pi þ ti % pj þ t j þ tji0 (Qi j ) with Qj > 0 and Qi ¼ 0},
460
Public Goods, Environmental Externalities and Fiscal Competition
where the equality, taken for Qi j ¼ 0, defines the boundary between T3 and T1 ; and the last expression, taken in equality form, defines the boundary between T3 and T5 . (iv) Finally, using (6) (for ui ¼ III): T4 ¼ {(ti , t j ) 2 Tj pj þ t j ^ pi þ ti þ tij0 (Qij )with Qij > 0 and Qjj ¼ 0}, where the inequality implies that region j is in equilibrium of type III, as well as that region i is in equilibrium of type I since it also implies (8). (v) Symmetrically: T5 ¼ {(ti ,t j ) 2 Tnpi þ ti ^ p j þ t j þ ti0j (Qji )with Qji > 0 and Qi i ¼ 0}, where now region i is in a no-production equilibrium and region j in an autarkic one. That the slopes of the boundaries between the subsets T1 and T2 and T1 and T3 are equal to þ1, is clear from the definition of these sets. As far as the other boundaries are concerned, the property that they have a positive slope is established as follows. Consider a tax pair (ti ,t j ) corresponding to a point on the boundary between T3 and T5 . By definition of these sets, these taxes induce for region i a RME such that Qii (ti , t j , Ri ) ¼ 0 [with Ri ¼ 0 and Qij (ti , t j ) ¼ Qi ]. Any change in taxes (dti , dt j ) that induces another RME with the same properties must satisfy @Qii i @Qii j dt þ j dt ¼ 0 @ti @t or dti @Qii =@t j ¼ : dt j @Qii =@ti Since by proposition 2 (for ui ¼ II) the numerator is positive and the denominator negative, this ratio is positive. On the other hand, since at any point on the boundary between T2 and T4 Qij ( ) ¼ 0 holds (with Rj ¼ 0 and Qji ¼ Qj ), a symmetric argument applies. Fig. 1 thus clearly identifies the five subsets we sought for. It also allows us to recognize immediately which regime, and hence which pair of market equilibria, are induced by any conceivable pair of taxes. 2.3.
Regional fiscal optima
We now consider the local political decisions whereby the fiscal variables of the model are determined. Specifically, we assume that within
Ch.19 Commodity Tax Competition between Member States of a Federation
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each region i the prevailing tax and public good quantity are ‘optimal’ ones; that is, they are chosen so as to maximize local welfare, taking into account both the other region’s tax and the local market equilibrium so induced. We call a ‘fiscal optimum for region i’ the pair (ti , Ri ) that has these properties. In order to characterize such public choices more precisely we need: Definition 3.
8i, an indirect utility function is the function
V i (ti , t j , Ri ) U i [Qi (ti , t j , Ri ), X i (ti , t j , Ri ), Ri ], where Qi ( ) ¼ Qii (ti ,t j ,Ri ) þ Qij (ti ,t j ), X i ( ) ¼ [( pi þ ti )Qii (ti ,t j ,Ri ) þ ( pj þ t j )Qij (ti ,t j ) þ tij (Qij (ti ,t j ))]: Definition 4. 8i, a fiscal choice is any pair (ti , Ri ) with ti ^ 0 and Ri ^ 0. Definition 5. 8i, given the tax t j chosen by the other region, a feasible fiscal choice is a pair (ti ,Ri ) that satisfies Ri % ti [Qii (ti , t j , Ri ) þ Qij (ti , t j )]:
(9)
Note that while the feasibility of region i’s fiscal choices is affected by the other region’s tax choice t j , it is not influenced by its public good choice Rj , due to the fact that in (9) @Qji =@Rj ¼ 0, according to proposition 2. Definition 6. Given t j , a fiscal optimum for region i (relative to t j ) is the fiscal choice [ti , Ri ] that solves the problem: max V i (ti , t j , Ri ),
{ti , Ri }
s.t. (9), ti ^ 0, Ri ^ 0, and t j ¼ constant:
(10)
We shall exhibit below explicit formulae for the optimal tax ti , which are obtained by means of usual first-order condition techniques. However, the peculiar structure of our model, as well as some of our subsequent developments, justify that we show here in some detail how we proceed. When solving problem (10), account must be taken of the fact that both the objective function V i and the feasibility constraint (9) take on different forms according to the alternative tax pairs (ti , t j ) being considered,
462
Public Goods, Environmental Externalities and Fiscal Competition
due to the fact that the latter may induce different regimes. Specifically [in all expressions below, recall that Qii ¼ Qii (ti , t j , Ri ) and Qij ¼ Qij (ti , t j )]: (i) For all pairs (ti , t j ) that belong to Tr , r ¼ 1, 3, we have: Vri (ti , t j , Ri ) ¼ U i [Qii þ Qij , ( pi þ ti )Qii ( pj þ t j )Qij t ij (Qij ), Ri ],
(11)
where Qij ¼ 0 if r ¼ 1, and Qij ^ 0 if r ¼ 3; and (9) is of the form Ri % ti Qii . Hence, the Lagrangean associated with problem (10) reads: Lir (ti , t j , Ri , cir ) ¼ Vri ( ) þ cir (ti Qii Ri ),
(12)
where cir ^ 0 is the Kuhn–Tucker multiplier. (ii) For all pairs (ti , t j ) that belong to Tr , r ¼ 2, 4, we similarly have: Vri (ti , t j , Ri ) ¼ U i [Qii , ( pi þ ti )Qii, Ri ];
(13)
and (9) is of the form Ri % ti (Qii þ Qji ). Hence, the Lagrangian in these cases reads: Lir (ti , t j , Ri , cir ) ¼ Vri ( ) þ cir [ti (Qii þ Qji ) Ri ]:
(14)
We can now state [in the expression below, g i ¼ (@U i =@X i )=(@U i =@Ri )]: Proposition 4. 8i, given t j , the tax ti at a fiscal optimum is of one of the following forms: (i) ti ¼
Qi (1 g i ) ii
, @Q @Qi @tii þ g i Qii @Rii
(15:1)
if the pair [t j , ti ] belongs to the interior of Tr , r ¼ 1, 3; (ii)
ti ¼
h
Qii (1 g i ) þ Qij
@Qii @ti
þ
@Qji @ti
@Qi
þ gi Qi @Rii
i,
(15:2)
if the pair [t j , ti ] belongs to the interior of Tr , r ¼ 2, 4. Finally, no fiscal optimum for i, with ti > 0, can be such that the pair [ti , t j ] belongs to T5 . Proof. See the appendix. The formulae (15.1) and (15.2) show that the optimal tax rate is lower the more the public and taxed goods are substitutes (@Qii =@Ri < 0),
Ch.19 Commodity Tax Competition between Member States of a Federation
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and the stronger the (negative) responsiveness of the demands for the private good to changes in the tax (@Qii =@ti and @Qji =@ti ).10 The optimal tax formulae in (15.1) and (15.2) can be shown to correspond to a Ramsey formula, although the Ramsey formula is arrived at under different assumptions. In Ramsey’s problem, government expenditure is kept fixed and the government selects optimally several commodity taxes (as in differential incidence analysis). In our problem, the government optimizes with respect to both the public good and a commodity tax (balanced budget analysis).11 Concerning proposition 4, a further point, of a more formal nature, is in order. To facilitate the equilibrium analysis in the next section, we introduce here the assumption that for each t j , the regional optimal tax ti is unique, at least within the subset Tr to which the pair [t j ,ti ] belongs. A sufficient condition for this property to hold would be that the Lagrangians Lir in (12) and (14) be strictly quasi-concave in ti , with Ri chosen optimally for each ti , within each one of the domains Tr . One example is the case where (i) utility is additive, quadratic in Qi , and linear in X i and Ri and (ii) the transport cost function is quadratic. It may be that in this particularly simple three-good model, uniqueness could hold without imposing such severe restrictions on these functions, but we have not been able to show it so far. 10
For ti ^ 0 it is sufficient that gi % 1 and i i @Qi i i @Qi @ti > g Qi @R in r ¼ 1 and 3, and gi % 1 þ Qij =Qii and @Qi @Q j @Qii i i i þ i ^ g i Qii @t @t @Ri
in r ¼ 2 and 4. To see the correspondence of (15.1) with a Ramsey formula, (15.1) can be rewritten in terms of compensated demand elasticities to obtain: 11
ti b , ¼ qi «~ gi (@Qii =@Ri ) where qi ¼ pi þ ti (consumer price),
(150 )
! ~ i qi ~ i @Q ~i @Q @Q i i i «~ ¼ i i compensated elasticity of demand, as i ¼ i , : @q Qi @t @q @Qii b ¼ 1 g i ti (where @Qii =@Iis the income effect) @I When the taxed commodity and the public good are independent in demand (@Qii =@Ri ¼ 0) and cross price effects on demands are zero for all goods, then (150 ) is the Ramsey formula for an optimal tax rate (Atkinson and Stiglitz (1980, p. 372)).
464
2.4.
Public Goods, Environmental Externalities and Fiscal Competition
Interregional fiscal externalities
The welfare of region i, as represented in definition 3, is not a function of its own fiscal and market decisions only: the tax t j chosen by the other region affects V i also, much in the same way as an externality. This is also apparent in formulae (15.1) and (15.2), where some of the values of Qii and Qji vary with t j , in view of proposition 2. A clear understanding of this phenomenon – already pointed out in the previous literature on fiscal federalism [see, for example Boskin (1973) or Gordon (1983)] – is provided by evaluating the partial derivative of V i with respect to t j , taking into account the feasibility constraint (9). Using again the Lagrangians Lir (ti ,t j ,Ri ,ci ) stated above, we get (a)
for r ¼ 1,
(b)
for r ¼ 2,4,
(c)
for r ¼ 3,
@Li1 ¼ 0, @t j j @Lir i i @Qi ¼ c t > 0, @t j @t j i @Li3 @U i i i i @Qi ¼ Q þ c t v 0: @t j @X i j @t j
(16:1) (16:2) (16:3)
The signs of these derivatives are established by referring to proposition 2 (@Qii =@t j ¼ 0 for ui ¼ I; @Qji =@t j > 0 for uj ¼ II which is the type of equilibrium prevailing in j in both regimes 2 and 4; and @Qii =@t j > 0 for ui ¼ II, i.e. the type of equilibrium prevailing in i in regime 3, and to proposition 4, the proof of which establishes why i 1 @U i i @Qi c ¼ 1t > 0: @Ri @Ri i
These expressions reveal that the fiscal externality potentially has two components. The first, which we call the ‘public consumption effect’ consists of the positive terms ci ti (@Qji =@t j ) (in regimes 2 and 4) and ci ti (@Qii =@t j ) (in regime 3). It is a beneficial effect for region i: indeed the products ti (@Qji =@t j ) and ti (@Qii =@t j ) just measure the increase in the public good Ri that region i can secure itself thanks to the increased purchases made in i either by the consumers of region j (Qji ) or by the consumers of region i (Qii ) in reaction to the increase in the tax t j ; and ci values this increase in terms of the social utility of region i. The other component of the fiscal externality, which we call the ‘private consumption effect’, consists of the negative term – (@U i =@X i )Qi . It appears in regime 3 only, i.e. when region i is in a mixed equilibrium. It is always detrimental for region i: indeed, a unit increase in the tax t j reduces the real income of the consumers of region i by the amount Qij and
Ch.19 Commodity Tax Competition between Member States of a Federation
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this reduction in income is valued in social utility terms by the marginal utility of income (@U i =@X i ) as X i is the numeraire. As the expressions (16.1)–(16.3) above make clear, only the beneficial component of the externality is present for i in regimes 2 and 4, whereas both effects are present under regime 3 (albeit the positive one is of a slightly different form), with the sign of their sum depending on the relative weights of private (@U i =@X i ) versus public (@U i =@Ri ) consumption in region i’s social utility function. Note in passing that when the private consumption effect dominates, an increase in j’s tax is disadvantageous for region i: but this is only natural in a regime where i is importing taxes.
3.
Fiscal competition
3.1.
Interregional noncooperative fiscal equilibrium: definition and motivation
Our analysis thus far has focused on a single region’s market and fiscal decisions, as functions of its own characteristics as well as of its environment. The study of competition, to which we now turn, essentially consists in considering simultaneously the decisions made by the two regions. A natural conceptual framework for this task is to associate with our two-region economy a two-person game, in which the players i ¼ {a, b} are the local governments, the strategies are the local taxes ti and expenditure levels Ri , and the payoffs are the regional welfare functions V i . Treating this game as a noncooperative one, the concept of Nash equilibrium suggests itself to describe and characterize any state of the economy where each region is at a fiscal optimum relative to the fiscal choice made by the other. Such a state will henceforth be called a ‘noncooperative fiscal equilibrium’ (NCFE). Clearly, the noncooperative assumption is an extreme one in the context of fiscal federalism. We shall therefore, in Section 4, also consider some cooperative aspects of this game in terms of its Pareto efficient outcomes. On the other hand, an explicit study of noncooperative equilibria has the intrinsic interest of providing a characterization of what may be expected when no cooperation is taking place. In proceeding to the formal specification of the game, one must recall that for each player i the two components ti and Ri of their strategies are linked by the feasibility condition (9). Hence, the payoff function must actually be formulated as the Lagrangian Lir (ti ,t j ,Ri ,cir ), as defined in (12) or (14). Moreover, the assumption of the welfare maximizing choice of ti and Ri , by implying that (9) is always satisfied as an equality at a fiscal optimum for i, also implies that Ri is determined as
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Public Goods, Environmental Externalities and Fiscal Competition
soon as any value of ti is chosen. Player i’s only relevant strategic variable is therefore the tax level ti , and his payoff function reduces to W i (ti ,t j ) ¼ max min Lir (ti ,t j ,Ri ,cir ), i def
Ri
cr
r ¼ 1, . . . ,5:
The game associated with our two-region economy is now completely specified by the triplet G ¼ ({a,b},T,(W a ,W b )), of which the first element is the players set, the second is the strategy space – namely R2þ – and the third is the pair of payoff functions just defined. We may then state: Definition 8. For the economy consisting of the two regions a and b, a NCFE is a Nash equilibrium of the game G, i.e. a pair of tax choices (ta ,tb ) 2 T such that 8i,j 2 {a,b},i 6¼ j,ti maximizes W i (ti ,tj ). Graphically, a NCFE is a point in the space T of possible tax levels (see fig. 1). More technically, it is also characterized [see, for example, Friedman (1977, pp. 149 and 152 ff.)] as a fixed point of the best reply mapping f that associates with every tax pair the pair of optimal taxes of the two regions; that is, writing the best reply mapping as f : T ! T;(ta ,tb ) ! [ta ¼ f a (tb );tb ¼ f b (ta )], where each f i , i ¼ a,b, is called region i’s fiscal reaction function, a NCFE is a pair (ta ,tb ) such that ta ¼ f a (tb ) and tb ¼ f b (ta ). A priori, there may be one or several equilibria in our model; and there may be none at all. Another question is also which regime of the economy is induced when an equilibrium does exist. The study of such questions requires a close examination of the best reply mapping just defined, i.e. of each region’s fiscal reaction function. 3.2.
A region’s fiscal reaction function
The graphs of various examples of region i’s fiscal reaction function f i are represented in figs. 2(a)–(c). They all illustrate the fact that in the context of the present model, such functions have the characteristic property of exhibiting a ‘downward jump’ at some point t j of its domain. This is formally established by Proposition 5. At least at one point, t j , of its domain, the fiscal reaction function f i takes on two values, ti and ti . Furthermore, one has lim«!0 f i (t j «) > lim«!0 f i (t j þ «). Proof. See the appendix. The economic interest of exhibiting this property of multivaluedness of the fiscal reaction function is to point out that in the
Ch.19 Commodity Tax Competition between Member States of a Federation
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ti
ti
T3
T3 T1
T2
T2
0
=
(a)
tj
=
0
tj
tj (b)
tj
ti
T3 T2
T1 0
=
tj
tj
(c)
Figure 2. Possible forms of a region’s fiscal reaction function.
present model12 there is always a value of the other region’s tax, t j , for which region i is socially indifferent between two optimal domestic taxes: a high one, which induces an autarkic or a mixed equilibrium for itself (and hence regime 1 or 3), and a low one, which induces a mixed equilibrium for j (and hence regime 2). Such a switch from a high to a low tax may indeed be neutral for region i, in social utility terms, if the welfare loss it incurs from the reduction in the level of the public good that it can 12
The multi-valuedness of the reaction functions could (but not necessarily) be eliminated by assuming that the taxed commodities purchased at home and abroad are not perfect substitutes: U i (Qii ,Qij ,Ri ) would be the utility (whereas previously Qi ( ¼ Qii þ Qij ) appeared in the utility function). If 8i,j,@U i =@Qij ! 1 as Qij ! 0, then both regions would always be in a mixed equilibrium (cross hauling being possible) and individuals would always consume taxed goods produced by each region. This, however, changes the economic character of the model, implying that propositions 2 and 4 would be amended. Indeed, the optimal choices made by region i would depend not only on t j but on Rj as well. In some cases it may be realistic to assume that Qii and Qij are imperfect substitutes (for example, commuting individuals have to eat both where they work and where they live, and meals at different locations may not be seen as perfect substitutes), but in many other cases it would not (individuals may treat as perfect substitutes gasoline bought in either of the jurisdictions).
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Public Goods, Environmental Externalities and Fiscal Competition
finance by itself is just compensated by the welfare gains it obtains from (i) the reduction of the deadweight loss on its own consumers associated with its tax, and (ii) the expansion in its tax base, and hence public good production, entailed by the fact that residents of j switch from autarky to a mixed equilibrium (i.e. start purchasing goods in region i). Figs. 2(a)–(c) illustrate the three typical cases that may arise. The figures differ in that in (a) and (b) the sets Tr are drawn under the assump0 0 tion that tij (0) > 0,8i,j; whereas in (c), the assumption is that t ij (0) ¼ 0, which implies, according to the above definition of T1 , that this set reduces to a line. In all three cases, the jump is downwards and entails a switch in regime. On the other hand, when the graph of f i contains points that belong to T1 [fig. 2(a)], the jump occurs in the neighborhood of the boundary between T1 and T2 ; otherwise, the jump necessarily takes place from a point in T3 to a point in T2 [figs 2(b) and (c)]. While the proposition establishes the existence of a jump at one particular point, t j , along the t j axis, more such jumps may conceivably arise at other values of t j . We shall assume, however, that this is not the case. Our justification for this is not only one of analytical tractability of the NCFE existence problem (see below); it also results from the fact that the model does not contain characteristics that for other values of t j would make this occurrence a necessary one, as it is the case for t j . A sufficient condition for our assumption to hold is that for every t j , the payoff function W i be strictly quasi-concave in ti ; this in turn implies that for every t j 6¼ t j , the regional optimal tax ti is unique, making the reaction function f i single-valued there.13 The latter is also continuous, since in (15.1) and (15.2) the taxes ti are continuous functions of their arguments.14 Note also that along f i , the (feasible) social utility level of region i, that is, the image of its payoff function W i , remains constant across T1 [in view of (16.1) above]; it is always increasing across T2 [in view of (16.2)]; and may increase, decrease, or stay constant in T3 [by (16.3)]. Finally, the sign of the slope of the fiscal reaction function is another property worth some attention: when a region raises its tax, is the other region’s best reply to lower its own, or to raise it too? We shall need to assume in proposition 6 below that the reaction function is nonincreasing. This property is indeed verified at point t j , since by proposition 5 the jump there is downwards. It is also verified for that part (if any) of the graph of the function that belongs to T1 , since f i is constant 13
Another way to state the sufficient condition is that the bordered Hessian of second order conditions of the optimal tax problem (definition 6) be negative definite. 14 Namely, the demand functions Qii ,Qij ,Qji , their first derivatives (which are in turn continuous with respect to t j , a property of the regional market equilibria; see proposition 1 above), and gi (which is the ratio of two partial derivatives of the utility function assumed continuously differentiable at the outset).
Ch.19 Commodity Tax Competition between Member States of a Federation
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there [see fig. 2(a)]. But we have no general argument to establish nonincreasingness elsewhere. We may consider, however, some intuitive explanations of when this property may hold by examining two polar cases. i of the Consider first the case in which region i desires a fixed level R 2 i i2 public good (which can be shown to occur if @ U =@R ¼ 1). Since the public budget must balance, then this requires i ¼ ti Qi , if r ¼ 1,3, R i and i ¼ ti (Qi þ Qj ), if r ¼ 2: R i i Suppose now the other region increases its tax. In regime 1, where there is no fiscal competition, clearly @ti =@t j ¼ 0. For the other regimes (r ¼ 2,3) differentiation of the above equations, taking into account the reaction of i, yields: @Qj
ti @t ji @ti (a) for r ¼ 2, ¼
; @t j Qi þ Qj þ ti @Qii þ @Qji i i @ti @ti @Qi
ti @t ji @ti (b) for r ¼ 3, ¼ : @t j Qi þ ti @Qiii i @t One can show that the denominator is positive, using the necessary and sufficient conditions for the optimal choice of ti . Since, by proposition 2, @Qij =@t j > 0 and @Qji =@t j > 0, @ti =@t j is unambiguously negative for r ¼ 2,3, and i’s reaction function is downward sloping. This is sensible: given that the demand for Ri is fixed, an increase in the tax of region j expands region i’s tax base, allowing i to reduce its tax. Now consider the case where the demand for the public good is not fixed. One can show that with separability between the taxed good and the public good and constant marginal utility of the latter, the reaction function may be upward sloping. An extreme, but important, example of this is when the region maximizes only its fiscal revenue R (which amounts to assuming that the indirect social welfare function introduced in Section 2.2.2 is such that @U i =@X i ¼ @U i =@Qi ¼ 0 and @U i =@Ri ¼ 1): if in addition the demand functions are linear, the reaction function is upward sloping in this case. Intuitively, if a region puts a higher evaluation on the public good compared to the private good, it may be led to take advantage of the other region raising its tax (thus reducing competition) to increase its own, so as to raise the level of the public good. These two polar cases suggest that, while nonincreasingness of the reaction function cannot be shown to hold as a general property, the latter
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Public Goods, Environmental Externalities and Fiscal Competition
does follow when a sufficient degree of inelasticity of the demand for the public good prevails.15
3.3.
Existence of the NCFEs and induced regimes
Consider alternative examples of the two reaction functions f i and f j , as illustrated in figs. 3(a)–(d). To satisfy definition 8 of a NCFE, a pair of taxes – a point in T – must be such that both curves pass through this point (intersecting there, as with points E on figs. 3(a), 3(b), and 3(c); or just touching, as with the points that make up the set « on fig. 3(c), a case of multiple equilibria). But, as fig. 3(d) shows, the reaction curves may be such that there exists no point in T where they intersect.16 Of course, this phenomenon is due to the (local) multivaluedness of the reaction functions. The existence of a NCFE in our model can thus be ensured only for a subclass of the conceivable reaction functions it implies. The widest such class we can identify at this stage is provided by the following proposition, inspired by a recent theorem of Bamon and Fraysse (1985):
15
If Qi and Ri are separable, the slope of the reaction function, 8i, j, is the following: ! i 2 i i 2 j @U i @Qji i @ Qi i @Qi @ U @R t þt i j @t j @Ri2 @ti @Ri @t j @t @t @ti ¼ , for r ¼ 2, j @t detH and 2 i @Qii @ 2 U i @Ri @U i @Qii i @ Qi þ t @ti @t j @Ri2 @ti @Ri @t j @ti @t j , for r ¼ 3, ¼ j detH @t where @Ri =@ti ( ¼ Qii þ Qji þ ti (@Qji =@ti þ @Qji =@ti )) is the marginal increase in revenues associated with an increase in ti , and det H is the determinant of the bordered Hessian of the second-order conditions of region i’s optimal tax problem (definition 6). The necessary and sufficient conditions imply @Ri =@ti > 0 and det H > 0, and by proposition 2 @Qji =@t j > 0 and @Qij =@t j > 0. The first term of the numerator in each expression is nonpositive when @ 2 U i =@Ri2 % 0 and the second term is negative so long as the cross derivatives of the demands for Qii and Qji with respect to ti and t j are sufficiently high in value. The sign of @ti =@t j thus depends on whether or not the first term dominates the second. As @ 2 U i =@Ri2 becomes more negative, then the first term dominates so that @ti =@t j < 0. The general case of nonseparability can be worked out similarly; a detailed derivation of the slope of reaction functions is available from the authors upon request. Recall, though, that the alternative specification of our model, with imperfect substitution between Qii and Qij , may not entail the jump exhibited by proposition 5. Standard tools can then be used to prove the existence of a NCFE in that case. ti
16
Ch.19 Commodity Tax Competition between Member States of a Federation fj
fj
ti
ti
471
E E
T3
T3
fi
E'
fi
T1 T2
T1
0
T2
0
tj
tj
(a)
(b)
ti
ti
fj
E ε T1
fi T1 0
fi
T3
T3 T2 (c)
T2
fj tj
0
(d)
tj
Figure 3. Noncooperative fiscal equilibria.
Proposition 6. If both reaction functions are nonincreasing, there exists at least one NCFE. Proof. See the appendix. This points to the importance of the slope of the reaction functions that was discussed in the preceding section. On the other hand, we also can record two facts that characterize the location in the T-space of a NCFE, when it exists. Proposition 7. (a) A NCFE never occurs in the interiors of the subsets T4 or T5 . (b) A NCFE with positive taxes never occurs on the boundaries of the subset T1 .
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Public Goods, Environmental Externalities and Fiscal Competition
Proof. See the appendix. Part (a) of proposition 7 refers to the interiors of T4 and T5 because it rests on the argument that it can never be the case that a region’s optimal commodity tax ti be positive with no tax revenue being raised (Ri ¼ 0); remember proposition 4. It is possible, however, that ti ¼ 0; this is indeed the case if the region values the public good very little compared to the private good. If, in addition, the price pi is substantially larger than pj (see fig. 1), the boundary between T3 and T5 then coincides with the t j axis; then, the NCFE point must lie on that axis, and hence on the boundary between T3 and T5 . The same argument may apply to region j, allowing a NCFE point to lie on the ti axis and hence on the boundary between T2 and T4 . Part (b) of proposition 7 refers to positive taxes for a similar reason. Finally, the special case where the two regions have identical preferences and endowments, and hence symmetric reaction curves as shown in fig. 3(b), deserves some attention. The symmetric NCFEs necessarily occur away from T1 , where identical taxes would lie. This case clearly shows how the taxes and expenditures pattern can differ between regions at an equilibrium, in spite of their similarity, for purely strategic reasons.
4.
Fiscal competition and efficiency
4.1.
Pareto efficient fiscal choices
The NCFE concept rests on the double assumption that (i) each region reaches its fiscal optimum without regard for the effect of its decisions on the welfare of the other; and (ii) the reacting behavior of the other region is of the same nature. Leaving aside a possible critique of these assumptions, let us now consider the alternative fiscal choices that would emerge if they were made according to the Pareto efficiency criterion, as applied to the two-region economy. It is important to note that, in this analysis, we shall keep unchanged the institutional setting whereby tax and expenditure decisions are made: our purpose here is to characterize fiscal decisions that are efficient within this setting, and not to compare the efficiency of alternative settings. In particular, we assume that the commodity taxes are levied in each region in such a way that the region’s budget is balanced (no interregional public transfers are considered). In the framework of our model, the usual notion of Pareto efficient public decisions may be expressed as follows:
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Definition 9. For the economy consisting of regions a and b, a Pareto efficient fiscal choice (PEFC) is a pair of regional feasible fiscal choices ^ a ), (^tb , R ^ b )] such that there exists no alternative such pair [(^ta , R [(ta , Ra ),(tb , Rb )] for which ^ i ), 8i, j, V i (ti , t j , Ri ) ^ V i (^ti , ^t j , R with strict inequality for at least one i. In a similarly classical fashion, analytic expressions for the taxes ^ta b and ^t that are economy-wide Pareto efficient are obtained from the firstorder conditions of the solution to the problem: {ta
max V a (ta , tb , Ra ), b a b ,t ,R,R }
s.t. Ra % ta [Qaa (ta , tb , Ra ) þ Qba (ta , tb )] (ca ) b V b (ta , tb , Rb ) ^ V (m) Rb % tb [Qbb (tb , ta , Rb ) þ Qab (tb , ta )] (cb ) ta , tb , Ra , Rb ^ 0, b is an appropriately chosen level of utility, and the Greek letters where V in parentheses are the Kuhn–Tucker multipliers. From the computations laid out in the appendix, we obtain: Proposition 8. The tax levels (^ta , ^tb ) implied by a Pareto efficient fiscal choice for the economy are of one of the following forms (the numbering of the regimes is from a’s viewpoint): (i) if (^ta , ^tb ) belongs to the interior of T1 , ^ti1 ¼
Qii (1 gi ) i , i ¼ a, b; @Qii i i @Qi þ g Q @ti @Ri
(17)
(ii) if (^ta , ^tb ) belongs to the interior of T2 , @Qb (1 ga )Qaa þ (1 fa )Qba þ tb da aa @t ^ta2 ¼ b a b a @Qa @Qa @Qa a a b a @Qb a b , þ þ g Q t d þ f Q a a @ta @ta @Ra @ta
(18:1)
@Qb (1 g b )Qbb þ ta db ba @t ^tb2 ¼ ; b @Qbb @Qbb b b a b @Qa þ g Qb t d @tb @Rb @tb
(18:2)
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Public Goods, Environmental Externalities and Fiscal Competition
(iii) if (^ta , ^tb ) belongs to the interior of T3 , @Qa (1 g a )Qaa þ tb da ab @t ^ta3 ¼ a a @Qa @Qaa a a b a @Qb ; þ g Qa t d @ta @Ra @ta
(19:1)
@Qa (1 g b )Qbb þ (1 fb )Qab þ ta db ba @t ^tb3 ¼ a @Qbb @Qba @Qbb b b b a a b @Qa , þ þ g Q þ f Q t d b b @tb @tb @Rb @tb
(19:2)
where gi ¼
@U i =@X i , @U i =@Ri
da ¼
cb cb b and d ¼ : @U a =@Ra m(@U b =@Rb )
fa ¼ m
@U b =@X b , @U a =@Ra
fb ¼
@U a =@X a , m(@U b =@Rb )
Proof. See the appendix. By comparing formulae (18) and (19) with their respective counterparts in (15) of proposition 4, we see that the optimal tax formulae under Pareto efficiency differ from those that determine the tax levels in a noncooperative equilibrium. There are two sources of differences. First, when the ith region increases its tax on purchases made by residents of the other region, the Pareto efficient taxes take into account the loss in the jth consumer’s utility due to the tax increase. This is the private consumption effect that appears when fa , fb > 0 in (18.1) and (19.2) and corresponds to the first term of eq. (16.3). In the corresponding noncooperative fiscal equilibrium, the ith region when exporting taxes ignores the private consumption effect [i.e. fi ¼ 0 in (18.1) and (19.2)]. Second, the Pareto efficient optimal tax takes into account the public consumption effect where one region alters the tax base and public good level of the other [corresponding to (16.2) and the second term of (16.3)]. With a noncooperative equilibrium, each region ignores this externality. This can be seen by letting da ¼ db ¼ 0 in (18.1), (18.2), (19.1), and (19.2). When di ¼ fi ¼ 0 for both regions, we see that (18.2) and (19.1) correspond to (15.1), and (18.1) and (19.2) correspond to (15.2). 4.2.
Competitive vs. efficient fiscal choices
Using the above apparatus, in this section we can make two evaluative statements on NCFEs from the efficiency point of view. We shall limit ourselves, however, to ‘interior’ equilibria in the following sense:
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Definition 10. An interior NCFE is an equilibrium such that its pair of taxes belong to the interior of the set Tr corresponding to the regime it induces. Proposition 9. (a) If an interior NCFE occurs in regimes 2 or 3, it cannot be Pareto efficient. (b) A NCFE in T1 which is not Pareto dominated by a tax pair in Tr , r ¼ 2, . . . , 5, is Pareto efficient. Proof. See the appendix. The economic reasons for the inefficiency of NCFEs in regimes 2 and 3 lies of course in the (reciprocal) fiscal externality phenomenon, which is present in these two regimes, and is not taken into account in the individual behaviors that determine these equilibria. Part (a) of proposition 9 is thus just another instance of the inefficiency of equilibria with externalities in general. The possibility of Pareto efficiency of a NCFE occurring in regime 1, on the other hand, is immediately seen as soon as we think of an 0 economy where t ij (0) is very large, making the set T1 span a good part of the tax space. 4.3.
Competitive vs. efficient tax levels
Whilst the likely inefficiency of a NCFE is not a very surprising fact, the question is less clear – and often debated – whether fiscal competition drives local taxes at a level which is higher or lower than efficient ones would be. Of course, we know that, as usual with Pareto optima, there are a whole range of efficient tax pairs, some of which being more to the benefit of one region, others being more to the benefit of the other. There is thus no such thing as ‘the’ optimal taxes – nor, for that matter, ‘the’ equilibrium ones since multiple equilibria can prevail. The question of comparing equilibrium taxes with efficient ones remains an interesting one, however. One way to formulate this comparison is to assume that some NCFE prevails and then to ask in what direction the taxes should be modified in the two regions if a Pareto efficient tax pair was sought for: should they be lowered or raised? We deal here with this last question in the following terms. Definition 11. 8i, a feasible fiscal change is a pair (dti , dRi ) 2 R2 such that " !# @Qii @Qji j i i dti : þ i dR ¼ Qi þ Qi þ ti @ti @t
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Public Goods, Environmental Externalities and Fiscal Competition
Definition 12. A Pareto improving fiscal change is any pair of fiscal changes [(dti , dRi ),(dt j , dRj )] such that dV i ¼
@V i i @V i i @V i j dt þ dR þ j dt ^ 0, 8i, j, @ti @Ri @t
(20)
with strict inequality for at least one i. Proposition 10. Feasible fiscal changes that are Pareto improving with respect to a NCFE interior to T2 or T3 never reduce both taxes. Proof. See the appendix. The logic of the proof is related to our earlier observations on the nature of interregional fiscal externalities. Clearly, when welfare is improved in a region by a tax reduction in the other, then the fiscal externality is of the negative type for this region. The impossibility of a welfare improvement in both regions by a joint tax reduction from a NCFE thus derives from the fact that, at such an equilibrium, the fiscal externality cannot be detrimental for both regions: indeed, due to the impossibility of cross-hauling of commodity Q, in any regime only one region at most can face the private consumption fiscal effect, which is an external diseconomy. On the other hand, when the public consumption effect dominates for both regions at the NCFE, Pareto improving tax changes must both be increasing ones. Since by proposition 10 every Pareto improving feasible fiscal change (w.r.t. an interior NCFE) is such that at least one of the two taxes increases, one may further ask whether a region can be identified where the raise surely occurs. An answer can be given as follows. Proposition 11. In feasible fiscal changes that are Pareto improving with respect to a NCFE interior to T2 or T3 , the tax of the region which is in mixed equilibrium is always increased. Proof. See the appendix. Finally, the tax of the other region (say j) – which by construction is in autarkic equilibrium at such a NCFE – is either raised or lowered depending upon whether in the first region’s social welfare function V i , the public (respectively private) consumption effect dominates in the sense of expression (16.3) above.
Ch.19 Commodity Tax Competition between Member States of a Federation
5.
477
Conclusions
In the foregoing analysis we have attempted to characterize a noncooperative equilibrium between regions choosing an origin-based tax levied on the same commodity. Four main results should be stressed. First, a noncooperative equilibrium amongst two regions choosing optimal tax rates and public good production may not always exist, due to the occurrence of a jump in the tax reaction functions. Second, at a noncooperative equilibrium, differences in the two regions’ size of public sector and level of tax may arise not only from differences in tastes and endowments, but also from purely strategic behavior. Third, in the absence of interregional public goods spillovers, the inefficiency of a noncooperative fiscal equilibrium arises from two types of externalities, due to the taxes only: (i) a (detrimental) private consumption effect, which occurs when a raise in a region’s tax affects the private good purchases of the other jurisdiction’s residents, and (ii) a (beneficial) public consumption effect, which occurs when a raise in a region’s tax increases the tax base of the other. Finally, it was also shown that in a two-region economy, Pareto improving tax changes cannot lower both of the tax levels that prevail at a noncooperative fiscal equilibrium; moreover, such changes are always tax increases for the tax importing region but are ambiguous for the tax exporting region. Both the positive analysis of noncooperative fiscal equilibria and the normative one of Pareto efficient fiscal structures can be extended in several directions, such as the following. (i) Which characteristics of preference structures and transportation costs would induce uniqueness of a noncooperative fiscal equilibrium? (ii) How can a noncooperative fiscal equilibrium be reached; or, more precisely, is this equilibrium stable with respect to some reasonable adjustment rule? (iii) Would a centralized uniform tax be better than an origin-based decentralized commodity tax system? (iv) Can revenue-sharing schemes be devised so that each region in its own interest would set Pareto efficient tax rates? The latter two questions require an analysis of cooperative arrangements, which are observed in most federal states. The acceptance of these arrangements by member states is an essential feature of fiscal federalism. On the other hand, the sovereignty granted to regional authorities can possibly lead to conflicts between national and regional priorities. Therefore, a search for cooperative policy measures improving upon the outcome
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Public Goods, Environmental Externalities and Fiscal Competition
of noncooperative competition may be seen as an appealing subject for further study. Perhaps more general models than the one used here would be necessary for reaching satisfactory answers to the above questions. But the present one has the merit of having helped to raise them, on the basis of the results it has yielded. Appendix: Proofs of propositions 1–11 Proof of proposition 1. Given the Lagrangian ui (Qi , Qii , Qij , X i , Ri , ri , li ) ¼ U i (Qi , X i , Ri ) þ ri (Qii þ Qij Qi ) li [X i þ ( pi þ ti )Qii þ ( pj þ t j )Qij þ tij (Qij )], where ri ^ 0, li ^ 0, and starring optimal values of the variables, the firstorder conditions for a maximum of ui are: @ui @U i if Qi > 0, i ¼ 0, ¼ r % 0, if Qi ¼ 0; @Qi @Qi @ui ¼ 0, if Qi i i i i i > 0, ¼ r l ( p þ t ) i % 0, if Qi @Qi i ¼ 0; ¼ 0, if Qi @ui j > 0, i i j j i0 ¼ r l ( p þ t þ t ) j i % 0, if Qi @Qj j ¼ 0; @ui @U i if X i < 0, i ¼ 0, ¼ l % 0, if X i ¼ 0: @X i @X i
(A:1)
(A:2) (A:3)
(A:4)
If X i < 0, then by (A.4) li ¼ @U i =@X i > 0; if Qi > 0, then by i (A.1), ri ¼ @U i =@Qi > 0 and by (1) either Qi i > 0 or Qj > 0 or both. i i Thus, if both Q > 0 and X < 0, (A.2) and (A.3) imply eq. (5) in the text. On the other hand, uniqueness follows from the strict quasi-concavity of U i and the strict convexity of t ij . Proof of proposition 2. The proposition is proved by differentiating the first-order conditions (5) that characterize RMEs. The second-order conditions, which appear in these differentiations, are easily established by substituting Qii þ Qij for Qi in the utility function – in view of constraint (1) which is satisfied as an equality at a RME; see proof of proposition 1 – as well as by replacing X i in this function by its value as determined by constraint (3).
Ch.19 Commodity Tax Competition between Member States of a Federation
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(i) For ui ¼ I (where Qi ¼ Qii > 0, Qij ¼ 0). Since @U i =@Qi @U i = @X i ( pi þ ti ) ¼ 0 in this type of equilibrium, differentiation of this condition with respect to ti and Ri yields successively: 2 i @Qi 1 @U i @ U @2U i i i ¼ þ ( p þ t ) Qi (a) @ti @Qi @X i @X i2 H @X i , (A:5) i @U i =@X i i @Q ¼ <0 Q H @I i where H¼
@2U i @2U i @2U i i i i 2 ( p þ t ) þ ( p þ ti )2 < 0 @Qi2 @X i @Qi @X i2
(A:6)
(which is negative at a maximum of ui ), and I i denotes lump-sum income. (A.5) is the Slutsky equation where the first term is the substitution effect (<0) and the second term is the income effect ( % as Qi is not inferior). @Qi 1 @2U i @2U i i i (b) ¼ þ ( p þ t ) v 0, @Ri H @Qi @Ri @X i @Ri where Qi and Ri are substitutes (complements) if the factor within brackets is positive (resp. negative), given (A.6). (ii) For ui ¼ II (where Qi ¼ Qii þ Qij , Qii > 0, Qij > 0). Since @U i @U i i @U i @U i j 0 i ( p þ t ) ¼ ( p þ t j þ tij ) ¼ 0 i i i i @Q @X @Q @X
(A:7)
in this type of equilibrium, the Hessian matrix of the second-order conditions for a maximum may be written as i H H i00 @U > 0, ¼ Ht i i j H H ti00 @X i j (@U =@X ) where H is defined in (A.6) after use was made of (A.7). Differentiating the conditions (A.7) with respect to ti , t j , and Ri , respectively, and using Cramer’s rule, we obtain the following comparative static conditions: 2 i @Qii 1 1 @U i @ U @2U i i i i ¼ i00 þ þ Qi (p þt) < 0, (c) @ti @Qi @X i @X i2 H @X i tj (d)
@Qij @ti
¼
1 > 0, ti00 j
i Qij @ 2 U i @Qii 1 @2U i i 1 j i0 i @Q v 0, ¼ þ ( p þ t þ t ) Q (e) j j i00 i00 @t j H @Qi @X i @X i2 @I tj tj
480
Public Goods, Environmental Externalities and Fiscal Competition
@Qij
1 < 0, ti00 j @Qii 1 @2U i @2U i i i ¼ þ ( p þ t ) v 0, (g) @Ri H @Qi @Ri @X i @Ri (f)
(h)
@t j
@Qij @Ri
¼
¼ 0:
Note that @Qi =@ti ¼ @Qii =@ti þ @Qij =@ti ¼ (1=H) @U i =@X i Qii (@Qi =@I), where @Qi =@I was defined above as the income effect. The interpretation of @Qii =@Ri ¼ @Qi =@Ri remains the same as in the earlier case (ui ¼ I). (iii) For ui ¼ III (where Qi ¼ Qij , Qii ¼ 0). Since @U i =@Qi @U i = @X i ( pj þ t j þ t i0j ) ¼ 0 in this type of equilibrium, differentiation with respect to ti and Ri , respectively, yields: 2 i @Qij @Qi 1 @U i @ U @2U i j i j i0 ¼ j ¼ þ Qj ( p þ t þ tj ) > 0, (i) ^ @X i @t j @t @Qi @X i @X i2 H @Qij @Qi 1 @2U i @2U i j j i0 ¼ ¼ þ ( p þ t þ t j ) v 0, ( j) ^ @Ri @Ri H @Qi @Ri @X i @Ri where 2 i 2 i 2 i i ^ ¼ @ U 2 @ U ( pj þ t j þ t i0 ) þ @ U ( pj þ t j þ ti0 )2 @U ti00 < 0: H j j @Qi2 @Qi @X i @X i2 @X i j
Proof of proposition 3. By (6) (for ui ¼ II) and (6) (for ui ¼ III), j i i j j i0 i j j Qi j > 0 ) p þ t ^ p þ t þ t j (Qj ) > p þ t . But if Qi > 0 also, (6.II) and (6.III) as applied to j imply similarly pj þ t j > pi þ t j , a contradiction. Proof of proposition 4. (i) From the Lagrangian (12), we get (for ti > 0): i @Lir @U i i i i i @Qi ¼ 0, ¼ Q þ C r Qi þ t @ti @X i i @ti i @Lir @U i i i @Qi ¼ þ Cr t 1 ¼ 0: @Ri @Ri @Ri
(A:8) (A:9)
(These expressions of the partial derivatives are obtained by using Roy’s theorem.) From (A.9) we have i 1 @U i i i @Qi 1t : (A:10) Cr ¼ @Ri @Ri
Ch.19 Commodity Tax Competition between Member States of a Federation
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Since @U i =@Ri > 0 by assumption, nonnegativity of Cir at an optimum is only possible if ti (@Qii =@Ri ) < 1, implying Cir > 0 there. Using (A.10) in (A.8), the latter may be solved for ti , yielding expression (15.1) in the text. (ii) From (14), we similarly have: ! j i @Lir @U i i @Q @Q (A:11) ¼ Q þ Cir Qii þ ti i i þ Qii þ ti ii ¼ 0, @ti @X i i @t @t @Lir ¼ identical to (A:9): @Ri Using (A.10) in (A.11) yields (15.2) in the text. Finally, to show that for any t j no fiscal optimum for i can be such that the resulting tax pair belongs to T5 (where ui ¼ III), it is sufficient to make the following observation. For every tax pair in T5 , region i is at an equilibrium which entails no production of the local public good Ri . This results from a tax level ti chosen so high that all purchases of commodity Qi are made in region j. However, region i can achieve that same level of public good by choosing ti ¼ 0. With this lower tax, the cost of acquiring an identical amount of Q is reduced, inducing more X i to be available for consumption since there is no change in Ri . Hence, utility can only increase, since @U i =@X i > 0 by assumption. But then with the tax pair in T5 , U i was not at a maximum, and such a pair thus cannot be part of a fiscal optimum for i. Proof of proposition 5. We first state the proof for the case depicted in Fig. 2(a) in the text, i.e. where the set T1 has interior points and contains some portion of the reaction curve. The other cases follow. (a) Suppose f i (t j ) were a continuous single-valued function; its graph would then be a smooth curve, running successively through the sets T3 , T1 , and T2 (see fig. 4). Within T1 the curve is flat because, according to (15.1), ti is a constant with respect to t j over this set. When this graph crosses the boundary between two sets, e.g. between T1 and T2 , the intersection point, say a ¼ (ti ,t j ) in fig. 4, is by definition of the reaction function such that @Li1 @Li2 ¼ i ¼ 0, (A:12) @ti (ti , tj ) @t (ti , tj ) since (ti , t j ) 2 T1 \ T2 . Now, expressions (A.8) and (A.11) of these derivatives, when they are evaluated at the same point, allow us to write, j @Li2 @Li1 i i @Qi ¼ i þCt , @ti @t @ti
(A:13)
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Public Goods, Environmental Externalities and Fiscal Competition
where the second term is strictly negative since Ci > 0 by (A.10) and @Qji =@ti < 0 by proposition 2. But then (A.12) cannot hold at (ti ,t j ); in particular, if @Li2 =@ti ¼ 0 at this point, then necessarily @Li2 =@ti < 0, which implies: (i) that ti does not maximize Li2 , given t j ; as a result, point a cannot belong to the graph of the reaction function; and (ii) that a value of ti lower than ti increases Li2 , for t j fixed; the optimal tax ti is thus lower than ti , and the reaction curve at t j passes below point a in fig. 4, at a point such as b that necessarily belongs to T2 (see fig. 5). Since we have assumed (at the end of Section 2.3) that within each set Tr there is a single value of ti that maximizes Lir for all the relevant values of t j , we may extend all the way to the boundary between T1 and T2 (say to point c) the curve just defined at point b. On the other hand, since by (16.2) @Li2 =@t j > 0, 8(ti ,t j ) 2 T2 , Li2 decreases along this curve when one takes successively smaller values of t j , starting from t j . Note, however, that at point c, Li2 cannot be larger (and in fact is strictly smaller) than the level Li1 at point a: indeed, Li1 is constant all along the flat portion of our initial curve, since @Li1 =@t j ¼ 0, 8(ti ,t j ) 2 T1 , according to (16.1); and since for t j ¼ tjc it is ti that maximizes Li1 , the pair (tjc ,tic ) can only yield a lower value for Li1 . As a result, point c cannot belong to the graph of the reaction function f i . But then, in view of the continuity of Li2 along the curve cb, there must be a point t j 2 ]tj ,t j [ where Li ¼ Li . At that point, there are thus two optimal taxes c 2 1 ti , namely ti and ti < ti (see fig. 6); at t j the reaction function f i is thus multi-valued. (b) [Case of fig. (2b) in the text.] If the point c is located in such a way that tjc is strictly smaller than tjd (see fig. 7), the argument according to which the maximum of Li2 over all values of ti such that (ti ,tjc ) 2 T2 is smaller than Li1 at point a is still valid, but it may be that this same value of Li2 is larger than the maximum of Li1 with respect to ti for (ti ,tjd ) 2 T1 , which is itself equal to Li1 at point a. In this case, the point t j must lie to the left of tjd and no point of T1 belongs to the graph of the reaction function f i . (c) [Case of fig. (2c) in the text.] This case can be treated as a combination of the two preceding ones. Proof of proposition 6. For the two nonincreasing reaction functions f i and f j , let t j (respectively t i ) be the points at which they are multi-valued; let also fþi (t j ) and fi (t j ) [respectively fþj (t i ) and fj (t i )] denote the two distinct images of these functions at these points, with the convention that fþi > fi and fþj > fj . Consider now the following possible situations.
Ch.19 Commodity Tax Competition between Member States of a Federation
483
ti
assumed
t i* (0)
fi(t j) a
ti T3
b
T2
T1 0
tj
tj
Figure 4. ti
t i* (0) a
ti T3
b
t ic
c T1
0
T2 t cj
tj
tj
Figure 5.
(1) Suppose that t i ^ fi (t j ) and t j % fþj (t i ) (see fig. 8). Then, consider along the t j axis, the closed interval ~ j ¼ [fþj (t i ), f j (0)], T which is well defined since t i ^ 0 and f j (ti ) is nonincreasing. As fig. 8 ~ j there must be at least one point t j for which shows, over the interval T the graph of the function f i has a point in common with that of the function f j or, equivalently, of its inverse (f j )1 . Intuitively, this is so 1 because, on the one hand, over this interval f j (t j ) – the inverse of a
484
Public Goods, Environmental Externalities and Fiscal Competition ti
f i(t j)
t i* (0) ti
a T3
ti
b c T1
T2
0
t cj
tj
tj
tj
Figure 6. ti
t i* (0) ti
a
f i(t j) T3
b
ti T1
c T2
0
t cj
tj
tj
d
tj
tj
Figure 7.
nonincreasing function – decreases itself from the value t i ( > 0) which it takes at t j ¼ fþj (t i ), to zero; and on the other hand, f i (t j ) is smaller than or equal to t i at t j ¼ fþj (t i ), because f i is nonincreasing, and it is larger than or equal to zero at t j ¼ f j (0). More formally, the necessity of an intersection between the two curves could be established by means at a fixed point argument, applied to an upper semi-continuous correspondence that takes ~ j into itself, namely the set T ~ j, f(t j ) ¼ F j (t j ) F i (t j ) þ t j , ti 2 T 1
where F j
1
and F i are suitable transformations of f j
1
and f i .
Ch.19 Commodity Tax Competition between Member States of a Federation
485
(2) Suppose next that t i < fi (t j ) and t j % fj (t i ). Then, depending upon the value of f i (t j ) at the point t j ¼ fj (t i ) (see figs. 9 and 10), consider along the t j axis the intervals ~ j ¼ [t j , f j (t i )], T A j j ~ ¼ [ fþ (t i ), f j (0)], T B
if f i (fj (t i )) ^ t i if f i ( f j (t i )) < t i
(fig:9), (fig:10):
In either case, an argument similar to the one made in (1) above 1 may be used for establishing that the two curves f j and f i do intersect for some point belonging to the relevant interval. (3) Suppose, further, that t i % fþi (t j ) and t j ^ fj (t i ). Then, along the ti axis, and for the interval ~ i ¼ [ f i (t j ), f i (0)], T þ the same argument as in (1) can be used, after permutation of the indices i ~ i , do intersect for and j, to establish that the two curves, defined over T i i ~. some t 2 T (4) Suppose, finally, that t i ^ fþi (t j ) and t j > fþj (t i ). Here also the argument used in (2) may be used, after permutation of the indices i and j. Since the above four cases exhaustively cover all the possible relative positions of two nonincreasing curves with one downward jump each, they are sufficient to establish the proposition. Proof of proposition 7. (a) That regime 5 is excluded is implied by the last sentence of proposition 4. On the other hand, remember from Section 2.2.2 that our numbering of regimes is from region i’s point of view. When region j’s viewpoint is taken, the regime that is excluded for the same reason (the fifth from j’s point of view) is actually the fourth one from i’s viewpoint. (b) Since the proof of proposition 5 shows that a point on the boundary between T1 and T2 cannot at the same time maximize Li1 and Li2 , and thus belong to the graph of region i’s reaction function, it cannot be a NCFE. When region j’s viewpoint is taken, the same argument applies to all points on the boundary between T1 and T3 . Proof of proposition 8. As in the proof of proposition 4, we derive the efficient taxes from the first-order conditions of a Lagrangian associated with the Pareto efficiency problem stated in Section 4.1. Since, depending upon the regime induced by (ta ,tb ), the indirect utility functions V i take on the different forms Vri , r ¼ 1, . . . ,5, given in (11) and (13), the Lagrangian and the efficient taxes differ accordingly. We thus have:
486
Public Goods, Environmental Externalities and Fiscal Competition ti
f j(t i) ti
f i(t j) fi
(t j) ~ Tj 0
tj
f+j
tj f j(0 )
(t i)
Figure 8. ti
f i (t j)
f j(t i)
f i(t j)
ti
~ Tj
A
tj
0
tj
f j (t i)
f+j (t i)
f
j(0 )
Figure 9. ti
f j(t i) f i (t j) ti
f i(t j) ~ Tj
B
0
tj tj f j (t i)
Figure 10.
f+j (t i)
f j(0 )
Ch.19 Commodity Tax Competition between Member States of a Federation
487
(i) For all (ta ,tb ) 2 int T1 , for each i ¼ a,b, the function V1i as well as the feasibility constraint (9) are both independent of t j . Hence, if a tax pair (^ta ,^tb ) in T1 is a Pareto efficient one, ^ta maximizes V a and ^tb maximizes V b in the sense of (10) in the text. But this yielded (15.1), which is identical to (17). (ii) For all (ta ,tb ) 2 int T2 , V2a is given by (13), where a is substituted for i, and V2b by (11), where b is substituted for i and a for j. The Lagrangian reads: b ) þ Ca [ta (Qa þ Qb ) Ra ] l2 (ta ,Ra ,tb ,Rb ,m,Ca ,Cb ) ¼ V2a þ m(V2b V a a þ Cb (tb Qbb Rb ): Using Roy’s theorem, as in (A.8), (A.9), and (A.11) above, the relevant first-order conditions are a @l2 @U a a @Qba a a b a @Qa ¼ Q þ C Qa þ Qa þ t þ a @ta @X a a @ta @t @Qb Cb ab tb @t
@l2 @Ra @l2 @tb @l2 @Rb
@U b b þ m Q ¼ 0, @X b a a @U a a a @Qa ¼ þC t 1 ¼ 0, @Ra @Ra b b @U b b a @Qa b b b @Qb m ¼C þ C Q þ t Q ¼ 0, b @tb @tb @X b b b @U b b b @Qb ¼m þC t 1 ¼ 0: @Rb @Rb
(A:14)
(A:15) (A:16) (A:17)
Substituting in (A.14) the value of Ca determined by (A.15), and solving for ta , we obtain (18.1); substituting in (A.16) the value of Cb determined by (A.17) similarly yields (18.2). (iii) For all (ta , tb ) 2 int T3 , V3a is given by (11), where i ¼ a, j ¼ b, and V3b is given by (13), with i ¼ b. A reasoning similar to the one just made yields (19.1) and (19.2). Proof of proposition 9. (a) Since expressions (18) and (19) are derived from the first-order conditions for an interior solution to the Pareto efficiency problem, they are necessary for efficiency. However, according to proposition 4, taxes at an interior NCFE are, instead, of the form (15.1) and (15.2). Therefore they cannot be efficient in regimes T2 and T3 .
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Public Goods, Environmental Externalities and Fiscal Competition
(b) Note first that for a NCFE occurring in T1 , the tax levels as specified by (15.1) are identical to those of (17) of a PEFC within T1 . Since these tax levels are unique by assumption and independent of each other, there is no alternative tax pair within T1 that Pareto dominates them. (Equilibrium tax pairs within T1 are thus always Pareto efficient relative to T1 .) But there might be such tax pair(s) in one or several of the other sets Tr . To rule out such cases, it is sufficient to 0 consider an economy where marginal transportation costs tij (0), 8i,j, are so large that they prevent the two regions from reaching both under any of the RMEs occurring in regimes 2 to 5, a social utility level higher than their NCFE one. Proof of proposition 10. By definition of a NCFE, the sum of the first two terms in (20) is equal to zero for each i at such an equilibrium, at least for fiscal changes that are feasible. Thus, any tax reduction dt j < 0 by the other region can induce dV i > 0 only if dV i =dt j < 0 (evaluated at the NCFE point). Now, from evaluations (16.1)–(16.3) of @Lir =@t j in each one of the possible regimes, we see that @V i =@t j < 0 can occur in regime 3 only, i.e. only when region i is in a mixed equilibrium. Thus, in order to have dV i ¼ (@V i =@t j )dt j > 0, with dt j < 0 in both regions, both of them should be in a mixed equilibrium. But this contradicts proposition 3 which excludes cross-hauling. Proof of proposition 11. By construction of the sets T2 and T3 , and for all tax pairs in these sets, we have that when one region is in mixed equilibrium, the other is in autarkic equilibrium. But for the latter, according to expression (16.2), only a rise in the other region’s tax can induce an increase in social welfare. Therefore, the tax of the region in mixed equilibrium must rise for the fiscal change to be Pareto improving.
References Arnott, R. and Grieson, R., 1981. Optimal fiscal policy for a state or local government, Journal of Urban Economics, 9, 23–48. Atkinson, A.B. and Stiglitz, J., 1980. Lectures on public economics. New York: McGrawHill. Bamon, R. and Fraysse, J., 1985. Existence of Cournot equilibrium in large markets, Econometrica, 53, 587–598. Boadway, R., 1982. On the method of taxation and the provision of local public goods: comment, American Economic Review, 72, 846–851. Boskin, M.J., 1973. Local government tax and product competition and the optimal provision of public goods, Journal of Political Economy, 81, 203–210. European Communities, Commission, 1980. Report on the scope for convergence of tax systems in the Community. Luxembourg: Bulletin of the European Communities, Supplement 1/80.
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Friedman, J.W., 1977. Oligopoly and the theory of games. Amsterdam: North-Holland. Gordon, R.H., 1983. An optimal taxation approach to fiscal federalism, Quarterly Journal of Economics, 98, 567–586. McLure, C.E., Jr., 1969. The inter-regional incidence of general regional taxes, Public Finance, 24, 457–483. Mieszkowski, P.M., 1966. The comparative efficiency of tariffs and other tax-subsidy schemes as a means of obtaining revenue or protecting domestic production, Journal of Political Economy, 74, 587–599. Musgrave, R.A., 1969. Fiscal systems. New Haven: Yale University Press. Nash, J.F., 1951. Non-cooperative games, Annals of Mathematics, 54, 286–295. Oates, W.E., 1972. Fiscal federalism. New York: Harcourt Brace Jovanovich. Sandmo, A., 1974. A note on the structure of optimal taxation, American Economic Review, 64, 701–706. Starrett, David, 1980. On the method of taxation and the provision of local public goods, American Economic Review, 70, 380–392. Westhoff, F., 1977. Existence of equilibria in economies with a local public good, Journal of Economic Theory, 14, 84–112. Wildasin, D., 1979. Public good provision with optimal and non-optimal commodity taxation, Economic Letters, 4, 59–64. Williams, A., 1966. The optimal provision of public goods in a system of local government, Journal of Political Economy, 74, 18–33.
Chapter 20 ON PARETO IMPROVING COMMODITY TAX CHANGES UNDER FISCAL COMPETITION
Alain de Crombrugghe, Henry Tulkens Journal of Public Economics 41(3), 335–350 (1990).
This paper shows that when a ‘noncooperative fiscal equilibrium’ of the Mintz–Tulkens 1986 model of tax competition is inefficient, Pareto improving changes are positive for the taxes of both regions, under concavity of the regions’ welfare functions. It also identifies a class of such equilibria that are always efficient.
1.
Introduction
In the often debated question of tax harmonization in free trade areas, especially with regard to the projects aimed at completing the European integration, a clear notion of what the absence of harmonization means is desirable. Such a notion is provided by recent analyses of tax competition. In particular, optimal taxation approaches [in the spirit of Atkinson and Stiglitz (1980)] of the latter have been presented by Gordon (1983) in fairly general terms, and by Mintz and Tulkens (1986) in a more detailed, albeit more specific model involving only a commodity tax. In the latter paper (hereafter referred to as MT), the authors present a model of fiscal competition between two regions in a federal state, in which a regionally optimal commodity tax is chosen independently in The authors are grateful to professors A.B. Atkinson, J.-F. Thisse, A. Mas-Colell and especially J. Mintz, as well as to the participants in the 1986–87 Public Economics Workshop at CORE for their helpful remarks. The first author acknowledges financial support, as Aspirant, from the Fonds National de la Recherche Scientifique, as well as from the British Council for a year at the London School of Economics under the auspices of the European Doctoral Program in Quantitative Economics, followed by a grant of Harvard University.
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Public Goods, Environmental Externalities and Fiscal Competition
each region, to finance a local public good. Competition arises from the fact that under the origin principle, which is assumed to prevail, the relative level of the two taxes induces exports or imports of the taxed commodity – which is the same in both regions – from one region to the other, and this alters the size of the two tax bases. The outcome of this competitive process is called a ‘noncooperative fiscal equilibrium’ (NCFE) between the two regions. For the cases where the tax levels at a NCFE are shown to be Pareto inefficient,1 the authors raise the natural question whether Pareto superior taxes – more ‘harmonious’ ones as interpreted by Keen (1988) – would be higher or lower than equilibrium ones; the answer they provide is formulated in terms of the signs of Pareto improving tax changes, the latter being evaluated at an equilibrium of the model. But it is only a partial answer, in the sense that while it shows that in certain important cases a Pareto improving tax change is unambiguously positive for one of the regions, the direction of such a change for the tax of the other region is left ambiguous (MT’s Proposition 11, pp. 160–161); more exactly, this direction is claimed to vary according to the relative weights of private vs. public consumptions in the partial derivative of the region’s welfare function with respect to the other region’s tax [see MT’s comments following Proposition 11, based on their expression (16.3) of this derivative]. The object of this paper is to remove this ambiguity, and to show that Pareto improving tax changes are always positive for both regions at equilibria belonging to the class of NCFEs considered in Proposition 11. As a by-product, we can also strengthen part (b) of MT’s Proposition 9, which applies to another class of NCFEs:2 we establish here that those equilibria are always Pareto efficient. In Section 2 we recall the main features of the model and introduce explicitly an additional tool, namely regional social indifference curves, that were not used in the MT paper. In Section 3 we introduce a new definition and state our results with their proofs. Economic interpretations and some implications are briefly discussed in the concluding Section 4.
2.
The main features of the model
The formal structure of the MT model may be summarily presented as follows. 1 In Proposition 9 (p. 159) of MT, circumstances are identified under which a NCFE may be efficient. 2 This class comprises those alluded to in the previous footnote. It consists of all the NCFEs occurring in a regime of ‘double autarky’, and formally designated as belonging to the set T1 (see below).
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 493
2.1
Noncooperative fiscal equilibria
A NCFE is defined as a Nash equilibrium of a noncooperative game in which the players are the local fiscal authorities (denoted by the indices i and j), and the strategy of each player is the level ti of a single local commodity tax (resp. t j ) which is levied on the domestic purchases Qii (resp. Q jj ) and on the exports Q ij (resp. Qij ) to the other region, of a single private good locally produced. The tax is used to finance a local public good, provided in each region in quantities Ri (resp. Rj ). Finally, the payoffs are the constrained maxima of continuous social welfare functions Li and Lj specific to the two regions, whose arguments are the local consumptions of private and public goods, subject to a constraint of regional budget balance; for region i, for instance (for region j, permute i and j), this constraint reads: ti [(Qii þ Q ij )] ^ Ri :
(1)
In fig. 1, a NCFE is represented in R2þ as an intersection of two curves, which are the graphs in the tax space of the two regions’ ‘fiscal reaction functions’, respectively denoted as f i (t j ) and f j (ti ). These curves, called here the ‘fiscal reaction curves’ of the regions, are further characterized in subsection 2.5 below. Note that there may exist several equilibria, as exemplified by points Ea and Eb in fig. 1. 2.2
The consumers’ behavior
In fig. 1 there also appears a partitioning of the tax space into five subsets, denoted T1 ,T2 , . . . ,T5 . These subsets are derived from a specification of the behaviors of the consumers in each region, given the fiscal choices ti ,Ri and t j ,Rj made by the public authorities, and given constant producer prices pi and pj . Specifically, the representative consumer of, say, region i, is assumed to maximize his utility by choosing his supply of labor X i , and his consumption of the private good Qi , the latter being home purchased (Qii ) or imported (Qij ), or both. This reads: max
{X i ,Qi ,Qii ,Qij }
U i (Qi ,X i ,Ri )
(2)
subject to the constraints: ( pi þ ti )Qii þ ( pj þ t j )Qij þ tij (Qij ) þ X i % 0,
(3)
Qi ¼ Qii þ Qij ,
(4)
Qi ,Qii ,Qij ^ 0, X i % 0,
494
Public Goods, Environmental Externalities and Fiscal Competition ti T5 T3 T1 T2 T4
Ea f i(t j)
A
Eb p j − p i + τ i ' (0)
f j(t i)
j
p j − p i =0
tj
A
tj
tj
Figure 1. Fiscal reaction functions and noncooperative fiscal equilibria.
where t ij (Qij ) is an increasing and convex transport cost function, and Ri ,ti ,t j are taken as given. For each region, the resulting consumer demand for the private good, and in particular its breaking down into home purchases and imports from the other region, depend only on the levels of the taxes, ti and t j , and not of the local public good. This is what induces the partitioning of the tax space into the five subsets mentioned above. The points in each subset correspond to configurations of the two taxes with the following properties: . The coordinates of all the points in T1 are tax pairs (ti ,t j ) whose levels are such that at the regional equilibria just defined all consumers of both regions buy the taxed commodity in their own region only. Thus, Q ji ¼ Q ij ¼ 0 under those taxes, and the overall state (or ‘regime’) of the economy is called ‘double autarky’. . All points in T3 correspond to tax pairs such that the tax differential between regions i and j (with ti > t j ) induces some of the consumers from i to go and buy the private commodity in region j (Q ji > 0), whereas no customers of region j do so in region i (Q ij ¼ 0). Region i is thus importing taxes, i.e. paying fiscal revenue to the other region. At the same time, region j is exporting taxes.
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 495
. The points in T2 correspond to the reverse situation, with ti < t j : here, Q ji ¼ 0 and Q ij > 0.3 . The points in T4 and T5 correspond to tax pairs such that the tax of one region (j in the case of T4 and i in the case of T5 ) is so high that no domestic purchases and only imports are taking place (Q jj ¼ 0 and Qj ¼ Q ij > 0 in T4 , while Qii ¼ 0 and Qi ¼ Q ji > 0 in T5 ); as a result, the tax base (of j in T4 , alternatively of i in T5 ) is totally depleted and no financing of the local public good is possible for the relevant region. In fig. 1, the subsets so defined are drawn assuming, for convenience, that the producer prices, pi and p j , of the taxed commodity are equal in the two regions. If they differ, the entire subset T1 moves upwards or downwards in the diagram, without altering the structure just described. Note also that the boundaries of T1 with T2 and T3 are straight lines with slopes equal to unity;4 this fact will be of crucial importance in the sequel. The boundaries lie at a constant distance from the 458 line, the distance being equal to t 0 (0), i.e. the marginal transport cost of the first unit of imports of private goods. 2.3
The public sector behavior and the interdependences between the regions
From the consumer’s utility maximizing behavior, an indirect utility function can be derived, namely: V i (ti ,t j ,Ri ) U i [Qii (ti , t j , Ri ) þ Q ji (ti , t j ), X i (ti , t j , Ri ), Ri ],
(5)
where the expressions Q( ) and X ( ) denote demand functions for the private goods and the supply function of labor, respectively. On the other hand, the public budget constraint (1) is now to be rewritten as: ti [Qii (ti , t j , Ri ) þ Q ij (t j , ti )] Ri ^ 0:
(6)
Then, the welfare that the region derives from both its private and public consumptions can be formulated in terms of a ‘feasible regional welfare function’ that reads: Li (ti , t j , Ri ) V i (ti , t j , Ri ) þ ci ti [Qii ( ) þ Q ij ( ) Ri ],
(7)
where ci is a Lagrange multiplier.
3 In the case where region i has a large price advantage ( pi < p j ) and transport costs are low, ti > t j is possible in T2 near the boundary with T1 . Conversely, if region j has a large price advantage, ti < t j can occur in T3 near the boundary with T1 . For shorthand notation, we disregard those price advantages, without loss of generality. 4 See pp. 142–144 of MT for arguments formally establishing this and what follows.
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Public Goods, Environmental Externalities and Fiscal Competition
Note, however, that, depending upon the subsets Tr to which the taxes ti and t j under consideration belong, some of the variables Q are identically zero in the function V i and/or in the constraint. Accordingly, the feasible welfare function Li is denoted as Lir , r ¼ 1, . . . ,5, with the zero arguments deleted, whenever the reasoning bears upon tax pairs restricted to a specific subset Tr . Within each such subset, it is assumed by MT that the function Lir is concave, allowing for a unique maximum of Lir in ti , given t j . Over the whole tax space the overall function Li need not be concave, however. Each region’s public sector is assumed to choose its tax level and public good quantity so as to maximize the regional feasible welfare, given the choice made by the other region on its own tax and public good levels. This behavior is characterized by the first-order conditions for a maximum of (7) with respect to ti and Ri , taking t j as given. As computed in expressions A.8 to A.11 of MT, these conditions read, using Roy’s identity: 8 i @U i i > i i i @Qi > for r ¼ 1, 3, (8a) Qi þ cr Qi þ t i ¼ 0, > @t @Lir < @X i " ! # ¼ i @U i i @Q ij @ti > j i i i @Qi > > ¼ 0, for r ¼ 2, (8b) Q þ c Q þ Q þ t þ : @X i i i r i @ti @ti i @Lir @U i i i i @Qi ¼ Q þ c t 1 ¼ 0, r @Ri @Ri i @Ri
for r ¼ 1, 2, 3:
(9)
Solving for ti and Ri , from (8a) and (9) if regime 1 or 3 prevails at the global maximum in ti of Li , alternatively from (8b) and (9) if regime 2 prevails, yields the optimal levels of the regional tax and of the local public good, given t j . On the other hand, the derivative of (7) with respect to the external j tax t measures the degree of interdependence between the regions. MT [(16.1) to (16.3)] evaluate it as follows: 8 0, for r ¼ 1, (10:1) > > > i > @Lir < cir ti @Qi > 0, for r ¼ 2, (10:2) ¼ @t j > @t j > i > > : @U i Q i þ ci ti @Qi , for r ¼ 3: (10:3) r @X i j @t j Clearly, in the ‘double autarky’ regime (r ¼ 1), the regions do not influence each other; in regime 2, where region i exports taxes, a tax increase in the other region always benefits the former; and in regime 3, where region i is importing taxes, expression (10.3) yields an ambiguous effect of changes in the external tax on the welfare of the importing region: a negative (first term) private consumption effect, due to the raise in the tax on its imports, can be counterbalanced by a positive (second term) public
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 497
consumption effect, due to the increased fiscal revenue this raise entails on the home purchases. The relative weights of those two effects, which could not be evaluated by MT, are the main issue of this paper.5 2.4
Regional social indifference curves
It will be useful in what follows to introduce at this point a graphical representation of the indifference curves induced by the function Li in the space (ti ,t j ). In fig. 2 a few of these curves are drawn, with arrows indicating the direction of increasing welfare. The curves differ radically in shape according to the regime which is considered. In the set T1 , for instance, these curves are clearly horizontal, because of (10.1). The highest welfare level Li1 in this set is reached for the value of ti that solves (8a) and (9), illustrated by tiA . In the set T2 , the curves are as drawn, because of both the assumed concavity of Li2 and the sign of (10.2). For any given t j , the highest level of welfare Li2 in this set is reached for the value of ti where an indifference curve has a vertical tangent (e.g. tiN for t j ¼ t j ). In the set T3 , the curves are given the shape that they would have if, in addition to the assumed concavity of Li3 , the sign of (10.3) were always positive. If this sign were negative on (some part of) T3 , however, concavity of Li3 would imply that the curvatures be reversed, as suggested on fig. 3 (below) for values of t j > tBj . Irrespective of this, for any t j , the highest level of welfare Li3 is reached in T3 at the value of ti for which the corresponding indifference curve has a vertical tangent (e.g. tiM for t j ¼ t j in fig. 2). 2.5
The fiscal reaction functions
Finally, the construction and some characteristics of the fiscal reaction curves deserve to be recalled. (i) The fiscal reaction curve of a region, say region i, is defined as the graph of the function fi that associates with every tax t j of the other region the tax ti that maximizes the feasible welfare Li of the region. In the tax space (ti ,t j ) the reader can think of the points of this curve as the highest ones on the region’s ‘welfare hill’, for each value of t j . It is thus a locus of vertical tangency points of the indifference curves just identified, and is drawn as the heavy curves on figs. 2, 3 and 4. (ii) Each region’s fiscal reaction curve exhibits one ‘downward jump’ which occurs for some value t of the other region’s tax. This jump is essentially due to the advantage for a region to start to export goods 5 The derivative of Li in regimes 4 and 5 need not be restated, as Proposition 7(a) of MT shows that no NCFE ever occurs in these regimes, and this paper is concerned with NCFEs only.
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ti Cm
T5
Dm
T3 T1 T2
f i(t j)
L2N
L3M M
ti
A
M ti A
F
ti
N
N
D0 0
T4
i
i
C0
D
f i(t j)
i
L2N Go
tj
Gm tj
Figure 2. Region i’s indifference and fiscal reaction curves.
and taxes by cutting down its own tax, when the other region’s tax reaches some threshold level t. As established by MT’s Proposition 5, this jump is necessarily located in such a way that, for region i, the reaction curve never touches6 the boundary between T1 and T2 . Symmetrically, the fiscal reaction curve of region j never touches the boundary between T1 and T3 . Notice in fig. 2 the indifference curve linking the points N and M, where the jump occurs: the welfare level Li3M at point M is equal to Li2N at point N, with Li2 increasing faster than Li3 for t j > t j . (iii) Within each subset Tr , and away from the jump just discussed, the fiscal reaction curves are proved by MT to be continuous. (iv) When a portion of a region’s fiscal reaction curve does belong to T1 (as it is the case for region i in figs. 1 and 4), it is flat there: its value remains a constant at the level of the optimal domestic tax, in view of the absence of interregional trade that defines T1 .
3.
The new results
The two results we establish in this section are (i) that at an inefficient NCFE a welfare gain obtains for both regions if and only if they 6
The curve may even never reach T1 , as illustrated in fig. 1 for region j, and in fig. 2 for region i.
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 499
simultaneously increase their tax (subsection 3.2); and (ii) that a NCFE occurring under double autarky is always Pareto efficient (subsection 3.3). The first result rests on a formal property of the model, exhibited in a preliminary lemma (subsection 3.1), and the second result follows as a corollary from the first. In the next two subsections, the analysis is carried out assuming that region i is the tax importing one at the NCFE under discussion. If, instead, this were the case for region j, the same analysis would apply, provided that the indices i and j are permuted, and the subset T2 is considered instead of T3 . 3.1
Preliminaries: a definition and a lemma
In MT, as well as in subsection 2.4 above, the fiscal reaction function of either region is defined in general terms over the whole tax space, and thus across the regimes r ¼ 1,2,3. Here, we look more closely at its determinants within each subset Tr , thereby gaining new insights that will drive our results. Specifically, we introduce the following: Definition. For any region, say i, and any regime r, the ‘regime-specific fiscal reaction function’ is the function fri : Rþ ! Rþ ; t j ! ti ¼ fri (t j ) such that ti maximizes Lir given t j , and (ti ,t j ) 2 Tr . The graph in the tax space of a function fri so defined will be called the ‘regime-specific fiscal reaction curve’. In fig. 2, region i’s fiscal reaction curves specific to regimes r ¼ 1, 2, 3 are drawn, respectively, as the dashed curves C0 ACm , D0 ADDm and G0 FGm . Notice that the constraint (ti , t j ) 2 Tr implies that for those parts of a regime-specific curve that lie on a boundary of its relevant set Tr , the first-order conditions (8) and (9) need not be satisfied with equality. Elsewhere, i.e. in the interior of the subsets Tr , these conditions are satisfied with equality. Note also that in regime 1, the tax tiA that maximizes Li1 must necessarily be lower than the one corresponding to the upper boundary7 T1 \ T5 . On the other hand, region i’s overall fiscal reaction curve is also drawn in fig. 2 as the solid line C0 MNGm . It appears to consist of successive portions of two of the region’s regime-specific reaction curves. Notice that no point of the curve specific to regime r ¼ 1 belongs to the overall fiscal reaction curve of the region. This results from the underlying structure of the indifference curves in this case: they induce the earlier mentioned downward jump between point M in T3 and point N in T2 , so that T1 is bypassed altogether. The other possible situation is the one of 7
This is in fact an application of MT’s Proposition 4 (last sentence).
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fig. 1, where the overall fiscal reaction curve contains points that belong to regime-specific curves for each one of the three regimes. We now present a lemma that establishes a necessary link between the fiscal reaction curves (hence, the functions) specific to the import regime (r ¼ 3) and to the autarky regime (r ¼ 1). Lemma 1. For region i, the fiscal reaction curve specific to regime 3 reaches the boundary T3 \ T1 exactly at a point where its fiscal reaction curve specific to regime 1 leaves this boundary. Fig. 2 illustrates our claim: we want to establish that the curves C0 ACm and D0 ADDm have one and only one point in common – namely point A – located on the boundary T1 \ T3 and having as coordinate ti the value tiA that maximizes Li1 over T1 . Proof. We proceed by showing that the following exhaustive possibilities are excluded: (i) the curve C0 ACm never reaches the boundary T1 \ T2 ; (ii) the curve C0 ACm reaches T1 \ T3 at a point (tiþ ,tjþ ) above A, i.e. such that tiþ > tiA ; and (iii) the curve C0 ACm reaches T1 \ T3 at a point (ti , tj ) below A, i.e. such that ti < tiA . To this effect, consider first the derivatives of the welfare functions, i L1 and Li3 , with respect to ti , and notice that for any point (ti ,t j ) belonging to the boundary T1 \ T3 , expression (8a) above allows one to write the equality:
@Li1 @Li3 ¼ i @ti @t
(11) (ti ,t j )2T1 \T3
Taking now the three cases: (i) If C0 ACm never reaches the boundary, then by definition of the function f3i (of which C0 ACm is the graph), @Li3 =@ti > 0 should hold everywhere along T1 \ T3 ; and, by (11), @Li1 =@ti > 0 should similarly hold. But this last claim leads to a contradiction: indeed, it implies that for Li1 , its maximum in ti within T1 is reached only at a very high value of ti , in fact a value so high that it corresponds to the boundary T1 \ T5 ; however, by Proposition 4 (last sentence) of MT, no maximum value of ti can be such that the corresponding tax pair (ti , t j ) belongs to T5 . The curve C0 ACm must therefore reach somewhere the boundary T1 \ T3 . (ii) If C0 ACm reaches the boundary T1 \ T3 at a point (tiþ , tjþ ) such iþ that t > tiA , then for all the points of this boundary for which ti < tiþ , one should have, again by definition of f3i , @Li3 =@ti > 0 and, by (11),
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 501
@Li1 =@ti > 0 also. But this last claim is now in contradiction with the fact that tiA maximizes Li1 in T 1 : since tiA must be lower than the boundary T1 \ T5 , for all points on T1 \ T3 such that ti > tiA , @Li1 =@ti < 0 must hold. (iii) Finally, an argument symmetric to the previous one applies if C0 ACm reaches the boundary T1 \ T3 at a point (ti , tj ) such that ti < tiA : then indeed, for all the points on this boundary such that ti < ti < tiA , @Li3 =@ti < 0 should hold as well as @Li1 =@ti < 0. But this contradicts the fact that @Li1 =@ti > 0 at such points since only tiA maximizes Li1 in T 1 . These three contradictions demonstrate the lemma. & Apart from the role it will play in the sequel, this lemma has the following economic implication, which is of interest in its own right for the understanding of fiscal competition under our hypothesis of regional welfare maximization. Essentially, the link that the lemma establishes between the two fiscal reaction curves just considered implies that when it is optimal for a region to import taxes, given the level of the other’s tax, the region cannot hope to reach an equal (or a fortiori higher) welfare level by abruptly cutting down its current optimal domestic tax, if this is to result in inducing double autarky in the economy. The reason for this may be seen in the following characteristic of the situation: an importing region partially substitutes foreign goods for home goods, a fact that also implies that there is a (partial) substitution of transport cost for the home tax. Now, if this home tax were lowered, consumers by importing less would substitute back tax payments at home for transportation expenditures. For the amount of private good involved in that substitution, no utility gain in terms of private consumption would be obtained, and thus, on the private good side, the utility gain from the tax reduction would only be a limited one. On the public good side, on the other hand, such a decrease in the tax level would of course increase the tax base; but the lemma establishes that this public good effect, together with the private good effect, cannot restore the welfare level that is achieved under tax importing. The lemma does not preclude, however, the possibility described in fig. 2 that an abrupt cut in the home tax of an importing region provides the latter with an equal welfare level if the induced regime is such that the region becomes a tax exporting one (i.e. ends up in regime 2). Such a case was already covered by MT’s Proposition 5 on the necessary existence of one downward jump towards T2 in this model. Together, the lemma and the proposition just recalled allow one to make the following claim: abrupt tax cuts can be beneficial, under appropriate circumstances, only if they induce tax exports for the region which
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engages in such actions; they cannot be so if they end up in double autarky, as might be the case if they were motivated only by a desire to get rid of a tax importing situation. 3.2
The Pareto improving tax increases We can now state our first result:
Proposition 1. Feasible fiscal changes that are Pareto improving with respect to a noncooperative equilibrium interior to the sets T2 or T3 always increase both taxes. Proof. Our reasoning is in four steps: (1) For a tax-exporting region, it is already established by expression (10.2) that an increase in the other region’s tax is always beneficial. For a NCFE point such as Ea in fig. 1, this readily applies to region j, as far as an increase in ti is concerned. We thus only have to show in this case that for the tax-importing region i an increase in t j is also beneficial, i.e. that @Li3 =@t j > 0. (At an equilibrium such as Eb , the same reasoning would apply, after permutation of the indices i and j and appropriate reindexing of the sets Tr .) (2) As the sign of the partial derivative @Li3 =@t j cannot be determined from expression (10.3), an argument of global nature must be used. It is provided by the following statement: Lemma 2. In the subset T3 , the regional welfare function Li3 of the importing region reaches its global maximum at the point where the region’s fiscal reaction curve specific to regime 3 reaches the boundary with T1 . In fig. 1, and in figs. 2 and 4 as well, the claim is that the function Li3 reaches its maximum over T3 (with respect to both ti and t j )8 at point A. Proof of the lemma. We proceed again by contradiction. Suppose that the global maximum of the function Li3 [i.e. (ti , t j ) 2 T3 such that @Li3 =@ti ¼ @Li3 =@t j ¼ 0] is reached at a point of the reaction curve which is interior to T3 , such as B for instance (see fig. 3). Then, by concavity of Li3 , indifference curves can be drawn around point B, corresponding to lower welfare levels than this maximum. One of these curves, labelled Lio 3, is necessarily tangent to the boundary between T3 and T1 . Let us call A the 8 An economic interpretation of this lemma may be given in the following terms: ignoring the possibility of exporting taxes (i.e. regime 2), a region would always choose an autarky optimum for itself rather than one with tax imports, if it could choose both the home and the foreign tax. The reason for this is that the transport costs, induced only by the tax differential in this model, are a pure waste.
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 503
ti
T3 L i0
f i(t j)
3
T1 C
B A
+1
0
tj
B
tj
A
tj
Figure 3. Point B is an impossible global maximum for Li3 .
tangency point. At this point, the slope of the indifference curve Lio 3 must be equal to þ1, since as recalled above (see subsection 2.2) this is also the slope of the line that determines the boundary between T3 and T1 . This in turn implies that @Li3 =@ti > 0 at point A. On the other hand, this same point A must also belong to the (flat) fiscal reaction curve that is specific to regime 1: if it did not, the indifference curve Li3 passing through it could not be tangent to the boundary between T3 and T1 (as suggested by the dotted indifference curves on fig. 3). Now, Lemma 1 proves that the fiscal reaction curve specific to r ¼ 3 reaches the boundary with T1 at point A. However, as recalled by point C in fig. 3, at every point of such a curve one necessarily has @Li3 =@ti ¼ 0 in general; thus also at point A. But this contradicts our conclusion (@Li3 =@ti > 0) in the next to last paragraph. Hence, the point where Li3 is maximized cannot be distinct from A, and this demonstrates the lemma. (3) As the function Li3 can only reach its global maximum at a point such as A on the boundary with T1 , its concavity then implies @Li3 =@t j > 0 at any point of the fiscal reaction curve specific to r ¼ 3 for which t j < tjA . This is also the case at any such point. (4) Recalling that the overall fiscal reaction curve is entirely composed of portions of regime-specific curves, @Li3 =@t j > 0 is true, in par& ticular, at a NCFE located in T3 , such as point Ea in fig. 1. Fig. 4 illustrates our result. Given the NCFE point Ea , all Pareto superior taxes are those denoted by the set of points belonging to
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Public Goods, Environmental Externalities and Fiscal Competition ti f j(t i)
T3 T1
f i(t j)
Ea
A
0
tj
Figure 4. Pareto superior taxes, relative to the NCFE Ea .
the shaded area between the indifference curves reached at Ea ; they are higher ones. 3.3
The Pareto efficiency of an equilibrium in double autarky Our second result is as follows:
Proposition 2.
A NCFE occurring in T1 is always Pareto efficient.
Proof. Given a NCFE in T1 , a Pareto superior tax pair cannot belong to T1 , because any other tax pair in this set yields a lower level of welfare for at least one region [MT Proposition 9(b)]. It cannot belong to T3 or T2 either, because in the first case the welfare of region i would be lower, as implied by the Lemma 2 just established; and in the second case, the same argument would apply to region j. &
4.
Concluding comments
It was shown by MT that for an exporting region, an increase in the other region’s tax is always beneficial, due to a pure ‘public consumption effect’: the only effect of the rise of the other region’s tax is to increase the domestic tax base, and hence public expenditure, in the exporting region.
Ch.20 On Pareto Improving Commodity Tax Changes Under Fiscal Competition 505
For an importing region, however, the authors claimed that this effect may be counter-balanced by an additional ‘private consumption effect’: indeed, the rise in the other region’s tax also affects (negatively in this case) the imports and hence the aggregate private consumption in the region. We have proved here that for an importing region, the public consumption effect always dominates, at least along its fiscal reaction curve. In making their own fiscal choices, noncooperating regions ignore these external effects; this was also pointed out by Gordon (1983) without determining their sign, however. Here we conclude unambiguously that at a NCFE with imports and exports the fight for the tax base always ends up with local tax and public expenditure levels being too low.9 Public consumption beneficial effects dominate in Pareto improving changes, because under fiscal competition, imports are the result of a tradeoff between the home tax and the transport cost that enables consumers to partly protect their private consumption. We should point out, in addition, that at an inefficient NCFE, Pareto improvement requires that the tax changes be simultaneous. If they are not – one region deviating by not changing its tax in the appropriate direction while the other does so – all benefits are for the deviating region and the cooperating one loses. Thus, a prisoner’s dilemma situation characterizes cooperative tax changes at an equilibrium. This remark is of interest when tax harmonization is defined in terms of Pareto improving changes, as in Keen (1988). Finally, the possibility of efficient noncooperative equilibria appears to be somewhat wider than suggested in the MT paper, since this property necessarily obtains for any double autarky equilibrium (our Proposition 2). There, the two regions have opposite advantages to exporting taxes, and they cannot Pareto improve on the situation.
9 A recent paper by Wildasin (1988) compares Nash fiscal equilibria in taxes (Tequilibrium) and in public goods (Z-equilibrium) in a two-region economy where capital, a production factor, is the taxed good. With absentee owners of capital, a Z-equilibrium is an institutional setting that makes private consumption effects dominated by public consumption effects. So his taxes in a Z-equilibrium are lower than in a T-equilibrium, and definitely lower than the Pareto efficient taxes. In terms of Gordon’s (1983) paper (pp. 578–579 and 583), Wildasin’s Z-equilibrium eliminates partly the ‘ignorance problem’: each jurisdiction takes account of the other’s tax reaction (@t j =@zi ) to its public goods decision.
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References Atkinson, A.B. and Stiglitz, J., 1980. Lectures on public economics. New York: McGraw-Hill. Gordon, R.H., 1983. An optimal taxation approach to fiscal federalism, Quarterly Journal of Economics, 98(3), 567–586. Keen, M., 1988. Welfare effects of commodity tax harmonization, Journal of Public Economics, 33(1), 107–114. Mintz, J. and Tulkens, H., 1986. Commodity tax competition between member states of a federation: equilibrium and efficiency, Journal of Public Economics, 29(2), 133–172. (Chapter 19 in this volume). Wildasin, D., 1988. Nash equilibria in models of fiscal competition, Journal of Public Economics, 35(2), 229–240.
Chapter 21 OPTIMALITY PROPERTIES OF ALTERNATIVE SYSTEMS OF TAXATION OF FOREIGN CAPITAL INCOME
Jack Mintz, Henry Tulkens Journal of Public Economics 60(3), 373–401 (1996).
Foreign source capital income taxes are examined from the point of view of optimal taxation. In the framework of a simple economy with international real capital flows, a taxonomy of alternative systems of such taxation is first presented, showing how crediting and other tax parameters induce what are called source-based, residence-based and related systems. Next, tax rates are determined that are optimal from a single country’s point of view, given those of the others. The achievability of these rates under the various systems is analyzed. Finally, tax rates that are optimal from an international point of view are considered. Again, achievability of an international optimum under the various systems is considered, leading to the main conclusions that (i) a pure residence-based system in all countries can achieve an international fiscal optimum; (ii) it cannot, however, be sustained as an equilibrium.
1.
Introduction
Foreign source capital income taxation is essentially multidimensional: from the (multinational) firm’s point of view, domestic and foreign taxes are to be taken into account, with the further possibility of credit at home for the latter; from a state’s point of view, tax rates are to be chosen to apply to national as well as foreign firms, in addition to possible credit rates for the foreign taxes paid by the former. Moreover, for each pair of countries, specific fiscal arrangements resulting from treaties further increase the variety of possible situations. This is a revised version of the paper prepared for the International Seminar in Public Economics (ISPE) meeting, Linz, Austria, 19–21 August 1993. This version was also presented at the European Economic Association meeting in Maastricht, September 1994. Its completion was done as part of the activities of the European Union Human Capital and Mobility research network ‘Fiscal Implications of the European Integration’.
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In this paper we attempt a normative analysis of these issues in terms of optimal taxation, i.e. from an economic efficiency point of view. The nature of the problem varies, however, according to whether efficiency is considered from a single country’s point of view only, or from a world point of view. We therefore deal with the issue first in terms of national fiscal decisions, made in a given international context, on the one hand, and then by taking into account the interactions that national decisions of two countries have on one another. In a companion paper (Mintz and Tulkens, 1990) we have presented a positive analysis of the subject, specifically in terms of fiscal competition concepts as modeled by the game-theoretic notion of a Nash equilibrium.1 Here we consider instead the efficiency (i.e. Pareto optimality) aspects of the issues. The model we use and present in Section 2 is that of two open economies, both of which export capital to each other. Each country has domestic corporations investing at home and domestically based multinational corporations that invest abroad, with the rents accruing only to their domestic resident owners who provide entrepreneurial skills. Crosshauling of direct investment arises in equilibrium to the extent that the plants of the multinational firms are not perfect substitutes. The multinationals are financed not only by equity provided by the residents of their country but also by portfolio equity that is assumed to be traded in a world market. Although the two countries are ‘large’ relative to each other in terms of direct investment, they are assumed to be small in the international portfolio markets. Thus, the net-of-corporate tax rate of return to portfolio capital is exogenous to both countries.2 The governments use corporate taxes, as well as resident lump-sum taxes, to pay for the cost of national public goods. In each country the corporate income tax is levied, at possibly different rates, on three different sources of income: (i) domestic source income earned by the domestic firms; (ii) domestic source income earned by foreign firms; and (iii) foreign source income earned by the domestically based multinational firms. In The model developed here was initially formulated in Section 4 of Mintz (1986) and further elaborated for the study of tax competition in Mintz and Tulkens (1990). Both these papers and its present extension benefited from remarks and comments made by participants at the above conferences and at various seminars. Special thanks are due to Louis Gevers, Guttorm Schjelderup, Jim Bergin, Agnar Sandmo, Robert Inman, Peter Birch Sørensen and Manuel Leite Montero. Financial support from the Colle`ge Interuniversitaire d’e´tudes doctorales dans les sciences du Management (C.I.M.), Brussels, and the Social Sciences and Humanities Research Council (S.S.H.R.C.) of Canada, for mutual working visits at CORE and at Toronto is gratefully acknowledged. 1 This approach was initiated by Hamada (1966), and used for other kinds of tax in Kolstad and Wolak (1983) as well as in Mintz and Tulkens (1986). Feldstein and Hartman (1979) instead used the Stackelberg equilibrium concept. 2 If the countries are ‘large’ and the rate of return on portfolio capital is endogenous, then we would introduce another motive for corporate taxation that raises other issues that we do not wish to examine here.
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
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the case of the latter, the government may in addition allow part of foreign corporate taxes paid by them to be credited against their tax liability owed at home on foreign source income. At this point a taxonomy is needed of the alternative ‘systems’ of foreign income taxation, as they are induced by alternative values given by the taxing authorities3 to the various tax and credit rates just mentioned. Such a taxonomy is presented in Section 3, where the ‘sourcebased’ and the ‘residence-based’ are described as the main such systems, followed by some variants. Within this framework we define (in Section 4) as optimal for each country the tax and credit rates that maximize the country’s social welfare, given the similar choices made by the other country. We examine whether such optimum can be achieved under the various systems just mentioned. We next characterize optimization at the world level (Section 5), and consider also the extent to which international optimality can be achieved under the alternative systems. Our results are summarized in the concluding Section 6. To conclude this introduction, let us put this paper in perspective visa`-vis some major predecessors. On the one hand, our main result on the international optimality of the residence-based system is akin to those stated by Musgrave and Musgrave (1984, p. 763), Sinn (1987, pp. 195– 197) and Razin and Sadka (1990), except for the point that while they all assert that with full credit given for foreign taxes, capital is efficiently allocated from a worldwide perspective, the optimality we claim is for the situation where all countries adopt the pure residence principle, i.e. abstain from taxing foreign firms, in which case crediting is immaterial because there is nothing to credit. On the other hand, we do not find that from a national optimality point of view governments should fully tax the income of their multinationals and allow for a deduction of foreign taxes as argued by Feldstein and Hartman (1979) and Bond and Samuelson (1989). Instead, governments in the interest of domestic welfare may use either the residence-based system with a range of crediting schemes or a source-based system, both with some form of exception. Our results also differ from Gordon (1992), who suggests an alternative role for crediting: encouraging foreign governments to impose taxes on capital income so that a residencebased capital income tax can be enforced in a world with costlessly mobile portfolio capital. Our model deals explicitly with the taxation of foreign direct investments rather than portfolio capital. 3
For example, Hong Kong exempts from taxation most foreign source income. The United States, United Kingdom and Japan unilaterally credit for the foreign tax on repatriated earnings and branch profits of resident multinationals. France, Netherlands, Belgium and Switzerland generally exempt branch profits and, except for Switzerland, subsidiary dividends. Canada and Germany exempt from taxation foreign source dividends of subsidiaries with qualifying ownership by the parent. See Alworth (1988) for further discussion.
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A simple economy with international real capital flows
We consider an economy with two countries, in each of which a private good is being consumed in two different periods (1 and 2), in amounts C1 $0 and C2 $0, respectively, as well as a local public good, produced in amount R$0 in period 2 only. The initial resources of each country, available in amount A > 0 in period 1, can be devoted either to immediate consumption C1 , or saved and invested to permit consumption C2 in the next period. We assume that such investment can take two alternative forms: physical capital, in amount represented by K$0, or portfolio capital, in amount k, the sign convention on this last symbol being that k < 0 denotes borrowing and k > 0 lending. Consumption in period 2 can then occur by means of revenues obtained from two possible sources: production revenues, i.e, coming from the firms that use the physical capital, and portfolio revenues. Two additional features of our economy give it an international character: on the one hand, we consider that the production activities may take place either domestically or abroad, or both; the technologies need not be identical, neither between domestic and foreign firms operating in the same country, nor between firms of a given country operating domestically or abroad. Both country-specific and ownership-specific fixed factors of production lead to each country having its own multinational firms that operate simultaneously in both countries. On the other hand, all portfolio investment is considered to be made through an international financial market on which a single rate of return, r, can be obtained. As the latter will be assumed as a constant throughout this paper,4 our overall setting may be seen as one of two fairly small countries in which the savers have the choice between making direct investments in domestic and/or foreign productive activities that they can directly control, or making portfolio investments through a world financial market on which they can only behave as price takers. Finally, the public good in each country is assumed to be financed out of two kinds of taxes, operating simultaneously: one is levied on the firms’ corporate incomes, and alternative forms of this tax will be speci-
4 This is a ‘small’ country assumption although it is also appropriate for large countries if multinational companies do not affect the rate of return to saving that may be used to finance other business and non-business assets. Extending the model to consider an endogenous interest rate on portfolio capital would need to take into account the international demand for and supply of capital. There are a number of complications with respect to corporate tax competition that are introduced by this and these are informally discussed in Mintz and Tulkens (1991).
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
511
fied in detail below; the other tax is the usual lump-sum tax, whose role in the analysis will be to serve mostly as a reference point rather than a central theme of concern. We now formally describe how the private sectors of our economy are assumed to behave. Denoting any one of the two countries by the superscript i and the other by j, and indexing accordingly the economic magnitudes defined so far, we let the strictly quasi-concave utility function U i (C1i , C2i , Ri ) obeying the Inada conditions,5 represent the aggregate preferences of country i’s residents over the commodities involved; let Ai > 0 denote the local initial resources, Kii $0 the amount of these resources invested as physical capital in domestic production, similarly Kji $0 the amount so invested abroad (i.e. in country j), and ki the amount of portfolio capital held. Let also the domestic production technologies, when operated by country i’s nationals, be represented by the strictly concave production function fii (Kii ), and the foreign ones – also when operated by i’s nationals – by the strictly concave function fji (Kji ). We assume that these functions also obey the Inada conditions.6 In this notation the superscript thus refers to the nationality of the firm (which is the same as that of the owners of the capital it employs), and the subscript refers to the location (i.e. the country) of its operations.7 Assuming perfect competition in the private sector of both economies, and taking as given the public sector variables Ri and Rj in the utility functions, the private decisions (C1i , C2i , Kii0 , Kji0 , ki ) in either one of them, say i, can be characterized as the solution of max
{C1i , C2i , Kii , Kji , ki }
U i (C1i , C2i , )
(1)
s.t. C1i þ Kii þ Kji þ ki #Ai ,
(2)
C2i #Kii þ fii (Kii ) þ Kji þ fji (Kji ) þ (1 þ r)ki :
(3)
5 These conditions require the partial derivatives U1i (0, C2i , Ri ) ¼ U2i (C1i , 0, Ri ) ¼ U3i (C1i , C2i , 0) ¼ 1 and U1i (1, C2i , Ri ) ¼ U2i (C1i , 1, Ri ) ¼ U3i (C1i , C2i , 1) ¼ 0. They are assumed here to ensure that strictly positive amounts of all goods are consumed in equilibrium. 0 0 6 Here the Inada conditions are that fii (0) ¼ fji (0) ¼ 0, fii (0) ¼ fji (0) ¼ þ1 and i0 i0 fi (1) ¼ fj (1) ¼ 0, ensuring that positive equilibrium levels of investment are made in each technology. 7 Notice that this notation is easily extended to a world with more than two countries, a property that will also apply to our results.
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Public Goods, Environmental Externalities and Fiscal Competition
It can be readily seen that by solving this problem, values of Kii , Kji and k are determined that satisfy the conditions i
0
0
fii (Kii ) ¼ fji (Kji ) ¼ r and U1i =U2i ¼ 1 þ r from which the optimal values of the consumption variables C1i and C2i are derived. From this exercise it also appears that in this model each country’s private sector equilibrium is completely independent of the equilibrium of the other. However, as we will show in what follows, this is only true for the case where there is no public sector: the model is indeed so conceived of as to allow us to identify as clearly as possible the exact nature of the interactions that take place when public sector taxation and expenditure variables are introduced – a task to which we now turn.
3.
Alternative systems of taxation of foreign capital income
For the economy just specified, in Sections 4 and 5 we examine the effects of corporate taxes when they are chosen optimally by each country; in particular we compare these effects under the many forms that corporate taxation can take when it bears on international operations. To this end we define, in this section, the various bases and rates to be considered, as well as the alternative tax ‘systems’ they imply. At the end of the section we also introduce a lump-sum tax whose role, which is only of an analytical nature, will be made clear at that point. 3.1.
Bases and rates
In our setting there are three possible (and not mutually exclusive) bases for corporate taxes that a government of a country can raise (say, country i’s): . income generated at home by domestic firms, i.e. fii (Kii ); . income generated abroad by domestically based multinationals, i.e. fji (Kji ); and . income generated at home by foreign-based multinationals, i.e. fij (Kij ). We can therefore define for country i four rates that its government has conceivably at its disposal for taxing corporate income: tidd ¼ the rate of the tax it levies on the base fii (Kii ) just defined; tidf ¼ the rate of the tax it levies on fji (Kji );
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
513
tifd ¼ the rate of tax it levies on fij (Kij ); and finally ai ¼ the rate at which country i credits its domestically based multinationals operating abroad for the corporate income tax tfdj they have paid there.8 All four magnitudes so defined are assumed to belong to the interval [0, 1]. From the point of view of a multinational firm based in country i and operating in country j, we can further define Tji tfdj þ tidf ai tfdj ¼
tfdj (1
ai ) þ tidf
(4:1) (4:2)
as the effective corporate tax rate paid by the firm, partly in i and partly in j, on the income generated by its operations in country j.9 With this notation, it appears that, given the foreign tax tfdj , the effective rate Tji faced by a country i’s multinationals results from two choices made by its home government: the credit rate, ai , and the nominal tax on foreign source income, tidf . However, for any given level of Tji , the pair (ai , tidf ) that achieves Tji is not unique: indeed any such pair satisfying (4.1) will do. There is thus substitutability between these two fiscal instruments. This is illustrated on Fig. 1, where the effective rate Tji is plotted as a function of the domestic nominal rate tidf , given some value t j of the foreign rate. fd Notice that when Tji ¼ tfdj , the government of country i does not collect any revenue from its multinationals operating in j; it collects revenue only if Tji > tfdj . The situations where Tji < tfdj appear as those in which the tax credit is refundable. In those situations, which are depicted by the shaded area in Fig. 1, it is in fact the government of country i who pays (part of) the taxes levied by the government of j on the foreign multinationals operating on its territory: i pays all of these taxes if Tji ¼ 0, part of them if 0 < Tji < tfdj . Of course, tax credits are most often not refundable;10 thus all pairs i i (a , tdf ) represented by points in the shaded area may be considered as unrealistic. We rule out refundable credits by rewriting Tji as follows: Tji ¼ tfdj min [ai tfdj , tidf ] þ tidf
(5:1)
or, equivalently,
8 For country j, the corresponding four rates and bases so defined read, respectively, tddj and fj j (Kj j ), tdfj and fi j (Ki j ), tfdj and fji (Kji ), and a j . 9 Symmetrically, for a firm based in country j and operating in country i, the effective tax rate is Ti j tifd þ tdfj a j tifd (1 a j ) þ tdfj . 10 When they are, it is usually in an indirect way.
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Public Goods, Environmental Externalities and Fiscal Competition
i
Tj
ai = 0
0 < ai < 1 ai = 1 j
t fd i
ai t fd
{
+1
0
i
tdf
Figure 1. Graph of Tji in (4.1) as a function of tidf .
Tji ¼ tfdj ¼ tfdj (1 ai ) þ tidf
iftidf < ai tfdj ,
(5:2)
iftidf $ ai tfdj :
(5:3)
The graph of Tji as a function of tidf now appears as one of the heavy lines in Fig. 2: ABC if ai ¼ 1, AD if ai ¼ 0, and AFG for some value of ai 2 ]0,1[. The non-refundability of credits thus puts a lower bound on the effective tax rate that a country can charge its multinationals for their foreign source income – specifically, Tji $ max [tfdj , tidf ]. 3.2
Systems of taxation of foreign income: a taxonomy
With the notation thus introduced, the system of taxation of foreign corporate income adopted by a country (say, i) is compactly described by the values it chooses for the three components of the vector (tidf ,ai ,tifd ), where the first two components refer to the country’s treatment of its home-based multinationals, and the third one refers to its treatment of foreign-based multinationals. Several systems can be distinguished. 3.2.1 Source-based vs. residence-based systems Specifically, a pure source-based (or exemption11) system of taxation is chosen if the values of the three components just defined are as follows: 11
Also sometimes characterized as capital import neutral.
Ch.21 Optimality Properties Of Alternative Systems Of Taxation D
i Tj
G ai = 0 C
and ai t j < t i
0 < ai < 1
j
{
A
{ or a t
fd
i j fd
df
> ti
df
ai = 1
F
t fd ait jfd
515
B +1
0
tdfi
Figure 2. Graph of Tji in (5.1) as a function of tidf , for various values of ai .
(sb-i) tidf ¼ 0,
i.e. full exemption from domestic tax of the income generated abroad by the country’s multinationals; i (sb-ii) a ¼ 0, i.e. no crediting of taxes paid abroad by these same multinationals; (sb-iii) tifd > 0, i.e. levying of a tax on the income generated on the country’s territory by foreign-based multinationals12. Under the source-based system, the effective tax rate faced by country i’s multinationals is, using (4.1) or (5.1): Tji ¼ tfdj : In Fig. 2, the system corresponds to point A. Another feasible choice for country i is that of a pure residence-based (or crediting13) system of taxation, which amounts to the following: (rb-i) tidf > 0, i.e. levying of a domestic tax on the income generated abroad by the country’s multinationals;14 (rb-ii) ai ¼ 1, i.e. fully crediting the taxes paid abroad by the same multinationals; (rb-iii) tifd ¼ 0, i.e. fully exempting from domestic tax the income generated within the country by foreign-based multinationals. Under the residence-based system, we have, with (4.1), Tji ¼ tidf (the diagonal on Fig. 1); but with (5.1), the effective tax rate in this case reads: Tji ¼ tfdj ,
iftidf # tfdj ,
¼ tidf , iftidf > tfdj : 12 13 14
A frequent example is tifd ¼ tidd , for reasons of non-discrimination. Or capital export neutral. Here also, a frequent example is tidf ¼ tidd ; see Subsection 3.2.3 below.
516
Public Goods, Environmental Externalities and Fiscal Competition
Thus, in Fig. 2 the residence-based system corresponds to the kinked line ABC. 3.2.2 Variants The above are stricto sensu definitions of the source-based vs. residence-based systems. Variants of each of them can be further characterized in either one of two ways. On the one hand, if we allow for exceptions on the tax rates which are zero in the definitions, we obtain: . the source-based system with exception, defined by keeping ai ¼ 0 and substituting tidf > 0 for tidf ¼ 0 in (sb-i) above. The resulting effective tax rate reads: Tji ¼ tfdj þ tidf : The exception thus bears on the home tax treatment of the foreign source income of the country’s multinationals, as is the case if, for instance, tidf ¼ tidd . Alternatively, we obtain: . the residence-based system with exception, defined by keeping ai ¼ 1 and substituting tifd > 0 for tifd ¼ 0 in (rb-iii) above. The effective rate Tji is left unaffected since (rb-iii) only bears on the treatment of foreignbased multinationals. This exception is in force in countries that chose the residence-based system and who also charge their foreign multinationals, often at a rate tifd ¼ tidd , for reasons of non-discrimination with their own domestic firms. On the other hand, further variants are obtained by considering values other than 0 or 1 for the credit rate ai . For all such values, we call the system . partially residence-based, defined by substituting 0 < ai < 1 for ai ¼ 1 in (rb-ii) above. The induced effective rate is then as specified in (5.1). 3.2.3 The full taxation after deduction system The two basic systems with all their variants just defined, together comprehensively cover all conceivable systems of international taxation. Let us take, for instance, the full taxation after deduction (f.t.a.d.) system analyzed in Feldstein and Hartman (1979). It consists of a country, say i, charging the foreign source income of its home-based multinationals, net of the tax paid by them abroad, at a rate tidf equal to the one applicable to domestically earned income, namely tidd . This amounts to an effective tax rate (faced by an i-based multinational) of
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
517
T ji 1
tfdj
G
A
0
1
tdfi = tddi
Figure 3. Tji under the f.t.a.d. system.
Tji ¼ tfdj þ tidd (1 tjfd ), which in terms of (4.1) results from choosing both tidf and ai equal to tidd and thus reads: Tji ¼ tfdj þ tidd tidd tfdj : The f.t.a.d. system thus appears to belong to the category of partially residence-based systems (with or without exception, depending upon whether tifd is positive or zero, this being left unspecified in the Feldstein– Hartman description of the system). Graphically, the system is represented in Fig. 3 by any point on the line AG. This illustration clearly shows that under the f.t.a.d. system the effective tax rate on foreign source income varies with the level of the nominal tax on domestic income. It also shows that the system implies an effective tax rate always higher than tfdj (except when tidd ¼ 0), and thus for the government an always positive fiscal yield from the foreign source income of its domestic multinationals. 3.3
Lump-sum tax
For the purposes of the analysis developed in the following sections, we introduce a lump-sum tax, denoted15 by ti , with an (arbitrary) upper bound ti > 0. Following Dixit (1985), the reason for this apparently artificial construct is to enable us to distinguish between two kinds of situations, differing in a very important way: 15
ti is unrestricted in sign, which means that when it is negative it denotes a lump-sum subsidy.
518
Public Goods, Environmental Externalities and Fiscal Competition
(i) The first one, where the upper bound ti is assumed to be very high (or, equivalently, not to exist), so that for revenue purposes – i.e. for the financing of the public good Ri – full advantage can be taken of the well-known and unique efficiency properties of the lump-sum tax. The corporate tax is then, in fact, unnecessary; but if it can be shown that it is nevertheless optimal for country i to have corporate tax rates different from zero in such cases, then it will have to be attributed to other reasons than the specific needs of the public sector; and these reasons will need to be identified. (ii) The other case to be considered is, by contrast, the one where t i is rather low (possibly equal to zero), which corresponds to a situation where non-zero corporate tax rates are also justified by revenue needs. The comparison between the expressions for optimal tax rates under either one of these circumstances will also prove instructive in many respects.
4.
Domestically optimal fiscal choices in one country
Given the aggregate form that we have adopted to describe consumption and production decisions within each country, optimal taxation at the domestic16 level can only be defined and characterized in terms of fiscal choices (i.e. tax rates and credit as well as expenditure level) that maximize aggregate domestic welfare, as represented by the functions U i (respectively, U j ) defined earlier. This will be done in subsection 4.2 below. Prior to that, account must be taken of the fact that the private sector consumption arguments in these functions, as well as the investment variables in the constraints these arguments are subject to, are influenced by these fiscal choices. Describing this dependence analytically is the subject matter of subsection 4.1. 4.1
Market equilibrium for given taxes
To describe the behavior of the private sector in each of the two ‘adjacent’ economies of Section 2 as the solution of the problem (1)–(3) above, the introduction of a public sector that finances its expenditure out of the corporate and lump-sum taxes defined in Section 3 requires that constraint (3) be modified as follows: C2i # Kii þ (1 tidd )fii (Kii ) þ Kji þ (1 Tji )fji (Kji ) þ (1 þ r)ki t i : 16
(6)
As will appear in Section 5, optimality at the international level will not be defined in this aggregate way.
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
519
Then, the optimization problem, whereby the private sector equilibrium is characterized, is to be reformulated with the constraints (2) and (6), for any given values of Ri and of the tax parameters tidd , tidf , ai , tfdj (hence, of Tji ), and ti . The first-order equilibrium conditions are of the form: 0
(1 tidd )fii (Kii ) ¼ r,
(7)
0 Tji )fji (Kji )
(8)
(1
¼ r,
U1i =U2i ¼ (1 þ r):
(9)
As a result, the three private investment decisions, Kii , Kji and ki , appear to be functions of the domestic tax parameters, tidd , tidf , ai and ti , of the public expenditure level Ri , of the foreign tax rate t j , and finally of fd r. Because some properties of these functions will play an important role in what follows, it is necessary to establish them here, from the comparative statics analysis of conditions (7)–(9). These properties are as follows, for non-zero effects on variables other than ki (all other comparative static effects are equal to zero):17 0
@Kii fii ¼ 00 < 0, @tidd (1 tidd )fji @Kji @Tji
¼
fji
(10)
0
(1 Tji )fji
00
< 0:
(11)
Given these properties, the three investment functions mentioned above, when restricted to their relevant arguments, read: Kii ( ) Kii (tidd ), Kji (
)
(12)
Kji (Tji ),
(13)
ki ( ) ki (tidd , Tji , ti , Ri ),
(14)
where Tji ¼ tfdj min [ai tfdj , tidf ] þ tidf : Introducing these functions into constraints (2) and (6), and then substituting (2) and (6) for C1i and C2i , respectively, in (1), the preferences of country i’s residents can be represented by the indirect utility function:
17
All the comparative statics effects on ki are ambiguous in sign. However, they will not be required for our subsequent analysis.
520
Public Goods, Environmental Externalities and Fiscal Competition
W i (tidd , Tji , ti , Ri ) V i (tidd , tidf , ai , tifd , ti , Ri ) U i [Ai Kii ( ) Kji ( ) ki ( ), Kii ( ) þ (1 tidd )fii (Kii ( )) þ Kji ( )
(15)
þ (1 Tji ) fji (Kji ( )) þ (1 þ r)ki ( ) t i , Ri ]: Roy’s identity then yields the following: @W i ¼ li fii ( ), @tidd
@W i ¼ li fji ( ), @Tji
with li ¼ (@W i =@Ai )=1 þ r) denoting the marginal utility of income consumed in the second period. 4.2
Fiscal choices
4.2.1 Nationally optimal tax formulae With corporate income taxes and lump-sum taxes being the only way to raise public revenue in the economies described above, the financing of the public good in each of them requires that its quantity, Ri , and the tax and credit rates, tidd , tidf , tifd , ai and ti , be chosen to satisfy the budget balancing condition, Ri #tidd fii (Kii ( )) þ tifd fij (Kij ( )) þ max [(tidf ai tfdj ), 0] fji (Kji ( )) þ ti ,
(16)
where the expression (tidf ai tfdj ) may be replaced by (Tji tfdj ) in view of (5.2) and (5.3). The upper bound on lump-sum tax income must also be satisfied, i.e. ti # ti :
(17) Kij (
Tij ,
Note that in ) in (16), the variable is i.e. the effective rate that country j’s multinationals are facing, as a result of applying to country j the reasoning that led to (5.1) above for country i. This rate reads: Tij ¼ tifd min [a j tifd , tdfj ] þ tdfj :
(18)
Because we momentarily need to know the effect on this rate of changes in country i’s tax rate, tifd , on foreign multinationals, we also record here that @Tij ¼ 0, if a j ¼ 1 and tifd < tdfj , @tifd
(19:1)
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
¼ (1 a j ), if 0 < a j < 1 and a j tifd < tdfj ,
521
(19:2)
¼ 1,
if 0 < a j #1 and a j tifd $tdfj ,
(19:3)
¼ 1,
if a j ¼ 0:
(19:4)
This is illustrated in Fig. 4. The optimal fiscal choices of country i can now be described as values of tidd , Tji ,tifd , ti and Ri that maximize the indirect utility function (15) subject to the constraints (16) and (17), taking a j , tdfj and tfdj as fixed, i.e. taking as given the other country’s similar fiscal choices that affect country i’s objective function (15) or budget constraint (16). For any value of Tji so chosen, recall that Eq. (5.1) yields the family of pairs (ai , tidf ), whereby Tji can be implemented, given tfdj . Letting ci and wi denote the multipliers associated with (16) and (17), the Lagrangian of the problem reads: Li (tidd , Tji , tifd , ti , Ri ; a j , tdfj , tfdj ; ci , wi ) ¼ W i (tidd ,Tji , ti , Ri ) þ ci [tidd fii (Kii ( )) þ tifd fji (Kij ( )) þ (Tji tfdj ) fji (Kji ( )) þ t i Ri ]
(20)
þ wi [ti t i ]: The necessary conditions for an optimum are (asterisks denote optimal values of the variables): i @Li @W i i i i i 0 @Ki ¼ 0, (21:1) ¼ i þ c fi þ tdd fi @tidd @tdd @tidd
Tj i
aj = 0
} 0 < aj < 1 tdfj
aj = 1
tfdi Figure 4. Graph of Tij as a function of tifd for alternative values of a j .
522
Public Goods, Environmental Externalities and Fiscal Competition
@Kji @Li @W i j i i i i0 ¼ þ c f þ (T t )f j j fd j @Tji @Tji @Tji ! j j @Li j i i j 0 @Ki @Ti ¼ 0, ¼ c fi þ tfd fi @tifd @Tij @tifd
! ¼ 0,
@Li @W i ¼ þ ci wi ¼ 0, @t i @t i @Li @W i ¼ ci ¼ 0, @Ri @Ri i @Li i i ¼ 0, if w > 0, ¼ t t i i # 0, if w ¼ 0, @w @Li j j j i i i i i i i i i ¼ [tdd fi (Ki ( )) þ tfd fi (Ki ( )) þ (Tj tfd ) fj (Kj ( )) þ t @c ¼ 0, if ci > 0, i R] $ 0, if ci ¼ 0:
(21:2)
(21:3) (21:4) (21:5) (21:6)
(21:7)
From (21.1)–(21.3) we derive the following rules for the optimal corporate tax rates: (1 li =ci )fii , 0 fii @Kii =@tidd
ti dd ¼ Tji ¼ ti fd ¼
(1 li =ci )fji 0
fji @Kji =@Tji 0
(22) þ tfdj ,
fij
fij (@Kij =@Tij )(@Tij =@tifd )
(23) :
(24)
Before searching for what these basic expressions allow us to say about crediting and alternative tax systems, three technical remarks are in order. (1) On lump-sum taxes. To see the role of the lump-sum tax, notice that the assumed property @W i =@Ri > 0 implies that at an interior solution ci > 0, by (21.5); and this, together with the fact that @W i =@ti ¼ li < 0 by (15), yields, through (21.4), li ¼ ci ci . Thus at an optimum, . if wi ¼ 0 (which by (21.6) implies ti # ti ), then we have that i i l =c ¼ 1: this equality characterizes an optimum where the upper bound ti is not binding; . if wi > 0 (implying ti ¼ ti ), then we have instead that li =ci < 1: this inequality thus characterizes an optimum where ti is binding.
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
523
(2) Elasticity formulae. The optimal tax rate formulae in expressions (22)–(24) may be interpreted more readily by presenting them as Ramsey tax rules.18 Let us take the first expression, (22). From Eq. (7) we 0 have fii rii ¼ r=(1 tidd ). The term rii can be interpreted as the ‘taxadjusted cost of capital’, which in equilibrium is equal to the marginal gross rate of return on capital. We can rewrite the comparative static condition (10) as @Kii 1 @Kii rii @Kii ¼ and ¼ : 00 @rii @tidd (1 tidd ) @rii fii Now we define «ii ¼ rii =Kii @Kii =@rii as the elasticity of the demand for capital with respect to the (tax-adjusted) cost of capital. We also define Gii ¼ fii =Kii as the average gross rate of return on capital. After substitution and rearrangement, expression (22) becomes: ti (1 li =ci ) dd ¼ : «ii ( rii =Gii ) (1 ti dd )
(25)
The more elastic the demand for capital («ii ! 1), and the closer that the average and marginal gross rates of return on capital are to each other (rii =Gii ! 1), the smaller is the tax rate, ti dd . Similar expressions may be derived for Tji and ti fd in Eqs. (23) and (24), respectively. Let «ij ¼ rij =Kji @Kji =@rij , «ij ¼ rij =Ki j @Ki j =@rij , rij ¼ r=(1 Tji ), rij ¼ r=(1 Ti j ), Gij ¼ fji =KjI and Gij ¼ fi j =Ki j , each corresponding to earlier expressions for the elasticity of capital demand, the cost of capital and average rates of return on capital, respectively. The Ramsey tax rules for Tji and ti fd are then the following: Tji tfdj (1 Tji ) ti fd (1
ti fd
¼
¼
(1 li =ci ) , «ij rij =Gij 1 j
«i rij =Gij
:
(26) (27)
(3) Implication of full crediting abroad for the domestic tax rate on foreign j firms. For ti fd as obtained in (24), we can state that if a ¼ 1, then we must have j ti fd $ tdf :
Indeed, going back to (21.3), as well as to (19.1) and (19.3) to evaluate the last derivative in the parentheses when a j ¼ 1, the expression @Li =@tifd 18
We thank Agnar Sandmo for pointing out to us the connection between these formulae and Ramsey tax rules.
524
Public Goods, Environmental Externalities and Fiscal Competition
remains strictly positive for any value of tifd < tdfj : for an optimum to be reached, tifd must thus reach at least tdfj . 4.2.2 Tax systems and the national optimum The above developments lead to the following interpretations of the three tax formulae just obtained. 4.2.2.1 With unrestricted lump-sum taxes If, at the optimum, the lump-sum tax level t i is less than or equal to its upper bound ti with the multiplier wi ¼ 0 (this means that corporate taxes are not required to finance the public good), then expressions (22) and (23) become, respectively, j i ti dd ¼ 0 and tj ¼ tfd ,
while (24), which implies Tfdi > 0, remains unchanged. In terms of the taxonomy developed above, these optimum values can be achieved: . under the source-based system, since with ai ¼ 0 the optimal effective tax rate Tji ¼ tfdj implies from (4.1) ti df ¼ 0, i.e. exemption for foreign > 0; source income, and ti fd . under the residence-based system (full or partial) with exception, i.e. by i j i i choosing ti df ¼ a tfd for any 0 < a #1 as well as tfd > 0 for the exception; . under the full taxation after deduction system (with the complement i ti fd > 0), which is automatically achieved since tdd ¼ 0. In all systems the characteristic of the domestic optimum, in the circumstances under consideration, is simply that foreign source corporate income is fully exempted by country i (irrespective of whether or not that income is taxed abroad). The reader should notice the artificiality of the residence-based system in this setting: with crediting, the optimum requires that a domestic tax tidf be levied, equal to the credited fraction of the foreign tax (i.e. tidf ¼ ai tfdj ). It is just as simple not to credit at all – thereby avoiding the pure waste of the compliance costs entailed by enforcing the credit on the one hand and levying a tax equal to it on the other. Finally, under all systems, the tax rate ti fd on foreign multinationals is always positive and set at the revenue-maximizing level (which may imply ti < 0). It is also independent of the government’s choice of Ri . These two properties are only natural since this tax rate bears on a basis with no domestic welfare effect.
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
525
4.2.2.2 With restricted lump-sum taxes If, at the optimum, the lump-sum tax t i is restricted to ti with a positive value for the multiplier wi (which means that corporate taxes are required to finance the optimal amount of the public good), then (22) and (23) imply that 0
tidd > 0 and Tji > tfdj , with ti fd > 0 still unchanged as in (24). Comparing with the previous case, we have that: . the now positive domestic tax ti dd is proportional to its revenuemaximizing level, as determined by the (inverse of the) elasticity of Kii with respect to tidd , and with the proportionality factor 1 (li =ci ) being less than 1 and determined by the relative implicit values of private (li ) vs. social (ci ) income; . the effective tax on foreign source income Tji is now strictly larger than the foreign nominal tax tfdj , by an amount which is also proportional to the revenue-maximizing rate (here determined by the elasticity of Kji with respect to Tji ) and with the same proportionality factor 1 (li =ci ); . the tax on foreign corporations ti fd is, as before, the purely revenue-maximizing one; again,19 it is independent of the government’s choice of Ri . To find the values of tidf and ai that implement Tji , we may combine (23) with (5.1) to obtain: Tji ¼ Yi þ tfdj ¼ tfdj min (ai tfdj , tidf ) þ tidf
where Yi ¼
¼ tfdj ,
if tidf < ai tfdj ,
(28:a)
¼ tfdj (1 ai ) þ tidf ,
if tidf $ ai tfdj ,
(28:b)
i 1 cli fji 0 fji
(28)
@Kji
:
(29)
@Tji
If ai ¼ 0 is chosen, then only case (28.b) can hold, which implies ti df ¼ Yi ;
(30)
choosing instead 0 < ai #1, case (28.b) implies i j ti df ¼ Yi þ a tfd : 19
(31)
This property thus holds irrespective of the restriction on lump-sum taxes, a result also obtained by Wildasin (1987).
526
Public Goods, Environmental Externalities and Fiscal Competition
Since case (28.a) can never hold, because it would imply that Yi ¼ 0 (it corresponds to the shaded area in Fig. 1), expression (31) completely characterizes the pairs (ai , tidf ) that achieve the optimal effective rate Tji prescribed by (23). Graphically (see Fig. 5), for the value of Tji determined by i i (23) as OA, ti df satisfying (31) must be OB if a ¼ 0 is chosen, OC if a ¼ 1 is i chosen, and some value between OB and OC (e.g. OD) if 0 < a < 1. Turning again to tax systems, we have under restricted lump-sum taxes that: . the source-based system with exception can implement the national optimum as follows: i ai ¼ 0, ti fd as in (24), and for the exception: tdf as in (30);
. the residence-based system with exception can also implement the national optimum, with i 0 < ai #1, ti df as in (31), and for the exception: tfd as in (24):
A remark similar to before can be made: since both systems can achieve the domestic optimum, why bother with crediting? The national optimum is not achievable, however, under the full taxation after deduction system, because this system links the value of Tji to the value of tidd – a link that Eqs. (22)–(24) do not contain. In the context of our model, the f.t.a.d. system is thus suboptimal from a national point of view. However, it would achieve optimality if the functions fii (Kii ) and fji (Kji ) were identical.
ai = 0
T ij
ai = 1 0 < ai < 1 A
{
{
Υ*i
j
(1 − ai) tfd
tfdj
0
B
D
C
Figure 5. Alternative values of tidf and ai that implement Tji ¼ OA.
tdfi
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
527
Similar considerations may be made concerning the non-discrimination principle put forward in model conventions on double taxation, such as proposed by OECD or the UN for instance, because it amounts to imposing tifd ¼ tidd as an additional constraint on the fiscal choices. Finally, let us note that when lump-sum taxes are restricted, i.e. are insufficient to finance public needs, the optimal policy under the sourcebased system (with ai ¼ 0 and ti df > 0) implies double taxation of foreign source income for country i’s multinationals. This finds its justification in the fact that it is the effective rate that matters for optimality, and not the number of steps through which taxes are levied. 4.3.
Fiscal externalities
The forms (15), and (16) and (17), of the objective function and of the constraints, respectively, that characterize a country’s fiscal choices also allow us to exhibit the extent to which the other country’s such choices affect the welfare of the former. Specifically, denoting by Li the value of the Lagrangian formed by the maximand (15) and the constraints (16) and (17), the effects on i’s welfare of the fiscal decisions of country j are found by differentiating Li with respect to tfdj and Ti j (resulting from changes in tdfj and/or a j ) and then applying the envelope theorem to (20) to yield: @Li @tfdj @Li @Ti
j
¼ [li (1 ai ) þ ci ai ] fji < 0, 0
j j j ¼ ci ti fd fi @Ki =@Ti < 0:
(32)
(33)
Unambiguously, the corporate income tax rates at which country j charges multinationals (both those of country i through tfdj and its own through Ti j ) appear to reduce the welfare of country i. In both cases the reason lies in the fact that the tax of country j reduces the basis that is available for country i’s own taxes. These are fiscal externalities which, because they are negative, suggest that these two tax rates are chosen too high by country j, since it does not take into account the loss of i’s welfare when choosing them.
5.
Internationally optimal tax rates
5.1
Optimality at the international level
In this section we consider the optimality of the countries’ fiscal choices from a world point of view. Since at this level it makes little sense
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Public Goods, Environmental Externalities and Fiscal Competition
to aggregate national welfare into an international welfare function, the notion of optimality is to be formulated in its strict Paretian form, which in the case of our model reads as follows. International optimality. A pair of fiscal choices made by the two countries is an international fiscal optimum if they are both feasible (i.e. satisfy each country’s public sector budget constraint), and such that there exists no other pair of feasible fiscal choices that induces at least as high national welfare levels in both countries and a strictly higher one in at least one country. However, some of our results below are obtained only for a weaker version of this definition, namely: Local international optimality. A pair of fiscal choices made by the two countries is a local international fiscal optimum if they are both feasible and such that no feasible infinitesimal change in the fiscal variables can induce an increase in the welfare of both countries, or an increase in the welfare of one without a reduction in the welfare of the other. Notice that while a ‘global’ (i.e. according to the former definition) fiscal optimum must also be a local one, the reverse need not hold in general; and if a pair of fiscal choices is not a local optimum, then it cannot be a global one either. In addition, there is of course no reason to believe that in our model either a global or a local international fiscal optimum be unique. Rather, our task will be to find some general characteristics of optima, or, if possible, to exhibit some of them.
5.2.
Internationally optimal tax rates and the corresponding foreign income taxation system In the spirit just described, we can now state:
jo jo jo io io Proposition 1. The pair of tax rates [(tio dd , Tj , tfd ), (tdd , Ti , tfd )] is an international local fiscal optimum if its components are:
(i)
i tio dd ¼ tdd as in expression (22) above, and
(34:i)
jo tdd
(34:j)
¼
j tdd
as in the corresponding expression for j;
(ii)
jo tio fd ¼ tfd ¼ 0;
(iii)
Tjio ¼ Tji as in expression (23) above, and
(36:i)
Tijo ¼ Tij as in the corresponding expression for j, taking (ii) into account in either case:
(36:j)
(35)
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
529
io Proof. On the one hand, given (i) and (iii), the tax rates tio dd and Tj for i jo jo and tdd and Ti for j are maximizing values of these variables for W i and W j , respectively. Hence, every local change in these values can only induce dW i ¼ dW j ¼ 0. i On the other hand, tio fd ¼ 0 is not a maximizing value for W since (24) is i i not satisfied, and indeed dtfd > 0 would induce dW > 0 at the fiscal choice io io (tio dd , Tj , tfd ). However, taking into account the fiscal externality effect identified in (32) and applying it to j, the same change would also induce dW j < 0. A similar reasoning applies to dtfdj > 0, which would induce dW j > 0 at the fiscal choice (tddjo , Ti jo , tfdjo ), but also dW i < 0. Introducing a positive tax rate on foreign firms’ income in one country at a time thus cannot Pareto improve upon the pair of fiscal choices under consideration. Finally, could perhaps a simultaneous introduction of such a tax in both countries have that property? The answer is negative because we would have in country i:
@Li i @Li j i j i i dt þ fd i j dtfd ¼ c fi c fj @tfd @tfd ¼ ci ( fi j fji ), using (21.3) with tifd ¼ 0 to evaluate the first term, and (32) to evaluate the second term. Similarly, we would have in country j: @L j @tfdj
dtfdj þ
@L j i dt ¼ c j fji ci fi j @tifd fd : ¼ c j ( fji fi j )
For a Pareto improvement, both of these expressions should be non-negative, with one at least strictly positive. But, knowing that both ci and c j are strictly positive in all relevant cases, this is impossible since the differences in the parentheses are of opposite sign. & The optimality of the pair of fiscal choices (34)–(36) stated above is valid for both the restricted and unrestricted lump-sum cases – the unrestricted case being quite simple since it implies that all the above tax rates should be zero, with both countries raising their public revenue from lump-sum taxes only. This case is nevertheless interesting in its own right as it points out in the clearest way the contrast between the nonoptimality at the international level of the taxation of foreign firms and its optimality at the national level, as established by (24) and commented upon in subsection 4.2.2 above.
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Public Goods, Environmental Externalities and Fiscal Competition
As a corollary of Proposition 1, let us note that at the international fiscal optimum under consideration, taking (ii) into account in statement (iii), and using the definition (4.1)–(4.2) of Tji as well as its equivalent for jo t j Ti j , leads to Tjio ¼ ti df and Ti ¼ df , i.e. the effective tax rate of each country’s multinationals is just equal to the country’s nominal tax rate on these firms; moreover, crediting by home countries becomes immaterial and ai and a j are thus useless instruments at this optimum. It is also interesting to remark20 that an implication of Proposition 1 is that if fii ¼ fji , i.e. if multinationals of a country operate with the same technology as the domestic firms of that country, then optimality requires identical effective tax rates on their incomes. In terms of tax systems, finally, Proposition 1 may be restated as follows: Proposition 2. If two countries adopt the pure residence-based system as defined by (rb-i)–(rb-iii) in Section 3 above (implying tifd ¼ tfdj ¼ 0), and choose tax rates tidf and tdfj that are optimal for themselves, then the resulting pair of fiscal choices is internationally optimal. 5.3.
National optima vs. international optimal tax rates
The above results are of course to be set in contrast to the fiscal choices that are national optima, as identified in Section 4. There, tifd was characteristically chosen as strictly positive by country i, and the same holds true for tfdj , as chosen by country j. A pair of such national optima (actually a Nash equilibrium21) is therefore different from the international optimum just exhibited. Can such a pair nevertheless also be an international optimum? An answer is provided by the following proposition. i i Proposition 3. The pair of nationally optimal tax rates [(ti dd , Tj , tfd ), j j j (tdd , ti , Tfd )], as defined in (22)–(24) for country i, and similarly for country j, is not an international optimum. j j j i i i i Proof. Given the pair of tax rates [(ti dd , Tj , tfd ), (tdd , Ti , tfd )], tdd , Tj , i and ti fd are maximizing values for W and local changes in these variables i i thus cannot change W ; but dTj < 0 and dtifd < 0 would both induce dW j > 0, and thus achieve a Pareto improvement. Similarly, dTi j < 0 and dtfdj < 0 would induce dW j ¼ 0 and dW i > 0, a Pareto improvement. Thus, the stated pair of tax rates cannot be an international fiscal optimum, be it local or global. & 20 21
We thank a referee for this suggestion. Nash equilibria in this model are studied in more detail in Mintz and Tulkens (1990).
Ch.21 Optimality Properties Of Alternative Systems Of Taxation
531
Again, this statement is independent of whether lump-sum taxes are restricted or unrestricted. In both cases, tax rates on foreign firms appear to be too high to be optimal at the international level. In our companion study of Nash equilibria just mentioned, we observe that neither the pure residence nor the pure source systems can sustain such an equilibrium. Conversely, the fiscal optimum under the pure residence-based system discussed earlier cannot be sustained as such an equilibrium either: each country has indeed an incentive to tax the income generated within its borders by foreign firms, at the rates ti fd and tfdj that we identified in formula (24).
6.
Conclusion
From a purely national point of view, our conclusions, as stated in subsection 4.2.2 with the optimal tax formulae (22)–(24), were that both the source-based and the residence-based systems can equally implement a national optimum in our model, provided that under each22 system an exception is introduced, that actually treats foreign multinationals differently from what the system specifies for domestic multinationals. The full taxation after deduction system, however, was found not to be compatible in general with the domestic optimality conditions we derived in this model. From the world point of view, things are different. The analysis of fiscal externalities first suggested that both the effective tax rates Tji and Ti j on foreign source income, and the nominal tax rates tifd and tfdj on foreign firms would be lower at an international optimum than at a (Nash equilibrium) pair of national optima. A direct analysis of international optima showed in addition that a pure residence-based system adopted by all countries can achieve an international fiscal optimum.
References Alworth, J., 1988. The finance, investment and taxation decisions of multinationals. Oxford: Basil Blackwell. Bond, E. and Samuelson, L., 1989. Strategic behavior and the rules for international taxation capital, The Economic Journal, 99, 1099–1111. Dixit, A., 1985. Tax policy in open economies, in A.J. Auerbach and M. Feldstein (eds.). Handbook of public economics, vol. I. Amsterdam: North-Holland. Feldstein, M. and Hartman, D., 1979. The optimal taxation of foreign source investment income, Quarterly Journal of Economics, 93, 613–624. Gordon, R., 1992. Can capital income taxes survive in an open economy? Journal of Finance, 47(3), 1159–1180.
22
We take only the restricted lump-sum case, which is the most realistic one.
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Public Goods, Environmental Externalities and Fiscal Competition
Hamada, K., 1966. Strategic aspects of taxation of foreign investment income, Quarterly Journal of Economics, 80, 361–375. Kolstad, Ch.D. and Wolak, F.A. Jr., 1983. Competition in international taxation: the case of Western Coal, Journal of Political Economy, 91, 443–460. Mintz, J., 1986. Corporate tax design in an international setting: tax competition and the openness of the economy. Unpublished manuscript. Mintz, J. and Tulkens, H., 1986. Commodity tax competition between member states of a federation: equilibrium and efficiency, Journal of Public Economics, 29, 133–192. (Chapter 19 in this volume). Mintz, J. and Tulkens, H., 1990. Strategic use of tax rates and credits in a model of international corporate income tax competition. CORE Discussion Paper 9073. Louvain-la-Neuve: Center for Operations Research and Econometrics, Universite´ catholique de Louvain. Mintz, J. and Tulkens, H., 1991. The OECD convention: ‘model’ for corporate tax harmonization? in R. Prud-Homme (ed.). Public finance with several levels of governments, proceedings of the 46th Congress of the International Institute of Public Finance (Brussels). The Hague: Foundation Journal Public Finance. Musgrave, R. and Musgrave, P., 1984. Public finance in theory and practice, 4th edn. New York: McGraw-Hill. Razin, A. and Sadka, E.,1990. Integration of international capital markets: the size of governments and tax co-ordination, in A. Razin and J. Slemrod (eds.). Taxation in the global economy. Chicago, IL: The University of Chicago Press, 331–348. Sinn, H.W., 1987. Capital income taxation and resource allocation. Amsterdam: NorthHolland. Wildasin, D., 1987. The demand for public goods in the presence of tax exporting, National Tax Journal, 40, 591–601.
Chapter 22 TAX INTERACTION DYNAMICS AMONG BELGIAN MUNICIPALITIES 1984–1997
Jean Franc¸ois Richard, Henry Tulkens, Magali Verdonck Translated from Economie et Pre´vision (Ministry of Finance, Paris), 156 (5), 1–14 (2002).
The purpose of this paper is to test econometrically the existence of fiscal interactions between Belgian municipalities. At the time of writing, the motivation was to provide scientific support to the lively debate on fiscal competition that took place among Belgian politicians in the late nineties. Two types of taxes are considered, for which Belgian municipalities have the decision power as to rates: the ‘‘centimes additionnels’’ on the personal income tax and the ‘‘pre´compte immobilier’’ which is a property tax. A dynamic adjustment model is specified and estimated using panel data for 598 municipalities over 15 years. The empirical results obtained bear upon two main points: (i) Some interaction definitely has prevailed between the municipalities’ fiscal choices made during the observation period, for both taxes; (ii) However, the adjustment reactions to the other municipalities’ fiscal choices have occurred over time at the very low yearly pace of 6% and 10%, respectively, of the discrepancy between the actual rates and the preferred rates.
1.
Introduction and review of the literature
Tax interaction occurs when the taxation decisions taken by one independent authority are influenced by the taxation decisions of neighboring authorities. We see two possible sources of this type of interaction. The first is tax competition. In this case the transmission channel for tax interaction is the geographic mobility of tax bases: by using tax instruments, governments attempt to acquire a mobile tax base. Their interest in doing this is to increase their income (or not to decrease it), and to benefit from the advantages linked to the presence of these tax bases (more investment, more jobs, etc.). The second source of tax interaction is tax mimicry for political reasons. In this case, it is not the mobility of tax bases that forms the transmission channel, but the fear of the political
534
Public Goods, Environmental Externalities and Fiscal Competition
decision makers of being penalised by their electorate if their tax decisions are seen as being worse than their neighbors. These two sources of tax interaction are analysed from a theoretical point of view, according to two distinct streams of thought in the literature: tax competition and tax mimicry, sometimes called ‘‘yardstick competition’’. The objective of this article is to empirically test the existence of tax interactions in Belgium, at the local government level, i.e. the municipalities1. It is thus not about determining the channel through which these interactions are transmitted2. There is nascent empirical literature aimed at testing the presence of tax interactions. We have listed eight articles that attempt to answer the following question: ‘‘Do political decision makers take account of the decisions of their neighbors when making their tax choices, and if yes, to what extent?’’ All these articles, except one, are based on North American data. The majority examine the choices in terms of tax rates, and one of them examines the choices in terms of expenditure (Case, Rosen and Hines (1993)). Different types of taxes are considered: individual income tax (Heyndels and Vuchelen (1996), Goodspeed (1999), and Case (1993)), corporate income tax (Masayoshi and Boadway (2000), wealth tax (Brueckner and Saavedra (1998)), excises (Besley and Rosen (1998)) and the tax burden as a whole (Ladd (1992)). The empirical studies always include a reminder of the theoretical foundations of the phenomenon studied. They cover the two aforementioned streams: tax competition3, and yardstick competition. When the authors consider taxes on mobile bases (wealth tax or indirect taxes), their empirical study is generally preceded by a theoretical model of tax competition. When they look at taxes on less mobile bases (tax on income from labor), their empirical study is sometimes preceded by a theoretical model of the yardstick competition type. Irrespective of the basic theoretical model adopted to study the existence of tax interaction, the specification of the econometric model to perform the test is the same. In fact, the variable to be explained is the rate chosen by the entities considered, and the equation to be estimated is a linear equation containing the socio-economic variables 1 This exercise was done in order to shed scientific and empirical light on the debate concerning tax competition that took place in Belgium during the institutional negotiations of October 2000. The issue of increased fiscal autonomy of the Regions was a central one. A detailed account of these negotiations and their results is given in Van der Stichele and Verdonck (2001). 2 The reader interested in empirical studies bearing specifically on the mobility of tax bases should read the articles of Feld and Kirschga¨ssner (2000), Goodspeed (1999) as well as Heyndels and Vuchelen (1989). 3 For a recent review of the theoretical literature on tax competition see Wilson (1999).
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
535
of the entity concerned, as well as the rates of the neighboring entities. The major variant among the estimated equations concerns the method for weighting the rates of neighboring entities, in order to account for the fact that an entity can be more strongly influenced by certain entities and less so by others. In these articles, the presence of tax interaction is accepted if significant coefficients are obtained for the explanatory variables, which are the rates of neighboring entities. The studies conclude by noticing the existence of significant interaction between entities at the same level of government, with this interaction being positive in the vast majority of cases. The results indicate that the estimated coefficient for the rates of neighboring municipalities is of the order of 0.6 on average. Several authors end their studies with the following interpretation: ‘‘If all neighbors of entity i increase their tax rates by 1%, entity i will increase its tax rate by 0.6%’’. For the following reasons, we believe that this interpretation is incorrect. The previously mentioned studies analyse the problem statically, i.e. one year at a time. The data themselves often only relate to a single year, and if several years are considered, the analysis is done for each year separately. This means that the econometric results obtained implicitly assume that each municipality is able to effectively adopt its preferred rate of tax in the year of observation, given the rates of its neighbors. To us, this hypothesis seems to be too strong, and above all untenable if, as we have been able to do here, the analysis is done over a large number of consecutive years. In this context we cannot ignore varying adjustment times in the decision making process. This thus has to be taken into account when evaluating the actual tax interaction phenomenon. That is why we specify a dynamic model that explicitly takes adjustment time into account. The observed changes in the tax rates will then be interpreted as trends towards a rate that we call the municipality’s ‘‘preferred rate’’ - a rate that is subject to external influences such as the rates of the other municipalities, or even other economic variables that play a role in fiscal decisions. Thus, each rate change observed will not be interpreted as resulting from tax interactions, but as a movement towards the preferred rate. It is in the level of this rate that the presence of tax interactions is explicitly acknowledged as a possibility. That this presence can be detected in the facts is precisely what this econometric estimate aims to verify. The analysis is organised as follows. A model of the assumed tax behavior of each municipality is specified in Section 2. The data used for the empirical analysis are described in Section 3. The econometric method used is contained in Section 4. The results of the estimates and their
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Public Goods, Environmental Externalities and Fiscal Competition
econometric interpretation are presented in Section 5. Section 6 presents a summary and draws conclusions from the study.
2.
Specification of the economic model
The model assumed to describe the behavior of municipalities in the choice of their tax rates is set out in two stages. We start by defining the tax rate that we call the ‘‘preferred rate’’ of a local entity, and then describe an adjustment process over time towards this preferred rate. We present a model in terms of a single tax. When more than one type of tax is used, the model is assumed to apply to each tax separately. 2.1
The preferred rate in the absence of tax interaction
First, suppose that for each municipality i there is a tax rate Ti,t that it would apply in year t if it did not have to take any account of the other municipalities. It would choose this rate according to the economic conditions that we will represent by a vector Zi,t of miscellaneous variables, and also by considering the tax rate set by a different level of authority on the same basis TF ,t . This last element takes account of any vertical tax interaction between the municipalities and the other levels of government (regional or federal for Belgium). We summarise these conditions by specifying the equation Ti ¼ b0 Zi þ aTF
(1)
where the time index has been omitted because it is not essential at this stage, a is a scalar and b a vector of the same dimension as Zi . We will call Ti the ‘‘preferred rate of the municipality i in the absence of tax interaction’’4. However, during the period studied, the relevant tax rates of the other levels of government did not change or only changed once, which does not allow their impact to be measured. We will thus ignore them and replace (1) by Ti ¼ b0 Zi 2.2
(2)
The preferred rate in the presence of tax interaction
In order to take account of any tax interactions between municipalities, we amend the concept of preferred rate that has just been presented, by alternatively taking ‘‘the preferred tax rate of municipality i, in 4
It is not an optimum rate as we do not derive it from a model of optimisation. Such an exercise would certainly be desirable, but would be part of a different study.
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
537
year t, in the presence of tax interactions’’, denoting it as T~i,t and writing the equation that determines it in the form (again without the index t) T~i ¼ fii Ti þ
n X
fij Tj
j¼1 j6¼i
¼
fii (Ti
Ti ) þ
n X
(3) fij Tj
j¼1
with n X
fij ¼ 1
(4)
j¼1
where Tj is the actual rate of tax of the same type applied by municipality j, while fii and fij measure the influence on the tax decisions of municipality i of, respectively, its preferred rate without tax interaction, and the tax decisions of the other municipalities. The numerical value of fij represents the degree of ‘‘neighborliness’’ between i and j. We suppose that the influence of the one on the other increases in relation to this. Without any tax interaction from the other municipalities on municipality i, one has fij ¼ 0 for all j and fii ¼ 1, and thus T~i ¼ Ti On the other hand, with a maximum interaction, one has fii ¼ 0, f j6¼i ij ¼ 1 and thus X T~i ¼ fij Tj
P
j6¼i
Entity i then only chooses its rate on the basis of the influence that it is subject to from the neighboring entities. If all 589 municipalities are considered, and thus equation (3) as a system of 589 equations, the set of coefficients fij is a square 589 589 matrix that we will call the ‘‘external rate weighting matrix for the choice of preferred rates of municipalities’’, or more briefly ‘‘weighting matrix’’. Depending on the definition of the term ‘‘neighboring’’, we will have different matrices of weights P made up of the corresponding fij . The formal condition nj¼1 fij ¼ 1 expresses the desired rate as a convex combination of the preferred rate without competition and the rates of the neighboring entities5. 5 This condition is an identification constraint imposed by the fact that the rates Ti are not observed, and must be estimated or reconstructed from single observations. We note that the model does not change if Ti is multiplied by an arbitrary constant k and fii is
538
2.3
Public Goods, Environmental Externalities and Fiscal Competition
Dynamic adjustment towards the preferred rate
We will now denote Ti,t as the rate actually applied by municipality i in year t, thus the observed rate. It could be considered that this rate is the preferred rate, i.e. that Ti,t ¼ T~i,t and this for all t. But this hypothesis would be very optimistic regarding the speed and flexibility of tax decision making in the municipal institution - too optimistic probably. To us, it would seem more realistic to describe the behavior of decision makers over time by incorporating the possibility of an adjustment time between the applied rate and the preferred rate. To this end we will adopt an error correction model (ECM). This type of model has been used with considerable success in the theoretical and empirical literature on dynamic adjustments in the direction of longterm equilibrium solutions, in the presence of adjustment costs6. We use the standard notation D to represent first differences (for example, DTi,t ¼ Ti,t Ti,t1 ). In the context of our problem, a formulation of the MEC type is written as follows: DTi,t ¼ a þ b0 DZi,t v(Ti,t1 T~i,t1 )
(5)
The central coefficient of the ECM model is v, which represents the instantaneous fraction (speed) of adjustment, at a previous disequilibrium, between the actual rate and preferred rate. For any value of v, strictly between 0 and 1, this model converges to a long-term equilibrium in the (hypothetical) scenario of stationarity (DZi,t ¼ 0), or even of balanced growth (DZi,t ¼ gi , where gi designates a growth rate). It is more a theoretical property than an empirical property, as in practice one does not expect to observe a balanced growth of DZi,t . Of course, as the preferred rates T~i,t are not observed, equation (5) cannot be estimated as such. Equations (2) and (3) have to be substituted for T~i,t . This substitution produces the following equation: # " n X fij Tj,t1 þ «i,t DTi,t ¼ a þ b0 DZi,t v (1 þ fii )Ti,t1 fii b0 Zi,t1 j¼1 j6¼i
(6) divided by the same constant. This implies that the estimators of the coefficients of Ti are only defined up to a constant of proportionality. An identification constraint thus has to be imposed for each fii , or equivalently for each line of the matrix of fij ’s. fii ¼ 1 could for example have been imposed, but it seemed more natural to us to impose that the sum of each line of fij is equal to 1. 6 The interested reader here can consult the fundamental contributions of Sargan (1964), Hendry and Anderson (1977), Davidson, Hendry, Sbra and Yeo (1978), or Nickell (1985).
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
539
where ei,t represents the usual error term, assumed to average zero and to be uncorrelated with the regressors (the discrete character of the data implies truncation effects to be discussed in Section 4.4 below). 2.4
The weighting matrix
It thus remains for us to model the 589 589 ‘‘weighting’’ matrix f ¼ (fij ). To do this, we adopt a standard logistic formulation in which the logarithms of the fij ’s are assumed to be proportional to a linear combination (with weights d) of ‘‘socio-economic distance’’ variables dij (note that dii ¼ 0 for all i) and are then normalised to unity, i.e: fij ¼
exp ( d0 dij ) n P exp ( d0 dik )
(7)
k¼1
The P form of the denominator allows for the normalisation constraint nj¼1 fij ¼ 1 to be satisfied. After optimisation with respect to d, it is possible to calculate fij for each pair ij, and fii is then calculated using P the operation 1 nj¼1=j6¼i fij . The choice of the quantities dij is discussed in Section 3.2 below. Two comments apply to the specification of the fij ’s: 1. Given the relatively short observation period (15 years), we have limited ourselves to measures of distance which are constant over time (even if this means using sample averages over the observation period), which explains the absence of the index t for dij and fij . There is no conceptual obstacle to using measures of distance that vary in time, but in practice it comes down to calculating a 589 589 matrix per year. In our opinion, this is not necessary considering that the subsequent variations selected to calculate dij would only lead to small changes in fij over time. 2. In the course of the analysis, we examined the question of whether or not the fact of belonging to regions with the same or different languages has an impact on our proximity measurements. There are at least two ways of examining this question. The first consists of adding a dichotomic variable Iij to the list of dij , which takes the value 0 if i and j belong to the same linguistic region, and 1 in the opposite case. A richer model would consist of introducing different values for d according to the value of Iij , i.e. d1 for Iij ¼ 1 and d2 for Iij ¼ 0. In practice, this comes down to replacing d in equation (7) with: dij ¼ d1 Iij þ d2 (1 Iij ) It turned out that the differences between d1 and d2 were not significant, and this formulation was thus abandoned in favour of a single d common to all pairs of municipalities.
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Public Goods, Environmental Externalities and Fiscal Competition
To summarise this section, the model to be estimated is completely characterised by equations (6) and (7).
3.
Data and choice of variables
As already mentioned in Heyndels and Vuchelen (1996), Belgium has a series of characteristics that make it an ideal subject for an empirical study of this type. Firstly, the number of municipalities in Belgium gives us a large sample, as we can work with 589 municipalities each year. Secondly, for all the municipalities only two types of tax, namely the supplements on the individual income tax (hereafter, IPP supplements) and the supplements on the property tax (hereafter, PrI supplements), make up the bulk of municipal tax resources7. Thirdly, as these two taxes are additional (‘‘piggyback’’) to a federal tax and to a regional tax, respectively, the definition of the tax base is entirely uniform because in these two cases it is determined at a higher level. Consequently, we can compare the rates without having to adapt the bases. Finally, the internal borderlines that demarcate the different responsibilities, including responsibility for tax, of the entities in question are exactly the same. The municipalities are thus institutionally homogeneous. Note that these last three elements facilitate intermunicipal comparisons, both for the taxpayers and for the political decision makers. All the data used in the estimate are available for all municipalities for the years 1983 to 1997. The main sources of these data are the National Institute of Statistics, the Ministry of Finance, and the Department of Geography of the Catholic University of Louvain. We will now describe the data used to estimate equation (6), successively for the IPP supplements and for the PrI supplements. 3.1
The tax variables
The IPP supplements are set by each municipality as a percentage of the tax due by the taxpayer to the federal authority8. The supplement is expressed in ‘centimes’ where one centime is one percent of the tax due. The tax base for this federal tax is all the comprehensive income of the
7 Tax revenue represents 40% of the total municipal income, and the two taxes mentioned above constitute more than 80% of this tax revenue. The rest of the revenue consists of non-tax revenue and grants. For further information, see Flohimont (1999). The acronym IPP stands for Impoˆt des Personnes Physiques in Belgian tax terminology; similarly, PrI stands for Pre´compte Immobilier. 8 A consequence of this form of municipal supplement is that when the tax burden changes at the federal level, the municipal tax revenue changes in the same proportion.
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
541
taxpayers. The supplements vary from one municipality to the next, between 0 and 10%, with an average of 6.7% for the whole country. By contrast, the PrI supplements are expressed by each municipality as a number of hundredths of the PrI rate. If the PrI rate is 1.25%, the supplement, expressed in terms of ‘centimes’, is here 0:01 0:0125, a figure that is added to the PrI rate applied to the taxpayer. The tax base for the PrI is the imputed income from the property that the taxpayer owns, an income that is calculated by the (federal) tax department on the basis of objective criteria such as area, year of construction, etc. The municipal PrI supplements vary from 200 to 4000 from one municipality to the next. The PrI rate itself varies according to the region (because it is a regional tax competence): 1.25% in Wallonia and Brussels, and 2.5% in Flanders. 200 centimes thus do not correspond to the same actual tax rate, as it depends on the region where they apply. Consequently, and in order to express the municipal PrI supplements in comparable units across the whole country, it was necessary to convert the figures of these centimes. The rate of municipal property tax, thus reexpressed as a percentage of the tax base (the imputed income), varies between 2.5% and 50%, with an average of 24.65%. 3.2
The non-tax explanatory variables
The introduction of non-tax explanatory variables enables us to distinguish, between the similarities in rates between municipalities, those which are due to socio-economic characteristics from those due to tax interactions. In order to choose explanatory socio-economic variables, we will use previous studies and make the hypothesis that the set of variables to be used9 is not changed by the dynamic aspect introduced here. In the first set of estimates, we introduced nine explanatory variables, identical for the IPP estimates and the PrI estimates. The estimates then enable us to select the variables which are most relevant for each tax. The first socio-economic variable taken into account is the municipal population. In 1997, the number of residents per municipality varied from 85 in Herstappe to 453,030 in Antwerp, with an average of 18,000 in Belgium. The second variable is the population density. The most densely populated municipality is in the Brussels region, with 191.8 residents per hectare is Saint-Josse and the least densely populated municipality is Le´glise, with a density of 0.209. The Belgian average is 6.7. We interpret 9
Of course this choice of socio-economic variables is also dictated by the availability of data.
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Public Goods, Environmental Externalities and Fiscal Competition
this measure as an approximation of the level of urbanisation of the municipality. The third variable is income per resident in each municipality. In 1983, the poorest municipality, Bertogne, had an annual taxable income per resident of BEF 137,100 (3,999 euros) and the richest was Kraainem with BEF 335,500 (8,317 euros). In 1998, the gap had widened, with the minimum being BEF 192,000 (4,762 euros) per resident in Saint-Josse, and the maximum being BEF 593,200 (14,705 euros) in Lathem-SaintMartin. The national average was BEF 394,800 (9,787 euros). Three regional dichotomic variables have also been added to observe whether the rates vary significantly from one region to the next, for reasons other than changes in the socio-economic characteristics included in the model. For the weights of neighboring municipalities, four types of ‘‘distances’’ have been introduced: the first is the geographic distance from center to center, the second is the income per resident differential between two municipalities i and j, the third is the population density differential between i and j, and finally the last is the population differential. The choice of these variables for the weights of the municipalities results from the hypothesis that if there is tax mimicry or tax competition, it is because there is, at a given point in time, a comparison of the rates by the political decision makers or by the taxpayers. This comparison is assumed to be more direct, not only between geographically close municipalities, but also between municipalities with similar characteristics in terms of level of urbanization, size of population, or income classes. Table 1 below gives the units in which the explained and explanatory variables are expressed.
Table 1.
Variables, symbols and units
Variable Rate of PrI supplement Rate of IPP supplement Distance from center to center Population density Population Income per resident Constant Flanders Wallonia Brussels Note: BEF 1,000 equals 24.79 euros.
Symbol
Unit
PrI IPP DIS DEN POP REV CST FLA WAL BXL
50 centimes 1/2 % 100km 100 residents/km2 100,000 residents BEF 1,000 1 1 1 1
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
4.
Econometric method
4.1
Estimation technique
543
The model defined by equations (6) and (7) is nonlinear in the parameters n, b and d. The nonlinearity in (n, b) is eliminated by reparametrising (6) as follows): ! n X DTi,t ¼ a þ b0 DZi,t þ c0 fii Zi,t1 n (1 þ fii )Ti,t1 fij Tj ,t1 þ «i, t j¼i
(8) where c ¼ n:b. We thus have a linear expression in coefficients (a, b, c, n), conditionally to d and thus to fij . Taking this property of the model into consideration will enable us to considerably simplify the calculation of nonlinear10 least squares estimators for all parameters of the model, including d. Indeed, for any given value of d, it is sufficient to calculate the fij ’s corresponding to it, to substitute them in equation (8), and to estimate the remaining coefficients by ordinary least squares. We then search for the numerical values of d that minimize the sum of the squares of the (estimated) residues of equation (8). This numerical optimisation programme is written in FORTRAN 77 and uses a very powerful optimisation subroutine based on the simplex method and adapted from Press et al (1966). A complete optimisation calculation takes the order of 25 Seconds on a SUNFIRE 3800 workstation. Before presenting in Section 5 the results of the estimate, we have to discuss in the following subsections three essential complementary aspects of the least squares analysis: in subsection 4.2, the statistical calibration of the estimators thus obtained in the form of standard deviations (or of confidence intervals); in subsection 4.3, a selection procedure for the explanatory variables, in order to eliminate the insignificant variables; and in subsection 4.4, the effects of truncation. 4.2
Confidence intervals
The uncertainty inherent in the random nature of the observed sample has to be suitably characterised. This is generally done in the 10 Under the hypothesis of normality of the residues ei,t , the method of nonlinear least squares coincides with the method of maximum likelihood. As we discuss below, the hypothesis of normality of residues is clearly not satisfied in this application.
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Public Goods, Environmental Externalities and Fiscal Competition
form of standard deviations that measure the statistical dispersion of the estimators. Given the nonlinearity of the model, there is no analytical expression for these standard deviations. It is certainly possible to calculate approximations of them, called large sample ones (or asymptotic), but this is at the cost of calculations that can turn out to be awkward. In this case, we have preferred to use simulation techniques, which with a minimum programming effort enable us to obtain standard deviations corresponding to the actual size of the samples used. This technique essentially consists of assigning their estimated values to the model parameters and then taking random samples of the error terms «i,t , and by substitution in equation (6) then deriving fictitious samples of variations of the tax rates (or more specifically, given the dynamic nature of the model, fictitious trajectories of tax rates over the 15 years of our sample). Of course the explanatory variables Zi,t , which are not part of the modeling and are exogenous, are kept at their observed value. The estimation technique described above was applied to each fictitious sample thus obtained (of the size 589 14 ¼ 8246). For each specification of the model we repeated this simulation procedure 1000 times, and thus obtained 1000 random realizations of the estimation process. Finally, the standard deviations of these 1000 realizations were calculated, thereby obtaining a reliable statistical measure of the statistical dispersion of the estimators (also verifying that the empirical averages of the simulated estimators do not significantly differ from the values selected for the simulation, i.e. the estimators are not significantly biased). The confidence intervals for the estimated coefficients are thus defined as half-length intervals corresponding to two standard deviations, centered on the estimated value of the coefficient. The validation of this simulation technique of course depends on hypotheses relating to the distribution of the error terms «i,t , as it is from this distribution that the simulated errors will be drawn. An examination of the estimated residues of the model leads us to reject the standard hypothesis of normality of residues. More specifically, we observe a strong concentration (80 to 90%) of low residues, accompanied by a small number of often more dispersed residues. Under these conditions, simulations under a hypothesis of normality would lead to excessive concentrations of small simulated unknowns and consequently an excessive ‘‘confidence’’ in the estimated values of the coefficients. Under these conditions we used a ‘‘bootstrap’’ simulation method, which consists of sampling the random variables "it in their empirical distribution, as obtained using estimators of the coefficients. More precisely, an estimate of the model would produce 8246 estimated residues, which are each allocated a probability of 1/8246 for the simulation procedure (selection with replacement).
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
4.3
545
Selection of the explanatory variables
We initially have a total of 12 explanatory variables: three for the short term (the DZi,t ’s), five for the long term (the Zi,t1 ’s, complemented by two indicator variables for Flanders and Wallonia - which given the constant term a, amount to introducing a specific constant for each of the three regions), and four variables for the weights fij (the four types of ‘‘distance’’). In order to simplify the statistical and economic interpretation of the results, it is generally recommended eliminating variables from the model that do not significantly contribute. In order to do this we start with the most general possible model by including all the available explanatory variables, and then gradually eliminate the least significant ones until a model is obtained where all the coefficients are statistically significant. This elimination procedure is of course done separately for the PrI and IPP, as it can be expected that the variables selected will be different in the two cases. The elimination criterion is the usual criterion of the ‘‘t statistic’’ (ratio between the estimator and its standard deviation) being less than two. This means that all the results presented below are significant in the sense of the t statistic. Finally, note that to accelerate the interactive selection procedure of variables, it is legitimate to use the least squares standard deviations rather than those obtained from simulation. The first are lower than the second as, in particular, they do not take account of the uncertainty of the d’s and thus of the fij ’s. It then follows that every variable eliminated on the basis of least squares standard deviations would also be eliminated on the basis of standard deviations from the simulation. Of course, with regard to the dij , it is appropriate to use the simulation standard deviations, which are the only ones available, or, as a first approximation, the R2 statistic associated with the least squares calculation. 4.4
Effects of truncation
A detailed examination of the data indicates that municipal tax rates are not continuous variables, but are often rounded to specific fractions of points or to whole numbers of additional centimes. The clearest demonstration of this phenomenon comes from the high percentages of zero changes (82% of observations for the PrI supplements, and 87% for the IPP supplements). More generally, the large majority of changes to the IPP rates are multiples of 1/2% (between the extremes of 6% and þ6%). Similarly, the majority of changes to the PrI are in multiples of 50 additional centimes (between the extremes of 1120 and þ 2120, both not multiples of 50, and thus exceptional
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Public Goods, Environmental Externalities and Fiscal Competition
cases). This implicit discretisation raises questions of interpretation and modeling. The most immediate question is whether it is liable to markedly affect the estimates of the model represented by equation (6), in particular in the form of a downward bias in the dynamic adjustment coefficient v (note that 100% zero adjustments would be perfectly explained by all coefficients in this equation being zero). The simulation techniques described above answer this question. It is sufficient to do parallel simulations where the simulated tax rates are themselves rounded in order to be representative of the sample observed, i.e. in multiples of 1/2% for the IPP supplements and 50 additional centimes for the PrI supplements. In particular, we did truncated simulations of the two final versions (PrI and IPP) of our model, as presented below. Without anticipating the detailed analysis of the results, we will see that the dynamic adjustment coefficient v is biased downwards as a result of the truncation (and more particularly, the high number of zero changes), but in a way that does not change the essence of our qualitative conclusions relating to the slowness of the dynamic adjustments towards the long-term equilibrium rates.
5.
The results and their econometric interpretation
For the PrI supplements and for the IPP supplements, three categories of results have to be considered. The first is made up of estimates of the parameters fij , after optimisation of the d coefficients. The value of these parameters determines the importance that a municipality attaches to its preferred rate without tax interaction Ti,t , as well as to the rates of other municipalities, in the determination of its preferred rate in the presence of tax interactions T~i,t , and thus also in the choice of its actual rate of taxation Ti,t . In addition, these coefficients identify what criteria determine the weight attached by one municipality to the rate of another. The second category of results relates to the estimate of the fraction v of the difference between the rate actually applied Ti,t , and the preferred rate with tax interaction T~i,t , which is corrected in each period. Finally, the third category of results relates to the coefficients b and b of the non-tax explanatory variables, with b indicating the effect of these variables on the actual annual variations, and b measuring their impact on the long-term preferred rate T~i,t . 5.1
Results for the property tax (PrI supplements)
Table 2 presents the values of the estimated coefficients in the equation for the property tax. It only takes account of the significant variables. The explanatory variables cited in Section 3.2 that do not
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997 Table 2.
547
Estimators for the property tax
Coefficient
Variable
a b c ( ¼ vb)
POP REV REV
def
b v d1 d2
DIS DDEN
Estimatorb
t statistica
0:2345D þ 00 0:2949D þ 02 0:2467D þ 02 0:4208D þ 03 0:6127D þ 01 0:1026D þ 02 0:5361D þ 01
2.962 2.396 6.310 6.850 16.67 7.888 5.449
R2 non-linear regressionc: 0.036 a The (simulation) t statistic is defined as the absolute value of the quotient between the simulation average and its standard deviation. b The floating decimal notation of FORTRAN guarantees 4 significant decimals for all estimators. c This is a gross R2 (not corrected for truncation effects).
appear in the table do not improve the quality of the estimate. We have thus eliminated them. Below, we rewrite P equation (6) with the estimated coefficients. Because the fii ’s and nj¼1=j6¼i fij ’s are specific to each municipality, we cannot give them a value in this general equation. That is why, for indicative purposes, we write it with the estimated coefficients for a given municipality, Ixelles, that we denote as xl: DPr Ixl ,t ¼ 0,2345 29,49:DPOPxl ,t 0,06127 " (1 þ fxl ,xl ) Pr Ixl ,t1 fxl ,xl (402,8REVxl ,t1 )
n X
# fxlj Pr Ij,t1 þ «xl ,t
j¼1
(9) where fxlj ¼
exp ( 10,26DISxlj 5,361DDENxlj )
n P
exp ( 10,26DISxlk 5,361DDENxlk )
k¼1
fxlxl ¼ 1
n X
(10) fxlj ¼ 0,254
j¼1 j6¼xl
Before going into a systematic examination of the coefficients, an important comment concerning the R2 of this estimate is in order. It shows that the predictive quality of the model is particularly low, of the order of 3.6%. This is due to several factors. First of all, panel data usually generate a low R2 . Secondly, the data are discrete while the model has been specified for continuous variables. Thirdly, first differences are used. Finally the list of variables Zi taken into consideration is clearly not exhaustive - it thus has to be completed.
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Public Goods, Environmental Externalities and Fiscal Competition
a) The value of the fij ’s For the property tax, the optimisation of the value of the fij ’s through the coefficients d, shows us that the geographic distance (DIS) and the population density differential (DDEN) have a significant impact on the weight attached to the rate of another municipality when choosing its own rate. This means that when deciding its supplement to the property tax rate, a municipality i takes account of the tax rate supplements of the municipalities that (18) are not too far away in physical distance and (28) are relatively comparable in terms of population density. In other words, the importance that municipality i attaches to the rates applied in j is inversely proportional to these two types of distance. The significant nature of these two coefficients shows the importance of going a step further than the studies mentioned in the review of the literature, and of not being limited to the single criterion of geographic distance. Geographic distance plays a certain role, but it is not the only one. Thanks to these coefficients, it is possible to calculate the fii of each municipality, as well as the fij of each pair of municipalities. The fii ’s vary from 0.06 to 0.70 depending on the municipality (with an average of 0.122). In the long term, 6 to 70% of the actual rate of property tax of a municipality depends on its preferred rate without tax competition, and 30 to 94% on the rate of the closest municipalities in distance and in population density. For municipality xl in particular (see equation (10)), we interpret the results in the following way: 25.4% of the preferred long term rate in the presence of tax interaction T~xl ,t depends on its preferred rate in the absence of tax interaction T~xl ,t (and thus on its specific socioeconomic characteristics), and 74.6% on the rate of neighboring municipalities. An examination of the individual fii ’s shows that the fii ’s of the municipalities of the Brussels region are relatively large on average (the Brussels average is 0.282): compared with the Flemish and Walloon municipalities, the municipalities of the capital are less influenced by their neighbors and they attach relatively more importance to their preferred rate without tax interaction. The reason for this specific Brussels characteristic could come from important externalities between these municipalities and the peripheral municipalities in terms of public goods and services. In fact a quarter of the daytime users of Brussels infrastructure are commuters from the suburbs11. Another specific aspect of the Brussels region that could explain this difference in behavior is the very dense structure as well as the value of its real estate. 11
See Lambert, Tulkens et al. (1999)
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
549
The specific character of the Brussels fii ’s reinforces our idea that an increase in the number of explanatory variables, and in particular those concerning the supply of public goods and services as well as the value of the real estate, would markedly improve the predictive value of our model. The estimated coefficients of equation (10) show the following: 1. With equal population density differentials (¼DDEN) and with distances of, say, 20 km between i and j, and 50 km between i and h, the weight attached by i to j is 21.7 times larger than the weight attached by i to h. 2. With equal distances, and with a population density differential between i and j equal to 10 residents=km2 and a similar differential between i and h equal to 200 residents=km2 , the weight attached by i to j is 124 times greater than the weight attached by i to h. Remember that the sum of the fij ’s is equal to 1 and that there are 589 fij ’s. Each one is thus very small, and the weight attached to the municipalities further away, in distance or in terms of population density differential, rapidly tends towards 0. b) The value of the adjustment coefficient v The estimated adjustment coefficient is 0.0612 and is highly significant. This value means that when in municipality i a difference is observed at time t between its actual rate Ti,t-1 and its preferred rate T~i,t1 , this municipality adjusts its actual rate at time t by 2.12% of the observed difference. The dynamic adjustment is thus very slow. One should also keep in mind that the preferred rate of the municipality T~i,t constantly varies in relation to changes in the socio-economic characteristics of municipality i and the rates of neighboring municipalities. It is also important to note that the specification of the model implies that this adjustment coefficient is an average of the adjustments over time. In reality, it is generally observed that municipalities keep their rate constant for a few years, and then make a change larger than 6.12%. Such behavior could be attributed to the fixed costs (administrative and political) involved in a change of tax. On the basis of these explanations, we can deepen the interpretation of equation (9) specific to municipality xl in the following way. Let us make the hypothesis that all socio-economic characteristics of municipality xl are constant over time (the preferred rate in the absence of tax interactions is consequently fixed) and that the number of centimes in the PrI supplements of all other municipalities is increased by 100 at time t 1. Then, the preferred long term equilibrium supplement of xl increases by 74.6 centimes, as the equation implies that 74.6% of it depends on what happens in neighboring municipalities. As to the actual
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Public Goods, Environmental Externalities and Fiscal Competition
supplements, on average, municipality xl will make at time t an adjustment of 6.12% towards this new equilibrium preferred rate, i.e. it will increase its PrI supplement by 4.55 centimes (¼ 6:12% 74:6). c) The value of the coefficients of the explanatory variables The socio-economic variables with significant coefficients in the equation for the PrI supplements are the income per resident (REV) and the number of residents (POP). However, these two variables act in different ways: the population variations influence the short-term decisions through the vector of coefficients b, while the income per resident determines, through the vector of coefficients b, the level of the long-term preferred rate T~i, t and the (very) gradual adjustment towards it. We will not dwell on the interpretation of the short-term coefficient b, as it is not a key parameter in our model. Moreover, as we have said, the predictive value of the model is low and makes the interpretation of short-term movements awkward. Instead, let us look at the coefficient b determining the longterm objective. If, for a municipality, the annual income per resident reaches, from time t on, a permanent level of 5,000 francs higher than before, its preferred rate T~i, t is changed and increases by fii 5 0:4026, all other things remaining equal. In the specific case of municipality xl this rate T~x, t increases by 0:254 5 0:4026 ¼ 0:51. In terms of additional centimes, this means that the preferred long-term rate increases by 25.5 (as the T ’s are expressed in units of 50 centimes for the PrI supplements). Given the estimated adjustment coefficient, the model predicts that the municipality will adapt its actual rate towards this new preferred rate, by 25:5 0:06 ¼ 1:53 centimes per year. The sign of this estimated coefficient can be seen as a confirmation of the idea that the demand for public goods and services increases with income, and that the greater supply of these goods and services requires additional financial resources, which the municipality can obtain through a higher tax burden.
5.2
Results for personal income tax (IPP supplements)
Table 3 shows values of the coefficients estimated in the equation for the IPP suppements. Just like for the PrI supplements, only significant variables are retained. The same comment as in the previous section thus applies to R2 , which, although it has now a double value, is nevertheless very low. The estimated coefficients for the IPP yield the following equation:
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997 Table 3.
551
Estimators for individual income tax Estimatorb
t statistica
DEN REV DEN
0.7836D01 0.8068Dþ01 0.4391D02 0.9495Dþ00
3.891 2.691 4.390 7.020
c2 ( ¼ nb2 )
POP
0.1.12Dþ01
4.285
c3 ( ¼ nb3 ) def b1 b2 b3 n d1 d2
WAL DEN POP WAL
0.1217Dþ01 0.9000Dþ01 0.9595Dþ01 0.1153Dþ02 0.1055Dþ00 0.6630Dþ01 0.5268D01
11.50 7.072 3.923 12.17 23.44 7.688 5.247
Coefficient
Variable
a b1 b2 c1 ( ¼ nb1 ) def def
DIS DREV
R2 nonlinear regressionc: 0.0637 a, b, c, see table 2.
DIPPi, t ¼ 0,07836 8,068DDENi, t þ 0,004391 DREVi, t 0,1055(IPPi, t1 fii (9,000DENi, t1 þ 9,592 POPi, t1 þ 11,53 WALi ) n X fij IPPj, t1 ) þ «i, t
(11)
j¼1
where fij ¼
exp ( 6,630DISij 52,68DREVij ) n P exp ( 6,630DISik 52,68DREVik )
(12)
k¼1
a) The value of the fij ’s For the IPP supplements, the estimate of d shows that it is the rates of the geographically close municipalities and those similar in terms of income per resident that influence most the municipal choices. As with the property tax, we see that it was useful to analyse in more detail the criteria to be considered in the definition of ‘‘neighborliness’’: indeed, while we observe that the geographic distance plays the same role as shown by previous studies, we see here too that it is not the only one. But our analysis shows that the additional criterion to be considered differs according to the tax studied: income per resident for the IPP supplements and population density for the PrI supplements. For the IPP, the fii ’s vary from 0.063 to 0.843, with an average of 0.140. In this case, just as with the PrI, the average over the municipalities
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Public Goods, Environmental Externalities and Fiscal Competition
of the Brussels region average is higher (0.173) than in the other regions but less markedly so. The estimated coefficients of equation (12) show the following: 1. With equal income differentials (¼DREV), and with distances of 20 km between i and j and 50 km between i and h, the weight attached by i to j is 7.3 times larger than the weight attached by i to h. 2. At equal distances, with an income differential between i and j equal to 20,000 BEF/resident, and the differential between i and h equal to 100,000 BEF/resident, the weight attached by i to j is 67 times greater than the weight attached by i to h. b) The value of the adjustment coefficient n With the IPP supplements, the adjustment coefficient is 0.1055. When there is a difference between its actual rate and its long-term equilibrium rate, municipality i adjusts its actual rate, annually and on average, by 10% of this difference. Just as with the adjustment coefficient for the PrI supplements, the coefficient for the IPP supplements is highly significant. It is interesting to note that the speed of adjustment is greater with IPP than with PrI supplements. Two interpretations are possible. On the one hand, it can be attributed to the fact that the tax base for the IPP, which is the income of the taxpayer12, is more mobile than with the PrI, the registered income of the building13. On the other hand, we can see there an indication of the fact that tax interaction in IPP supplements and tax interaction in PrI supplements do not occur through the same transmission channels, and thus do not give rise to the same rates of adjustment. c) The value of the coefficients of the explanatory variables A distinction again has to be made between the socio-economic characteristics that influence the short-term tax rate and those that influence the long-term preferred rate. Among the variables with a shortterm effect, are the population density and the income per resident. Among the variables with an effect on the long-term equilibrium rate, are the population density and the number of residents, and being in Wallonia. For the same reasons as those given for the PrI supplements, we will not look at the short-term aspect. However, we do have to make one comment concerning the explanatory variable ‘‘population density’’, which has both a short-term and long-term effect with coefficients of opposite signs. When the density increases, the model predicts that the
12 13
The IPP is located in the municipality where the taxpayer lives. The PrI is located in the municipality where the building is located.
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
553
IPP rate adjusts downwards in the short-term. In the long-term however, the pressure on the equilibrium rate is upwards. The population density thus plays an ambiguous role on public needs and goods: a higher density can have a downward impact on taxation because of economies of scale in the supply of services, but a high density can also lead to needs linked to congestion, security, etc. Among the socio-economic variables with an impact on the preferred long-term rate, three are significant: the population density, the number of residents and the regional dichotomic variable for Wallonia. When the first two incur a permanent level change, the preferred longterm rate is changed. Thus, with all other things remaining equal, when DENi increases by 100 resident/km2 , T~i, t increases by fii 1 9. In the case of municipality xl, which has a fxl , xl of 0.38, T~i, t increases by 3.5. This means that if its preferred rate was 6%, the new preferred rate after an increase in the population density goes to 6 þ 1:7 ¼ 7.7% (note: the units for the IPP supplements are 1/2%, which is why T~i has to be divided by 2 for the interpretation). Note, it is the new preferred long-term rate; in fact, the municipality will only adjust annually at a rate of 10%, as the estimated n is 0.1055. On the other hand, with all other things equal, if POPi increases by 1000 residents, then T~i, t increases by fi, i 0:01 9:59. If municipality i has an average fi, i , i.e. 0.38, the T~i, t increases by 0.038. This means that if the preferred rate was 6%, the new preferred rate after the population increase goes to 6.019% (same comment as above). Again, the municipality will only adjust to this new long-term rate at a pace of 10% per year. For the interpretation of the regional dichotomic variable coefficient, remember that it is the difference between a Walloon municipality and a municipality of another region. We can then say that on average, if municipality i is Walloon, and has an average fi, i , i.e. 0.14, then its preferred long term rate is (0:14 1 11:53)=2 ¼ 0:807 higher than the rate of a non-Walloon municipality that has the same fi, i . The introduction of regional dichotomic variables thus shows that the fact of belonging to the Walloon region systematically implies higher rates than the other regions, with all other things being equal. It could be that DEN and POP are lower on average for the Walloon municipalities than for the Flemish and Brussels municipalities. In this case, the effect of WAL could only serve to counterbalance these differences. This result could also indicate that other characteristics specific to the Walloon region have not been taken into account in our explanatory variables, such as specific needs of these municipalities relating to social protection, for example.
554
6.
Public Goods, Environmental Externalities and Fiscal Competition
Summary and conclusions
The objective of this article was to econometrically test the existence of tax interactions between Belgian municipalities. Two types of tax that the municipalities have control over have been considered: supplements to personal income tax (IPP) and supplements to the property tax (PrI). A dynamic model has been specified and estimated from panel data of 589 Belgian municipalities over a 15 year period. The results obtained essentially relate to two points. On the one hand, they show evidence of interaction between municipalities in the choices of the rates of the two taxes during the period studied. This interaction is all the more important when the municipalities are close and similar (a new finding from this study). With regard to IPP supplements, it is the similarity in terms of income per resident that matters, whereas for the PrI supplements it is the similarity in terms of population density that counts. On the other hand, thanks to the dynamic specification introduced here, it has been possible to see that adjustments in response to the rates of other municipalities only occur slowly. For the property tax supplements, the difference between the applied rate and the desired rate due to interactions is reduced by only 6% per year. Thus 15 years is required before the preferred rate is reached, which in the meantime is liable to have changed. For the IPP supplements, this gap is reduced at 10% per year. It can be concluded that while tax interaction between municipalities - the only one observable in Belgium - is not absent in this country with IPP and PrI supplements, its importance is low, arguably too low to constitute a clear and major obstacle to the implementation of tax decentralisation policies in general. In this perspective, it should be pointed out that the model is open to other uses, in particular for the purpose of simulating coordinated municipal tax policies. It could be a major extension of the work presented here, that our methodological results be used to prepare cooperation decisions aimed at tax harmonisation between the institutions involved, a perspective that seems to be sorely missing from the current debates. Finally, the specification of the model could be improved, mainly by better modeling the ‘‘discrete’’ (time) nature of the tax decision process, and also by using a simultaneous estimate of the equations that model the two taxes. These extensions would increase the explanatory value of the model.
Ch.22 Tax Interaction Dynamics among Belgian Municipalities 1984–1997
555
References Ashworth, J. and Heyndels, B., 1997. Politicians preferences on local tax rates : an empirical analysis, European Journal of Political Economy, 13(3), 479–502. Besley, T. and Case, A., 1990. Incumbent behavior: vote seeking, tax setting and yardstick competition, American Economic Review, 85, 1, 25–45. Besley, T. and Rosen, H., 1998. Vertical externalities in tax setting : evidence from gasoline and cigarettes, Journal of Public Economics, 70(3), 383–398. Boadway, R. and Hayashi, M., 2000. An empirical analysis of intergovernmental tax interaction: the case of business income taxes in Canada. Ontario, Canada: Queen’s University, mimeo. Brueckner, J.K., 1998. Testing for strategic interaction among local governments: the case of growth controls, Journal of Urban Economics, 44, 438–467. Brueckner, J.K. and Saavedra, L.A., 1998. Do local governments engage in Strategic Tax Competition? Chicago: Department of Economics, University of Illinois at UrbanaChampaign, mimeo. Case, A., 1993. Interstate tax competition after TRA86, Journal of policy analysis and management, 12(1), 136–148. Case, A., Rosen, H.S. and Hines, J.R., 1993. Budget spillovers and fiscal policy interdependence: evidence from the states, Journal of Public Economics, 52, 285–307. Davidson, J.E.H., Hendry, D.F., Sbra, F. and Yeo, S., 1978. Econometric modelling of the aggregate time-series relationship between consumer’s expenditure and income in the United Kingdom, Economic Journal, 88, 661–692. Feld, L. and Kirschgassner, G., 2000. Income tax competition at the state and local level in Switzerland. Munich: CESifo Working Paper Series, 238. Flohimont, O., 1999. Les finances des pouvoirs locaux sous l’angle re´gional, Bulletin du Cre´dit Communal (Bruxelles), n8 210, 1999/4. Genser, B. and Weck-Hannemann, H., 1993. Fuel taxation in EC countries: a politicoeconomy approach. Konstanz, Germany: University of Konstanz, mimeo. Goodspeed, T. J., 1998. Tax competition, benefit taxes and fiscal federalism, National Tax Journal, LI(3), 579–586. Goodspeed, J. T., October 1998. Tax structure in a federation. New York: Hunter College, mimeo. Goodspeed, J. T., 1999. Tax competition and tax structure in open federal economies: evidence from OECD countries with implications for European Union. New York: Hunter College, mimeo. Hendry, D.F. and Anderson, G.J., 1977. Testing dynamic specification in small simultaneous systems: an application to a model of building society behaviour in the United Kingdom, in M.D. Intriligator (ed.). Frontiers in quantitative economics. Amsterdam: North Holland, vol. 3. 361–383. Hendry, D.F. and Von Ungern-Sternberg, T., 1981. Liquidity and inflation effects on consumers expenditures, in A.S. Deaton (ed.). Essays in the theory and measurement of consumers behaviour. Cambridge: Cambridge University Press, chapter 9. Heyndels, B., 1990. De financiering van de gewesten via een aanvullende personenbelasting en de betekenis hiervan voor de Belgische gemeenten, Cahiers E´conomiques de Bruxelles, 127(3), 307–336. Heyndels, B. and Smolders, C., 1994. Fiscal illusion at the local level: empirical results for the Flemish municipalities, Public Choice, 80, 325 – 338. Heyndels, B. and Vuchelen, J., 1989. Gemeentenbelastingen en belastbaar inkomen, Tijdschrift voor Economie en Management, XXXIV(1), 57–76. Heyndels, B. and Vuchelen, J., 1998. Tax mimicking among Belgian municipalities, National Tax Journal, LI(1), 89–101.
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Hines, J.R., 1993. Budget spillovers and fiscal policy interdependence: evidence from the states, Journal of Public Economics, 52, 285–307. Hines, J., 1999. Lessons from behavioral responses to international taxation, National Tax Journal, 52(2), 305–322. Hugounenq, R., Le Cacheux, J. et Madies, Th.,1999. Les risques de la concurrence fiscale en Europe, Proble`mes e´conomiques, 15 de´cembre 1999. Ladd, H. F., 1992. Mimicking of local tax burdens among neighboring countries, Public Finance Quarterly, 20(4), 450–467. Lambert, J.-P., Tulkens, H. et al., 1999. Les modes alternatifs de financement de Bruxelles, rapport de recherche re´alise´e a` la demande de Monsieur GRIJP, Ministre de la recherche scientifique de la Re´gion de Bruxelles-Capitales, Faculte´s Universitaires Saint-Louis, Bruxelles. Maddala, G.S., (ed.), 1993. The econometrics of panel data, 2 volumes. The international library of critical writings in econometrics. Cheltenham: Edward Elgar. Nickell, S.J., 1985. Error correction, partial adjustment and all that: an expository note, Oxford bulletin of economics and statistics, 47, 119–130. Phillips, A.W., 1954. Stabilization policy in a closed economy, Economic Journal, 64, 290– 323. Phillips, A.W., 1957. Stabilization policy and the time form of lagged responses, Economic Journal, 67, 265–277. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T., 1986. Numerical recipes: the art of scientific computing. Cambridge: Cambridge University Press. Sargan, J. D., 1964. Wages and prices in the United Kingdom: a study in econometric methodology (with discussion), in P.E. Hart, G. Mills J.K. and Whitaker (eds.). Econometric analysis for national economic planning. London: Butterworth, vol. 16 of Colston Papers, 25–63. Van Der Stichele, G. and Verdonck, M., 2001. Les modifications de la loi speciale de financement dans l’accord du Lambermont, Courrier hebdomadaire du CRISP (Bruxelles), n81733. Wilson, J. D., 1999. Theories of tax competition, National Tax Journal, LII(2), 269–304.
BIBLIOGRAPHY OF HENRY TULKENS (THROUGH 2004)
A. Books B. Articles, Reports and Other I. Resource Allocation Processes II. Environment III. Taxation and Federalism IV. Efficiency Analysis V. Further work on Public Utilities, Public Goods and Cost Benefit Analysis
558
A.
Public Goods, Environmental Externalities and Fiscal Competition
Books
a) as author or co-author 1. Programming Analysis of the Postal Service: A Study in Public Enterprise Economics, Collection de la Faculte´ des Sciences e´conomiques, sociales et politiques n8 34, Louvain: Librairie Universitaire, 1968 (316 pp.). 2. Fondements d’Economie Politique, 1st edition, La Renaissance du Livre, Bruxelles, 1970 (250 pp.); 2nd edition 1986, De Boeck Universite´ and Editions Universitaires, Bruxelles-Paris (420 pp.); 3d edition 2001, same publisher, 530 pp.). (with Alexis Jacquemin and Philippe Mercier). 3. Public Expenditure Management in Small Developing Economies, Pacific Technical Assistance Centre, IMF and the University of the South Pacific, Suva, Fiji, 2004 (with Anne Morant and Luc Leruth).
b) as editor or co-editor 4. Economie Publique, French edition of the proceedings of the Joint Conference held by the International Economic Association and the Centre National de la Recherche Scientifique (C.N.R.S.) at Biarritz, France, September 1966, published by C.N.R.S., Paris, 1968 (599 pp.) (with Henri Guitton). 5. Finances Publiques et Re´gionalisation, Proceedings of the Conference held at LouvainLa-Neuve by the Institut Belge de Finances Publiques, Cabay, Louvain-La-Neuve, 1982 (252 pp.) 6. The Performance of Public Enterprises Concepts and Measurement, North-Holland Publ. Co., Amsterdam, 1984 (296 pp) (with Maurice Marchand and Pierre Pestieau). 7. La performance e´conomique des socie´te´s de transports urbains, Centre international de recherche et d’information sur l’e´conomie publique, sociale et coope´rative (C.I.R.I.E.C.), Lie`ge, 1988 (371 pp.) (with Bernard Thiry). This book has obtained the ‘‘Prix Ferdinand de Lesseps’’, awarded by the Fe´de´ration Franc¸aise des Travaux Publics, on the 24th of April, 1990. 8. Efficacite´ et Management, Proceedings of the 5th Commission of the Huitie`me Congre`s des Economistes Belges de Langue franc¸aise, Centre Interuniversitaire de Formation permanente (CIFOP), Charleroi, Belgium,1989 (293 pp.). 9. Contributions to Operations Research and Economics: the Twentieth Anniversary of CORE, Proceedings of the international symposium held at Louvain-la Neuve, The M.I.T. Press, Boston, 1989 (560 pp.) (with Bernard Cornet). 10. Mode´lisation et De´cisions e´conomiques, collection Ouvertures Economiques, De Boeck-Universite´, Bruxelles, 1990 (228pp.) (with Bernard Cornet). 11. Guest editor of Journal of Productivity Analysis vol. 10 n8 1, July 1998 (Special issue devoted to the Fourth European Workshop on Productivity and Efficiency Analysis held at CORE, November 1997).
B
Articles, Reports and Other Boldfaced items are reprinted in this volume.
I.
RESOURCE ALLOCATION PROCESSES
Bibliography of Henry Tulkens 1968–2004
559
Articles ‘‘Dynamic Processes for Public Goods: An Institution-Oriented Survey’’, Journal of Public Economics, 9 (2), 163 –201 (April 1978). ‘‘Surplus-Sharing Local Games in Dynamic Exchange Processes’’, Review of Economic Studies, XLVI (2), n8 143, 305 –314, (April 1979) (with S. Zamir). ‘‘Commodity Exchanges as Gradient Processes’’, Econometrica 48 (2), 387– 400 (March 1980), (with Claude d’Aspremont). ‘‘A Planning Process for the Efficient Allocation of Resources to Transportation Infrastructure’’, in Thisse, J. and Zoller, H. (eds.) Locational Analysis of Public Facilities, North-Holland Publishing Co., Amsterdam, 1983, 127–152 (with Tomasikila Kioni Kiabantu). ‘‘Exchange Processes, the Core and Competitive Allocations’’, chapter 2 (pp. 15 –32) .in Gabszewicz, J.J., Richard, J.F. and Wolsey, L. (eds), Economic Decision Making: Games, Econometrics and Optimization. Contributions in honour of Jacques Dre`ze, North-Holland, Amsterdam, 1990 (with Parkash Chander). ‘‘Politica ambiental: a utilizac¸ao de taxas pigouvianas no caso dinaˆmico’’, Revista die Econometria (The Brazilian Review of Econometrics) 15 (2), 87–105, 1996 (with Francisco de Sousa Ramos). ‘‘Strategically Stable Cost Sharing in an Economic-Ecological Negotiation Process’’, chapter 4 in A. Ulph. (ed.) Environmental Policy, International Agreements, and international Trade, Oxford University Press 2002 (with Parkash Chander).
II.
ENVIRONNEMENT
Articles . General ‘‘Traitement au moindre couˆt des effluents industriels et tarification d’une station d’e´puration’’, Recherches Economiques de Louvain, XXXVIII, 365 –388 (November1972). ‘‘L’imputation des couˆts et le choix des instruments d’action en matie`re de pollution des eaux’’, Reflets et Perspectives de la Vie Economique, XII (5) 349–359 (December 1973). ‘‘Stability Analysis of an Effluent Charge and the ‘Polluters Pay’ Principle’’, Journal of Public Economics 4 (3), 245 –270 (August 1975) (with Franc¸oise Schoumaker); translated in Japanese, 1976. ‘‘L’environnement peut-il eˆtre appre´hende´ comme un bien e´conomique?’’, Chapter 1 (pp. 22–31) in C. Stoffaes and M. Richard (eds), Environnement et choix e´conomiques de l’entreprise, Cahiers de Prospective, InterEditions, Paris, 1995. ‘‘The Core of an Economy with Multilateral Environmental Externalities’’, International Journal of Game Theory 26, 379– 401, 1997 (with P. Chander).
. International externalities ‘‘An Economic Model of International Negotiations Relating to Transfrontier Pollution’’, in Krippendorff, K. (ed.), Communication and Control in Society, Gordon and Breach, New York - London, 1979, 199–212.
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‘‘Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems’’, European Economic Review, 36 (2/3), 288 –299 (1992) (with Parkash Chander). ‘‘Aspects strate´giques des ne´gotiations internationales sur les pollutions transfrontie`res et du partage des couˆts de l’e´putation’’, Revue Economique 43(4), 755 –768, July 1992 (with Parkash Chander). ‘‘The acid rain game as a resource allocation process, with application to negotiations between Finland, Russia and Estonia’’, The Scandinavian Journal of Economics, 97(2) 325 –343, 1995. (with V. Kaitala and K.G. Ma¨ler). Reprinted in Carlo Carraro, Ed., Governing the Global Environment, The Globalization of the World Economy, An Elgar Reference Collection, Cheltenham 2003. ‘‘A core-theoretic solution for the design of cooperative agreements on transfrontier pollution’’, International Tax and Public Finance 2 (2), 279–294, 1995 (with Parkash Chander). Reprinted as chapter 5 in A. Ulph. (ed.) Environmental Policy, International Agreements and International Trade, Oxford University Press 2002 , and as chapter 7 in Michael Hoel (ed.), Recent Developments in Environmental Economics, Vol.II, The International Library of Critical Writings in Economics, Elgar, Cheltenham 2004. ‘‘International negotiations on acid rains in Northern Europe: a discrete time iterative process’’, chapter 10 (pp. 217–236) in in Xepapadeas, A. (ed.), Economic Policy for the Environment and Natural Resources, Elgar, London, 1996. (with Marc Germain and Philippe Toint). ‘‘Calcul e´conomique ite´ratif pour les ne´gociations internationales sur les pluies acides entre la Finlande, la Russie, l’Estonie’’, Annales d’Economie et de Statistique (Paris) n843 (July-September), 101–127, 1996.(with Marc Germain and Philippe Toint). ‘‘Cooperation vs. free riding in international environmental affairs: two approaches’’, chapter 2 (pp. 330– 44) in N. Hanley and H. Folmer (eds), Game Theory and the Environment, Elgar, Londres, 1998. Translated in French. ‘‘Financial transfers to sustain cooperative international optimality in stock pollutant abatement’’, chapter 11 in Faucheux, S., Gowdy, J. and Nicolaı¨, I. eds, Sustainability and Firms: Technological Change and the Changing Regulatory Environment, Edward Elgar, Cheltenham 1998 (with Marc Germain and Philippe Toint). ‘‘Stabilite´ strate´gique en matie`re de pollution internationale avec effet de stock : le cas line´aire’’, Revue Economique (Paris) 49 (6), 1435 –1454 (with Marc Germain and Aart de Zeeuw). ‘‘The Kyoto Protocol: An Economic and Game Theoretic Interpretation’’, chapter 6 (pp. 98 –117) in Kristro¨m, B., Dasgupta P. and Lo¨fgren K.-G. (eds), Economic Theory for the Environment : Essays in Honor of Karl-Go¨ran Ma¨ler, Edward Elgar, Cheltenham 2002 (with P. Chander, J.P. van Ypersele and S. Willems). ‘‘Coope´ration versus ‘‘free-riding’’ dans le domaine des affaires internationales sur l’environnement: deux approaches’’, chapter 2 (pp. 47–62) in: Gilles Rotillon ed., Re´gulation environnementale. Jeux, coalitions, contrats, Economica, Paris 2002. ‘‘Transfers to Sustain Dynamic Core-Theoretic Cooperation in International Stock Pollutant Control’’, Journal of Economic Dynamics and Control 28, 79–99 (2003) (with Marc Germain, Philippe Toint and Aart de Zeeuw). ‘‘Simulating with RICE Coalitionally Stable Burden Sharing Agreements for the Climate Change Problem’’. Resource and Energy Economics 2003. (with Johan Eyckmans). «Stable International Agreements on Transfrontier Pollution with Ratification Constraints », chapter 1 (pp. 9–36) in C.Carraro and V. Fragnelli (eds), Game Practice and the Environment, FEEM/Elgar Series in Environment, 2004 (with Sergio Currarini).
Reports and Other
Bibliography of Henry Tulkens 1968–2004
561
- ‘‘Preferences, Distribution and Objectives’’, Panel Discussant on Urbanization and Environment (with I. Heggie, R. Thoss, K.G. Ma¨ler, and E.S. Mills), (pp. 253 –255) and ‘‘Discussion of Paper by I. Hoch on Inter-urban Differences in the Quality of Life’’, (pp. 91–95), both published in Rothenberg, J. and Heggie, I. (eds.), Transport and the Urban Environment, proceedings of the conference held by the International Economic Association at Lyngby, Denmark, 1972, published by Mcmillan, London and St Martin’s Press, New York, 1974. - ‘‘Mode`les de de´cision pour le choix de la qualite´ des eaux et de la re´partition des charges de l’e´puration’’, Final report of the research project CB1: ‘‘Analyse e´conomique de la lutte contre la pollution des eaux’’, presented to the Commission Interministe´rielle de la Politique Scientifique, Brussels, December 1974. (300 pp., mimeo). - ‘‘Externalite´s et biens collectifs’’, in Deschamps, R., Le marche´ dans l’Economie Contemporaine, Report for the Commission I of the 3rd Congre`s des Economistes Belges de Langue Franc¸aise, published by the Centre Interuniversitaire de Formation Permanente (CIFOP), Charleroi, November 1978, pp. 34 –39. - ‘‘Limits to Climate Change’’, paper presented at the Sixth CORE – FEEM - GREQAM – CODE Coalition Formation Workshop held at Louvain-la-Neuve, January 26 –27, 2001; CLIMNEG-CLIMBEL paper n8 42 (with Parkash Chander).
III. TAXATION AND FEDERALISM Articles La the´orie e´conomique peut-elle justifier la re´sorption de la fraude fiscale et le choix des mesures compensatoires?’’ in Frank, M., ed., L’exacte perception de l’impoˆt, Actes de la confe´rence de ‘Institut Belge de Finances Publiques, Bruylant, Bruxelles, 1973. ‘‘Analyse e´conomique de la concurrence entre juridictions fiscalement souveraines’’, Bulletin de Documentation (Ministe`re des Finances, Bruxelles), September 1985, 44 –60. ‘‘Commodity Tax Competition Between Member States of a Federation: Equilibrium and Efficiency’’, Journal of Public Economics 29 (3), 133 –172 (June 1986) (with Jack Mintz). ‘‘On Pareto Improving Commodity Tax Changes under Fiscal Competition’’, Journal of Public Economics 41 (3) 335 –350 (April 1990) (with Alain de Crombrugghe). ‘‘The OECD convention: a ‘‘model’’ for corporate tax harmonization?’’, in Prud’homme, R. (ed.), Public Finance with Several Levels of Governments, proceedings of the 46th Congress of the International Institute of Public Finance (Brussels), Foundation Journal Public Finance, The Hague, 1991 (with Jack Mintz). ‘‘Financement des communaute´s et re´gions, coordination des politiques budge´taires et fe´de´ralisme fiscal : a` propos des communications de MM. Savage et Deschamps’’, in R. Deschamps, J. Ch. Jacquemin et M. Mignolet (eds). Finances publiques re´gionales et fe´de´ralisme fiscal. Journe´es d’e´tudes du 11 mars 1994, Namur, Presses Universitaires de Namur, 1994. ‘‘Economie et fe´de´ralisme’’, chapter 4 in Centre Interuniversitaire d’Etudes du Fe´de´ralisme (ed.), Manuel du Fe´de´ralisme, De Boeck-Universite´, Bruxelles, 1994 (with Gonzales d’Alcantara). ‘‘Optimality Properties of Alternative Systems of Taxation of Foreign Capital Income’’, Journal of Public Economics 60(3), 373 – 401, 1996 (with Jack Mintz). ‘‘Fiscaliteit, solidariteit en federalisme: vergelijking van zes landen’’, in Christine Vanderveeren en Jef Vuchelen (eds.) Een Vlaamse fiscaliteit binnen een economische en monetaire unie. Referaten voor het Vlaams Symposium van 4 –5 december 1998, Antwerpen, Intersentia Rechtswetenschappen, 235 –273, 1998 (with Philippe Cattoir).
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‘‘Fiscale autonomie: stand van zaken en een voorstel voor Belgı¨e’’, pp. 167–180 in: Koninklijke Vlaamse Academie van Belgie¨ voor Wetenschappen en Kunsten ed., Colloquim Belgium: Quo Vadis ?, Turnhout, Brepols, 1999,. Fiscalite´, solidarite´ et fe´de´ralisme. Comparaison de six pays, pp. 171–216 in: Groupe Avenir, Universite´ catholique de Louvain ed., Des ide´es et des hommes. Pour construire l’avenir de la Wallonie et de Bruxelles, Louvain-la-Neuve, Academia Bruylant, Louvain-la-Neuve, Brussels 1999, (with Philippe Cattoir). L’autonomie fiscale: e´tat de la question et une proposition pour la Belgique, pp. 159–169 in: Groupe Avenir, Universite´ catholique de Louvain ed., Des ide´es et des hommes. Pour construire l’avenir de la Wallonie et de Bruxelles, Academia Bruylant, Louvain-laNeuve, Brussels 1999,. Le financement de Bruxelles: enjeux et voies possibles, Reflets et perspectives de la vie e´conomique, XXXIX, 2–3, 2000, 197–212 (with Lambert J.P. , Cattoir P. , Taymans M., Van Der Stichele G. , Verdonck M). ‘‘Quelles frontie`res pour Bruxelles ? La re´ponse d’un exercice statitique, ge´ographique et e´conomique’’, pp. 73 –88 in: XIVe Congre`s des Economistes Belges de Langue Franc¸aise, Rapport Commission 3, P.-P. Combes, I. Thomas ed., CIFOP, Charleroi, 2000. ‘‘Coope´ration et design des institutions: une perspective e´conomique’’, pp. 467– 487 in: Autonomie, solidarite´ et coope´ration. Quelques enjeux du fe´de´ralisme belge au 21e`me sie`cle, Philippe Cattoir, Philippe de Bruycker, Hugues Dumont, Henry Tulkens and Els Witte eds., Bruxelles, De Boeck & Larcier, 2002 (with Jack Mintz). ‘‘Refinancement, autonomie et loyaute´ fiscale, ou du bon usage des mots en politique institutionnelle’’, Administration publique. Revue du droit public et des sciences administratives, T 2–3 – 4, 2003, 265 –267. ‘‘Dynamique des interactions fiscales entre communes belges (1983 –1996)’’, in E´conomie et Pre´vision (Journal of the Direction de la Pre´vision, Ministry of Finance, Paris), 156, 2002/5, 1–14 (with Jean-Franc¸ois Richard and Magali Verdonck). ‘‘Pourquoi le fe´de´ralisme ?’’, Revue E´conomique (Paris) 54 (3), 469– 476, May 2003. ‘‘On Cooperation in Musgravian Models Of Externalities Within a Federation’’, chapter 14 (pp.451– 465) in S. Cnossen and H.W.Sinn (eds), Public Finances and Public Policy in the New Century (festschrift published at the occasion of the 90th birthday of Richard Musgrave and of the 10th anniversary of CESifo, Munich, MIT Press, 2003.
Reports and Other - Discussion of the paper ‘‘Personele Inkomensverdeling - Beleid’’ by professors Bienstman, Carrin, Kesenne and Proost (K.U. Leuven), presented at the Veertiende Vlaams Wetenschappelijk Economisch Congres: Inkomens-en Vermogensverdeling, pp. 59–65 of the Proceedings, published by Vereniging voor Economie, Brussels, March 1979. - Discussion of the papers ‘‘The choice between income and consumption tax bases’’ by professor David Bradford, and ‘‘Un impoˆt progressif sur les de´penses de consommation: alternative a` l’impoˆt sur le revenu des personnes physiques?’’ by professor Max Frank, presented at the colloquium of the Institut Belge de Finances Publiques on ‘‘l’Impoˆt des personnes physiques’’, Brussels, March 26, 1982, published in Bulletin de Documentation (Ministe`re des Finances, Bruxelles) November 1982, pp. 189–198. - ‘‘Strategic use of tax rates and credits in a model of international corporate income tax competition’’, CORE Discussion Paper n8 9073, Center for Operations Research and Econometrics, Universite´ Catholique de Louvain, Louvain-la-Neuve, December 1990 (with Jack Mintz) (paper presented at the Norwegian Ministry of Finance Conference on Taxation and International Integration, Solstrand Fjord, Norway, June 7, 1990. - ‘‘Modes alternatifs de financement de Bruxelles’’, Recherche re´alise´e pour le compte de Mr Rufin Grijp, Ministrre de la Recherche scientifique de la Re´gion de Bruxelles-
Bibliography of Henry Tulkens 1968–2004
563
Capoitale, Centre d’Etudes Re´gionales bruxelloises, Faculte´s Universitaires SaintLouis, Bruxelles, April 1999 (with Jean-Paul Lambert, Philippe Cattoir, Miche`le Taymans, Ge´raldine Van der Stichele and Magali Verdonck - ‘‘Perequatiemechanismen en aansporinggen tot hervorming’’, samenvatting van het eindrapport ‘‘Me´canismes de pe´re´quation et incitations aux re´formes’’, Recherche re´alise´e pour le compte de Mme Wivina Demeester, Vlaamse Mionister voor Financie¨n, Begroting en Gezondheidsbeleid, Center for Operations Research and Econometrics (CORE), Universite´ Catholique de Louvain, June-July 1999 (with Philippe Cattoir, Ge´raldine Van der Stichele and Magali Verdonck). - ‘‘Crite`res pour la de´termination des frontie`res de Bruxelles’’, Rapport adresse´ a` Mr Herrve´ Hasquin, Ministre de l’Ame´nagement du Territoire, des Travaux Publics et des Communications de la Re´gikon de Bruxelles-Capitale, Center for Operations Research and Econometrics (CORE), Universite´ Catholique de Louvain, May 1999 (with Isabelle Thomas and Pierre Berquin).
IV. EFFICIENCY ANALYSIS Articles ‘‘Measuring Labor-Efficiency in Post Offices’’, Chapter 10 in Marchand, M., Pestieau, P. and Tulkens, H. (eds.), The Performance of Public Enterprises: Concepts and Measurement, North-Holland Publishing Co., Amsterdam, 1984, (243 –268) (with D. Deprins and L. Simar). ‘‘The Performance Approach in Public Enterprise Economics’’, Annales de l’Economie Publique, Sociale et Coope´rative (C.I.R.I.E.C., Lie`ge), 74(4), 429– 444 (december1986). ‘‘La performance productive d’un service public: de´finitions, me´thodes de mesure et application a` la Re´gie des Postes en Belgique’’, L’Actualite´ Economique, Revue d’Analyse Economique (Montre´al) 62(2), 306 –335 (June 1986). ‘‘Performance e´conomique des socie´te´s de transports urbains: concepts, me´thodes et synthe`se des re´sultats de l’ouvrage’’, part I (pp. 13 a` 38) of Thiry and Tulkens, H. (eds), La performance e´conomique des socie´te´s de transports urbains, Centre international de recherche et d’information sur l’e´conomie publique, sociale et coope´rative (C.I.R.I.E.C.), Lie`ge, 1988 (with Bernard Thiry). ‘‘Mesure de l’efficacite´ productive: me´thodologies et applications aux socie´te´s de transports intercommunaux de Lie`ge, Charleroi et Verviers’’, chapter 2 (pp. 81 a` 136) in Thiry, B. and Tulkens, H. (eds), La performance e´conomique des socie´te´s de transports urbains, Centre international de recherche et d’information sur l’e´conomie publique, sociale et coope´rative (C.I.R.I.E.C.), Lie`ge, 1988 (with Bernard Thiry and Alaxis Palm). ‘‘Mesure de l’efficacite´ productive: applications a` la socie´te´ de transports intercommunaux de Bruxelles’’, chapter 3 (pp. 137 a` 170) in Thiry and Tulkens, H. (eds), La performance e´conomique des socie´te´s de transports urbains, Centre international de recherche et d’information sur l’e´conomie publique, sociale et coope´rative (C.I.R.I.E.C.), Lie`ge, 1988 (with Bernard Thiry and Charles Nollet). ‘‘La modernisation, une ne´cessite´; la privatisation un moyen parmi d’autres. Re´flexions sur les aspects e´conomiques de la gestion du secteur public en Belgique’’, Reflets et Perspectives de la Vie Economique, XXVII (2), 111–114, 1988. ‘‘Mesures de la performance productive d’une entreprise publique: Application a` la STIB’’, chapter 4.1 (pp.297–320) in CORE (ed.), Gestion de l’Economie et de l’Entreprise, De Boeck Universite´, Bruxelles, 1988 (with Antoinette d’Aspremont and Charles Nollet).
564
Public Goods, Environmental Externalities and Fiscal Competition
‘‘Efficacite´, management et science e´conomique’’, general introduction (pp.5 –15) to Efficacite´ et Management, H. Tulkens, ed., Proceedings of the 5th Commission of the Huitie`me Congre`s des Economistes Belges de Langue franc¸aise, Centre Interuniversitaire de Formation permanente (CIFOP), Charleroi, Belgium,1989. ‘‘Productivite´, efficacite´, et progre`s technique: notions et mesures dans l’analyse e´conomique’’, chapter 2, (pp.19– 47) in Efficacite´ et Management, H. Tulkens, ed., Proceedings of the 5th Commission of the Huitie`me Congre`s des Economistes Belges de Langue franc¸aise, Centre Interuniversitaire de Formation permanente (CIFOP), Charleroi, Belgium,1989 (with Bernard Thiry). ‘‘Mesures de l’efficacite´ a` la Re´gie des postes’’, chapter 6 (pp.129–140) in Efficacite´ et Management, H. Tulkens, ed., Proceedings of the 5th Commission of the Huitie`me Congre`s des Economistes Belges de Langue franc¸aise, Centre Interuniversitaire de Formation permanente (CIFOP), Charleroi, Belgium,1989. ‘‘Une mesure de l’efficacite´-couˆt de 253 communes francophones’’, chapter 9 (pp.177–208) in Efficacite´ et Management, H. Tulkens, ed., Proceedings of the 5th Commission of the Huitie`me Congre`s des Economistes Belges de Langue franc¸aise, Centre Interuniversitaire de Formation permanente (CIFOP), Charleroi, Belgium,1989 (with Philippe Vanden Eeckaut). ‘‘Productivity, Efficiency, and Technical Progress: Concepts and Measurement’’, Annals of Public and Cooperative Economics/Annales de l’Economie Publique, Sociale et Coope´rative 60 (1), 9– 42 (Jan.-March 1989) (with Bernard Thiry). ‘‘Allowing for Technical Efficiency in Parametric Estimates of Production Functions: with an Application to Urban Transit Firms’’, Journal of Productivity Analysis 3 (1/2), 41– 65, June 1992 (with Bernard Thiry) ‘‘Mesures de l’efficacite´-couˆt de 235 communes belges’’, Cahiers de l’Institut Belge de Finances Publiques, Bruxelles, n8 2, 1992 (with Philippe Vanden Eeckaut and MarieAstrid Jamar). ‘‘A study of Cost Efficiency and Returns to Scale for 235 Municipalities in Belgium’’, chapter 12 (pp. 300–334) in Fried, H., Lovell, C.K. and Schmidt, S. (eds), The Measurement of Productive Efficiency: Techniques and Applications, Oxford University Press, 1993 (with Philippe Vanden Eeckaut and Marie-Astrid Jamar). ‘‘Comment’’ on the paper ‘‘Measuring the contribution of public infrastructure capital in Sweden’’ by Berndt and Hansson, Scandinavian Journal of Economics 94 (Supplement), S-169 - S-172, 1992. ‘‘On FDH Efficiency Analysis: Some Methodological Issues and Applications to Retail Banking, Courts and Urban Transit’’, Journal of Productivity Analysis 4 (1/2), 183 –210, 1993. ‘‘Assessing and explaining the performance of public enterprises: some recent evidence from the productive efficiency viewpoint’’, Finanzarchiv Neue Folge 50 (3), 293 –323, 1993.(with Pierre Pestieau). ‘‘Les performances e´conomiques de la STIB au mois le mois’’, Revue Suisse d’Economie Politique et de Statistique 130 (4), 627– 646, 1994. ‘‘L’efficacite´ au service de l’e´quite´ en e´conomie, in Faculte´s universitaires Saint-Louis (ed.), Variations sur l’e´thique. Hommage a` Jacques Dabin, Publications des Faculte´s universitaires Saint-Louis, Bruxelles, 721–729, 1994. ‘‘Non-frontier measures of efficiency, progress and regress’’, International Journal of Production Economics 39, 83 –97, 1995 (with Philippe Vanden Eeckaut). ‘‘Non-Parametric Efficiency, Progress and Regress Measures For Panel Data: Methodological Aspects’’, European Journal of Operations Research 80, 474 – 499, 1995 (with Philippe Vanden Eeckaut). ‘‘Nonparametric Approaches to the Assessment of the Relative Efficiency of Bank Branches’’, chapter 10 (pp. 223 –244) in D.G.Mayes ed., Sources of Productivity Growth, Cambridge University Press, Cambrige U.K., 1996 (with Amador Malnero).
Bibliography of Henry Tulkens 1968–2004
565
‘‘Evaluation de l’arrie´re´ judiciaire a` la lumie`re d’une mesure de l’efficacite des activites des tribunaux’’, Bulletin de documentation, Ministere des Finances, 58 (5), 1–27, September 1998. ‘‘Mesurer l’efficacite´ :avec ou sans frontie`res?’’, chapter 3 (pp.75 –100) in P.Y.Badillo and J.C. Paradi (eds), La me´thode DEA: analyse des performances, Hermes Science Publications, Paris, 1999 (with Philippe Vanden Eeckaut). ‘‘Evaluating the Financial Performance of Bank Branches’’, Annals of Operations Research, Forthcoming 2006 (with J.T. Pastor and C.K. Lovell). ‘‘Efficiency Dominance Analysis: Basic Methodology’’, manuscript, 2004.
Reports and other - ‘‘Mesure de l’efficacite´ de l’activite´ des tribunaux et e´valuation de l’arrie´re´ judiciaire’’, Document de travail du Fonds de Documentation Statistique et Economique sur les Services Publics Belges, c/o CORE, Universite´ Catholique de Louvain, Louvain-laNeuve, avril 1990 (with Marie Astrid Jamar). (paper presented at the Septie`mes Journe´es Microe´conomie Applique´e, held at Universite´ du Que´bec a` Montre´al, Canada, May 26, 1990). - ‘‘Non-Parametric Efficiency Analyses in Four Service Activities: Retail Banking, Municipalities, Courts and Urban Transit’’, CORE Discussion Paper n8 9050, july 1990 (paper prepared for at the Third Franco-American Economic Seminar on Productivity Issues in Services at the Micro Level, National Bureau of Economic Research, Cambridge, Massachusetts, USA, July 23 –26 1990) - ‘‘Assessing the performance of public sector activities: some recent evidence from the productive efficiency viewpoint’’, CORE Discussion Paper n8 9060, Center for Operations Research and Econometrics, Universite´ Catholique de Louvain, Louvain-laNeuve, November 1990 (with Pierre Pestieau). - ‘‘Efficiency Dominance Analysis (E.D.A.): A Frontier Free Efficiency Evaluation Method’’, paper presented at the World Congress of the International Federation of the Operations Research Societies (IFORS), Lisbon (July 1993) and at the Third European Workshop on Efficiency and Productivity Analysis, CORE, Louvain-laNeuve (October 1993), mimeo. - ‘‘How to measure efficiency and productivity with special reference to banking’’, introductory presentation at the Nordic Workshop in Productivity and Efficiency, held at the inauguration of Handelsho¨gskolan, Go¨teborg University , September 19–20, 1995. Go¨teborg, September 1995 (mimeo). - ‘‘Frontier or not frontier ? That is the question . . . ’’ A lecture at the Historical Overview session of the Fifth European Workshop on Efficiency and Productivity Analysis held at The Royal Veterinary and Agricultural University Department of Economics and Natural Resources Copenhagen, Denmark, October 9, 1997 - ‘‘Efficiency Analysis: Review of Methodology and of Some Applications in the Public Sector and in Macroeconomics’’, Handout for two lectures given at the University of Naples, 26 –27 May 1998
V.
FURTHER WORK ON PUBLIC UTILITIES, PUBLIC GOODS AND COST BENEFIT ANALYSIS
Articles
566
Public Goods, Environmental Externalities and Fiscal Competition
‘‘L’e´volution historique de la tarification et de la demande des services postaux en Belgique (1840–1961)’’, Recherches Economiques de Louvain, XXXI, 633 – 673 (December 1965). ‘‘Proble`mes the´oriques et pratiques de la tarification dans les Services publics’’, Bulletin de Documentation (Ministe`re des Finances, Bruxelles), July 1969, 5 –23. ‘‘La lutte contre la de´linquance: les initiatives prive´es et leurs rapports avec le secteur public’’, Revue de Droit pe´nal et de criminologie (Bruxelles), 51, 879–897 (June 1971) (with Alexis Jacquemin). ‘‘The Publicness of Public Enterprise’’, chapitre 2 (pp. 23 –32) in W.G. Sheperd and Associates, Public Enterprise: Economic Analysis of Theory and Practice, Lexington, Massachussetts, 1976. ‘‘Quel sens donner aux tarifs publics ?’’, Annales de l’Economie Publique, Sociale et Coope´rative (Lie`ge) (special issue on electricity pricing), 67 (3), 3 –8 September 1979 (with Maurice Marchand), ‘‘Calcul des be´ne´fices’’, chapter 3 (15 – 43) in Blauwens, G., Tulkens, H., Thys-Clement, F. and Anselin, M., Analysee couˆts-be´ne´fices de l’extension future du re´seau hydraulique belge, Secre´tariat Ge´ne´ral du Ministe`re des Travaux Publics, Bruxelles, March 1982. ‘‘Entreprises et Services Publics: e´volution institutionnelle ge´ne´rale et dix-neuf e´tudes de cas (1950–1980), Bulletin de Documentation (Ministe`re des Finances, Bruxelles), September 1983, pp. 5 –126 (with Fr. Robert, F.X. de Donne´a and M. Marchand); reprinted as chapter 10 in Frank, M. (ed.) Histoire des Finances Publiques en Belgique, Tome IV (2 volumes), 1988, pp 819–930. ‘‘The Performance of Public Enterprises: Normative, Positive and Empirical Issues’’, Chapter 1 in Marchand, M., Pestieau, P. et Tulkens, H. (eds.), The Performance of Public Enterprises: Concepts and Measuremant, North-Holland Publishing Co., Amsterdam, 1984, (1– 40) (with M. Marchand and P. Pestieau). ‘‘Investissement, emploi, et calcul e´conomique’’, Rapport ge´ne´ral de la Commission n8 3 du Sixie`me Congre`s des Economistes Belges de Langue Franc¸aise (Bruxelles), in Actes du Congre`s published by C.I.F.O.P., Charleroi, November 1984, 9–25. ‘‘Un coup de pouce au financement des acade´mies de musique’’, Reflets et Perspectives de la Vie Economique, XXXII (2), 143 –152, 1993.
Reports and Other - ‘‘The Cost of Delinquency: A Problem of Optimal Allocation of Private and Public Expenditure’’, CORE Discussion Paper 7133, Center for Operations Research and Econometrics, Universite´ Catholique de Louvain, August 1971 (with Alexis Jacquemin). - ‘‘Analyse couˆts-be´ne´fices des nouvelles centrales e´lectriques belges’’, Rapport a` M. le Ministre des Affaires Economiques Eyskens et a` MM. les Secre´taires d’Etat a` l’Energie Knoops et a` l’Environnement Aerts, January 1984, pp. 92 (with G. Blauwens, P. de Meester and A. Jaumotte). - ‘‘Pre´sentation et discussion de faisabilite´ d’un mode`le des activite´s portaires du Benelux, prenant en compte l’impact de ces activite´s sur l’e´conomie des pays membres’’, Annex (bijlage) F (pp. F-1 - F-42) in Denduyver, J., Tulkens, H., Goemans, T., and Klaassen, L., ‘‘Huidige en toekomstige sociaal-economische betekenis van de Benelux zeehavens’’, Feasibility study commissioned by the BENELUX Economic Union, Gent, Louvain-la-Neuve en Rotterdam, april 1986 (mimeo). - ‘‘Note relative a` la de´termination du nombre et de la valeur des parts d’associe´ dans l’I.E.C.B.W. a` attribuer a` l’occasion de l’apport de nouvelles sections par une commune associe´e et/ou lors de l’entre´e de nouveaux associe´s’’ adresse´e a` Mr Gillis, Directeur de l’Intercommunale des Eaux du Centre du Brabant Wallon, 15 january 1991.
Bibliography of Henry Tulkens 1968–2004
567
- ‘‘Evaluation de douze entreprises publiques belges dans une optique de privatisation — Waardebepaling van twaalf belgische publieke ondernemingen vanuit een privatiseringsoptiek’’, PETERCAM Securities, Brussels and Hill Samuel, London (with Paul De Grauwe, Andre´ Farber, Wim Moesen and Jozef Vuchelen), march 1992. - Discussion of the paper ‘‘Een Inleiding tot de theoretische grondslagen van de sociale kosten-batenanalyse’’ by professor Wilfried Pauwels (St Ignatius, Anvers), presented at the conference of the Institut Belge de Finances Publiques, published in Bulletin de Documentation (Ministe`re des Finances, Bruxelles) January 1980, pp. 41–50. - Discussion of the paper ‘‘Information Problems and the Efficient Management of the Supply of Local Services’’ by Professor Horst Hanusch (University of Augsburg), presented at the colloquium of the Institut Belge de Finances Publiques on ‘‘Les finances communales’’, Brussels, October 5,1983, published in Bulletin de Documentation (Ministe`re des Finances, Bruxelles) April 1985, pp.137–142.
AUTHOR INDEX1
Roman numerals in bold refer to the four parts of the book. Aigner, D. III 286, 292, 314 Alchian, A.A. III 352, 353 Allais, M. I 64 Alworth, J. IV 509 Amann, M. II 140 Amir, R. II 251 Anderson, G.J. IV 538 Aoki, M. I 40 Arnott, R. IV 450 Arrow, K. II 127, 136 Athanassopoulos, A. III 432 Atkinson, A.B. III 335, 418; IV 463, 491 Bahn, O. II 253 Bamon, R. IV 470 Banker, R. III 314, 349, 390, 396, 397, 398, 399, 432 Bardhan, I. III 436 Barla, P. III 358, 365 Barrett, S. II 172, 178, 218, 253 Barrow, M. III 344 Basar, T. II 255 Bauer, P.W. III 348 Berg, S.A. III 419 Bergstrom, Th I 13, 28 Bernheim, D. II 159, 183 Bertrand, C. II 221 Besley, T. IV 534 Billera, L. I 56, 65 Birch, S. III 344 Bloch, F. II 153 Boadway, R. IV 450, 534 Bo¨hm, V. I 49 Bond, E. IV 509 Bonnisseau, J.-M. I 63 1
Borcherding, T.E. III 356 Boskin, M.J. IV 450, 451, 464 Bowen, H.R. I 13, 35, 36 Bowlin, W.F. III 397 Boyer, J. II 272 Brueckner, J.K. IV 534 Buchanan, J. I 10–11, 13, 28, 40–43, 45; II 127 Burgat, P. III 360, 363 Byrnes, P. III 357 Carraro, C. II 165, 172, 177, 178, 186, 217, 218, 253, 561 Case, A. IV 534 Caves, R.E. III 363 Chaffai, M. III 362 Champsaur, P. I 13, 28, 40, 49, 64–65, 77, 81, 90, 94; II 154, 161, 165 Chander, P. I 63–79; II 101–105, 123–134, 136, 145, 146, 150, 151, 153–175, 195–215, 252, 257–258, 260–261, 265 Charnes, A. I 58; III 314, 315, 316, 398, 400, 401, 403, 431, 432 Cheng Zhong Qin I 63 Chwe, M.S.-Y. II 246 Ciscar, J.C. II 217 Cooper, W.W. III 314, 316, 349, 390, 396, 397, 398, 399, 400, 401, 403, 431, 432, 436 Cordella, T. II 177 Cornes, R. II 127 Cornet, B. I 40, 63, 64 Cote, D. III 358, 362 Cubbin, J. III 357, 360, 363
Index prepared by Indexing Specialists (UK) Limited, HOVE, BN3 2DW, UK.
570
Dasgupta, P. II 190 d’Aspremont, A. III 334, 377, 390 d’Aspremont, C. I 9, 49, 64, 76, 81–95; II 178, 251 Davidson, J.E.H. IV 538 de Alesi, L. III 353 de Carvalho, F. I 9 de Crombrugghe, A. IV 491–506 de la Valle´e Poussin, D. I 13, 15, 20–23, 40, 50; II 114, 127, 136 de Zeeuw, A. II 202, 219, 247, 251–274 Debreu, G. I 69; II 211; III 344, 395, 397 Decaux, A. II 201, 204, 211–212 Defourny, J. III 362 Delhausse, B. III 360, 363 Deprins, D. III 275, 279, 284, 285–309, 312, 313, 314, 316, 318, 398, 432, 440 Distexhe, V. III 360, 363 Dixit, A. IV 517 Dre`ze, J. I 3–7, 13, 15, 20–23, 28, 40, 49, 50, 58, 63, 77, 81; II 114, 127, 136, 153, 217 Easley, D. III 353 Edgeworth, F.Y. II 211 Edmonds, J. I 49 Ellerman, D. II 195, 201, 204, 211–212 Eyckmans, J. II 217–249, 270 Fankhauser, S. II 204 Fa¨re, R. III 313, 339, 358, 378, 396, 403, 404, 415, 418, 432 Farrell, M.J. III 285, 286, 287, 290, 292, 293, 307, 314, 316, 339, 348, 375, 376, 398, 400, 432, 433 Fecher, F. III 360, 363, 365 Feldstein, M. IV 508, 509, 516 Ferrantino, M.J. III 362 Filar, J.A. II 264 Filippini, M. III 360 Finus II 246 Flohimont, O. IV 540 Foley, D. I 13; II 101, 154, 161, 165, 185
Author Index
Førsund, F. III 286, 292, 329, 348, 378, 399 Fourgeaud, C. I 13 Fraysse, J. IV 470 Fried, H. III 432, 440 Gabszewicz, J. I 63; II 195 Gathon, H.-J. III 366, 367 Gerard-Varet, L. II 178 Germain, M. II 174, 192, 202, 217–218, 227, 247, 251–274 Gevers, - I 9 Gillies, D. I 55 Good, H.D. III 358, 365 Goodman, R.P. I 13 Goodspeed, J.T. IV 534 Gordon, R. IV 450, 451, 464, 491, 505, 509 Green, J. I 64, 75 Greenberg, J. I 9 Grieson, R. IV 450 Grosskopf, S. III 311, 313, 339, 357, 358, 360, 377, 378, 395, 396, 403, 404, 415, 418, 432 Groves, T. I 13 Hackl, F. II 241 Hahn, F. I 64 Hamada, K. IV 508 Hammond, P. II 190 Hart, O.D. III 354 Hartley, K. III 344 Hartman, D. IV 508, 509, 516 Haurie, A. II 253 Hayek, F.A. III 354 Hendry, D.F. IV 538 Henry, C. I 15, 28, 33, 58, 63, 77, 81 Heyndels, B. IV 534, 540 Hicks, J.R. III 352 Hines, J.R. IV 534 Hjalmarsson, L. III 358 Hoel, M. II 124, 178, 219, 252 Hori, H. I 14 Houba, H. II 256 Hougaard, J. III 432 Hurwicz, L. I 64; II 127, 136
Author Index
Jamar, M.A. III 311, 329, 373 Johansson, M. II 140 Jones, D. III 354 Jordan, J.S. I 65, 71, 72 Jouvet, P.-A. II 251 Kaitala, V. II 135–152, 190, 219, 252 Kalai, G. I 65 Kaneko, M. II 155, 167 Kay, J.A. III 344 Keen, M. IV 492, 505 Kohlberg, E. I 58 Kolstad, Ch.D. IV 508 Koopmans, T. III 285, 344, 395 Kortanek, K. I 58 Kranich, L. II 264 Kverndokk, S. II 252 Kwerel, E. II 190 Kypreos, S. II 253 Ladd, H.F. IV 534 Laffont, J.J. I 9, 35, 75; II 154, 161, 165, 186; III 355 Land, K.C. III 314 Lange, Oskar I 3 Ledyard, J. I 13, 64, 67 Leibenstein, H. III 344, 352 Lewin, A. III 329 Lindahl, Eric I 3, 10, 18–20 Lioukas, S. III 366 Loute, E. I 49, 81 Lovell, C. A. K. III 279–284, 286, 292, 307, 313, 314, 343, 348, 352, 373, 395, 396, 399, 404, 432, 440, 565 Luenberger, D.G. I 84, 88 MacFadden, D. III 314 McLure, C.E. IV 450 Ma¨ler, K.-G. II 101, 102, 124, 126, 135–152, 153–155, 159, 161, 165, 167, 172, 174, 177–178, 183, 185, 190–191, 219, 252, 255, 265 Malinvaud, E. I 3, 13, 15, 16, 18–23, 25–27, 33, 36, 39, 42, 52, 53, 64, 74, 81, 87; II 114, 127, 136 Manzini, A. III 358
571
Marchand, M. III 343, 345 Mas-Colell, A. II 155, 167 Masayoshi IV 534 Maschler, M. I 14, 65 Maskin, E. II 190, 255 Meeusen, W. III 314 Michel, Ph. II 251 Mieszkowki, P.M. IV 450 Milleron, J.C. I 13, 14, 15, 36, 82 Mintz, J. II 177; IV 445–489, 507–532 Moulin, H. I 49; II 154, 161, 165 Mueller, D. I 10–11, 38; II 127 Musgrave, P. IV 509 Musgrave, R. I 15, 17, 38; IV 450, 509 Nalebuff, B. III 354 Negishi, T. I 64 Nemytskii, V.V. I 58 Nesterov, Y. II 251 Newbery, D. II 124, 130 Nickell, S.J. IV 538 Nihoul, J.C.J. II 110 Nordhaus, W. II 202–203, 217–218, 221–222, 227, 233, 238, 240–241, 272 Oates, W.E. IV 450 Olesen, O. III 314 Olsder, G.-J. II 255 Osborne, A. II 260 Oum, T.H. III 360, 366 Owen, G. I 65 Parikh, A. II 127 Peacock, A. I 17 Peleg, B. I 14; II 159, 183 Perea, A. II 264 Perelman, S. III 343, 360, 365, 366 Pestieau, P. III 293, 343–372 Peters, H. II 264 Peters, W. III 351 Petersen, N.C. III 321, 339, 396, 400, 404 Petrosjan, L.A. II 252, 264 Pint, E.M. III 355
572
Pohjola, M. II 135–136, 139, 140, 143, 219, 252 Pruckner, G.J. II 241 Qin, Cheng Zhong I 63 Radner, R. I 64 Ray, D. II 165 Razin, A. IV 509 Rees, R. III 345, 346 Reiter, S. I 64 Respaut, B. III 326 Rhodes, E.L. III 358 Richard, J.-F. I 63, 81; IV 533–556 Roberts, J. I 5, 33, 38, 39, 50, II 132 Rosen, H.S. IV 534 Rubinstein, A. II 260 Ruys, P. I 13, 36 Saavedra, L.A. IV 534 Sadka, E. IV 509 Samuelson, L. IV 509 Samuelson, P.A. I 4, 12, 13, 20, 38, 126, 197, 219, 228 Sandler, T. II 127 Sandmo, A. IV 453, 508, 523 Sbra, F. IV 538 Scarf, H. I 69; II 154, 165, 185, 186, 211 Scherer, F.M. III 346 Schmeidler, D. I 55, 56 Schmidt, P. III 286, 292, 293, 307, 314, 348 Schoumaker, F. I 30, 33; II 127, 130 Shapiro, C. III 355 Shapley, L. I 55, 57; II 165, 166, 167, 186, 209–210 Shephard, R.W. III 294, 375, 376 Shiao, L. I 65 Shubik, M. II 165, 167, 186, 210 Sickles, R. III 365 Silvestre, J. II 155, 167 Simar, L. III 275, 279, 284, 285–309, 312, 313, 314, 316, 318, 373, 395 Siniscalco, D. II 165, 172, 178, 186, 218, 253 Sinn, H.W. IV 509
Author Index
Smale, S. I 43, 64, 77, 81, 95 Smets, H. II 118, 121 Solow, R.M. III 408 Starrett, D. II 154; IV 450 Stears, R.E. I 65 Steenbeckeliers, G. I 49 Stepanov, V.V. I 58 Sterner, T. III 362 Stigler, G. III 351 Stiglitz, J. IV 463, 491 Ta¨htinen, M. II 140 Tahvonen, O. II 136, 140, 143, 219, 252 Thiry, B. III 312, 313, 334, 378, 390, 397, 403, 410, 413 Thisse, J. II 135, 450, 491 Thomson, S. III 353 Tideman, T.N. I 9 Tirole, J. II 255 Toint, P. II 174, 192, 202, 218, 227, 247, 251–274 Ulph, A. II 135, 251 Uzawa, H. I 60, 64, 118, 119 Van Den Broeck, J. III 314 van der Ploeg, F. II 219, 252, 255 van Ypersele, J.-P. II 195–215, 251, 267, 272 Vanden Eeckaut, P. III 275, 282, 283, 284, 312, 321, 322, 338, 343, 373–394, 395–429, 431 Varian, H. I 75 Verdonck, M. IV 533–556 Vial, J.-P. II 253 Vickers, J. III 344 Vind, K. I 37 Vohra, H. II 165 Vuchelen, J. IV 534, 540 Wang, Y. I 65 Westhoff, F. IV 450 Whinston, M.D. II 159, 183 Wicksell I 10 Wildasin, D. IV 525
Author Index
573
Willems, S. II 195–215 Wolak, F.A. Jr IV 508 Wolsey, L. I 63
Yarrow, G. III 344, 356 Yen, P.Y. II 132 Yeo, S. IV 538
Yaari, M. I 9 Yang, Z. II 202–203, 218, 221–222, 227, 233, 235, 238, 240–241, 272
Zaccour, G. II 252 Zamir, S. I 9, 32, 49–62, 64, 67, 68–69, 71, 81; II 118
SUBJECT INDEX1
Roman numerals in bold refer to the four parts of the book. abatement costs, pollution II 177–93, 204–5 absence of tax interaction IV 536 acid rain game II 135–52, 154–5, 159, 165, 174 cooperation II 146–8 Finland, Russia and Estonia example II 139–51 increasing marginal damages II 149–51 optimality II 136–42 rationality II 142–6 transfer payments II 146–8, 151 active dominance, EDA III 437–8 adjustments exchange processes I 73 to preferred rates IV 538–9 prices I 61 taxation IV 538–9, 549–50, 552 airline efficiency studies III 358, 365 algorithms, dynamic cooperative games II 265–7 allocative efficiency, public sector III 347, 350–1 alpha-characteristic function II 160–1 alpha-core, imputation II 186–9 alternative optimality conditions, pure exchange economy I 82–4 Arrow–Debreu economy II 125–6, 178, 180, 191 atmospheric carbon concentrations II 236–7 atmospheric temperature changes II 268 autarkic equilibria IV 455, 467 autonomous fiscal authorities IV 449–89 1
backlog evaluation, Belgian courts III 329–33 banking III 322–9 FDH analysis III 281, 322–9, 357, 364 private sector III 326–9, 357–8, 364 productive efficiency III 357–8, 361–2, 364 public sector III 322–9 barter processes I 60–1 bases, taxation IV 512–14 BAU see business-as-usual behavioral assumptions, pollution II 111–13 Belgium banking III 322–9, 357, 361–2 courts III 329–33 insurance mutuals III 363 local taxation IV 533–56 post office labor efficiency III 279–80, 285–309 public sector performance III 343–4 refuse collection III 363, 365 STIB III 336–8, 390–3 benchmark observations, efficiency measures III 378–89 benchmark production correspondence III 380–9, 421–4, 427–8 Bowen–Laffont process I 35–6, 45 BPR indexes III 423–4 business-as-usual (BAU) II 235–41 capital import neutral IV 514–15 capital income taxation see foreign capital income taxation carbon emissions II 218, 235–6
Index prepared by Indexing Specialists (UK) Limited, HOVE, BN3 2DW, UK.
576
CDM see clean development mechanism centimes additionnels IV 533, 540 see also personal income tax Chander–Tulkens transfers II 146–7, 150–1, 213 clean development mechanism (CDM) II 197 climate change II 195–215, 217–49 CLIMNEG world simulation (CWS) model II 217–49 computing equilibria II 234–5 contrast to RICE model II 221–2 equation listing II 248 Nash equilibria II 223–5, 229–30, 234 numerical specification II 233–4 Pareto efficiency II 219, 225–9 partial agreement Nash equilibrium II 229–30, 234 scenarios II 235–40 statement II 220–1 transfers and time II 246–7 closed-loop transfer schemes II 270–1 coalitional stability II 217–49 climate change agreements II 217–49 emissions II 199–201, 205–9, 211–14 simulations II 233–47 transfers II 230–3 coalitions coalitional rationality II 230–3, 260–5 equilibria II 162–4, 183 PANE II 243–4 payoffs II 243–4, 269–70 players II 154–5 stability II 199–201, 205–9, 211–14, 217–49 Cobb–Douglas production function III 285–6, 292, 305, 307, 312 collective decision mechanism II 108 commodity exchanges I 81–95 commodity tax autonomous fiscal authorities IV 449–89
Subject Index
competition IV 449–89 efficiency IV 449–89 equilibrium IV 449–89 federation member states IV 449–89 new results IV 498–504 Pareto improving changes IV 491–506 competitive aspects emissions trading II 210–11 exchange processes I 63–79 fiscal choices IV 474–5 productive efficiency III 351–6, 363–5 resource allocations I 63–79 taxation IV 449–89, 533 computations dynamic cooperative games II 265–71 FDH measures III 316–20, 340 confidence intervals IV 543–4 consumer behavior, fiscal models IV 493–5 consumption, world-wide discounted II 240–1 contemporaneous analysis mathematical programming III 404–8 non-frontier efficiency measures III 377–8 progress/regress III 408–17, 421–2, 425–7 reference production sets III 401–2 convergence example, competitive core allocations I 71–7 convex polyhedral method, labor efficiency III 285, 287, 293–4, 307–8, 311–12 convexity data envelopment analysis III 407–8, 414 FDH analysis III 399–400, 407–8, 421, 432 cooperation acid rain game II 146–8 climate change II 195–7, 217–49
Subject Index
cooperative international optimum II 263–4 institutional issues I 43–6 organization types III 362–3 surplus sharing local games I 52–4 transfrontier pollution II 177–93 value over noncooperation II 240–1 cooperative games II 160–5, 173, 180–1, 184–6, 260 cost sharing II 131–3 international pollution control II 251–74 see also dynamic cooperative games core of economies pure exchange I 63–79 with multilateral environmental externalities II 153–75 cost sharing acid rain game II 145–6 pollution models II 130–3, 189–90 costs abatement costs II 177–93, 204–5 Bowen–Laffont process I 35, 45 marginal costs I 35; II 204–5 pollution damages II 191–2, 204 transport IV 453 courts III 329–33 backlog evaluation III 329–33 Belgium III, 329–33 data III 329–31 FDH analysis III 281–2, 329–33 inefficiency III 332–3 CWS model see CLIMNEG world simulation model damage costs, pollution II 149–51, 191–2, 204 data envelopment analysis (DEA) III 280–1, 283, 414 convexity III 407–8, 414 efficiency dominance III 433 FDH analysis III 312, 314–19, 321–2, 327–8, 338–40 Malmquist indexes III 420–1 non-frontier efficiency measures III 390
577
nonparametric efficiency III 396–400, 402, 404, 406–7, 414–18 progress/regress III 414–18, 427 reference production sets III 396–400, 402, 404, 406–7 DDEN see population density differential DEA see data envelopment analysis decentralized resource allocation processes I 1–95 commodity exchanges I 81–95 exchange processes I 49–95 gradient processes I 81–95 private/public goods I 1–95 decision-making, FDH analysis III 339 degrees, efficiency III 404, 406 delivery points, post office data III 300 deregulation III 355–6, 365–6 dichotomic variables, regional IV 542 DIS see geographic distance discharges into North Sea II 108 discrete time Malinvaud–Dre`ze–de la Valle´e Poussin process I 33–5 distribution profiles, local games I 49, 54 domestic tax rates IV 523–4 domestically optimal fiscal choices IV 518–27 elasticity formulae IV 523 fiscal externalities IV 527 lump-sum taxation IV 520, 522 in one country IV 518–27 dominance III 283–4, 431–42 benchmark observations III 378–80 efficiency dominance III 431–42 FDH measurement III 320–2 nonparametric efficiency III 398, 407–9, 413 pairwise dominance III 376–7, 407, 434–7, 439 progress/regress III 409, 413 public bank branches III 323–5 urban transit III 335, 338
578
double autarky regime IV 496 dynamic cooperative games II 251–74 closed-loop vs. open-loop II 252–3, 270–1 components II 260–1 computability issues II 265–7 cooperative international optimum II 263–4 economic model II 260 equations and parameters II 272 imputations II 262–3 international pollution control II 251–74 numerical example II 267–70 dynamic resource allocation processes I 9–48 institutional issues I 37–46 interpretive comments I 37–46 modifications/extensions I 23–36 public goods I 9–48 E-dominance III 434–40, 441 ECM see error correction model econometric methods IV 543–53 confidence intervals IV 543–4 estimation techniques IV 543 explanatory variable selection IV 545 personal income tax IV 550–3 results/interpretations IV 546–53 taxation IV 543–6 truncation effects IV 545–6 economic performance III 345–7 economic variables, pollution models II 110 economic-ecological system II 125–6 economies with environmental externalities II 153–75 economies with public goods one public/one private good I 11–13 processes I 51–2 quasi-linear utilities I 75–7 solution concepts I 13–14 states I 11–13 EDA see efficiency dominance analysis Edgeworth barter process I 60–1
Subject Index
EFF see Pareto efficiency efficiency analysis III 275–444 efficiency dominance analysis III 431–42 FDH analysis III 280–4, 311–42 non-frontier measures III 373–94 nonparametric efficiency III 395–429 post office labor III 279–80, 285–309 progress/regress III 373–94 public sector performance III 343–72 with time series data III 373–94 with panel data see panel data see also Pareto efficiency efficiency dominance analysis (EDA) III 283, 431–42 definitions III 433–5 E-dominance III 434–40 measurement III 435–40 motivation III 431–3 elasticity formulae IV 523 emissions abatement II 135–52, 217–49, 268 control rates II 237–9 international cooperation II 217–49 Kyoto Protocol II 197 marginal abatement costs II 204–5 optimal abatement rates II 268 quotas II 205–6 scenarios II 235–41 trading II 206–11 E.N.E. see environmentally nationalistic equilibrium energy prices II 202–5 environmental externalities II 153–75 environmental variables, pollution II 109–10 environmentally nationalistic equilibrium (E.N.E.) II 112–13 equilibria regional markets IV 451–65 tax competition IV 449–89
Subject Index
see also Nash equilibria; noncooperative equilibrium; noncooperative fiscal equilibrium error correction model (ECM) IV 538 estimation econometric methods IV 543 personal income tax IV 551 property tax IV 547 Estonia II 139–51 exchange processes adjustment direction I 73 competitive allocations I 63–79 convergence example I 71–7 core convergence I 63–79 definition I 67–8 formulation I 71–4 monotonicity I 74–5 replicable processes I 68–70 exemption systems, taxation IV 514–15 explanatory variables, taxation IV 545, 550, 552–3 extreme dominance indexes III 438–40 fallback noncooperative equilibrium II 258 Farrell method see convex polyhedral method FDH see free disposal hull analysis feasibility, of a negotiation process II 116–18 feasible sets, input-efficiency III 289, 291–4 federation member states, taxation IV 449–89 Feldstein–Hartman description of the f.t.a.d. system IV 517 Finland II 139–51 fiscal choices Pareto efficient IV 472–4 fiscal competition IV 443–506 consumer behavior IV 493–5 efficiency IV 472–6 models IV 492–8 noncooperative fiscal equilibrium IV 465–6
579
optimality IV 443–506 Pareto improving commodity tax changes IV 491–506 fiscal externalities IV 451–65, 527 Fiscal Politics Conference 1976 I 9 fiscal reaction functions IV 466–70, 497–502 foreign capital income taxation IV 507–32 alternative systems IV 507–32 bases/rates IV 512–14, 528–30 internationally optimal tax rates IV 528–30 taxonomy IV 514–17 variants IV 516 fossil fuels II 203–5 France III 363, 365 free disposal hull (FDH) analysis III 280–4, 311–42 banking III 281, 322–9, 357, 364 convexity III 399–400, 407–8, 421, 432 courts III 281–2, 329–33 four viewpoints III 339–40 frontiers III 313–22, 339–40 Malmquist indexes III 420 measurement III 313–22, 334–8 methodological issues III 313–22 nonparametric efficiency III 396–400, 402, 404, 406–7, 409–14 progress/regress III 409–14, 422–3, 425–7 reference sets III 311–16, 396–400, 402, 404, 406–7 sequential measurement III 334–8 STIB data III 390 urban transit III 281–2, 334–8 free riding I 11; II 119–20, 173, 190–1 frontier analysis data envelopment analysis III 414–18 EDA versus frontier efficiency III 431–2, 440–2 FDH analysis III 313–22, 334–40, 409–14
580
frontier analysis (continued ) Malmquist indexes III 418–21 mathematical programming III 404–8 non-frontier measures III 374–94 nonparametric efficiency III 404–21 progress/regress III 408–21 shifts III 408–21 full crediting abroad, taxation IV 523–4 full dominance indexes III 438–40 full taxation after deduction (f.t.a.d.) systems IV 516–17, 524, 526 games see cooperative games; dynamic cooperative games; local games; noncooperative games gamma-characteristic functions, cooperative games II 164–5 gamma-core of economies with environmental externalities II 165–73 in dynamic cooperative games II 232–3, 242–5, 260–3 linear payoff functions II 165–9 nonlinear payoff functions II 169–73 geographic distance (DIS) IV 542, 548 global approaches, local games I 58–9 global E-dominance III 435, 440 goodness of fit III 339, 432 gradient processes commodity exchanges I 81–95 M70 processes I 87–9 MDP process I 89–94 projections I 84–7 welfare maximization I 84–7 group rationality, acid rain game II 145–6 impoˆt des personnes physiques (IPP) see personal income tax imputations, games I 54; II 186–9, 262–3 income per resident, taxation IV 542
Subject Index
indexes BPR indexes III 423–4 efficiency dominance III 435–40 extreme indexes III 438–40 full indexes III 438–40 integer indexes III 438–40 Malmquist indexes III 378, 418–21 nonparametric efficiency III 404, 418–21, 423–4 progress/regress III 380–9 regime indexing IV 458 individual income tax see personal income tax individual rationality acid rain game II 142–5 emissions agreements II 230–1 transfer schemes II 256–9 transfrontier pollution II 116–18 induced regimes, fiscal equilibrium IV 470–2 inefficiency Belgian courts III 332–3 input-efficiency III 290–1 noncooperative fiscal equilibrium IV 491–506 X-inefficiency III 351–2 see also efficiency analysis input-efficiency III 289–97 feasible sets III 289–94 induced measures III 294–7 measuring inefficiency III 290–1 observed data III 291–7 post office labor efficiency III 289–97 input/output disposability method, labor efficiency III 285, 288, 294, 308 institutional issues interpretive comments of resource allocation processes I 37–46 planning I 38–40, 43–6 public goods literature I 10–11 voluntary exchange I 43–6 insurance III 360, 362–3, 365 integer dominance indexes III 438–40
Subject Index
integrated economy-climate world (CWS) model II 220–2 interaction dynamics, taxation IV 533–56 international aspects behavioral assumptions II 111–13 climate change II 195–215 cooperation II 135–52, 217–49 environmental agreements II 251–74 environmental externalities II 153–75 negotiations II 107–21, 195–215 optimality II 252–6, 268; IV 527–31 pollution control II 251–74 real capital flows IV 510–12 tax rates IV 527–31 transfers II 191, 218–19, 231–2, 244–5 transfrontier pollution II 107–21 international pollution control II 251–74 closed/open-loop models II 270–1 coalitional rationality II 260–5 individual rationality II 256–9 international optimality II 252, 253–6 time factors II 256–9 transfer schemes II 251–74 interpretive comments dynamic resource allocation I 37–46 econometric methods IV 546–53 public goods processes I 40–3 transfrontier pollution models II 128–9 interregional fiscal externalities IV 451–65 interregional noncooperative fiscal equilibrium IV 465–6 intertemporal analysis mathematical programming III 404–8 non-frontier efficiency III 377, 390, 392 reference production sets III 402–3
581
IPP (impoˆt des personnes physiques) supplements see personal income tax IRPE see Pareto efficiency Japan II 203–5 Justices of the Peace (JPs) III 329–33 Kuhn–Tucker multiplier IV 462, 473 Kyoto Protocol II 195–215, 229 coalitional stability II 205–6, 245–6 economics II 198–202 efficiency II 205–6 main features II 197 noncooperation II 202–5 post-Kyoto policies II 245–6 quotas II 205–6 world coalitionally stable optimum II 211–14 L (Lindahl) process I 15–18, 36 labor efficiency, post offices III 279–80, 285–309 Lindahl equilibrium II 173, 191 Lindahl (L) process I 15–18, 36 Lindahl–Malinvaud (LM) process I 18–20 linear payoff functions II 165–9 linear programming, FDH measurement III 316–20 Lipschitz property I 77; II 118 literature review, taxation IV 533–6 LM (Lindahl–Malinvaud) process I 18–20 local games I 49–62 definition I 54 distribution profiles I 54 imputations I 54 local/global approaches I 58–9 MDP process I 33 solution concepts I 54–6, 59 strategically stable processes I 57–8 surplus sharing I 33, 49–62 lump-sum taxation IV 517–20, 522 optimal fiscal choices IV 520
582
lump-sum taxation (continued ) restricted taxes IV 525–7 unrestricted taxes IV 524 Lyapunov function I 77, 87 M – m process I 59–61 M70 process I 87–9 macroeconomic magnitudes II 239–40 mail handling III 299 Malinvaud’s theory of planning I 39 Malinvaud–Dre`ze–de la Valle´e Poussin (MDP) process I 20–3, 32–5, 49, 89–94 Malmquist indexes III 378, 418–21 marginal costs acid rain game II 149–51 Bowen–Laffont process I 35 emissions abatement II 204–5 market equilibrium, taxation IV 518–20 Markov perfectness II 255 Marshallian process I 67–70, 72 mathematical models, pollution II 107–21, 123–34 mathematical programming, nonparametric efficiency III 404–8 MDP see Malinvaud–Dre`ze–de la Valle´e Poussin process measurement efficiency dominance III 435–40 FDH analysis III 313–22, 334–8, 340 mathematical programming III 404–8 non-frontier efficiency III 373–94 nonparametric efficiency III 395–429 post office labor efficiency III 279–80, 285–309 productive efficiency III 347–51 progress/regress III 373–429 Mintz–Tulkens model IV 491–506 monetary transfers, pollution II 184–5 monotonicity I 74–5
Subject Index
motivation, fiscal equilibrium IV 465–6 multi-region economy–climate (RICE) model II 218, 221–2 multidimensional objectives, public sector performance III 345–7 multilateral environmental externalities II 153–75 cooperative games II 160–5, 173 economy model II 155–7 linear payoff functions II 165–9 noncooperative games II 157–60 municipal population variable IV 541 mutual insurance organizations III 362–3, 365 Nash equilibria acid rain game II 143–4, 148, 150 carbon emissions II 218 cooperative games II 160–2, 167 cost-sharing II 189 CWS model II 223–5 emissions II 199–201, 218, 223–5, 235–41 foreign capital taxation IV 508, 531 gamma-core properties II 173, 189 noncooperative games II 157–60, 182–3; IV 493 PANE II 229–30, 234, 243–4 see also equilibria national optimum tax systems IV 520–7, 530–1 negotiations II 195–215, 217–49 climate change II 195–215, 217–49 CWS model II 217–49 transfrontier pollution II 107–21, 123–34 noncooperation, Kyoto Protocol II 202–5 noncooperative equilibrium (NCE) emissions II 199 fiscal equilibrium IV 465–6, 470–2, 491–506 Nash equilibrium II 255 negotiations II 113, 126, 130
Subject Index
transfers II 253–6 see also equilibria noncooperative fiscal equilibrium (NCFE) definitions IV 449, 451, 493 existence IV 470–2 fiscal competition IV 465–6 fiscal reaction functions IV 494 induced regimes IV 470–2 inefficiency IV 491–506 reaction curves IV 471 see also equilibria noncooperative games II 157–60, 180–3; IV 493 non-frontier efficiency measures III 373–94 benchmark observations III 378–89 benchmark production correspondence III 380–9 dominance III 376–80 nonparametric limitations III 377–8 pairwise dominance III 376–7 progress/regress III 373–94 set-frontier efficiency III 374–6 STIB data III 390–3 time series data III 373–94 nonlinear payoff functions II 169–73 nonparametric efficiency III 395–429 contemporaneous frontiers III 404–8 data envelopment analysis III 396–400 FDH analysis III 396–400, 404, 406–7 intertemporal frontiers III 404–8 mathematical programming III 404–8 non-frontier measures III 374, 377–8 panel data III 395–429 production sets III 395–404 progress/regress measures III 395–429 sequential frontiers III 404–8
583
see also data envelopment analysis; productive efficiency nonprofit organizations III 353 non-tax explanatory variables IV 541–2 normative interpretations, processes I 40–3 North Sea, pollution II 107–21 observations input-efficiency III 291–7 production sets III 396–400 reference subsets III 404–6, 412–13 time-related observations III 401–4, 412 one region equilibria types IV 454–7 open-loop Nash equilibrium II 218, 223–5 open-loop transfer schemes II 270–1 optimality acid rain game II 136–42 fiscal choices in one country IV 518–27 fiscal competition IV 443–506 foreign capital taxation IV 507–32 international aspects II 252–6, 268; IV 527–8 public goods I 36 pure exchange economy I 82–4 regional market equilibria IV 451–65 tax formulae IV 520–4 outliers, FDH measurement III 320–2 output input/output disposability method III 285, 288, 294, 308 post office data III 299–300 ownership III 344, 351–7, 362 pairwise dominance E-dominance III 434–7, 439 non-frontier efficiency III 376–7 nonparametric efficiency III 407 PANE see partial agreement Nash equilibria w.r.t. coalition
584
panel data mathematical programming III 404–8 nonparametric efficiency III 395–429 progress/regress III 395–429 reference production sets III 401–4 parametric methods, productive efficiency III 348–50, 359, 361, 363, 365 Pareto efficiency CWS model II 219, 225–9 double autarkic equilibrium IV 504 fiscal choices IV 472–4 IRPE I 64, 66 without transfers II 235–44 Pareto efficiency without transfers (EFF) II 235–44 Pareto efficient fiscal choices (PEFC) IV 472–4 Pareto improving commodity tax changes IV 491–506 partial agreement Nash equilibria w.r.t. coalition (PANE) II 229–30, 234, 243–4 passive dominance III 436–9 payoffs, emissions II 268–70 PEFC see Pareto efficient fiscal choices performance banking efficiency III 322–9 economic performance III 345–7 indexes III 380–9 indicators III 344, 347, 350 public sector III 282, 284, 343–72 personal income tax (IPP supplements) IV 533, 540–1, 545–6, 549–53 adjustment coefficients IV 549–50, 552 econometric methods results IV 550–3 estimator table IV 551 explanatory variable coefficients IV 552–3 parameter values IV 551–2 planning, institutional issues I 38–40, 43–6
Subject Index
players, coalitions II 154–5 pollution international control II 251–74 North Sea model II 107–21 ‘polluters pay’ principle II 117, 120–1 transboundary air pollution II 135–52 transfer schemes II 251–74 transfrontier models II 107–21, 123–34, 177–93 population IV 541–2, 548–9 population density differential (DDEN) IV 548–9 positive interpretations, processes I 40–3 post office labor efficiency III 279–80, 285–309 Cobb–Douglas function III 285–6, 292, 305, 307 computations/results III 300–7 convex polyhedral method III 285, 287, 293–4, 307–8, 311–12 data III 298–300 input-efficiency III 289–97 input/output disposability method III 285, 288, 294, 308 measurement III 285–309 technical efficiency III 285–6 pre´compte immobilier (PrI) see property tax preferred rates, taxation IV 536–9 presence of tax interaction IV 536–7 PrI (pre´compte immobilier) supplements see property tax price adjustment function I 61 M – m process I 59–61 public goods I 28–32 price-based processes see Lindahl–Malinvaud process private goods decentralized resource allocation processes I 1–95 dynamic resource allocation processes I 23–8
Subject Index
Walrasian approach I 27–8 private sector banking III 326–9, 357–8, 364 ownership III 351–7, 362 privatization III 351, 353 productive efficiency III 351–64 regulation III 355–6 privatization III 351, 353 processes decentralized resource allocation processes I 1–95 definitions I 13–14 dynamic exchange processes I 49–62 Edgeworth barter process I 60–1 institutions I 37–8, 40–3 Lindahl process I 15–18, 36 Lindahl–Malinvaud process I 18–20 M70 processes I 87–9 Marshallian process I 67–70, 72 MDP process I 20–3, 32–5, 49, 89–94 normative versus positive interpretations I 40–3 price formation I 42–3 public goods economy I 11–23, 51–2 surplus sharing local games I 49–62 see also Bowen–Laffont process; exchange processes; gradient processes production, benchmark production correspondence III 380–9, 421–4, 427–8 production frontiers III 313–22, 334–40 production plans III 376–7, 396–400 production sets III 395–404, 406–8, 420 productive efficiency III 343–72 competition III 351–6, 363–5 empirical studies III 356–67 frontier method III 347–50 ownership III 351–7, 362 public enterprises III 343–72 regulation III 351–6, 365–6 productivity indexes III 378
585
profit III 345, 353 progress/regress III 373–429 benchmark production correspondence III 421–4, 427–8 data envelopment analysis III 414–18, 427 definitions III 409–10 FDH analysis III 409–14, 422–3, 425–7 frontier shifts III 408–21 indexes III 380–9, 393 measures III 373–429 non-frontier efficiency III 373–94 nonparametric efficiency III 395–429 panel data III 395–429 in progress/regress observations III 380 time series data III 373–94 proofs, taxation propositions IV 478–88 property tax (PrI supplements) IV 533, 540–1, 545–6 adjustment coefficients IV 549–50 estimators IV 547 explanatory variables IV 550 parameters IV 548 results IV 546–50 proportionality, FDH analysis III 320–2 public goods decentralized resource allocation processes I 1–95 environmental externalities II 97–274 many producers I 28–32 public sector banking III 322–9 commodity tax changes IV 495–7 performance III 282, 284, 343–72 post office labor efficiency III 279–80, 285–309 productive efficiency III 343–72 pure exchange economy I 63–79, 82–4
586
quantity-based processes see Malinvaud–Dre`ze–de la Valle´e Poussin process quasi-linear utilities, core economies I 75–7 quotas, emissions II 205–6 railway efficiency studies III 366–7 Ramsey formula IV 463 Ramsey–Keynes rule II 218–19, 225 rates, taxation IV 512–14, 530–1 rationality, individual acid rain game II 142–6 emissions agreements II 230–3 transfrontier pollution II 116–18 reaction curves IV 471 reference sets FDH analysis III 311–16 Malmquist indexes III 420 nonparametric efficiency III 396–414 observations subsets III 404–6, 412–13 production sets III 396–404, 406–8, 420 progress/regress III 409–14 refuse collection III 360, 363, 365 Re´gie des Postes, Belgium III 285–309 regimes definition IV 457 double autarky IV 496 indexing IV 458 regional market equilibria (RME) fiscal optima IV 451–65 interregional fiscal externalities IV 464–5 model IV 452–4 one region types IV 454–7 regimes IV 457–60 regions dichotomic variables IV 542 fiscal optima IV 460–3 fiscal reaction functions IV 466–70 interdependence, public sector behavior IV 495–7 market equilibria IV 451–65
Subject Index
social indifference curves IV 497 regress see progress/regress regulation III 351–6, 365–6 replicability, FDH analysis III 320–2 replicable exchange processes I 68–70 residence-based taxation systems IV 514–15 resource transfers II 205–6 restricted lump-sum taxation IV 525–7 retail banking see banking RICE model see multi-region economy–climate model Rio de Janeiro framework convention 1992 II 196 RME see regional market equilibria Roy’s identity IV 496, 520 Russia II 139–51 Samuelson rule II 228, 231 selecting explanatory variables, taxation IV 545 self-managed firms III 353–4, 362–3 sequential analysis FDH measurement III 334–8 mathematical programming III 404–8 non-frontier efficiency III 377–8, 390, 392–3 progress/regress III 408, 410, 413–14, 417–18, 421 reference production sets III 402–3 setwise E-dominance III 434–5, 437–40 simulations, emissions II 233–47, 267–70 ‘small country’ assumption IV 510 social welfare see welfare Socie´te´ des Transports Intercommunaux de Bruxelles (STIB) III 336–8, 390–3 source-based taxation systems IV 514–15 Southern Bight II 107–21 states, economies I 11–13 statistics, emissions II 202
Subject Index
STIB (Socie´te de Transports Intercommunaux de Bruxelles) III 336–8, 390–3 strategically stable processes, local games I 57–8 strategically stable solutions, MDP process I 32–3 sulphur emissions II 139–42 surplus sharing local games see local games surpluses, MDP process I 93 survey I 9–48 taxation IV 443–556 absence of tax interaction IV 536 alternatives systems IV 507–32 bases IV 512–14 Belgian municipalities IV 533–56 choice of variables IV 540–2 commodity tax IV 449–89 competition IV 449–89 countries IV 509 data IV 540–2 econometric methods IV 543–53 economic model specification IV 536–40 foreign capital income IV 507–32 formulae IV 520–4 internationally optimal rates IV 527–31 literature review IV 533–6 lump-sum taxation IV 524–7 market equilibrium IV 518–20 mimicry IV 533–4 national optimum systems IV 520–7, 530–1 Pareto improving commodity tax changes IV 491–506 preferred rates IV 536–7 rates IV 512–14, 527–31, 536–7 tax competition IV 533 tax-space subsets IV 459 tax interaction dynamics IV 536–40 technical efficiency III 285–6, 347, 350–1 technical progress
587
STIB III 336–8 see also progress/regress temperature changes II 237, 268 theory of tournaments III 354–5 time series non-frontier efficiency III 373–94 progress/regress III 373–94 reference production sets III 403 urban transit III 334–8 time-related observations, efficiency analysis III 401–4, 412 Tizio–Caio economy I 28–9 tournaments theory III 354–5 trading, emissions II 197, 206–11 transaction rule, Uzawa I 61 transboundary air pollution abatement II 135–52 transfers Chander–Tulkens transfers II 146–7, 150–1, 213, 230–2, 246–7, 256–63, 269 in CWS model II 246–7 of resources II 205–6 transfrontier pollution models II 107–21, 123–34, 177–93 basic models II 109–16, 124–6, 178–9 behavioral assumptions II 111–13 cooperative games associated with II 180–1, 184–6 cost sharing II 130–3, 189–90 damage-cost function II 191–2 economic/ecological components II 181–2 formal properties II 128 free riding II 190–1 gamma-core imputations II 186–9 international transfers II 191 interpretations II 128–9 modelling negotiations II 126–7 monetary transfers II 184–5 Nash equilibria II 182–3 negotiations II 113–20, 126–8 noncooperative games associated with II 182–3 notation II 109–11
588
transfrontier pollution models (continued ) ‘polluters pay’ principle II 117, 120–1 statement of process II 127 strategic considerations II 130–3 transport IV 453 see also urban transit truncation effects, econometric methods IV 545–6 two-region economies IV 451–65 United States of America (USA) II 203–5, 234, 238, 240–1, 242–246 unrestricted lump-sum taxation IV 524 urban transit III 334–8, 390–3 data III 336 FDH analysis III 281–2, 334–8 production frontier shifts III 334–6 sequential FDH III 334–5
Subject Index
STIB III 336–8, 390–3 technical progress III 336–8 time series III 334–8 USA see United States of America variables, taxation IV 540–2 voluntary exchange institutions I 43–6 Walrasian price adjustment approach I 27–8, 61, 77–8 water quality, North Sea II 107–21 welfare functions IV 491–506 maximization problems I 84–7 public sector performance III 345–6 social indifference curves IV 497 window analysis III 403–4 world optimum, emissions abatement II 198, 217–18 X-inefficiency III 351–2
