CONTENTS FRANCISCO J. BUERA, ALEXANDER MONGE-NARANJO, AND GIORGIO E. PRIMICERI: Learning
the Wealth of Nations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TOMASZ STRZALECKI: Axiomatic Foundations of Multiplier Preferences . . . . . . . . . . . . . . . FERNANDO ALVAREZ AND ROBERT SHIMER: Search and Rest Unemployment . . . . . . . . . . . .
1 47 75
The Distribution of Wealth and Fiscal Policy in Economies With Finitely Lived Agents . . . . . . . . . . . . . . . . . . . . . . . . . . TAVNEET SURI: Selection and Comparative Advantage in Technology Adoption . . . . . . . . JOHN KENNAN AND JAMES R. WALKER: The Effect of Expected Income on Individual Migration Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . THOMAS J. HOLMES: The Diffusion of Wal-Mart and Economies of Density . . . . . . . . . . . .
211 253
ANNOUNCEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FORTHCOMING PAPERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REPORT OF THE SECRETARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REPORT OF THE TREASURER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REPORT OF THE EDITORS 2009–2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ECONOMETRICA REFEREES 2009–2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REPORT OF THE EDITORS OF THE MONOGRAPH SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES . . . . . . . . . .
303 307 309 319 327 331 341 345
JESS BENHABIB, ALBERTO BISIN, AND SHENGHAO ZHU:
VOL. 79, NO. 1 — January, 2011
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Econometrica, Vol. 79, No. 1 (January, 2011), 1–45
LEARNING THE WEALTH OF NATIONS BY FRANCISCO J. BUERA, ALEXANDER MONGE-NARANJO, AND GIORGIO E. PRIMICERI1 We study the evolution of market-oriented policies over time and across countries. We consider a model in which own and neighbors’ past experiences influence policy choices through their effect on policymakers’ beliefs. We estimate the model using a large panel of countries and find that it fits a large fraction of the policy choices observed in the postwar data, including the slow adoption of liberal policies. Our model also predicts that there would be reversals to state intervention if nowadays the world was hit by a shock of the size of the Great Depression. KEYWORDS: Learning, dynamic of policies, economic growth, market-oriented policies, neighborhood effects.
1. INTRODUCTION THIS PAPER EXPLORES THE ABILITY of a learning model to explain the adoption of market-oriented policies over time and across countries. Whether market allocations outperform state intervention has always been the subject of a heated debate. Not only do people around the world hold conflicting views about this issue, but these views change over time as countries learn from their own past experience, as well as the experience of other countries. We show that the evolution of beliefs about the relative desirability of free markets can be a major driving force behind countries’ transitions between regimes of state intervention and market orientation. The notion that past experience shapes beliefs and policy decisions is well rooted in policy circles. For instance, Fischer (1995, p. 102) suggested the rise of socialist ideas in the 20th century as a clear case of interaction of outcomes, ideas, and policies: It is not hard to see why views on the role of the state changed between 1914 and 1945 [. . . ] A clear-headed look at the evidence of the last few decades at that point should have led most people to view the market model with suspicion, and a large role for the state with approbation—and it did.
In a similar vein, Krueger (1997) argued that the eventual dismantlement of these ideas was the consequence of the change in views that “resulted from a combination and interaction of research and experience with development and development policy.” The testimony of a central figure in this dismantlement (Margaret Thatcher in Cran (2002)) also hints at the importance of 1 We would like to thank the editor, Graziella Bertocchi, V. V. Chari, Chad Jones, Ramon Marimon, Joel Mokyr, Juan Pablo Nicolini, Monika Piazzesi, Andrés Rodríguez-Clare, Martin Schneider, Ruben Segura, Neil Wallace, Tao Zha, four anonymous referees, and participants in many seminars and conferences. We are also grateful to Alonso Alfaro for excellent research assistance and to James Fearon for providing us with an updated series of the data on civil wars (Fearon and Laitin (2003)).
© 2011 The Econometric Society
DOI: 10.3982/ECTA8299
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cross-country information spillovers: I remember the foreign minister and the finance minister from another country saying to me: “You’re the first prime minister who has ever tried to roll back the frontiers of socialism. We want to know what’s going to happen. Because if you succeed others will follow.
We formalize this connection between ideas, policies, and economic development in the context of a model where the performance of alternative policy regimes is uncertain. Policymakers start with some priors about the growth prospects of market-oriented and state-interventionist regimes, and use Bayes law to update these priors with the arrival of new information from all countries in the world. They may find it optimal to discount information from other countries because the effect of policy regimes on economic growth may be heterogeneous and spatially correlated. A country decides to pursue marketoriented policies if the perceived net impact of these policies on its gross domestic product (GDP) growth exceeds their political cost. We assume that this cost has a country-specific component, and we let it also be a function of the economic and political characteristics of the country and of its neighbors, including the exposure to military conflicts. We estimate the model (initial prior beliefs and parameters of the political cost) with data on the market orientation, GDP, and a number of geographical, political, and civil war indicators for a panel of 133 countries over the years 1950–2001. We discipline our estimation procedure by imposing that countries’ initial beliefs in 1950 must be consistent with the data available for the 1900– 1950 period. Our first question is whether such a learning mechanism can explain the data on the adoption of market-oriented policies. We indeed find that the baseline model correctly predicts almost 97% of the policy choices observed in the data. A more interesting challenge, however, is whether the model is able to predict policy switches, given the persistence of policy regimes, as reflected by the fact that only about 2% of the observations correspond to a policy switch in our data set. The baseline model is also strong in this dimension. It accounts for 258%, 538%, 742%, and 774% of the actual policy switches within windows of ±0, 1, 2, and 3 years, respectively. The empirical success of the model is mainly driven by the evolution of beliefs, and not by the countries disparate and changing economic and political conditions. Indeed, if we shut down the learning mechanism—and estimate fixed beliefs, but otherwise allow the economic, political, and military controls to vary as in the data—then our model loses more than two-thirds of its predictive power. This holds despite the fact that our estimates suggest that countries are more likely to pursue market-oriented policies when they are more democratic, have higher per capita income, and are not hosting a civil war, among other things. Therefore, the inclusion of these variables is supported by the data, but does not undermine the importance of evolving beliefs as determinants of policy changes.
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The main challenge of our model is to account for the observed slow and late trend toward market orientation for most developing countries. The model explains it as follows. First, the information in the pre-1950 data led us to a calibration of priors consistent with most countries believing that governments outperform markets. Second, data from the 1950s and early 1960s—the beginning of our estimation sample—apparently consolidated these views, since state-interventionist economies grew on average faster than expected and, particularly, faster than market-oriented economies. In agreement with our quotation from Fischer, our learning model would then predict policymakers to favor state intervention, at least until new information would lead them to revert their beliefs. In fact, in agreement with our quotation from Krueger, the growth collapse of the late 1970s and the 1980s triggered a switch in policymakers’ beliefs. Through the lens of our model, the more dismal performance of state-interventionist countries during this period led some policymakers to downgrade their optimism for state intervention and move to market orientation. The superior performance of market-oriented countries during the 1990s consolidated these views and led even more countries to switch. The empirical strength of our model allows us to tackle a number of other important questions about the relationship between beliefs formation and policy decisions. First, we ask what features of the model are crucial for the slow adoption of market-oriented policies. We find that the resistance to markets is a very robust property of our learning model, as long as policymakers entertain the possibility that the impact of various economic policies is heterogeneous across countries. Faster diffusion of market regimes would have required policymakers to be fully convinced that the effects of policies are the same in every country. Ruling out a priori the possibility of heterogeneity seems at odds with the observed behavior of policymakers (see, for example, Di Tella and MacCulloch (2007)). Second, we investigate whether the world has now reached a point in which the vast majority of policymakers are convinced that market-oriented regimes are beneficial for growth. According to our results, we have not reached that point. Our estimated end-of-sample beliefs indicate that a considerable number of policymakers hold negative views about the merits of market economies. Moreover, many of those with favorable views are either quite uncertain or believe that the gains are quantitatively small. This result motivates a third question: “Would a global recession induce some policy reversals to interventionism?” Our answer is “Yes.” More specifically, we estimate that approximately 5–10% of market-oriented countries would become state interventionist if the world economy experienced a negative growth shock similar to the Great Depression of the 1930s. In this paper, we focus on the formation and evolution of beliefs as a key force driving the choice of economic policies, as advocated by Kremer, Onatski, and Stock (2001). To this end, we abstract from many other interesting determinants of policy decisions, such as redistributional issues, interest-
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group politics, and the role of multilateral institutions, which are the subject of an extensive political economy literature (see Grossman and Helpman (1995), Krusell and Rios-Rull (1996), Acemoglu, Johnson, and Robinson (2005), and Persson and Tabellini (2002), among many others.) Within this literature, our paper is perhaps most closely related to Alesina and Angeletos (2005), Piketty (1995), Mukand and Rodrik (2005), and Strulovici (2008), because policy choices are the outcome of decisions of rational agents learning from past experience. Relative to these papers, we abstract from voting and redistributive considerations to attain instead an empirically tractable multicountry learning model. Our study is also connected to some recent work that emphasizes the role of historical factors on policy choices. For instance, Acemoglu, Johnson, and Robinson (2001, 2005), Sokoloff and Engerman (2000), and Engerman and Sokoloff (2002) documented a very strong correlation of countries’ initial endowments and colonial histories with their current institutions and level of development. While these papers concentrate on the determinants of political regimes, our interest in explaining economic policies leads us to take political regimes as given. This modeling choice is in line with the results of Acemoglu, Johnson, Robinson, and Yared (2008, 2009), who argued that a country’s development does not affect the choice of political regimes if we control for long-run factors (which, in our model are estimated on a shorter sample, are captured by country-specific effects). Our approach is also consistent with Giavazzi and Tabellini (2005), who found strong evidence about the effect of political regimes on economic policies and much weaker evidence of the converse. However, with the exception of the recent contribution of Persson and Tabellini (2009), none of these papers considers the impact of a country’s neighbors’ experience on policies or institutions. Models of policymakers as learning agents have been successfully applied to explain the rise and fall of U.S. inflation (see, for instance, Sargent (1999), Cogley and Sargent (2005), Primiceri (2006), and Sargent, Williams, and Zha (2006)). Differently from this literature, policymakers in our model do not face a complex trade-off between alternative policy objectives. Our interest is instead in the role of learning spillovers among countries. From this point of view, our paper is also related to the literature on social learning and information spillovers in technology adoption and diffusion (see, for instance, Besley and Case (1994), Foster and Rosenzweig (1995), and Conley and Udry (2010)). This paper is also linked to a fast growing body of work on the empirical behavior of beliefs (see, for example, Aghion, Algan, Cahuc, and Shleifer (2008), Alesina and Giuliano (2009), Alesina and Fuchs-Schundeln (2007), Alesina and Glaeser (2004), Di Tella and MacCulloch (2007), Mayda and Rodrik (2005), and Landier, Thesmar, and Thoenig (2008)). These studies have the appealing feature of using direct measures of beliefs in the data. In this paper, instead, we infer policymakers’ beliefs from a structural learning model to overcome the problem of the lack of a consistent measure of beliefs for
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the many countries and periods that we consider. However, we find that our implied beliefs correlate mostly in the correct direction with the available data from the two leading cross-country surveys (the World Value Survey and the Latinobarometro). Moreover, Aghion et al. (2008), Alesina and Glaeser (2004), and Di Tella and MacCulloch (2007) found support for the correlation between beliefs and actual policies adopted across countries, which is the central theme of our paper. Furthermore, Landier, Thesmar, and Thoenig (2008) found that beliefs are slowly moving, home biased, and path dependent, which is qualitatively consistent with our results. Finally, our work draws on the empirical literature on policy decisions and growth (e.g., Jones (1997)) and the impact of trade policies and market orientation (e.g. Dollar (1992), Sachs and Warner (1995), Edwards (1998), or Rodriguez and Rodrik (2000)). Notice, however, that our focus is exactly on the converse, that is, understanding the adoption of market-oriented policies. This is somewhat similar to Blattman, Clemens, and Williamson (2002), Clemens and Williamson (2004), and, to a lesser extent, Sachs and Warner (1995), although none of these papers stresses the importance of past growth performances for current policy choices. The rest of the paper is organized as follows. Section 2 presents our theoretical model. Section 3 describes the estimation methodology. Section 4 examines the dynamics and geography of market orientation and economic growth during the postwar period. The estimation results and counterfactual exercises are discussed in Sections 5–8, and Section 9 concludes. Data and programs are available in the Supplemental Material (Buera, Monge-Naranjo, and Primiceri (2010)). 2. MODEL In this section we present a simple model in which the sequential arrival of information reshapes policymakers’ beliefs and affects policy choices in each of the n countries of the world economy. 2.1. The Perceived Data-Generating Process and the Policy Problem We simplify the choice of policies to the dichotomic case of countries that are either market oriented or heavily rely on state intervention. Policies are chosen period by period and we let θit be an indicator variable that equals 1 if country i is characterized by a market-oriented economy in period t and equals 0 otherwise. Let Yit denote per capita GDP in country i at time t and let yit ≡ log Yit − log Yit−1 denote its growth rate. Policymakers are in power for one period and choose θit to solve (2.1)
max Eit−1 [log Yit − θit Kit ] θit
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
where the term Kit corresponds to the political and social costs of marketoriented policies at time t. By imposing a policy objective as in (2.1), we are implicitly abstracting from voluntary experimentation, an assumption that we discuss further in Section 2.2. The maximization is subject to policymakers’ perceived relationship between policy choices and GDP growth, (2.2)
yit = βSi (1 − θit ) + βM i θit + εit
where βSi and βM i represent the average growth rates of country-i GDP under policies of state intervention and market orientation, respectively.2 The term εit is an exogenous shock to the growth rate of GDP in country i that is uncorrelated over time, but potentially correlated across countries. More precisely, the vector εt ≡ [ε1t εnt ] is distributed according to (2.3)
iid
εt ∼ N(0 Σ)
where N denotes the Normal distribution. We assume the following information structure: Policymakers do not know the value of βSi and βM i , but have perfect knowledge of all the remaining model parameters, including the covariance matrix of the growth shocks Σ. The timing of events is as follows: At the end of time t − 1, policymakers in country i observe the data on policy choices and GDP growth of all n countries, and update their beliefs about βSi and βM i . At the beginning of time t, they observe the realization of Kit and then decide what policy θit to adopt. Finally, while the political cost is directly observable by policymakers, it is unobservable to the researcher. For estimation purposes, we assume that Kit is a function of a country-specific term fi and a vector Πit of observable political and economic variables of country i (mainly, a political and a civil war indicator, and lagged per-capita GDP, as discussed below): Kit = fi + ξΠit + kit iid
kit ∼ N(0 2i ) Under this formulation, the political costs Kit can inherit the autocorrelation and cross-country correlation properties from the variables in Πit , even if the shock kit is uncorrelated over time and across countries. Notice also that the volatility of the political costs 2i is allowed to vary across countries. 2
In Buera, Monge-Naranjo, and Primiceri (2008), we experimented with alternative specifications of (2.2), particularly with one that allows the growth rate of GDP to also depend on the gap between a country’s GDP and the technological frontier, as in the convergence regressions of Barro and Sala-i-Martin (1995). Moreover, in Sections 3 and 7, we discuss the relationship between the perceived law of motion (2.2) and the actual law of motion.
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2.2. Optimal Policy Optimal policy at time t is given by (2.4)
S θit = 1{Eit−1 (βM i ) − Eit−1 (βi ) > Kit }
where 1{·} is the indicator function. The optimal policy decision depends on the expected average growth rates: policymakers choose to pursue marketoriented policies if the expected gain in growth rates exceeds the political cost. By imposing a policy objective as in (2.1), we have implicitly followed Sargent (1999) and assumed that policymakers do not intentionally experiment. Abstracting from voluntary experimentation rules out the possibility that a country pursues a policy regime that is perceived to be, on average, detrimental to growth, with the sole purpose of learning about it. We rule out experimentation for a number of reasons. First, while from a normative perspective experimenting might be beneficial, from a positive perspective it remains an open question the extent to which governments conduct social experimentation on the grand scale. Economists like Lucas (1981) or Blinder (1998) have argued against such behavior in policymaking. Moreover, standard voting schemes seem to reduce the occurrence of collective experimentation (see Strulovici (2008)). Second, the gains from experimentation are reduced in our setting because of two forces: (i) our model resembles a two-armed bandit problem (Berry and Fristedt (1985)), but the presence of random variation in the political cost Kit circumvents the incomplete learning results typical of the literature on multiarmed bandit problems (see El-Gamal and Sundaram (1993)); (ii) in our multicountry learning setting, policymakers benefit from the variation in the political costs of other countries, which should lower the incentives to experiment. Third and most important, the joint decision problem of n countries is extremely hard to solve, because it involves strategic experimentation motives.3 2.3. Learning We assume that in period t = 0, policymakers start off with a Gaussian prior M density on the vector of unknown coefficients β ≡ [βS1 βSn βM 1 βn ] : (2.5)
β ∼ N(βˆ 0 ; P0−1 )
where βˆ 0 and P0 represent the prior mean and precision matrix, respectively. For the prior covariance matrix, we choose the following parameterization (2.6)
P0−1 = I2 ⊗ (V · R · V )
3 Much simplified versions of this problem have been solved by Bolton and Harris (1999) and Keller, Rady, and Cripps (2005).
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
where V = diag([ν1 σ1 νn σn ]) and R denotes the a priori correlation matrix. The Kronecker structure implies that we are simplifying our problem by assuming that policymakers start with a prior that assigns zero correlation and the same degree of uncertainty to the impact of market-oriented and stateinterventionist policies on economic growth. The coefficients {νi }ni=1 parameterize this uncertainty, which is also normalized by the standard deviation of the growth shocks in each country ({σi }ni=1 ), that is, the diagonal elements of Σ. Priors are recursively updated with every new vintage of data. Our timing assumptions and the application of Bayes law deliver simple formulas for the optimal updating of the mean and precision matrix of the distribution of β: Pt = Pt−1 + Xt Σ−1 Xt βˆ t = βˆ t−1 + Pt−1 Xt Σ−1 (yt − Xt βˆ t−1 ) gt
where yt ≡ [y1t ynt ] , Xt ≡ [diag(1 − θt ) diag(θt )], θt ≡ [θ1t θnt ] and the recursion is initialized at the prior mean and precision matrix. The gain, gt ≡ Pt−1 Xt Σ−1 , determines the impact of new observations on the posterior mean of β. There are three important sets of coefficients that affect the structure of the gain of this learning problem. First, everything else equal, the higher is the initial uncertainty (νi ) the higher is the relative precision of new observations and the larger is the gain. Second, a higher international correlation among growth shocks (off-diagonal elements of Σ) reduces the total informational content of the vector of growth rates in each period and slows down the updating of beliefs. Third, and crucially, the a priori correlation matrix (R) determines how much weight is given to data from other countries in updating the beliefs about βi . In one extreme, if the effects of policy regimes are perceived to be entirely country specific, then beliefs about each country will be updated using data from that country only. In the opposite extreme, if policymakers believe that the growth effects of policy regimes are the same across countries, then beliefs for each country are updated using data from all other countries. More plausibly, policymakers might have a prior that the growth effect of market orientation in a country is more correlated with that of nearby countries and less so with more distant countries. To capture this idea in a flexible and tractable way, we assume that the prior correlation between the growth effects of government controls (or market friendliness) in countries i and j, Rij , is a parametric function of a vector of covariates zij , (2.7)
γ] Rij = exp[−zij
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γ to be nonnegative.4 This formulation is similar where we constraint zij to that adopted in the literature on geographically weighted regressions (Fotheringnam, Brunsdon, and Charlton (2002)). The vector zij may include various measures of geographic, cultural, or economic distance between country i and country j.
3. INFERENCE Like the agents of our model, we (the econometricians) are also Bayesian and wish to construct the posterior distribution for the unknown parameters of the model. These unknown coefficients are5 {βˆ Sj0 }nj=0 expectation of initial beliefs about the effect of state intervention, n {βˆ M expectation of initial beliefs about the effect of market orientaj0 }j=0 tion, {νj }nj=0 standard deviation of initial beliefs about the effect of state intervention and market orientation, standard deviation of the exogenous component of the political { j }nj=1 cost, fixed effect of the political cost, {fj }nj=1 ξ coefficients of the political cost, γ coefficients parameterizing the correlation of initial beliefs. If we collect the set of unknown coefficients in the vector α, and denote by DT ≡ {yt θt Πt }Tt=1 the entire set of available data on policies, growth, and countries’ political and economic characteristics, then standard application of Bayes rule delivers p(α|D) ∝ L(D|α) · π(α) where p(·), L(·), and π(·) represent the posterior, sampling, and prior densities, respectively, and ∝ denotes the proportionality relation. We now turn to the description of the priors and the construction of the likelihood function. 3.1. Priors Since our model has many parameters, we use informative priors to prevent overfitting problems. For instance, we would like to avoid cases in which we 4 The fact that bivariate correlations are between 0 and 1 does not generally guarantee that a correlation matrix is positive definite if the dimension of the matrix exceeds 2. However, (2.7) belongs to the Matérn family of covariance functions (Matérn (1960)), which guarantees the positive definiteness of R. 5 For simplicity, we set Σ to its estimated value obtained using the entire set of data (see Section 7).
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
fit the data well, but only due to implausible estimates of policymakers’ initial beliefs. As an example, consider the literature on macroeconomic forecasting: highly parameterized models do well in sample, but perform poorly out of sample. The use of priors considerably improves the forecasting performance of these models (see, for instance, Doan, Litterman, and Sims (1984), Litterman (1986), or, more recently, Banbura, Giannone, and Reichlin (2010)). The role of priors is similar in our context since we also aim to use the model to conduct a set of counterfactual experiments. We assume the prior densities βˆ Si0 ∼ N(β¯ S0 ω2β )
i = 1 n
¯M 2 βˆ M i0 ∼ N(β0 ωβ )
i = 1 n
νi ∼ IG(sν dν )
i ∼ IG(s d ) fi ∼ N(f¯ ω2f )
i = 1 n i = 1 n i = 1 n
ξ ∼ Uniform γ ∼ Uniform These prior densities are parameterized as follows: • We set β¯ S0 = 00275 and β¯ M 0 = 00125. We have chosen these numbers using the Maddison (2006) data. First of all, the average annual growth rate of per capita GDP of countries in the Maddison data set in 1901–1950 (excluding the years corresponding to the two wars) is approximately 2%. Then we split the countries of the Maddison data set into two groups according to the value in 1950 of an indicator that measures the degree of market orientation; this indicator was constructed by Sachs and Warner (1995) as we describe below. We find that, between 1946 and 1950, the state-interventionist countries grew on average 15% faster than the market oriented. The values of β¯ S0 and β¯ M 0 ¯S ¯M are then chosen so that β¯ S0 + β¯ M 0 = 2% and β0 − β0 = 15%. Notice that starting with a prior that most countries believed that state intervention fostered growth is consistent with the fact that only about 30% of the countries were market oriented in 1950. These priors are also consistent with the evidence in Clemens and Williamson (2004). The value of ωβ is set to 002, which implies a quite agnostic view about the mean of initial beliefs. • We select sν and dν so that νi has an a priori mean and a standard deviation equal to 0264. The prior on νi is potentially important because it affects the speed of learning, especially for those countries for which fewer data are available. In calibrating this prior, we first observe that σi2 ·νi2 should be approximately equal to var(y¯i ), the variance of the average growth rate of GDP. We obtain an estimate of var(y¯i ) = 001752 as the variance of the average growth rates of the countries present in the Maddison data set between 1901 and 1950
LEARNING THE WEALTH OF NATIONS
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(excluding the wars).6 To obtain an estimate of σi2 based on presample observations, we again use the Maddison data and run a regression of GDP growth on time and fixed effects. We then compute the variance of the residuals for each country and calculate the mean of these variances (which equals 00044). Therefore, we set the mean of the prior for νi to [(00175)2 /00044]1/2 = 0264. • We select s and d so that i has an a priori mean and standard deviation equal to 001. The idea here is to discourage the model from fitting the data using very large variances of the exogenous component of the political cost kit . This prior distribution implies that if policymakers believe that growth under market orientation is 1% higher than under state intervention, they will adopt market-oriented policies with probability 87% on average (standard deviation 10%). • We set f¯ = 0 so that market orientation does not entail a priori higher average political costs than state intervention. The standard deviation of the fixed effect is set to ωf = 002 so that the model can capture particularly low or high average political costs if they are needed to fit the data. • We use a flat prior for the coefficients ξ and γ, which are common to all countries. 3.2. The Likelihood Function To derive the posterior distribution of the unknown coefficients, we update these priors with the likelihood information. In Section 2 we have fully specified the policy objective and the perceived data-generating process (DGP), that is, policymakers’ perceived relation between their policy choice and growth outcomes. Notice, however, that we have refrained from postulating what might be the true relation between policy choices and GDP growth. We will specify a “true” DGP in Section 7 to make sure that our results are not due to the fact that policymakers’ perceived model is very far off from a plausible true DGP and to perform some out-of-sample simulations. Here we only impose one condition about the true DGP, that is, that, in reality, GDP growth and the determinants of the political cost depend on actual policies and are not directly affected by the beliefs that led to those policies. This natural assumption is very convenient because it makes the inference about our parameters of interest (α) independent of the details of the true DGP.7 In other words, our inference about the coefficients of the policy objective and policymakers’ perceived model, as well as the results that we present in the next section, are not affected by the details of the way policy affects growth outcomes and the determinants of the political cost in reality. 6 There is a huge outlier in the distribution of the average growth rates across countries. Therefore, this variance is estimated with a robust method (squared average distance from the median of the 16th and 84th percentiles). 7 It is important to note that even in those cases in which this assumption is undesirable, our methodology can still be interpreted as a limited-information estimation strategy.
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With a slight abuse of notation in which L is used generically to denote an arbitrary density function, the likelihood function can be written as a product of conditional densities:
L(DT |α) = L(D1 |α)
T
L(Dt |Dt−1 α)
t=2
Using the assumption mentioned in the previous paragraph, each term of the product equals
L(Dt |Dt−1 α) = L(yt |θt Πt Dt−1 )L(θt |Πt Dt−1 α)L(Πt |Dt−1 ) so that
L(DT |α) = c ·
n i=1
L(θi1 |Πi1 α) ·
T
L(θit |Πit Dt−1 α)
t=2
where c is a constant term independent of α. Since the policy decision is given by (2.4), it follows that the conditional density L(θit |Πit Dt−1 α) can be written as
ˆM βit−1 − βˆ Sit−1 − fi − ξΠit θit t−1 L(θit |Πit D α) =
i
1−θit ˆM βit−1 − βˆ Sit−1 − fi − ξΠit · 1−
i where (·) denotes the cumulative distribution function (cdf) of a standard Gaussian density. 4. POSTWAR DYNAMICS AND GEOGRAPHY OF MARKET ORIENTATION In the rest of the paper, we explore the ability of our learning model to explain the observed adoption of economic policies across countries and over time. To this end, we use data from a large panel of liberalizations (133 countries, 1950–2001) originally constructed by Sachs and Warner (1995), and revised and extended by Wacziarg and Welch (2008). In this section, we describe these data and present evidence on the connection between policy choices and countries’ past growth performances. The findings of this section motivate the structural estimation exercise undertaken in Section 5. 4.1. Measuring Market Orientation To investigate the dynamics of market orientation, we require data on the policy orientation of a large panel of countries and years. This entails two mea-
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surement challenges: policies are intrinsically multidimensional, and comparable measures across countries and time are scarce. With these restrictions in mind, as a proxy for market orientation, we use the Sachs and Warner (1995; hereafter SW) indicator of liberalizations. This indicator was originally constructed as a measure of openness to international trade. In particular, SW argued that a government disposes of a variety of mechanisms to intervene in the economy and restrict international trade. The most direct mechanism, of course, is to impose tariffs and other barriers on imports. Other mechanisms include taxing, restricting or monopolizing exports, and limiting or blocking the convertibility of the country’s currency. Finally, a socialist government is likely to have significant distortions on international trade. Following this logic, SW required the following five criteria to classify a country as “open”: (i) The average tariff rate on imports is below 40%. (ii) Nontariff barriers cover less than 40% of imports. (iii) The country is not a socialist economy (according to the definition of Kornai (2002)). (iv) The state does not hold a monopoly of the major exports. (v) The black market premium is below 20%. The resulting indicator is a dichotomic variable. If in a given year a country satisfies all of these five criteria, SW call it open and set the indicator to 1; otherwise, the indicator takes the value of 0.8 Since most of these criteria (especially (iii)–(v)) capture forms of government intervention that go clearly much beyond restrictions on international trade, the SW indicator is better interpreted as a broader, albeit stark, measure of market orientation. In this respect, we follow Rodriguez and Rodrik (2000) by viewing the “SW indicator [as one that] serves as a proxy for a wide range of policy and institutional differences,” where “trade liberalization is usually just one part of a government’s overall reform plan for integrating an economy with the world system. Other aspects of such a program almost always include price liberalization, budget restructuring, privatization, deregulation, and the installation of a social safety net” (Sachs and Warner (1995)). As a consequence, we dub a country as “market-oriented” (MO) if the SW indicator is equal to 1 and as “state interventionist” (SO) if it is equal to 0. In our study, we use the revision of the SW indicator carried out by Wacziarg and Welch (2008), who expanded the sample of countries and years, and re8
Since a complete panel data set with all these criteria is not available for the entire postwar period, SW and Wacziarg and Welch (2008) performed a complementary country by country analysis of large reform episodes to find out whether large changes in policies led a country not to meet (or start meeting) some of these criteria. This extended data set is referred to as the panel of liberalization dates.
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
vised the information used to classify the countries.9 The data are an unbalanced panel, partly because separations and unifications have led to starting or stopping reporting some countries as independent entities. It is important to recognize that the dichotomic nature of the SW indicator is an important limitation. Countries with very different degrees of state intervention end up being classified equally. Moreover, the indicator fails to capture reforms if they do not simultaneously move countries in all five criteria (e.g., China in later years). Unfortunately, richer indicators, such as those produced by the World Bank or the Heritage Foundation, are only available for a reduced sample of countries and a handful of recent years. The large coverage of countries and years in the SW indicator is essential to study changes in the orientation to markets, which, as we now show, happen only so often. 4.2. The Dynamics of Market Orientation Figure 1 plots the evolution of the world fraction of market-oriented countries according to the SW indicator (thin, solid line with scale on the left-hand side axis). In 1950, only about 30% of the countries qualified as market ori-
FIGURE 1.—World average of market orientation and growth. 9
For example, SW applies criterion (iv) only for African countries, while Wacziarg and Welch extended it to other regions. In general, for the same years as SW, the revision by Wacziarg and Welch led more countries to be classified as market oriented relative to the original SW measure.
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ented. The spike in the late 1950s reflects the creation of the European Economic Community, followed by the inclusion in the sample of a number of state-interventionist developing countries. In the years from 1963 to 1984, the share of market-oriented economies did not exhibit a clear trend. However, in 1985 we observe the beginning of a global movement toward market orientation that continues until the end of the sample. By 2001, the fraction of market economies had increased to almost 80%. The worldwide average hides interesting regional differences. Figure 2 displays the fraction of market economies (solid line with scale on the left-handside axis) within eight regions of the world. Some of these regions started the sample as state interventionists and began a transition to market economies only later on. Earlier transitions toward market orientation include Western Europe, with the creation of the European Economic Community, and the Asian–Pacific region. On the contrary, North and Central America started rela-
FIGURE 2.—Regional patterns of market orientation and growth.
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
tively market oriented. However, after the establishment of the Central American Common Market and the diffusion of economic policies based on import substitutions, essentially only the United States and Canada remained market friendly between 1960 and 1985. After the crisis in the early 1980s, the region began its transition to market orientation with Costa Rica and Mexico. 4.3. Market Orientation, Growth, Wars, and the Political Environment To relate policy choices and past growth performances, Figure 1 also plots the (unweighted) average growth rate of per capita GDP (thick, solid line with scale on the right-hand-side axis).10 Interestingly, many countries opted for market orientation only after the growth collapse of the late 1970s and early 1980s, which was even more severe for state-interventionist economies (dashed line). A similar pattern can be observed at the regional level (Figure 2): most regions began a transition to market orientation after the deterioration of their growth performances. Our model of belief formation interprets these low growth episodes as the driving forces of the policy changes. However, it is important to recognize that also the world’s political and social landscape changed considerably during our sample period. According to our model, this might contribute to explain the evolution of observed policy choices. To investigate this possibility, Figure 3 displays the behavior of two important determinants of the political cost of market orientation in our structural model: the world average of Polity2 scores and the fraction of countries engaged in a civil war. Data on the political environment and on military conflicts were taken from the Polity IV data set and from Fearon and Laitin (2003), respectively. The Polity IV data set is the most widely used data set in empirical work on political science and international relations. The Polity2 score aggregates democratic and autocratic qualities of institutions and classifies countries in a spectrum on a 21-point scale, ranging from −10 (least democratic) to +10 (most democratic, see Marshall and Jaggers (2005) for further details). We use the dummy variable on civil wars from Fearon and Laitin (2003). Fearon and Laitin classified a country to be in a civil war (indicated by a value of 1 for the variable) if it was engaged in a military conflict that satisfied some minimum criteria.11 Figure 3 shows a remarkable trend toward democracy during the second part of the sample, which is likely to be an important force behind the run-up 10
GDP data are obtained from Penn World Table 6.2. The primary criteria are (i) the conflict involves fighting between agents of (or claimants to) a state and organized, nonstate groups who sought either to take control of a government, to take power in a region, or to use violence to change government policies; (ii) the conflict killed at least 1000 over its course, with a yearly average of at least 100; (iii) at least 100 were killed on both sides (including civilians attacked by rebels). For further details, including secondary criteria and comparison with other indicators on wars, see Fearon and Laitin (2003). 11
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FIGURE 3.—World average of market orientation and political variables.
toward market orientation. As for the civil war indicator, it also exhibits an improvement in the most recent years, although it appears to be less pronounced. Below we show the importance of our learning mechanism in explaining the switches in policy regimes, even after controlling for the observed changes in the political environment. 4.4. Reduced-Form Regressions Before turning to structural estimation, we run reduced-form regressions to further examine the relationship between policy decisions and observed growth rates, and highlight salient (partial) correlations in the data. In particular, we show that a country’s choice of being market oriented or state interventionist is correlated with the fraction of neighboring countries following these policies and their past growth performances, even after controlling for other political and economic variables, as well as country and time fixed effects. Specifically, we consider the linear probability model (4.1)
E(θit | · · ·) = φi + φt + φ1 θit−1 +φ2 Eˆ it−1 (y|θ = 1) + φ3 Eˆ it−1 (y|θ = 0)+φ4 θ¯ it−1
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
Here, the policy decision of country i in period t (θit ) is a function of its own past policy (θit−1 ), a distance-weighted measure of other countries’ policies (θ¯ it−1 ), and the distance-weighted average growth rate over the previous 3 years of other countries under the two policy regimes (Eˆ it−1 (y|θ = 1) and Eˆ it−1 (y|θ = 0)).12 Our theory predicts policies to be persistent (φ1 > 0) due to the persistence of beliefs implied by Bayesian updating. It also predicts that countries are more likely to pursue market-oriented policies in periods in which market-oriented neighbors grow faster (φ2 > 0), as a consequence of information spillovers. Similarly, in periods of faster growth of state-interventionist neighbors, the probability of being market oriented should decline (φ3 < 0). Finally, the precision of the information about the success of market economies depends on the number of countries following those policies. Therefore, our theory predicts a positive correlation among policies (φ4 > 0), provided that on average countries update their beliefs about the benefits of market orientation upward (which will be the case according to the estimation results of our structural model). It is important to recognize, however, that finding φ4 > 0 can also be symptomatic of a force of “fads and fashion” in policy choices, which may also be at play in the data (see Banerjee (1992) or Bikhchandani, Hirshleifer, and Welch (1992)). Columns 1–4 of Table I present the estimation results of equation (4.1) on our data. These results can be interpreted either as maximum likelihood estimates or as posterior modes with uninformative priors. In column 1, we estimate a simplified version that excludes θ¯ it−1 from the regressors and sets δ = 2500, which fixes the effective neighborhood of the median country, de−dij /δ , to be 20 countries. In columns 2–5 we progressively fined as j=i e add various features to the model, by including country fixed and time effects (columns 2–4); adding θ¯ it−1 —the effect of the average policy choices of neighbors—and estimating the effective country neighborhoods when measuring past growth performance and policy choices, that is, optimizing over δ (columns 3 and 4); and, finally, controlling for the political and economic environment, as measured by the Polity2 score, log-relative GDP, and the Fearon and Laitin indicator for civil wars (column 4). Table I reveals three robust features of the data that which are consistent with the predictions of our structural model. First, policies are very persistent (second row). The probability that a country that was market oriented in period t − 1 keeps the same policy in period t is 85–96% larger than that of countries that were interventionist in period t − 1. Second, policy choices are highly correlated to past performance of policy regimes (third and fourth rows). For each additional point of per capita GDP growth of market-oriented Formally, we define Eˆ it−1 (y|θ) = 3s=1 j:θjt−s =θ e−dij /δ yjt−s / 3s=1 j:θjt−s =θ e−dij /δ , θ = 0 1, −d /δ −d /δ and θ¯ it−1 = j=i e ij θjt−1 / j=i e ij . 12
19
LEARNING THE WEALTH OF NATIONS TABLE I ESTIMATION RESULTS OF THE REDUCED -FORM LINEAR PROBABILITY MODEL AND PROBIT MODEL (COLUMN 5)
Constant θit−1 Eˆ it−1 [y|θ = 1] Eˆ it−1 [y|θ = 0] θ¯ it−1
(1)
(2)
(3)
0037∗∗∗ (0005) 0958∗∗ (0004)
088∗∗∗ (001)
087∗∗∗ (001)
039∗ (022) −069∗∗∗ (019)
060∗ (035) −077∗∗∗ (029) 013∗∗∗ (003)
004 (016) −058∗∗∗ (011)
Polity IV Log-relative GDP Civil wars δ Country-specific effect Time effect
2500
2500
No No
Yes Yes
3849.8∗∗∗ (1112.0) Yes Yes
(4)
086∗∗∗ (001) 054∗ (034) −059∗∗ (028) 013∗∗∗ (003) 00019∗∗∗ (00005) −00014 (00086) −00107 (00076) 3280.4∗∗∗ (987.6) Yes Yes
(5)
043∗∗∗ (002) 099∗∗ (045) −102∗ (055) 008 (006) 0001 (00008) −00092 (00205) −00207 (00129) 2449∗∗ (1093) Yes Yes
Note: *, **, and *** denote significant at the 10%, 5%, and 1% levels.
(state-interventionist) countries in the neighborhood of country i, the long-run probability that country i is market oriented increases (decreases) by approximately 5% (6%) in our specification with fixed and time effects. Finally, countries are more likely to be market oriented when their neighbors followed the same policies (fifth row) or when they have a high Polity2 score (sixth row). We have opted for a linear probability model as the simplest framework to highlight some key (partial) correlations in the data. To check robustness, we have also experimented with a Probit specification and obtained similar results, as illustrated by column 5 of Table I, although the estimation of nonlinear models with fixed effects suffers from the incidental parameters problem (see Neyman and Scott (1948) or Chamberlain (2010)). Available approaches to tackle the incidental parameters problem in dynamic discrete choice models tend not to apply or to be feasible in our framework with lagged dependent explanatory variables, not strictly exogenous regressors, unbalanced panels, and a relatively small cross-sectional dimension (see Lancaster (2000), Honore and Kyriazidou (2000), or Arellano and Hahn (2007)). We conclude this section by noting that these results must be interpreted with caution in light of the “reflection” problem emphasized by Manski (1993).
20
F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
Specifically, the identification of endogenous interactions effects (φ4 ) and contextual effects (φ2 and φ3 ) is problematic due to the collinearity of the two effects. In our model, however, this problem is much less severe because neighborhoods are individual-country specific. In other words, all countries discount the information from all other countries differently, depending on their geographic location. This feature breaks the symmetry that causes the collinearity problem. On the other hand, the fact that we estimate the structure of groups adds a nontrivial element of complexity relative to more standard linear-inmean models. Finally, the structural learning model that we study in the rest of the paper belongs to a class of binary choice models whose nonlinearity makes identification less problematic, although, to the best of our knowledge, dependent on the specific nonlinear structure that we postulate (Brock and Durlauf (2001a, 2001b)). We now turn to the estimation of this model. 5. RESULTS This section presents the estimation results and assesses the goodness of fit of our learning model. In our baseline specification, the cross-country correlation of initial beliefs about the effect of market orientation and state intervention depends on the geographic distance (in thousands of kilometers (km)) between countries’ capitals (dij ), which is highly correlated with both economic and cultural distance. The political cost of being market oriented (Kit ) is a function of four explanatory variables: (i) a country-specific effect (fi ), (ii) the logarithm of an average (over the past 5 years) of a country’s GDP relative to the United States (rYi[t−5t−1] ), (iii) the Polity2 score (πit ), and (iv) the civil war dummy of Fearon and Laitin (2003) (cwit ). From each of these variables, we subtract their pooled means (except, of course, to the country-specific effects) to normalize the expected political cost, of market orientation to be equal to zero for the average country/year. We also present some alternative specification of the political cost including additional time-varying explanatory variables. 5.1. Parameter Estimates Table II reports the posterior estimates of the coefficients parameterizing the structure of the prior correlation (γ) and the political cost (ξ). The second column of the table refers to the baseline model. According to our estimates, the coefficient γ is positive, so that the cross-country correlation of initial beliefs decreases with geographic distance. This correlation is approximately equal to 085 for countries that are 1000 km apart, while it becomes only about 05 for countries 5000 km distant from each other. The a priori cross-country correlation of beliefs plays an important role in our analysis because it affects the weight policymakers put on the experience of other countries and, therefore, the speed of learning. In the next sections we will analyze its contribution to
21
LEARNING THE WEALTH OF NATIONS TABLE II
COEFFICIENT ESTIMATES IN THE BASELINE MODEL AND VARIOUS ALTERNATIVE MODELS WITH DIFFERENT SPECIFICATIONS OF THE STRUCTURE OF THE POLITICAL COST Baseline (M1 )
Prior Correlation (γ) dij Geographic distance Political Cost (ξ) rYi[t−5t−1] Log-relative GDP πit Polity2 cwit Civil wars πnit Polity2 (neighbors) wnit Civil wars (neighbors) scit Soviet controlled infi[t−5t−1] Inflation tdit−1 Trade deficit eduit Education Country-specific effect Time effect Number of observations
0152 (0021) −00035 (00015) −00007 (00001) 00048 (00011)
Yes No 4641
M2
M3
M4
M5
M6
0163 (0022)
0150 (0018)
0181 (0055)
0265 (0032)
0155 (0022)
−00031 −00042 −00051 00097 −00157 (00015) (00018) (00017) (00019) (00021) −00006 −00005 −00006 −00005 −00001 (00001) (00001) (00001) (00001) (00001) 00045 00048 00054 00047 00094 (00013) (00012) (00012) (00006) (00015) −00012 (00002) −00025 (00008) 0169 (0096) −00003 (00001) 00054 (00047) −00088 (00006) Yes No 4641
Yes No 4641
Yes No 3903
Yes No 3951
Yes Yes 4641
the evolution of policymakers’ beliefs about the effect of state intervention and market orientation on growth. Our results suggests that the political costs of being market oriented are reduced with a higher relative per capita income, with more democracy, and with the lack of a civil war. Finding lower political costs for being market friendly for relatively more developed countries is consistent with the theoretical results of Acemoglu, Aghion, and Zilibotti (2006), who argued that countries are likely to switch from state intervention to market orientation as they develop. To evaluate the economic significance of the estimated coefficient of −00035 on (rYi[t−5t−1] − rY ), the deviation of log-relative GDP from the pooled mean, Figure 4(a) plots the model-implied probability of being market oriented for countries with different levels of income. The figure is constructed for the case in which a country holds similar beliefs about the effects of market orientation and state intervention, and assumes that they have a common standard deviation of the political cost, i = 053%, which corresponds to the mean estimate
22
F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI (a)
(b)
FIGURE 4.—Probability of being market oriented as a function of (a) the logarithm of GDP relative to the United States and (b) the Polity2 score for the average country.
of i . Clearly, poor countries are substantially less likely to be market oriented than more developed economies. For instance, everything else equal, a country with a level of GDP similar to the Unites States has a 90% probability of being market oriented, while a country with only 10% of the U.S. per capita GDP (e.g., Zambia) has only a probability of 10% of being market oriented. We also find that more democratic countries seem to have a lower political cost of being market oriented. Figure 4(b) plots the implied probability of being market oriented as a function of the deviation of the Polity2 score from the pooled mean. Everything else equal, a country with low Polity2 score is substantially less likely to be market oriented relative to more democratic countries. Finally, the presence of a civil war increases the political cost of market orientation. In particular, the average country is 30% less likely to be market oriented if it hosts a civil war. The third through seventh columns of Table II refer to models that differ from the baseline for the structure of the political cost. For instance, in model M2 we control for the presence of a civil war in at least one of the neighboring countries (wnit ) and for the average Polity2 score of the neighboring countries (πnit ). Model M3 allows the political cost to depend on a dummy variable equal to 1 if a country was a member of the Warsaw Pact and, therefore, under Soviet Union military control (scit ). Model M4 includes the aver-
LEARNING THE WEALTH OF NATIONS
23
age level of inflation over the past 5 years (infi[t−5t−1] ) and the trade deficit (tdit−1 ) of a country as proxies for its macroeconomic stability. In model M5 , we control for the average years of schooling in the country’s total population (eduit ), measured using the data set of Barro and Lee (2001).13 Finally, in model M6 , we augment our baseline specification of the political cost of market orientation with a time effect. First, notice that the average Polity2 score of neighboring countries has an even larger negative effect on the political cost than the domestic Polity2 score. Perhaps a bit surprisingly, the presence of a civil war in a neighboring country decreases the political cost of market orientation at home, while the Soviet control dummy has a very large positive impact on this cost. However, the statistical significance of this coefficient is weak, which is also the case for the coefficients on inflation in M4 . A higher level of education seems to be associated with a lower political cost of market orientation, consistent with the findings of Mayda and Rodrik (2005). Second, notice that the inclusion of these extra variables hardly changes the values of the coefficient estimates of the common determinants of the political cost. One exception is M5 , where the addition of the education variable perversely reverts the sign of the coefficient on relative GDP. Finally, the seventh column indicates that the inclusion of a time effect decreases the importance of the Polity2 score, increases the relevance of logrelative GDP and the civil war dummy, and does not change the estimate of the prior spatial correlation. 5.2. Model’s Fit In this subsection we use different criteria to assess how well our model fits the data. We find that the model explains quite well the observed dynamics of market-oriented policies over time and across countries. Figure 5 reports the actual (solid line) and predicted (dashed–dotted line) fractions of market-oriented economies present in our sample. The modelpredicted series corresponds to the sequence of one-step-ahead predictions of the baseline model, absent any shock to the political cost. For example, the value for 1990 represents what the estimated model predicts for the world share of market-oriented countries, given the information on GDP growth and policies up to 1989, the information about the political environment and civil wars in 1990, and assuming that in 1990 the shock to political costs is zero for each country. The figure makes clear that the model captures fairly well the high fraction of state-interventionist economies in the first part of the sample and the run-up toward market orientation starting in the mid 1980s. 13 Due to the limited coverage of the data set of Barro and Lee (2001) relative to ours, we used an extrapolation procedure to obtain data for the years 1950–1959 and 2000–2001.
24
F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
FIGURE 5.—Actual and model-predicted fraction of market-oriented countries.
Figure 6 provides a more disaggregated picture of the fit of the model, by comparing the actual and predicted series for eight regions of the world economy. Notice that the model captures well episodes of policy reversals (e.g., Central and South America) and the heterogeneous timing of liberalizations (e.g., the earlier liberalizations in Asia and the more protracted reforms in Africa). Finally, Table III presents a quantitative assessment of the fit of the model. Focusing on the second column, the baseline model correctly predicts almost 97% of the policy choices observed in the data. Moreover, it accounts for 258%, 538%, 742%, and 774% of the switches in policies, within ±0-, 1-, 2-, and 3-year windows, respectively. Explaining changes in policies is a very challenging test, because we only observe 101 policy switches in our data set (2% of the total policies). We are therefore quite satisfied with the performance of the model along this dimension. Table III also reports the fit of the five alternative specifications of the political cost that we presented above. Notice that models M2 , M3 , and M4 fit the data essentially as well as M1 , validating our choice of the baseline. More substantive fit improvements can be seen for the model including education (M5 ) and the one with a time effect (M6 ). In particular, M6 performs better especially in terms of explaining policy switches at short horizons, which should not be surprising since the time effect is bound to exactly capture the average evolution of policies over time. It is important to notice, however, that our baseline model is preferred to models with a time effect by selection criteria
25
LEARNING THE WEALTH OF NATIONS
FIGURE 6.—Actual and model-predicted fraction of market-oriented countries across regions. TABLE III MEASURES OF FIT FOR THE BASELINE MODEL AND VARIOUS ALTERNATIVE MODELS WITH DIFFERENT SPECIFICATIONS OF THE STRUCTURE OF THE POLITICAL COST Baseline (M1 )
Log-likelihood Share of correct predictions Share of correctly predicted switches ±0-year window ±1-year window ±2-year window ±3-year window
−419.49 96.6% 25.8% 53.8% 74.2% 77.4%
M2
M3
M4
M5
M6
−384.61 −413.33 −381.11 −310.71 −275.21 97.0% 96.6% 96.3% 97.1% 97.9% 28.0% 58.1% 77.4% 81.7%
26.9% 57.0% 74.2% 78.5%
26.7% 51.9% 74.7% 77.2%
29.1% 68.3% 81.0% 84.8%
58.1% 75.3% 80.6% 87.1%
26
F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
that take into account the increased number of parameters (i.e., the Bayesian information criterion).14 5.3. The Role of Learning and the Evolution of Beliefs The good empirical performance of our model is due to two ingredients: (i) the estimated evolution of policymakers’ beliefs and (ii) the estimated structure of the political cost. The objective of this subsection is disentangling the relative importance of these two factors for the fit of the model and demonstrating that learning is crucial to explain the observed dynamics of marketoriented policies. To evaluate the contribution to fit of the learning component of the model, we simply reestimate the five versions of the model presented above without learning. In particular, we estimate policymakers’ initial beliefs, but force them to remain at the level of 1950 and not to be updated throughout the sample. We note that this restricted version of our model can be interpreted as a reduced-form representation of policies, corresponding to a standard panel Probit model. The second column of Table IV summarizes the results of this experiment for our baseline specification. The model without the learning mechanism is still capable of explaining a fair amount of the policy choices that we observe in the data. This is not surprising, however, considering that many countries in our sample kept the same policy in place for the entire postwar period. As we argued earlier, a much more challenging task is predicting policy switches. As is clear from Table IV, the model without learning fails dramatically along this dimension, accounting for only 54%, 14%, 226%, and 247% of the switches in the data, within ±0-, 1-, 2-, and 3-year windows, respectively. We conclude that the success of our baseline model at predicting policy switches (second TABLE IV MEASURES OF FIT WHEN THE LEARNING MECHANISM IS TURNED OFF Baseline (M1 )
Log-likelihood Share of correct predictions Share of correctly predicted switches ±0-year window ±1-year window ±2-year window ±3-year window
−1282.5 85.5% 5.4% 14.0% 22.6% 24.7%
M2
M3
M4
M5
M6
−1114.8 −1280.4 −1066.4 −701.11 −484.7 88.7% 85.5% 88.9% 92.0% 95.8% 10.7% 20.4% 26.9% 31.2%
6.4% 15.0% 22.6% 24.7%
7.6% 21.5% 24.1% 25.3%
26.6% 41.8% 46.8% 49.4%
50.5% 65.6% 66.7% 69.9%
14 Of course, a formal model comparison exercise in a Bayesian framework would be based on the marginal data densities. However, the computation of the marginal data density is extremely challenging in this setting due to the very high number of parameters.
LEARNING THE WEALTH OF NATIONS
27
column of Table III) is mainly due to the presence of the learning mechanism that we emphasize in this paper. The third through seventh columns of Table IV conduct a similar exercise for the various alternative specifications of the political cost. The inclusion of neither the neighbors’ characteristics nor the Soviet control dummy or the indicators of macroeconomic stability (M2 , M3 , and M4 ) alter the result that learning is essential. The addition of average years of schooling (M5 ) slightly improves the performance of the model without learning, although the model with learning still fits the data substantially better. Moreover, it is important to recall that this model is estimated on a smaller data set relative to the baseline. Finally, as one would expect, the use of time effects enhances the performance of the models without learning (M6 ), whose fit becomes comparable to our baseline. We want to stress that, by construction, the time effect in M6 captures the average evolution of market-oriented policies over time. This is not our preferred specification because it is fundamentally against the spirit of our learning model, which can be interpreted as providing a structural explanation and interpretation for the time effect. To demonstrate the importance of this feature of our model, we perform an out-of-sample forecasting exercise—the Achilles’ heel of most highly parameterized models. Specifically, we estimate our baseline model using data until 1990 and then use the estimated model parameters to generate policy forecasts between 1991 and 2001. Figure 7 reports the world share of market-oriented policies in the data and predicted by our model. Until 1990, the model’s predicted series represents one-step-ahead insample predictions, while after 1990 they are out-of-sample forecasts. Between 1990 and 2001, the model predicts an increase in the share of market-oriented economies of 177%, about three-fourths of the increase observed in the data (242%). On the contrary, the model with time effects, but without learning, fails in this out-of-sample forecast exercise, as it predicts only a 56% increase in the world share of market-oriented countries.15 The intuition for the good empirical performance of the learning model is due to two main features of the data. First, learning is capable of explaining the initial low share of market-oriented economies due to the information in the pre-1950 data, which led to a calibration of priors consistent with most countries believing that the state outperforms the market. In addition, the first decade of our estimation sample is consistent with this idea, because stateinterventionist economies exhibited higher average growth rates than marketoriented ones. This reinforced preferences for government intervention and, consistent with the data, prevented a faster diffusion of market orientation. Second, the learning mechanism is able to explain the wave of liberalizations that started in the mid-1980s because of the low GDP growth observed in the 1970s and early 1980s, especially for state-interventionist countries. Through 15 For the model with a time effect, we have assumed that the value of the time effect remains unchanged relative to its estimate in 1990.
28
F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
FIGURE 7.—Actual and model-predicted fraction of market-oriented countries.
the lens of the model, past growth experiences are important determinants of beliefs and policy dynamics. To make this intuition more precise, Figure 8 presents a summary of the estimated evolution of beliefs over time and across countries for our baseline model. In particular, Figure 8 plots the time evolution of the empirical distriˆ S∗ n bution of {βˆ M∗ it − βit }i=1 , that is, the difference between the posterior mode of the mean of policymakers’ beliefs about the effect of market-oriented and state-interventionist policies on growth. The lightest shaded areas correspond to the 10–90th percentiles of the distribution, while the darkest shade denotes the 40–60th percentiles. As mentioned earlier, initial beliefs about the success of market economies were quite negative and characterized by considerable dispersion across countries. According to our estimates, in 1950 the vast majority of countries believed that market-oriented policies were inferior to state intervention in terms of economic growth. These beliefs evolved, slowly shifting toward favoring market-oriented regimes. By 2001, the histogram of imˆ S∗ n plied {βˆ M∗ iT − βiT }i=1 has moved upward, with a perceptible majority of countries believing that market-oriented policies are growth enhancing. However, quite interestingly, the dispersion of beliefs across countries has declined, but certainly not disappeared. Figure 9 summarizes the precision of these beliefs across countries and over time by plotting the evolution of the empirical distribution of the uncertainty about the difference between average growth under market orientation and state intervention. For most countries, the initial uncertainty about the differ-
LEARNING THE WEALTH OF NATIONS
29
ˆ S∗ n FIGURE 8.—Evolution over time of the empirical distribution of {βˆ M∗ it − βit }i=1 , that is, the difference between the posterior mode of the mean of policymakers’ beliefs about the effect of market-oriented and state-interventionist policies on growth.
FIGURE 9.—Evolution over time of the empirical distribution of the uncertainty about the difference between average growth under market orientation and state intervention.
30
F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
ential effect of these policies is substantial. By 2001, countries have sharpened their views considerably, although significant doubts remain. In fact, the median country believes that, on average, market orientation leads to an approximately 05% faster GDP growth than government intervention (Figure 8), but it also attaches to this estimate a standard deviation of about 065%. Given our results, an obvious question to ask is whether the world has reached a point at which the vast majority of policymakers are convinced that a marketoriented regime is beneficial for economic growth? Figures 8 and 9 tell us that this is certainly not the case. There still exists a considerable amount of negative views about the merits of market economies. Moreover, many of those countries with a favorable view either remain quite uncertain or believe that the gains are quantitatively minimal. As a matter of fact, in 2001 only 47 countries in our data set would estimate average growth under market orientation to be statistically significantly (at the 95% confidence level) higher than average growth under state intervention. As one would expect, the United States, Western Europe, and countries like Hong Kong or Singapore belong to this small group, while many developing countries belong to the rest. 5.4. The Model’s Implied Beliefs and Survey Data In the previous sections, we have argued that our estimated learning model fits the policy data well. As an additional validation exercise, in this subsection we show that the beliefs implied by our model are positively correlated with available cross-country survey data on beliefs. The World Value Survey (WVS) and the Latinobarometro interview a large number of individuals in different countries on their views and expectations on many social, cultural, and economic issues. The WVS has conducted five “waves” of interviews in all the regions of the world. Of these waves, only the first four overlap with our sample, that is, those conducted in 1981–1984, 1989– 1993, 1994–1999, and 1999–2004. The Latinobarometro has collected opinions in 16 Latin American countries since 1995 and from this survey we can use responses for the years 1995, 1998, 2000, and 2001. The comparison between our model’s implied beliefs and these survey data faces two challenges. First there are no questions in the survey that exactly match the model’s notion of beliefs about the growth implications of policies, ˆ S∗ that is, βˆ M∗ it − βit . From the many questions asked, we selected those that, ex ante, in our opinion, were conceptually closest. Table V lists them in descending order according to this criterion. The second challenge is data availability, since the most relevant questions are usually available only for a small subset of countries and years, as also indicated in Table V. For each question, we compute country/year averages of the individual answers. Table V reports the correlation of the resulting indicator with our ˆ S∗ model’s implied beliefs, that is, βˆ M∗ it − βit . The last column of the table also
31
LEARNING THE WEALTH OF NATIONS TABLE V CORRELATIONS OF THE MODEL’S IMPLIED BELIEFS WITH AVAILABLE SURVEY DATA Variable Survey World Values Survey (WVS) Code
E127 E062 E036 E037 E039 E042 E040
ˆ S∗ Correlation With βˆ M∗ it − βit
Availability
Description
Market economy right for country Allow or restrict imports Private or state owned business Government must provide Competition: good or harmful State should control or free firms Success: hard work or connections
Waves # Obs.
I
10 49 98 100 90 30 66
II
III
x x x
x x x x x x x
x
Latinobarometro
IV
Estimate
Fit?
x x x
−0.76 −0.13 −0.06 0.20 0.06 0.03 −0.13
Yes Yes Yes Yes Yes Yes No
01
Estimate
Fit?
x x
−0.26 −0.25 −0.29 −0.26 0.14 −0.37
Yes Yes Yes No No Yes
Years
Code
Description
# Obs.
C602 C604 C601 C506 C504 C603
Market economy is best for country Private enterprise is beneficial Prices determined by competition Government must solve problems Pay better more efficient workers Encourage foreign investment
32 16 48 16 8 32
95
98
00
x x x
x x
x x
x
specifies whether these correlations have the expected sign, depending on the numerical scale and the structure of each question. The most closely related questions in both surveys ask whether markets are perceived to be beneficial for the country. In both cases, we find that the responses are aligned with the predictions of our model. Indeed, even for a small number of observations, the correlation is surprisingly strong in the WVS. The other indicators capture views about the desirability of government intervention, the operation of firms, and whether private enterprise, competition, or international trade (imports) are perceived as desirable. We also include C603 (encouragement of foreign investment) from the Latinobarometro, since incentives to foreign investment have been strongly associated to pro-openness policy agendas in Latin America during the 1990s. It is important to keep in mind that these indicators, and many others in the two surveys, are less relevant for our purposes since they capture views on specific dimensions of government intervention or redistributive beliefs, which can be perfectly compatible with a country having a SW indicator of 1, for example, the welfare state in Western Europe. Moreover, other questions such as E042 (whether firms should be free or state controlled) in the WVS, refer to current policies as opposed to absolute views on regulation of firms. Nevertheless, with a few exceptions, the results for these other indicators are also in line with the predictions of our
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
model. However, we view these positive results as merely suggestive, due to the data limitations highlighted above. 6. A COMPETING HYPOTHESIS An alternative explanation for the adoption of market-oriented policies is that instead of learning, policymakers perfectly know the model’s parameters and their drift over time. Fitting the data would then require that marketoriented policies were inferior during the 1950s, 1960s, and 1970s, and since the mid 1980s these policies have become the most convenient option. A natural rationalization of such a story could be based on trade complementarities, because some of the benefits of market-oriented policies come from trade with other countries. Since trade expands when other countries are also market oriented, free markets might have been suboptimal until the 1980s— because potential trade partners were interventionists—while they might have become optimal in the 1990s, once worldwide trade openness enhanced trade opportunities for all countries. To evaluate the empirical merit of this alternative hypothesis, in this subsection we assume that policymakers base their policy choices on a modified version of model (2.2): (6.1)
MM ¯ yit = βSi (1 − θit ) + (βM θ−it−1 )θit + εit i +β
where θ¯ −i represents an average of other countries’ policies and the benefits of market orientation vary endogenously over time with the policy decisions of other countries. Presumably, βMM > 0 and policy decisions are strategic complements.16 We estimate equation (6.1) on our data, using four different proxies for θ¯ −i : (i) the average policy of the other countries in country-i’s region; (ii) the distance-weighted average policy of other countries, setting the effective neighborhood to 20 countries (similar to the formulation used in Section 4.4); (iii) the average volume of trade, (imports + exports)/GDP, of the other countries in country-i’s region; and (iv) the distance-weighted average volume of trade of other countries. In all four cases, we experiment with pooled, fixed effect, and fixed and time effect panel regressions. Not one of our estimates of βMM is statistically significantly greater than zero. Even worse, using proxies (iii) and (iv), we obtain negative point estimates for βMM . Therefore, these results do not provide support for the conjecture that market orientation was useless in the first part of the sample, due to the scarcity of trade partners. Moreover, even if we pick the highest point estimate, βMM = 00096 (achieved with proxy (ii) in the regression with fixed and time effects), and give our policymakers the knowledge of this coefficient, the model without learning can only explain 269% of the policy switches within a ±3-year window. 16 We have modeled strategic interaction with a lag to abstract from the possibility of multiple equilibria and the related complications in evaluating the likelihood function (Tamer (2003)).
LEARNING THE WEALTH OF NATIONS
33
To explain at least about 50% of the policy switches, we need βMM = 003, the highest point estimate plus 2 standard deviations. This suggests that an explanation of the data along these lines requires a theory not only of economic, but also (geo)political complementarities, which might be interesting to explore in future research. Moreover, integrating these complementarities into our learning framework seems a promising development. In such a model, learning the benefit of market-oriented policies could provide impetus to the flows of liberalizations, while strategic complementarities could constitute an important propagation mechanism. 7. A “TRUE” DATA-GENERATING PROCESS AND THE MOVEMENT TOWARD MARKET ORIENTATION
In this section, we relate our learning model to a flexible statistical representation of the data on growth and market orientation, which we call the true DGP. We estimate it using the entire set of available data from 1950 to 2001 and compare these estimates to the real-time evolution of policymakers’ beliefs implied by our learning model. We have two goals. First, we want to evaluate the extent to which our model of beliefs formation is consistent with the statistical information available in our sample. Since our policymakers are perfectly rational Bayesian agents, this task boils down to making sure that their initial priors are not unreasonable. Such a comparison between a plausible true DGP and policymakers’ initial priors also serves our second purpose, that is, examining what factors—if any—are responsible for the slow adoption of market orientation around the world, which our model fits well. 7.1. The True DGP We assume that the true DGP is a hierarchical linear model, that is, the flexible statistical representation of the postwar growth data is (7.1) (7.2)
yit = bSi (1 − θit ) + bM i θit + eit
i = 1 n
et ∼ N(0 S · Q · S)
M where b ≡ [bS1 bSn bM 1 bn ], S is an n × n diagonal matrix with standard deviations on the main diagonal, and Q is an n × n correlation matrix. This formulation allows market-oriented and state-interventionist policies to have a country-specific effect on economic growth. These effects correspond to the realization of a draw from a population with distribution
(7.3)
¯ ζ 2 · I2 ⊗ (S · W · S)) b ∼ N(b
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F. J. BUERA, A. MONGE-NARANJO, AND G. E. PRIMICERI
where b¯ ≡ [1n · b¯ S 1n · b¯ M ] and 1n is an n × 1 vector of ones, while b¯ S , b¯ M , and ζ are scalar parameters. The elements of the n × n correlation matrix W are modeled as Wij = exp[−dij · λ] which allows for the possibility of spatial correlation.17 Specifying the true DGP as a hierarchical linear model is appealing for several reasons. First, equation (7.1) is identical to (2.2), implying that the true and perceived DGP are absolutely continuous with respect to each other. Indeed, if growth rates are generated by equations (7.1)–(7.3), policymakers endowed with model (2.2) and priors (2.5) would eventually learn the truth. Second, had policymakers at some previous time t0 < 1950 known the population distribution of b, then they would have adopted it as their prior at that initial time. Consequently, beliefs in 1950 can be interpreted as the outcome of the evolution of these beliefs in the pre-1950 period, in response to various shocks, including, for instance, the Great Depression. In practice, of course, it is unreasonable to assume that policymakers knew beforehand the population distribution of b, which, as econometricians, we can estimate only because we have access to the entire set of postwar data. Nevertheless, a comparison between the estimates of (2.5) and (7.3) is useful to understand the extent to which policymakers’ initial beliefs in 1950 were sensible, in light of the postwar data. Consistent with our learning model, policies are predetermined with respect to et . Therefore, the likelihood function can be evaluated using the procedure described in Gelman, Carlin, Stern, and Rubin (2004). The estimation results, reported in Table VI, can be interpreted either as maximum likelihood estimates or as posterior modes and standard deviations under uninformative priors. Observe that we find no difference between the population means of the growth effect of market-oriented and state-interventionist policies (b¯ S and b¯ M ). However, our estimate of the realized vector b indicates that the growth impact of market orientation is greater than that of state intervention in most counTABLE VI ESTIMATES OF THE HIERARCHICAL LINEAR MODEL b¯ S
b¯ M
ζ
λ
a
κ
0.021 (0.003)
0021 (0004)
0317 (0064)
0388 (0179)
0247 (0037)
0317 (0067)
17 For symmetry, we also allow the growth shocks to be spatially correlated by assuming that Qij = a · exp[−dij · κ].
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n FIGURE 10.—Empirical distribution of {bM∗ − bS∗ i i }i=1 , that is, the difference between the true DGP maximum likelihood estimate of average growth under market orientation and government ˆ S∗ n intervention, and {βˆ M∗ iT − βiT }i=1 , that is, the difference between the posterior mode of the mean of policymakers’ beliefs about average growth under market orientation and state intervention.
tries. This suggests that countries underperforming under state intervention are scattered around the world, while countries performing well under market orientation tend to be clustered together. Therefore, when the model is allowed to account for this spatial correlation structure (captured by the estimate of λ), each of these individual observations tends to be overweighted (for the scattered countries) or underweighted (for the clustered countries). Figure 10 n − bS∗ illustrates this point by plotting the empirical distribution of {bM∗ i i }i=1 , that is, the difference between the estimate of average growth under market orientation and government intervention across countries. Most of the distribution is located on the right with respect to zero, with the 10th and 90th percentiles equal to 0% and 28%, respectively. To understand why the effectiveness of these policies exhibits geographical dispersion, we think that it may be useful to interpret our DGP as a reducedform representation of a more general framework in which the interaction of market orientation and possibly other policies and institutions is important for growth. For instance, it could be the case that being market oriented is more beneficial in combination with other institutions that favor the persistent development of the private sector (e.g., rule of law, good enforcement of contracts, or efficient safety nets). If so, the “reduced-form” effectiveness of market orientation will inherit the possibly strong cross-country dispersion and spatial correlation of these other policies and institutions.
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For comparison, Figure 10 also plots the empirical distribution of {βˆ M∗ iT − S∗ n ˆ βiT }i=1 , that is, the difference between the mean of policymakers’ beliefs about the effect of market orientation and state intervention on growth at the end of the sample. While the estimates based on the true DGP suggest that policymakers are more uncertain and more pessimistic about the net impact of market-oriented policies, it is interesting to note that the two distributions are remarkably close to each other. Indeed, a test of the hypothesis that S 18 ˆ M∗ ˆ S∗ bM i − bi = βiT − βiT fails to reject the null for 119 countries out of 133. This suggests that the evolution of policymakers’ beliefs and implied policy decisions are not driven by extreme initial priors, but rather by the observed joint dynamics of growth and policies observed in the data. Finally, given the estimates of the heterogeneous effects of policies in the true DGP and those of the political cost of market orientation, we ask which countries should efficiently choose to be interventionists by the end of our sample. Our results suggest that only two countries would experience large gains under state-interventionist regimes: Iraq and Sri Lanka. This is due to their high country-specific political cost, their low level of development, the presence of a civil war (in Sri Lanka), and a dictatorship (in Iraq). We also find that five additional countries would have some more modest benefits under state intervention: Belarus, India, Moldova, Pakistan, and Syria. With the exception of Moldova, this is explained by the fact that market orientation in these countries entails high political costs, despite leading to faster growth. In all of these cases, however, the gains of state-interventionist policies are likely to disappear as these countries develop, their governments become more democratic, and their civil wars come to an end. 7.2. The Late Movement Toward Market Orientation In this subsection, we perform a number of counterfactual experiments to understand the sensitivity of our results to alternative specifications and estimates of our model of beliefs’ formation. Specifically, we investigate whether reasonable changes in policymakers’ priors in 1950 or their perceived model would have speeded up the adoption of market-oriented policies. First, we consider the role of the initial uncertainty and the mean of policymakers’ priors. Second, we analyze the evolution of beliefs for alternative assumptions about the prior spatial correlation. Finally, deviating more substantially from the baseline and our true DGP, we examine a model where policymakers believe that the effect of policies is common across countries, but they have heterogeneous priors about these common effects. We begin by asking whether the slow diffusion of market friendliness can be attributed to the fact that policymakers’ priors in 1950 were too dogmatic in 18 These tests would fail to reject even more often if we took into account the uncertainty about the remaining coefficients of the true DGP, which we ignore for computational reasons.
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favor of state intervention. To answer this question, we modify the 1950 initial beliefs of policymakers by increasing their initial uncertainty, {νi }ni=1 , by about 50%. With this increase, the mean of {νi }ni=1 equals the estimate of ζ, that is, the standard deviation of the population distribution of b in the true DGP. We then simulate the evolution of policymakers’ beliefs under this new prior, given the observed data. Under this counterfactual scenario, we find that the median country would have preferred market orientation starting in 1989, which is very similar to 1991, our result with our baseline model. Our general conclusion is that reasonable changes in policymakers’ initial uncertainty would not have substantially speeded up the adoption of market-oriented policies. In a second experiment, we shift the mean of policymakers’ initial beliefs in 1950 so that the prior mean of the average country is equal to our estimate of b¯ S and b¯ M , obtained using the entire post-1950 sample. In other words, we ask what would have happened if policymakers’ initial beliefs in 1950 had been on average as favorable toward market orientation as suggested by the fullsample estimation. Obviously, with these counterfactual initial priors, many more countries should have been market oriented in the early 1950s. However, the superior performances (and the prevalence) of interventionist regimes in the 1950s would have very quickly shifted policymakers’ beliefs back toward favoring state intervention and the median country would have become market oriented in 1989. Indeed, from approximately the mid 1950s, the baseline and counterfactual models imply a similar evolution of policymakers’ beliefs. In a third experiment we set the mean and standard deviation of policymakers’ initial beliefs equal to their estimated values, but we consider alternative values of the parameter γ governing the perceived spatial correlation of the effect of policies. Figure 11 reports the time in which the median country starts preferring market orientation, as a function of the prior correlation coefficients at a distance of 3000 km (our baseline estimated value is 0634 and the correlation implied by the true DGP is 0312). Notice that the median country never chooses to become market oriented before 1991, no matter what the prior correlation is. When this correlation is low, state intervention persists even longer. This is because, in updating beliefs about the growth effect of policies in one country, the experience of the rest of the world is perceived to be not very relevant and, as a consequence, information accumulates slower. However, this effect is not monotone because high spatial correlations also imply very slow adoption of market-oriented policies as we explain below. Our general conclusion is that the slow diffusion of market orientation is a robust result of very different values of policymakers’ perceived spatial correlation of the effect of policies on growth. To understand why the prior spatial correlation has a nonmonotone effect on the timing of adoption of market-oriented policies, it is instructive to consider a special case of our model of beliefs formation. In particular, assume that (i) growth shocks have the same variance and are uncorrelated across countries (i.e., Σ = σ 2 In ), (ii) policymakers’ priors are characterized by the same
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FIGURE 11.—Year in which the median country becomes market oriented, as a function of the prior spatial correlation at a distance of 3000 km.
degree of uncertainty (i.e., V = σ 2 ν 2 In ), and (iii) the prior mean on the effect of policies on growth is still heterogeneous, but the prior spatial correlation across all countries is equal to 1. Under these assumptions, the evolution of beliefs about average growth under market orientation is given by the formula ˆM βˆ M it = βit−1 +
1 + ν2 ·
ν2 · n n t j=1
s=1
θjs
¯ ¯t − β¯ M × [cov(θt yt − βˆ M t−1 ) + θt (y t−1 )]
i = 1 n
where the upper bar stands for the cross-sectional means and cov denotes the cross-sectional covariance. Notice that if y¯t = β¯ M t−1 , beliefs about market orientation are updated upward at time t only if cov(θt yt − βˆ M t−1 ) is positive, that is, if market-oriented countries perform better than expected in t − 1. In other words, beliefs are not driven by the naive correlation between policies and growth, but by the correlation between growth and the deviations from the expected heterogeneous effects of policies. According to our model, this is what prevented an earlier diffusion of market orientation around the world. Apparently, at least until the late 1980s, the only countries growing fast under market orientation were those that were expected to do so. The fact that policy effects are perceived to be heterogeneous plays a key role for this result. In fact, if policymakers thought that the effect of policies ˆM were common across all countries (i.e., βˆ M it−1 = βjt−1 for all i and j), they would only have to learn about this common effect, about which the data of all other countries would become fully informative. In such a case cov(θt yt − βˆ M t−1 )
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would boil down to cov(θt yt ), so that beliefs about the benefits of market orientation would improve for the good performance of any market-oriented country, not just those not expected to perform well. As a consequence, learning is greatly accelerated and the diffusion of market orientation is much faster. We illustrate this point in a fourth counterfactual experiment in which we consider a model where policymakers believe that the effect of policies is common across countries, but they have heterogeneous priors about these common effects.19 In particular, for each country i, we set the initial prior mean of the common effect of policies to the estimated value of βˆ Si0 and βˆ M i0 , and set the initial standard deviation to the estimated value of νi . However, we now assume that policymakers of country i believe that the effect of policies on growth in all other countries is the same as in their country.20 Figure 12 plots the time evolution of the empirical distribution of {βˆ M∗ it − S∗ βˆ it }ni=1 , that is, the difference between the posterior mode of the mean of policymakers’ beliefs about the effect of market-oriented and state-interventionist policies on growth. Observe that in this model with perceived common effects, the median country would have embraced market orientation in the
ˆ S∗ N FIGURE 12.—Evolution over time of the empirical distribution of {βˆ M∗ it − βit }i=1 , that is, the difference between the posterior mode of the mean of policymakers’ beliefs about the effect of market-oriented and state-interventionist policies on growth. 19
The important feature of this experiment is the perceived common effect, not the heterogeneous priors, which we adopt only for convenience. 20 This is a special case of the model of Buera, Monge-Naranjo, and Primiceri (2008).
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1960s, soon after market-oriented countries started growing faster than stateinterventionist economies. To conclude, the slow adoption of market-oriented policies that we observe in the data is a very robust feature of our model with potentially heterogeneous effects of economic policies on GDP growth. Substantially faster diffusion of market orientation would have taken place only if policymakers had ruled out a priori the possibility of heterogeneity and had been fully convinced that the effect of policies is common across countries. This is in contrast to the estimates of our true DGP and, more importantly, to the casual observation about policy debates in different countries (see, for instance, Di Tella and MacCulloch (2007)). 8. POLICY REVERSALS: ANOTHER GREAT DEPRESSION One natural application of our model is trying to answer the question, “Would we observe policy reversals toward state intervention if the world was hit by another Great Depression?” The answer of our estimated model is “yes.” We obtain this answer by conducting a final set of counterfactual simulations. Differently from the previous section, here we keep all model parameters at their estimated posterior mode, but generate artificial paths of GDP growth so as to induce a large global recession. More specifically, these simulations are constructed as follows. Suppose that time τ is the starting point of the counterfactuals. Policymakers update their beliefs with time-τ information. We construct the political cost based on the value of the Polity2, civil war and relative GDP variables at time τ + 1, and a draw of the exogenous shock from the estimated distribution.21 The combination of beliefs and political cost determines policymakers’ choice of the policy variable, which, in turn, contributes to the realization of GDP growth in period τ + 1. A new vintage of data is now available and policymakers form time-τ + 1 beliefs by updating their priors with the new information, and so on for H periods. To generate GDP growth as a function of the policy decision, we use a slightly modified version of the true DGP estimated in Section 7. The modification consists of the introduction in (7.1) of a forcing variable, Fis , which we calibrate to match the size of the Great Depression in the 1930s. Using the Maddison data set, we compute {Fis }H s=1 = [−00047 −00517 −00867 −00687 00023 0 0] i = 1 n which corresponds to the average deviation across countries of the growth rate in 1929–1933 relative to the average growth between 1919 and 1928. 21 Going forward, we assume that the Polity2 and civil war variables remain unchanged relative to their value at time τ.
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FIGURE 13.—Change in the share of market-oriented countries as a consequence of a severe worldwide recession in 2002, as a function of the differential impact of the recession on market-oriented and state-interventionist countries.
Figure 13 depicts the change in the share of market-oriented countries within 5 years in response to the global recession that starts in 2002. Negative numbers reflect that countries, on net, are switching from being marketoriented to being state-interventionist. The change in the share of marketoriented countries is plotted as a function of the impact of the depression on market-oriented relative to state-interventionist economies, keeping the average impact equal to {Fis }H s=1 . For instance, the fraction of market-oriented countries reverting to interventionism would be about 10% if the depression affected market-oriented economies twice as much as state-interventionist ones. We find that this effect would take almost 40 years to wash out. Part of the intuition for this result is straightforward: if policymakers perceive that state intervention leads to faster growth, they will update their beliefs accordingly and, in some cases, switch policy. However, as evident from the intercept of Figure 13, a depression that has the same impact on market-oriented and state-interventionist economies would also induce some policy reversals. This is because policymakers would update downward the beliefs about both the benefits of markets and governments. However, since most countries in 2002 are market oriented, the signal about the deterioration in growth performance of market-oriented economies would be much more precise and these beliefs would evolve faster. This intuition suggests also that had such an equal recession hit the world economy in the 1960s or 1970s, when beliefs were less precise and when most countries were state interventionists, it would have spawned an even larger reversal in the opposite direction, that is, from state intervention to market orientation.
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9. CONCLUDING REMARKS We have explored the ability of a learning model to explain the observed transition of countries between regimes of state intervention and market orientation. In our model, the crucial determinants of policy choices are policymakers’ beliefs about the relative merits of the markets versus the state. These beliefs, in turn, are influenced by past experience. The estimated model fits well the data of 133 countries for the postwar period. Our results indicate that countries around the world were slow to adopt market-oriented policies because policymakers take seriously the possibility that the impact of these policies is country-specific. Moreover, our findings suggest that by the end of the sample many countries remain agnostic about the growth prospects of market economies. Indeed, according to the model, a global GDP growth shock of the size of the Great Depression would induce between 5 and 10% of countries in the world to revert to state intervention. While slow in our context, we should stress that the speed of learning is typically influenced by the strength of the effectiveness of policies. Therefore, policies and institutions with more visible benefits on countries’ growth (e.g., Acemoglu, Johnson, and Robinson (2005)) are more likely to be regarded as successful and adopted widely around the world provided that they can be easily implemented. Similarly, learning would presumably be faster if these policies also had an effect on other relevant economic variables. However, in this case policymakers might face a more complex trade-off between alternative policy objectives, which could complicate their learning process (Sargent (1999)). In this paper, we have focused on the evolution of beliefs as the driving mechanism of policy changes and have abstracted from other forces that are undoubtedly important. For example, in our learning model, we have not considered other reasons that may lead countries to herd behind the policy choices of other countries. Similarly, it would be interesting to enrich our model with explicit political economy dimensions and examine how they interact with the mechanism of beliefs formation that we have presented in this paper. These are our priorities for future research. REFERENCES ACEMOGLU, D., P. AGHION, AND F. ZILIBOTTI (2006): “Distance to Frontier, Selection, and Economic Growth,” Journal of the European Economic Association, 4 (1), 37–74. [21] ACEMOGLU, D., S. JOHNSON, AND J. A. ROBINSON (2001): “The Colonial Origins of Comparative Development: An Empirical Investigation,” American Economic Review, 91 (5), 1369–1401. [4] (2005): “Institutions as the Fundamental Cause of Long-Run Growth,” in Handbook of Economic Growth, ed. by P. Aghion and S. Durlauf. Amsterdam: Elsevier/North Holland, chap. 6, 385–472. [4,42] ACEMOGLU, D., S. JOHNSON, J. A. ROBINSON, AND P. YARED (2008): “Income and Democracy,” American Economic Review, 98 (3), 808–842. [4] (2009): “Reevaluating the Modernization Hypothesis,” Journal of Monetary Economics, 56 (8), 1043–1058. [4]
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MANSKI, C. F. (1993): “Identification of Endogenous Social Interaction: The Reflection Problem,” Review of Economic Studies, 60 (3), 531–542. [19] MARSHALL, M. G., AND K. JAGGERS (2005): “Polity IV Project: Dataset Users’ Manual,” Mimeo, George Mason University. [16] MATÉRN, B. (1960): “Spatial Variation,” Meddelanden från Statens Skogsforskningsinstitut, 49 (5), 1–144. [9] MAYDA, A. M., AND D. RODRIK (2005): “Why Are Some People (and Countries) More Protectionist Than Others?” European Economic Review, 49 (6), 1393–1430. [4,23] MUKAND, S. W., AND D. RODRIK (2005): “In Search of the Holy Grail: Policy Convergence, Experimentation, and Economic Performance,” American Economic Review, 95 (1), 374–383. [4] NEYMAN, J., AND E. L. SCOTT (1948): “Consistent Estimates Based on Partially Consistent Observations,” Econometrica, 16 (1), 1–32. [19] PERSSON, T., AND G. TABELLINI (2002): Political Economics: Explaining Economic Policy. Cambridge, MA: MIT Press. [4] (2009): “Democratic Capital: The Nexus of Political and Economic Change,” American Economic Journal: Macroeconomics, 1 (2), 88–126. [4] PIKETTY, T. (1995): “Social Mobility and Redistributive Politics,” The Quarterly Journal of Economics, 110 (3), 551–584. [4] PRIMICERI, G. E. (2006): “Why Inflation Rose and Fell: Policymakers’ Beliefs and US Postwar Stabilization Policy,” The Quarterly Journal of Economics, 121 (3), 867–901. [4] RODRIGUEZ, F., AND D. RODRIK (2000): “Trade Policy and Economic Growth: A Skeptic’s Guide to the Cross-National Evidence,” NBER Macroeconomics Annual, 15, 261–325. [5,13] SACHS, J. D., AND A. WARNER (1995): “Economic Reform and the Process of Global Integration,” Brookings Papers on Economic Activity, 1995 (1), 1–118. [5,10,12,13] SARGENT, T. J. (1999): The Conquest of American Inflation. Princeton, NJ: Princeton University Press. [4,7,42] SARGENT, T. J., N. WILLIAMS, AND T. ZHA (2006): “Schocks and Government Beliefs: The Rise and Fall of American Inflation,” American Economic Review, 96 (4), 1193–1124. [4] SOKOLOFF, K. L., AND S. L. ENGERMAN (2000): “History Lessons: Institutions, Factors Endowments, and Paths of Development in the New World,” Journal of Economic Perspectives, 17 (3), 217–232. [4] STRULOVICI, B. (2008): “Voting and Experimentation,” Mimeo, Northwestern University. [4,7] TAMER, E. (2003): “Incomplete Simultaneous Discrete Response Model With Multiple Equilibria,” Review of Economic Studies, 70 (1), 147–165. [32] WACZIARG, R., AND K. H. WELCH (2008): “Trade Liberalization and Growth: New Evidence,” World Bank Economic Review, 22 (2), 187–231; Working Paper 10152, NBER. [12,13]
Economics Dept., University of California, Los Angeles, 8283 Bunche Hall, Los Angeles, CA 90095, U.S.A. and NBER;
[email protected], Dept. of Economics, Pennsylvania State University, 502 Kern Graduate Building, University Park, PA 16802, U.S.A.;
[email protected], and Dept. of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208, U.S.A., CEPR, and NBER;
[email protected]. Manuscript received December, 2008; final revision received June, 2010.
Econometrica, Vol. 79, No. 1 (January, 2011), 47–73
AXIOMATIC FOUNDATIONS OF MULTIPLIER PREFERENCES BY TOMASZ STRZALECKI1 This paper axiomatizes the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001). The axiomatization relates multiplier preferences to other classes of preferences studied in decision theory, in particular, the variational preferences recently introduced by Maccheroni, Marinacci, and Rustichini (2006a). This paper also establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent. This link enables measurement of the parameters on the basis of observable choice data and provides a useful tool for applications. KEYWORDS: Ambiguity aversion, model uncertainty, robustness.
1. INTRODUCTION THE EXPECTED UTILITY CRITERION ranks payoff profiles f according to (1) V (f ) = u(f ) dq where u is a utility function and q is a subjective probability distribution on the states of the world. A decision maker with such preferences behaves as if he is certain that the state is distributed according to the probabilistic model q. To model situations where the decision maker does not have enough information to formulate a single probabilistic model and have full confidence in it, for example, when it is hard to statistically distinguish between similar probabilistic models, Hansen and Sargent (2001) formulated the criterion V (f ) = min u(f ) dp + θR(p q) (2) p
where θ ∈ (0 ∞] is a parameter and the function R(p q) is the relative entropy of p with respect to q. Relative entropy, otherwise known as Kullback–Leibler divergence, is a measure of “distance” between two probability distributions. 1
I am indebted to my advisor, Eddie Dekel, for his continuous guidance, support, and encouragement. I am grateful to Peter Klibanoff and Marciano Siniscalchi for many discussions which resulted in significant improvements of the paper. I would also like to thank Jeff Ely, Todd Sarver, and seminar audiences at Berkeley, Bocconi, CEMFI, Chicago, Collegio Carlo Alberto, Columbia, Duke (Fuqua), Harvard, Hebrew University, LSE, NYU (Econ and Stern), Northwestern, Penn, Princeton, Stanford (Econ and GSB), UCL, University of Iowa, University of Minnesota, Warwick, Washington University (Econ and Olin), Yale, and the Hansen–Sargent conference for graduate students. This project started after a very stimulating conversation with Tom Sargent and was further shaped by conversations with Lars Hansen. I am very grateful to the co-editor and three anonymous referees for their insightful and helpful comments. All errors are my own. © 2011 The Econometric Society
DOI: 10.3982/ECTA8155
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An interpretation of equation (2) is that the decision maker has some best guess q of the true probability distribution, but does not fully trust it. Instead, he considers many other probabilities p to be plausible, with plausibility diminishing proportionally to their “distance” from q. The role of the proportionality parameter θ is to measure the degree of trust of the decision maker in the reference probability q or, in other words, the concern for model misspecification. Higher values of θ correspond to more trust; in the limit, with the convention that 0 · ∞ = 0 when θ = ∞, the decision maker fully trusts his reference probability q and uses the expected utility criterion (1). Multiplier preferences (2) also belong to the more general class of variational preferences studied by Maccheroni, Marinacci, and Rustichini (2006a), which have the representation (3)
V (f ) = min p
u(f ) dp + c(p)
The interpretation of (3) is like that of (2), and multiplier preferences are a special case of variational preferences with c(p) = θR(p q). In general, the conditions that the function c(p) in (3) has to satisfy are very weak, which makes variational preferences a very broad class. In addition to expected utility preferences and multiplier preferences, this class also nests the maxmin expected utility preferences of Gilboa and Schmeidler (1989), as well as the mean-variance preferences of Markowitz (1952) and Tobin (1958). An important contribution of Maccheroni, Marinacci, and Rustichini (2006a) was to provide an axiomatic characterization of variational preferences. Because variational preferences are a very broad class of preferences, it is desirable to establish an observable distinction between multiplier preferences and other subclasses of variational preferences. This is, for example, the case with the maxmin expected utility preferences of Gilboa and Schmeidler (1989): a strengthening of the Maccheroni, Marinacci, and Rustichini (2006a) axioms restricts the general function c(p) to be in the Gilboa and Schmeidler (1989) class. The main finding of this paper is that the sure-thing principle of Savage (imposed on the Anscombe–Aumann domain) characterizes the class of multiplier preferences within the class of variational preferences. This is possible because, as the main theorem shows, the class of multiplier preferences is precisely the intersection of the class of variational preferences and the class of second-order expected utility (SOEU) preferences with representation (4)
V (f ) =
φ(u(f )) dq
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FIGURE 1.—Relations between classes of preferences: VP, variational preferences; MP, multiplier preferences; SOEU, second-order expected utility preferences; EU, expected utility preferences; MEU, maxmin expected utility preferences; CP, constraint preferences.
for some real-valued function φ.2 Figure 1 depicts the relationships between these classes.3 The sure thing principle axiom used in the characterization is standard in the literature; in particular, it is not in any way related to the very specific functional form of relative entropy—it is the interaction between the axioms that delivers the representation. The proposed axiomatic characterization is important for three reasons. First, it provides a set of testable predictions of the model that allow for its empirical verification. This will help evaluate whether multiplier preferences, which have already proved useful in modeling behavior at the macro level,4 are an accurate model of individual behavior. Second, the axiomatization establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent. This link enables measurement of the parameters on the basis of observable choice data alone, without relying on unverifiable assumptions. Finally, the axiomatization is helpful in understanding the relation between the multiplier preferences and the axiomatic models of ambiguity aversion motivated by the Ellsberg (1961) paradox, where people exhibit a preference for choices involving objective rather than subjective probabilities. 2
For axiomatic characterizations of such preferences, see Neilson (1993, 2010), Nau (2001, 2006), Ergin and Gul (2009), and Grant, Polak, and Strzalecki (2009). 3 Hansen and Sargent also introduced a closely related class of constraint preferences, repre sented by V (f ) = min{p|R(pq)≤η} S (u ◦ f ) dp, which are a special case of Gilboa and Schmeidler’s (1989) maxmin expected utility preferences; see Figure 1. Due to their greater analytical tractability, multiplier—rather than constraint—preferences are used in applications. 4 See Barillas, Hansen, and Sargent (2009), Benigno and Nisticò (2009), Hansen, Sargent, and Tallarini (1999), Hansen, Sargent, and Wang (2002), Karantounias, Hansen, and Sargent (2009), Kleshchelski and Vincent (2009), Li and Tornell (2008), Maenhout (2004), and Woodford (2006).
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The paper is organized as follows. Section 2 introduces some notation and basic concepts, as well as the definition of multiplier preferences. Section 3 presents an axiomatic characterization of multiplier preferences within the class of variational preferences in the classic setting of Anscombe and Aumann. Section 4 studies another choice domain, introduced by Ergin and Gul (2009), and presents a fully subjective axiomatization of multiplier preferences. Section 5 concludes. 2. PRELIMINARIES 2.1. Setup Decision problems considered in this paper involve a set S of states of the world, which represents the possible contingencies that may occur. One of the states, s ∈ S, will be realized, but the decision maker has to choose the course of action before learning s. Let Σ denote a sigma-algebra of events in S.5 The set of all finitely additive probability measures on (S Σ) is denoted Δ(S) and endowed with the weak∗ topology, where a net {pd }d∈D converges to p if pd (A) → p(A) for all A ∈ Σ; the set of all countably additive probability measures is denoted Δσ (S); its subset that consists of all measures absolutely continuous with respect to q ∈ Δσ (S), is denoted Δσ (q). The set Z denotes the possible consequences and Δ(Z) denotes simple probability distributions on Z. An element of Δ(Z) is called a lottery. A lottery paying off z ∈ Z with probability 1 is denoted δz . For any two lotteries π π ∈ Δ(Z) and a number α ∈ (0 1), the lottery απ + (1 − α)π assigns probability απ(z) + (1 − α)π (z) to each prize z ∈ Z. The alternatives that the decision maker faces, called acts, are mappings from S to Δ(Z).6 Formally, an act is a finite-valued, Σ-measurable function f : S → Δ(Z); the set of all such acts is denoted F (Δ(Z)). If f g ∈ F (Δ(Z)) and E ∈ Σ, then fE g denotes an act with fE g(s) = f (s) if s ∈ E and fE g(s) = g(s) if s ∈ / E. If f g ∈ F (Δ(Z)) and α ∈ [0 1], then αf + (1 − α)g denotes an act that assigns the lottery αf (s) + (1 − α)g(s) in each state s ∈ S. The choices of the decision maker are represented by a preference relation , where f g means that the act f is weakly preferred to the act g. A functional V : F (Δ(Z)) → R represents if for all f g ∈ F (Δ(Z)), f g
if and only if
V (f ) ≥ V (g)
An important class of preferences are the expected utility (EU) preferences, where the decision maker has a probability distribution q ∈ Δ(S) and a utility function that evaluates each consequence u : Z → R. A preference relation The set S may be infinite or finite. When S is finite, it is assumed that Σ = 2S . This setting was introduced by Fishburn (1970). Settings of this type are usually named after Anscombe and Aumann (1963), who were the first to work with them. 5 6
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has an EU representation (u q) if a functional V : F (Δ(Z)) → R represents , where u(z)f (s)(z) dq(s) V (f ) = S
z∈Z
In each state of the world s, the decision maker computes the expected utility of the lottery f (s) and then averages those values across states. By slightly abusing notation, define the affine function u : Δ(Z) → R by u(π) = z∈Z u(z)π(z). Using this definition, the expected utility criterion can be written as (5) V (f ) = u(f (s)) dq(s) S
Risk aversion is the phenomenon where sure payoffs are preferred to payoffs that are stochastic but have the same expected monetary value. If Z = R, that is, lotteries have monetary payoffs, then risk averse EU preferences have concave utility functions u. Likewise, one preference relation is more risk averse than another if it has a “more concave” utility function. Formally, an EU preference represented by (u1 q1 ) is more risk averse than one represented by (u2 q2 ) if and only if q1 = q2 and u1 = φ ◦ u2 , where φ : R → R is a strictly increasing concave transformation. A special role will be played by the class of transformations φθ , indexed by θ ∈ (0 ∞]: ⎧ u ⎨ − exp − for θ < ∞, (6) φθ (u) = θ ⎩ u for θ = ∞. Lower values of θ correspond to more concave transformations, that is, more risk aversion. 2.2. Sources of Uncertainty and the Ellsberg Paradox Observe that every act f : S → Δ(Z) involves two sources of uncertainty: first, the payoff of f is contingent on the state of the world, for which there is no objective probability given; second, given the state, f (s) is an objective lottery. The existence of two sources of uncertainty enables a distinction between purely objective lotteries, that is, acts which pay the same lottery π ∈ Δ(Z) irrespective of the state of the world, and purely subjective acts, that is, acts that in each state of the world pay off a degenerate lottery δz for some z ∈ Z, which possibly depends on s. With a slight abuse of notation, let Δ(Z) denote the set of purely objective lotteries. Note that given q ∈ Δ(S), each purely subjective act f induces a purely objective lottery πf ∈ Δ(Z) defined by πf (z) = q(f −1 (z)) for all z ∈ Z.
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An EU decision maker has the same attitude toward objective lotteries and toward subjective acts. From the representation (5), it follows that for any two purely objective lotteries π π if and only if u(z)π (z) ≥ u(z)π(z) z∈Z
z∈Z
and for any two purely subjective acts f f if and only if u(z)πf (z) ≥ u(z)πf (z) z∈Z
z∈Z
In particular, any purely subjective act f is indifferent to the objective lottery πf that it induces. However, more general preferences need not have such a uniform decision attitude and they may be source sensitive, that is, exhibit more aversion to one source than the other. This is illustrated by the Ellsberg (1961) paradox, which demonstrates that most people prefer choices involving risk (i.e., situations in which the probability is well specified) to choices involving ambiguity (where the probability is not specified).7 EXAMPLE 1—Ellsberg Paradox: Consider two urns containing colored balls. The decision maker can bet on the color of the ball drawn from each urn. Urn I contains 100 red and black balls in unknown proportion, while urn II contains 50 red and 50 black balls. In this situation, most people are indifferent between betting on red from urn I and on black from urn I; this reveals that, in the absence of evidence against symmetry, they view those two contingencies as interchangeable. Moreover, most people are indifferent between betting on red from urn II and on black from urn II; this preference is justified by their knowledge of the composition of urn II. However, most people strictly prefer betting on red from urn II to betting on red from urn I, thereby displaying ambiguity aversion. Ambiguity aversion cannot be reconciled with the EU model. To see that, let the state space S = {R B} represent the possible draws from urn I. Betting $100 on red from urn I corresponds to an act fR = (δ100 δ0 ), while betting $100 on black from urn I corresponds to an act fB = (δ0 δ100 ). On the other hand, betting $100 on red from urn II corresponds to a lottery πR = 1 δ + 12 δ0 , while betting $100 on black from urn II corresponds to a lottery 2 100 πB = 12 δ0 + 12 δ100 . These correspondences reflect the fact that betting on urn I involves subjective uncertainty, while betting on urn II involves objective risks. Note in particular that πR = πB . 7 Other experimental evidence on source sensitivity includes Abdellaoui, Baillon, Placido, and Wakker (2010), Chipman (1960), Curley and Yates (1989), Einhorn and Hogarth (1985), Fox and Tversky (1995), and Heath and Tversky (1991).
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Suppose that the subjective probability of drawing red from urn I is q and drawing black from urn I is 1 − q. Observe that V (πR ) = V (πB ) = 12 u(100) + 1 u(0), whereas V (fR ) = qu(100) + (1 − q)u(0) and V (fB ) = (1 − q)u(100) + 2 qu(0). Because of the indifference V (fR ) = V (fB ), it follows that q = 12 ; hence, V (fR ) = V (fB ) = 12 u(100) + 12 u(0). It follows that πB ∼ πR ∼ fR ∼ fB , contradicting the typical Ellsberg choices. As the above example shows, the Ellsberg pattern of choices cannot be explained by a model with a unique probability measure and with uniform aversion to both sources. In the literature there have been two main approaches to this problem. The first one replaces the probability measure q with some other measure of belief that captures the decision maker’s lack of information about the source; see, for example, the Choquet model of Schmeidler (1989), maxmin model of Gilboa and Schmeidler (1989), variational preferences of Maccheroni, Marinacci, and Rustichini (2006a), or smooth preferences of Klibanoff, Marinacci, and Mukherji (2005). The other approach is to keep q but to introduce another parameter that captures the higher aversion toward variability coming from one source than another, as in the SOEU model (Ergin and Gul (2009), Grant, Polak, and Strzalecki (2009), Gul and Pesendorfer (2010), Nau (2001, 2006), Neilson (1993, 2010)). As this paper shows (see Section 3.3), the multiplier preferences have both representations and for this reason they belong to both families of models. 2.3. Multiplier Preferences Hansen and Sargent (2001) and Hansen, Sargent, Turmuhambetova, and Williams (2006) introduced the class of multiplier preferences where the decision maker does not know the true probabilistic model p, but has a “best guess” or approximating model q, also called the reference probability. The decision maker thinks that the true probability p is somewhere near the reference probability q. The notion of distance used by Hansen and Sargent is relative entropy. DEFINITION 1: Let a reference measure q ∈ Δσ (S) be fixed. The relative entropy R(· q) is a mapping from Δ(S) into [0 ∞] defined by ⎧ dp ⎨ dp if p ∈ Δσ (q), log R(p q) = dq ⎩ S ∞ otherwise. A decision maker who is concerned with model misspecification computes his expected utility according to all probabilities p, but he does not treat them equally. Probabilities closer to his best guess have more weight in his decision.
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DEFINITION 2: A relation has a multiplier representation if it is represented by V (f ) = min u(f (s)) dp(s) + θR(p q) p∈Δ(S)
S
where u : Δ(Z) → R is a nonconstant affine function, θ ∈ (0 ∞], and q ∈ Δσ (S). In this case, is called a multiplier preference. The multiplier representation of may suggest the following interpretation. First, the decision maker chooses an act without knowing the true distribution p. Second, “Nature” chooses the probability p so as to minimize the decision maker’s expected utility. Nature is not free to choose, but it incurs a “cost” for using each p. Probabilities p that are farther from the reference measure q have a larger potential for lowering the decision maker’s expected utility, but Nature has to incur a larger cost to select them. This interpretation suggests that a decision maker with such preferences is concerned with model misspecification and makes decisions that are robust to such misspecification. He is pessimistic about the outcome of his decision which leads him to exercise caution in choosing the course of action. Such cautious behavior is reminiscent of the Ellsberg paradox, as in Example 1 above. In fact, as Example 2 in Section 3.4.2 shows, multiplier preferences can be used to model such behavior. 3. AXIOMATIZATION WITH OBJECTIVE RISK 3.1. Variational Preferences To capture ambiguity aversion, Maccheroni, Marinacci, and Rustichini (2006a) (henceforth MMR) introduced the class of variational preferences, with representation V (f ) = min u(f (s)) dp + c(p) (7) p∈ΔS
S
where c : ΔS → [0 ∞] is a cost function. Multiplier preferences are a special case of variational preferences where c(p) = θR(p q). The variational criterion (7) can be given the same interpretation as the multiplier criterion (2): Nature wants to reduce the decision maker’s expected utility by choosing a probability distribution p, but she is not entirely free to choose. Using different p’s leads to different values of the de cision maker’s expected utility S u(f (s)) dp, but comes at a cost c(p). To characterize variational preferences behaviorally, MMR used the following axioms. AXIOM A1—Weak Order: The relation is transitive and complete.
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AXIOM A2—Weak Certainty Independence: For all f g ∈ F (Δ(Z)), π π ∈ Δ(Z), and α ∈ (0 1), αf + (1 − α)π αg + (1 − α)π ⇒
αf + (1 − α)π αg + (1 − α)π
AXIOM A3—Continuity: For any f g h ∈ F (Δ(Z)), the sets {α ∈ [0 1] | αf + (1 − α)g h} and {α ∈ [0 1] | h αf + (1 − α)g} are closed. AXIOM A4—Monotonicity: If f g ∈ F (Δ(Z)) and f (s) g(s) for all s ∈ S, then f g. AXIOM A5—Uncertainty Aversion: If f g ∈ F (Δ(Z)) and α ∈ (0 1), then f ∼g
⇒
αf + (1 − α)g f
AXIOM A6—Nondegeneracy: f g for some f g ∈ F (Δ(Z)). AXIOM A8 —Weak Monotone Continuity: If f g ∈ F (Δ(Z)), π ∈ Δ(Z), {En }n≥1 ∈ Σ with E1 ⊇ E2 ⊇ · · · and n≥1 En = ∅, then f g implies that there exists n0 ≥ 1 such that πEn0 f g. MMR showed that the preference satisfies Axioms A1–A6 if and only if is represented by (7) with an affine and nonconstant u : Δ(Z) → R and c : ΔS → [0 ∞] that is convex, lower semicontinuous, and grounded (achieves value zero). Moreover, Axiom A8 guarantees that function c is concentrated only on countably additive measures (observe, that Axiom A8 holds trivially if S is finite). The conditions that the cost function c satisfies are very general. For example, if c(p) = ∞ for all measures p = q, then (7) reduces to (5), that is, preferences are expected utility. Similarly, setting c(p) = 0 for all measures p in a closed and convex set C and c(p) = ∞ otherwise, denoted c = δC , reduces (7) to the representation of the maxmin expected utility preferences of Gilboa and Schmeidler (1989). As mentioned before, multiplier preferences also are a special case of variational preferences. They can be obtained by setting c(p) = θR(p q). The next section shows that pinning down this functional form is possible with Savage’s P2 applied to all Anscombe–Aumann acts.8 8 If the existence of certainty equivalents of lotteries is assumed, that is, for any π ∈ Δ(Z) there exists z ∈ Z with z ∼ π, then P2 can be weakened and imposed only on purely subjective acts.
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3.2. Axiomatization of Multiplier Preferences AXIOM P2 —Savage’s Sure-Thing Principle: For all E ∈ Σ and f g h h ∈ F (Δ(Z)), fE h gE h
⇒
fE h gE h
DEFINITION 3: An event E ∈ Σ is nonnull if there exist f g h ∈ F (Δ(Z)) such that f Eh gEh. THEOREM 1: If S has at least three disjoint nonnull events, then Axioms A1– A6, A8, and P2, are necessary and sufficient for to have a multiplier representation (2). Moreover, in this case, two triples (θ u q ) and (θ u q ) represent the same multiplier preference if and only if q and q are identical, and there exist α > 0 and β ∈ R such that u = αu + β and θ = αθ . The two cases θ = ∞ (lack of concern for model misspecification) and θ < ∞ (concern for model misspecification) can be distinguished on the basis of the independence axiom.9 In the case when θ is finite, its numerical value is uniquely determined, given u. A positive affine transformation of u changes the scale on which θ operates, so θ has to change accordingly. This is reminiscent of the necessary adjustments of the constant absolute risk aversion coefficient when units of account are changed. In addition, it should be mentioned that there exists an axiomatization by Wang (2003) of a class of preferences that includes multiplier preferences as a special case. However, his result is formally unrelated and it assumes different primitives: preferences are defined on triples (f C q), where f is an act with monetary payoffs, C ⊆ Δ(S) is a set of probability measures, and q ∈ Δ(S) is a reference measure. In particular, his axioms impose consistency conditions as the elements C and q are varied exogenously. In contrast, here C = Δ(S), and q is fixed and derived from preferences. 3.3. Sketch of the Proof The following variational formula (see, e.g., Proposition 1.4.2 of Dupuis and Ellis (1997)) plays a critical role in the analysis of multiplier preferences. For any bounded and Σ-measurable function ξ : S → R and q ∈ Δσ (S), min ξ dp + θR(p q) = φ−1 (8) φ ◦ ξ dq θ θ p∈ΔS
9
S
S
The weaker certainty independence axiom is also sufficient for making such a distinction. Alternatively, Machina and Schmeidler’s (1995) axiom of horse/roulette replacement or Grant and Polak’s (2006) axiom of betting neutrality could be used.
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This formula implies that the variational min u(f (s)) dp + θR(p q) (9a) p∈ΔS
S
and the SOEU
φθ u(f (s)) dq(s) (9b) S
representations are ordinally equivalent, establishing the necessity of the MMR and the P2 axiom of Savage.10 The sufficiency argument relies on the fact that the MMR axioms guarantee the existence of an affine utility function u : Δ(Z) → R and a functional I that maps utility-valued acts to reals, such that the functional f → I(u(f )) represents the preferences. For simplicity, assume that u(Δ(Z)) = R. MMR showed that the functional I has the translation invariance property that I(u ◦ f + k) = I(u ◦ f ) + k for any act f and any k ∈ R. On the other hand, Axiom P2 together with the MMR axioms implies that the preferences have a SOEU representation
(10) φ u(f (s)) dq(s) S
for some strictly increasing function φ.11 Moreover, Axioms A3 and A5 imply that φ is continuous and concave, and Axiom A8 implies that q is countably additive. This representation, together with translation invariance, implies that
φ u(f (s)) dq(s) ≥ φ u(g(s)) dq(s) S
S
if and only if (iff)
φ u(f (s)) + k dq(s) ≥ φ u(g(s)) + k dq(s) S
S
which by uniqueness of the SOEU representation implies that there exist α(k) > 0 and β(k) ∈ R such that φ(x + k) = α(k)φ(x) + β(k) for all x k ∈ R. 10 The fact that multiplier preferences rank purely subjective acts according to the EU criterion has been observed before in various levels of generality by Jacobson (1973), Whittle (1981), Skiadas (2003), and Maccheroni, Marinacci, and Rustichini (2006b). Because of this fact, it is not possible to distinguish multiplier preferences from the EU preferences based on the preferences over purely subjective acts alone: a setting with multiple sources of uncertainty, like the Anscombe–Aumann or Ergin–Gul ones, is needed. 11 Recall that Savage’s P2 axiom is imposed on F (Δ(Z)), not just the purely subjective acts, as in Savage (1972). The fact that imposing all Savage’s axioms on F (Δ(Z)) implies SOEU was first shown by Neilson (1993). I am grateful to Peter Klibanoff for this reference.
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This is a generalized Pexider equation, whose only solutions are φ = φθ for θ ∈ (0 ∞], which establishes the sufficiency of the axioms. 3.4. Discussion 3.4.1. Second-Order Expected Utility It follows from the above proof that the class of multiplier preferences is precisely the intersection of the class of variational preferences and the class of SOEU preferences. When viewed as a SOEU preference, multiplier preferences impose the condition φ = φθ . When viewed as a variational preference, multiplier preferences impose the condition c(p) = θR(p q). It is worthwhile to notice that this means that no other variational preferences have a SOEU representation, that is, assuming that c is a statistical distance other than the relative entropy leads to models which do not have a SOEU representation for any φ. Conversely, no other SOEU preference has a variational representation, that is, assuming that φ is a function other than the negative exponential leads to models which do not have a variational representation for any c. 3.4.2. Source Sensitivity of Multiplier Preferences Focus on the case θ < ∞ and notice that from the SOEU representation (9b), it follows that, for any two purely objective lotteries, π π if and only if u(z)π (z) ≥ u(z)π(z) z∈Z
z∈Z
On the other hand, for any two purely subjective acts, f f if and only if φθ (u(z))πf (z) ≥ φθ (u(z))πf (z) z∈Z
z∈Z
This means that the decision maker has a different attitude toward objective lotteries and toward subjective acts, while behaving according to EU in each subdomain. In particular, he is more averse toward subjective uncertainty (as captured by φθ ◦ u) than toward objective risk (as captured by u). This phenomenon is called second-order risk aversion.12 What lead to the Ellsberg-type behavior are violations of EU across those domains. The following example shows that, because of this property, multiplier preferences can be useful for modeling Ellsberg-type behavior. EXAMPLE 2 —Ellsberg’s Paradox Revisited: In the context of Example 1, consider a multiplier preference with a parameter θ. Observe that V (πR ) = 12 This notion was introduced by Ergin and Gul (2009) in a setting with two subjective sources of uncertainty (see Section 5).
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V (πB ) = φθ ( 12 u(100) + 12 u(0)), whereas V (fR ) = qφθ (u(100)) + (1 − q) × φθ (u(0)) and V (fB ) = (1 − q)φθ (u(100)) + qφθ (u(0)). Because of the indifference V (fR ) = V (fB ), it follows that q = 12 ; hence, V (fR ) = V (fB ) = 1 φ (u(100)) + 12 φθ (u(0)). By Jensen’s inequality, πB ∼ πR fR ∼ fB for all 2 θ θ < ∞. This means that the decision maker prefers objective risk to probabilistically equivalent subjective uncertainty, displaying behavior typical in Ellsberg’s experiments. 3.4.3. Measurement of Parameters Ellsberg’s paradox provides a natural setting for the experimental measurement of the parameters of the model because the intensity of the preference for betting on the first versus the second urn, that is, the premium that the decision maker is willing to pay to switch between these two bets, is directly related to the value of the parameter θ. EXAMPLE 3: In the context of Examples 1 and 2, consider a multiplier preference with a constant relative risk aversion utility function u(z) = (w + z)1−γ , where w is the initial level of wealth, and parameter θ. Observe that V (πR ) = V (πB ) = φθ ( 12 w1−γ + 12 (w + 100)1−γ ), whereas V (fR ) = V (fB ) = 12 φθ (w1−γ ) + 1 φ ((w + 100)1−γ ). Let x denote the certainty equivalent of πR and πB , that is, 2 θ the amount of money that, when received for sure, would be indifferent to πR and πB . Formally, x solves (11)
1 1 (w + x)1−γ = w1−γ + (w + 100)1−γ 2 2
The observed value of the certainty equivalent x allows computation of the curvature parameter γ using Equation (11); let γ(x) be the solution to this equation.13 Similarly, let y be the certainty equivalent of fR and fB , that is, the amount of money that, when received for sure, would be indifferent to fR and fB . Formally, y solves (12)
1
1
φθ (w + y)1−γ(x) = φθ w1−γ(x) + φθ (w + 100)1−γ(x) 2 2
The observed value of the certainty equivalent y makes it possible to compute the parameter θ using equation (12); let θ(x y) be the solution to this equation. The procedure described above suggests that simple choice experiments could be used for empirical measurement of both u and θ. Such measurement 13 If the utility function u belongs to some higher-dimensional family of utility functions, more certainty equivalents need to be elicited so as to infer all of its parameters.
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of parameters would be very useful in applied settings, where it is important to know the numerical values of parameters. For example, the macro-finance literature devotes a lot of attention to the discrepancy between the micro- and macro-level estimates of the curvature of u. By analogy, it would be valuable to know the micro-level estimate of θ to be able to compare it to the value calibrated from the macro-level data. The above procedure provides a simple “revealed-preference” method of comparison that is complementary to the heuristic method of “detection error probabilities” developed by Anderson, Hansen, and Sargent (2003) and Hansen and Sargent (2007). 4. AXIOMATIZATION WITH TWO SUBJECTIVE SOURCES This section discusses a choice domain that does not rely on the assumption of objective risk: instead, there are two sources of subjective uncertainty, toward which the decision maker may have different attitudes. This type of environment was discussed by Chew and Sagi (2008), Ergin and Gul (2009), Gul and Pesendorfer (2010), and Nau (2001, 2006). 4.1. Subjective Sources of Uncertainty Assume that the state space has a product structure S = Sa × Sb , where a and b are two separate issues, or sources of uncertainty, toward which the decision maker may have different attitudes. In comparison with the Anscombe– Aumann framework, where objective risk is one of the sources, here both sources are subjective. Let Aa be a sigma algebra of subsets of Sa and let Ab be a sigma algebra of subsets of Sb . Let Σa be the sigma algebra of sets of the form A × Sb for all A ∈ Aa , let Σb be the sigma algebra of sets of the form Sa × B for all B ∈ Ab , and let Σ be the sigma algebra generated by Σa ∪ Σb . Let F (Z) be the set of all simple Σ-measurable acts f : S → Z; moreover, let Fa (Z) be the set of acts that are Σa -measurable, and likewise for Fb (Z). Let U denote the set u(Z). Ergin and Gul (2009) axiomatized preferences which are general enough to accommodate probabilistic sophistication and even second-order probabilistic sophistication. An important subclass of those preferences are second-order expected utility preferences represented by (13) φ u(f (sa sb )) dqa (sa ) dqb (sb ) V (f ) = Sb
Sa
where the measures qa ∈ Δ(Sa ) and qb ∈ Δ(Sb ) are convex-ranged,14 u : Z → R, and φ : Dφ → R is a strictly increasing and continuous function with domain 14 A measure q is convex-ranged if for every E ∈ Σ and every α ∈ (0 1) there exists Σ E ⊆ E with q(E ) = αq(E). It is well known that this requirement is equivalent to nonatomicity for q ∈ Δσ (S). Any measure on R that has a density with respect to the Lebesgue measure has this property; in applications of multiplier preferences, q is most often a Normal distribution.
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Dφ := { Sa u(f (sa sb )) dqa (sa ) | f ∈ Fa (Z)}. To characterize preferences represented by (13), Ergin and Gul (2009) used the following axioms. AXIOM P1 —Weak Order: The preference is complete and transitive. AXIOM P2 —Sure-Thing Principle: For all events Ea ∈ Σa and Eb ∈ Σb , and ˜ h˜ ∈ F (Z), the following relationships exist: ˜ h acts f g h h ∈ Fa (Z) and f˜ g (a) fEa h gEa h iff fEa h gEa h . (b) f˜Eb h˜ g˜ Eb h˜ iff f˜Eb h˜ g˜ Eb h˜ . AXIOM P3 —Eventwise Monotonicity: For all z z ∈ Z, f ∈ F (Z), and all nonnull events E ∈ Σ, zE f zE f iff z z . AXIOM P4 —Weak Comparative Probability: (a) For all x y and x y , and A A ∈ ΣA , xA y xA y iff x A y x A y . (b) For all f g f g ∈ Fa (Z) such that f g and f g , and B B ∈ Σb , fB g fB g iff fB g fB g . AXIOM P5 —Nondegeneracy: There exist x y ∈ Z such that x y. AXIOM P6 —Small Event Continuity: For all f g ∈ F (Z) with f g, and all z ∈ Z, the following statements hold: (a) There exists a partition E1 En ∈ Σa of S such that for all i, zEi f g and f zEi g. (b) There exists a partition F1 Fm ∈ Σb of S such that for all j, zFj f g and f zFj g. There is a close relationship between representations (13) and (10). The role of objective risk is now taken by a subjective source: issue a. For each sb , the decision maker computes the expected utility of f (· sb ) and then averages those values using function φ. 4.2. Second-Order Risk Aversion In the Anscombe–Aumann framework, the concavity of the function φ is responsible for second-order risk aversion, that is, higher aversion toward subjective uncertainty than toward objective risk. This property is a consequence of the axiom of uncertainty aversion (Axiom A5). Similarly, in the present setup, the concavity of the function φ is responsible for higher aversion toward issue b than toward issue a. This property was introduced by Ergin and Gul (2009), who formally defined it in terms of mean-preserving spreads. However, this
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definition refers to the probability measures obtained from the representation and hence is not expressed directly in terms of observables.15 In the presence of other axioms, the following purely behavioral axiom is equivalent to Ergin and Gul’s (2009) definition. AXIOM A5 —Second-Order Risk Aversion: For any f g ∈ Fb (Z) and any E ∈ Σa , if f ∼ g, then fE g f . This axiom is a direct subjective analog of Schmeidler’s (1989) axiom of uncertainty aversion (Axiom A5). THEOREM 2: Suppose satisfies Axioms P1 –P6 and certainty equivalents exist, that is, for any f ∈ F (Z), there exists z ∈ Z with z ∼ f . Then Axiom A5 is satisfied if and only if the function φ in (13) is concave.16 4.3. Axiomatization of Multiplier Preferences The additional axiom that delivers multiplier preferences in this framework is weak Fa independence, which is the direct subjective analog of the weak certainty independence axiom (Axiom A2) of MMR. AXIOM A2 —Weak Fa Independence: For any event E ∈ Σa for all f g ∈ F (Z) and h h ∈ Fa (Z), fE h gE h
fE h gE h
⇒
In addition, a technical axiom, similar to Axiom A8, is needed. Continuity: If f g ∈ F (Z), x ∈ Z, {En }n≥1 ∈ Σb AXIOM A8 —Fb -Monotone with E1 ⊇ E2 ⊇ · · · and n≥1 En = ∅, then f g implies that there exists N ≥ 1 such that xEN f g. THEOREM 3: Axioms P1 –P6 , A2 , A5 , and A8 are necessary and sufficient for to be represented by V , where V (f ) = min u(f (sa sb )) dqa (sa ) dpb (sb ) + θR(pb qb ) pb ∈Δ(Sb )
Sb
Sa
and u : Z → R, θ ∈ (0 ∞], and qa ∈ Δ(Sa ) qb ∈ Δσ (Sb ) are convex-ranged measures. 15 Theorems 2 and 5 of Ergin and Gul (2009) characterize second-order risk aversion in terms of induced preferences over induced Anscombe–Aumann acts and an analog of Axiom A5 in that induced setting. However, just as with mean-preserving spreads, those induced Anscombe– Aumann acts are constructed using the subjective probability measure derived from the representation. 16 The full analysis that does not rely on the existence of certainty equivalents is contained in Appendix A.2.
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4.4. Sketch of the Proof By Theorem 3 of Ergin and Gul (2009), Axioms P1 –P6 are equivalent to being represented by (13). Assume for simplicity that certainty equivalents exist and that u(Z) = R. By Theorem 2, Axiom A5 is equivalent to φ : R → R being concave. Moreover, it is easy to see that Axiom A8 is equivalent to qb being countably additive. A direct verification establishes that Axiom A2 is necessary. To establish the sufficiency of Axiom A2 , observe that by the convexrangedness of qa , there exists an event E ∈ Σa such that qa (E) = 12 . For all k ∈ R and f g ∈ Fb (Z), Axiom A2 (applied to f g h, and h such that u(f ) = 2u(f ), u(g) = 2u(g ), u(h) = 0, and u(h ) = 2k) implies that
φ u(f (sb )) dqb (sb ) ≥ φ u (g(sb )) dqb (sb ) Sb
iff
Sb
φ u(g (sb )) + k dqb (sb )
φ u(f (sb )) + k dqb (sb ) ≥ Sb
Sb
which, by the same argument as in the proof of Theorem 1, implies that φ = φθ . 5. CONCLUSION One of the challenges in decision theory lies in finding decision models that would do better than expected utility in describing individual choices, but would at the same time be easy to incorporate into economic models of aggregate behavior. This paper studies the model of multiplier preferences which is known to satisfy the latter requirement. By obtaining an axiomatic characterization of this model, the paper studies its individual choice properties, which will help to determine whether it also satisfies the first requirement mentioned above. The axiomatization provides a set of testable implications of the model, which will be helpful in its empirical verification. The axiomatization also enables measurement of the parameters of the model on the basis of observable choice data alone, thereby providing a useful tool for applications of the model. APPENDIX: PROOFS Let B0 (Σ) denote the set of all real-valued Σ-measurable simple functions and let B0 (Σ K) be the set of all functions in B0 (Σ) that take values in a convex set K ⊆ R. In the course of the proof of Theorem 1, a result of Grant, Polak, and Strzalecki (2009) will be invoked that delivers a SOEU representation on each finite
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partition of S. The following theorem shows that these representations can be “patched” together to obtain an overall SOEU representation on S. Let Ξ denote the set of all finite partitions of S that are composed of events in Σ; let Ξ3 ⊆ Ξ denote the set of all such partitions that contain at least three nonnull events. For any G ∈ Ξ, let A(G ) be the algebra generated by G . For any G ∈ Ξ3 , let FG (Δ(Z)) denote the set of acts in F (Δ(Z)) that are measurable with respect to A(G ). DEFINITION 4 —Ξ 3 -SOEU: A preference on F (Δ(Z)) is Ξ3 -SOEU iff for any G ∈ Ξ 3 , the restriction of to FG (Δ(Z)) is represented by f → E∈G φG (uG (f (E)))pG (E) with a nonconstant affine function uG : Δ(Z) → R with range UG , a strictly increasing, continuous, and concave function φG : UG → R and measure pG : A(G ) → [0 1] such that at least three events in G have nonzero probability. THEOREM 4: Suppose that Ξ3 = ∅ and that satisfies Axioms A1, A4, and P2. The preference is Ξ 3 -SOEU if and only if there exists a measure p ∈ Δ(S) and a nonconstant affine function u : Δ(Z) → R with range U , and a strictly increasing, concave, and continuous function φ : U → R such that is represented by f → S φ(u(f (s))) dp(s). PROOF: The sufficiency of the representation is straightforward. For necessity, let G G ∈ Ξ 3 . The restrictions of to FG (Δ(Z)) and to FG (Δ(Z)) coincide on constant acts Δ(Z). Thus, from the uniqueness of the von Neumann– Morgenstern utility, it follows that uG and uG are identical up to a positive affine transformation. Fix any two prizes z z and for each G ∈ Ξ 3 , normalize uG so that uG (z) = 1 and uG (z ) = 0. Define u to be the common utility function for all G ∈ Ξ 3 . For any G G ∈ Ξ, define G ≥ G iff G is finer than G , that is, for every E ∈ G there exists F ∈ G with E ⊆ F . For any G G ∈ Ξ, let G ∨ G be their coarsest common refinement and let G ∧ G be their finest common coarsening. LEMMA 1: If E ∈ Σ is nonnull, then for any finite Σ-measurable partition {F1 Fn } of E at least one of the sets F1 Fn is nonnull. PROOF: CLAIM 1: For any nonnull E ∈ Σ, there exist π ρ σ ∈ Δ(Z) such that ρE σ πE σ. PROOF: There exist f g h ∈ F (Δ(Z)) such that fE h gE h. By P2 choose h to equal to some σ ∈ Δ(Z) different than any of the prizes given by f and g. Let {E1 En E c } ∈ Ξ be a partition of S with respect to which both fE σ and gE σ are measurable. Let ρ be the most preferred element among {f (Ei ) | i =
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1 n} and let π be the least preferred element among {g(Ei ) | i = 1 n}. Q.E.D. By A4, ρE σ fE σ and gE σ πE σ. Thus ρE σ πE σ. CLAIM 2: For any nonnull E ∈ Σ and any two-element Σ-measurable partition {F1 F2 } of E, at least one of the sets F1 F2 is nonnull. PROOF: Suppose that there exists a two-element Σ-measurable partition {F1 F2 } of E such that both sets F1 and F2 are null. Then ρFi h ∼ πFi h for any h ∈ F (Δ(Z)) and all i = 1 2. In particular, ρE σ = ρF1 (ρF2 σ) ∼ πF1 (ρF2 σ) = ρF2 (πF1 σ) ∼ πF2 (πF1 σ) = πE σ. This is a contradiction with Claim 1. Q.E.D. CLAIM 3—Inductive Step: If for any nonnull E ∈ Σ and any n-element Σmeasurable partition {F1 Fn } of E , at least one of the sets F1 Fn is nonnull, then for any nonnull E ∈ Σ and any n + 1-element Σ-measurable partition {F1 Fn+1 } of E, at least one of the sets F1 Fn+1 is nonnull. PROOF: By Claim 2, at least one of the sets F1 ∪ · · · ∪ Fn =: E and Fn+1 is nonnull. If the latter is true, this concludes the proof. If E is nonnull, then the premise of this claim applied to the set E and its partition {F1 Fn } concludes the proof. Q.E.D. LEMMA 2: G ∨ G ∈ Ξ 3 for any G ∈ Ξ3 and G ∈ Ξ. PROOF: By assumption, there are at least three disjoint nonnull sets in G . By Lemma 1 for any such set E ∈ G , there is at least one nonnull member of {E ∩ F | F ∈ G }. Thus, there are at least three nonnull members of G ∨ G . Q.E.D. LEMMA 3: The functions {φG }G ∈Ξ3 and measures {pG }G ∈Ξ3 can be chosen in such a way that there exists φ : U → R such that φG = φ for any G ∈ Ξ 3 , and for any G G ∈ Ξ 3 , the restrictions of measures pG and pG to A(G ∧ G ) coincide. PROOF: For each G ∈ Ξ 3 , normalize φG so that φG (u(z)) = 1 and φG (u(z )) = 0. First, let G H ∈ Ξ3 such that G ≥ H. Observe, that FH (Δ(Z)) ⊆ FG (Δ(Z)), so both (φG pG ) and (φH pH ) represent preferences on FH (Δ(Z)). By the uniqueness of the expected utility representation, the restriction of pH to G coincides with pG and the functions φH and φG are identical up to a positive affine transformation, which by the above normalization assumption implies that they are equal. Second, let G G ∈ Ξ 3 . By Lemma 2, G ∨ G ∈ Ξ 3 . Furthermore, G ∨ G ≥ G and G ∨ G ≥ G . From the above paragraph, it follows that φG = φG ∨G = φG . Let φ be this common function. Also the restriction of pG ∨G to A(G ) coincides with pG ; hence the restriction of pG ∨G to A(G ∧ G ) coincides with the restriction of pG to A(G ∧ G ). Likewise, the restriction of pG ∨G to A(G ) coincides with pG ; hence the restriction of pG ∨G to A(G ∧ G ) coincides with Q.E.D. the restriction of pG to A(G ∧ G ).
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LEMMA 4: There exists p ∈ Δ(S) such that p(E) = pH (E) for any E ∈ Σ and any H ∈ Ξ3 with E ∈ H. PROOF: Let G¯ ∈ Ξ3 be some fixed element of Ξ 3 . For any E ∈ Σ, define the partition GE := G¯ ∨ {E E c }; by Lemma 2, GE ∈ Ξ 3 . Define the function p : Σ → [0 1] by p(E) := pGE (E). Let E F ∈ Σ be such that E ∩ F = ∅. Let GEF := G¯ ∨ {E F (E ∪ F)c }. By Lemma 2, GEF ∈ Ξ 3 and by Lemma 3, pGE (E) = pGEF (E) because E ∈ A(GE ∧ GEF ). Likewise, pGF (F) = pGEF (F) because F ∈ A(GF ∧ GEF ). Also, pGE∪F (E ∪ F) = pGEF (E ∪ F) because E ∪ F ∈ A(GE∪F ∧ GEF ). By definition, p(E ∪ F) = pGE∪F (E ∪ F) = pGEF (E ∪ F) = pGEF (E) + pGEF (F) = pGE (E) + pGF (F) = p(E) + p(F). Hence, p ∈ Δ(S). Suppose H ∈ Ξ3 and E ∈ H. Then E ∈ A(H ∧ GE ) and by Lemma 3, Q.E.D. pH (E) = pGE (E). Hence, by definition, p(E) = pH (E). CONCLUSION OF THE PROOF OF THEOREM 4: For any act f ∈ F (Δ(Z)), define V (f ) := φ(u(f (s)) dp(s). To verify that V represents , let f g ∈ F (Δ(Z)). Let Gfg ∈ Ξ be a partition such that both f and g are measurable with respect to A(Gfg ). Let G¯ ∈ Ξ3 be some fixed element of Ξ3 and let G := Gfg ∨ G¯ ; by Lemma 2, G ∈ Ξ 3 . By assumption, f g iff E∈G φG (uG (f (E))pG (E) ≥ E∈G φG (uG (g(E))pG (E). Since uG = u, φG = φ (by Lemma for all E ∈ G (by Lemma 4), it follows that 3), and p(E) = pG (E) f g iff E∈G φ(u(f (E))p(E) ≥ E∈G φ(u(g(E))p(E). Q.E.D. A.1. Proof of Theorem 1 The necessity of the axioms was shown in the sketch of proof (Section 3.3); this section focuses on sufficiency. Uniqueness follows from Corollary 5 of MMR. A.1.1. Niveloidal Representation By Lemmas 25 and 28 of MMR and Lemma 22 in Maccheroni, Marinacci, and Rustichini (2004), Axioms A1–A6 imply that there exists a nonconstant affine function u : Δ(Z) → R and a concave functional I : B0 (Σ U ) → R such that f g iff I(u ◦ f ) ≥ I(u ◦ g), where U := u(Δ(Z)). Moreover, within this class, u is unique up to positive affine transformations. Without loss of generality (wlog), U ∈ {R R+ R− [0 1]}; the inclusion of the endpoints does not matter for further analysis. For any y ∈ int U , define U y := {x ∈ U | x + y ∈ U }. The functional I has the property that I(ξ + y) = I(ξ) + y for all y ∈ U and ξ ∈ B0 (Σ U y ). A.1.2. Utility Acts For each act f , define the utility act associated with f as u ◦ f ∈ B0 (Σ U ). The preference on acts induces a preference on utility acts: for any ξ ξ ∈
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B0 (Σ U ), define ξ u ξ iff f f , for some ξ = u ◦ f and ξ = u ◦ f . The choice of particular versions of f and f is irrelevant, because of Axiom A4. From Section A.1.1, it follows that ξ u ξ iff I(ξ ) ≥ I(ξ ) iff I(ξ + y) ≥ I(ξ + y) iff ξ + y u ξ + y for all y ∈ int U and all ξ ξ ∈ B0 (Σ U y ). A.1.3. Second-Order Expected Utility By Proposition 7 of MMR the preference is ambiguity averse in the sense of Ghirardato and Marinacci (2002). Observe that for any G ∈ Ξ 3 , there is a natural bijection between FG (Δ(Z)) and (Δ(Z))|G | that preserves Axioms A1–A6, P2, and the ambiguity aversion in the sense of Ghirardato and Marinacci (2002). Thus, by Theorem 2 of Grant, Polak, and Strzalecki 3 (2009), for any G ∈ Ξ , the restriction of to FG (Δ(Z)) has an additive representation f → E∈G φG (uG (f (E)))pG (E) with a nonconstant affine function uG : Δ(Z) → R with range UG , a strictly increasing, continuous, and concave function φG : UG → R, and measure pG : A(G ) → [0 1] such that at least three events in G have nonzero probability. By Theorem 4 above, the preference on F (Δ(Z)) has an additive representation
φ u(f (s)) dq(s) S
where the measure q ∈ Δ(S) and the function φ : U → R is strictly increasing, concave, and continuous. By Theorem 1 in Villegas (1964, Section 1), Axiom A8 implies that q ∈ Δσ (S). A.1.4. Proof That φ = φθ For any y ∈ int U , define φy (x) φ(x + y) for all x ∈ U y . It follows from := y that S φ ◦ ξ dq ≥ S φy ◦ ξ dq iff S φ ◦ ξ dq ≥ Sections A.1.2 and A.1.3 φ ◦ ξ dq for all ξ ξ ∈ B0 (Σ U y ). Thus, (φ q) and (φy q) are EU repS resentations of the same preference on B0 (Σ U y ). By the uniqueness (up to positive affine transformation) of the EU representation, it follows that φ(x + y) = α(y)φ(x) + β(y) for all y ∈ int U and all x ∈ U y . This is a generalization of Pexider’s equation (see equation (3) of Aczél (1966, Section 3.1.3, p. 148)). If U is unbounded, then by Corollary 1 in Aczél (1966, Section 3.1.3), up to positive affine transformations, the only strictly increasing concave solutions are of the form φθ , for θ ∈ (0 ∞]. If U is bounded, then wlog assume that int U = (0 1) and define the set R := {(x y) ∈ R2 | x > 0 y > 0 x + y < 1} and functions m l n k : (0 1) → R by m := α, l := β, n := φ|(01) , and k := φ|(01) . The functional equation k(x + y) = m(y)n(x) + l(y) holds for all (x y) ∈ R. It follows from the corollary in Aczél (2005) that either k(x) = Cx + B + Pω or k(x) = ωδeCx + B for some arbitrary parameters B P C ω, and δ with Cωδ = 0. It follows that φ is an exponential function up to positive affine transformations; by concavity, φ = φθ in the interior of U . By continuity of φ, this extends to the whole set U .
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A.1.5. Conclusion of the Proof
Combining the results of Sections A.1.3 and A.1.4, f g iff (φθ ◦ u ◦ S σ f ) dq ≥ S (φθ ◦ u ◦ g) dq. Because q ∈ Δ , by the variational formula (8), it follows that f g iff minp∈ΔS S (u ◦ f ) dp + θR(p q) ≥ minp∈ΔS S (u ◦ g) dp + θR(p q). Q.E.D. A.2. Proof of Theorem 2 To relax the assumption of existence of certainty equivalents, the following definition will be used. DEFINITION 5: Act f ∈ Fa (Z) is symmetric with respect to E ∈ Σa if for all z ∈ Z, fE z ∼ zE f Symmetric acts have the same expected utility on each “half” of the state space.17 AXIOM A5 —Second-Order Risk Aversion: If acts f g ∈ Fa are symmetric with respect to E ∈ Σa , then for all F ∈ Σb , fF g ∼ gF f
⇒
fE g fF g
The proof of Theorem 2 follows from the proof of the following stronger theorem. THEOREM 5: Suppose satisfies Axioms P1 –P6 . Then Axiom A5 is satisfied if and only if the function φ in (13) is concave. PROOF: By Theorem 3 of Ergin and Gul (2009), Axioms P1 –P6 are equivalent to being represented by (13). Q.E.D. A.2.1. Necessity Suppose f ∈ Fa (Z) is symmetric with respect to E ∈ Σa . Let α = qa (E). Axiom P5 and representation (13) imply that there exist z z ∈ Z with z z . 17
Symmetric acts are acts that can be “subjectively mixed,” that is, mixed using states rather than probabilities. Such subjective mixtures are different from subjective mixtures studied by Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), whose construction relies on rangeconvexity of u. In the present setting, subjective mixtures are not needed under range-convexity of u.
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Thus, fE z ∼ zE f and fE z ∼ zE f imply that (14) (u ◦ f ) dqa + (1 − α)u(z ) = αu(z ) + (u ◦ f ) dqa Ec
E
(u ◦ f ) dqa + (1 − α)u(z ) = αu(z ) +
(15) E
Ec
(u ◦ f ) dqa
By subtracting (15) from (14), (1 − α)[u(z ) − u(z )] = α[u(z ) − u(z )]; thus, α =
1 2
and therefore
(u ◦ f ) dqa =
E
Ec
(u ◦ f ) dqa
Let f g ∈ Fa (Z). Denote U(f ) = Sa (u ◦ f ) dqa and U(g) = Sa (u ◦ g) dqa . If f and g are symmetric with respect to E ∈ Σa , then 1 (u ◦ f ) dqa = (u ◦ f ) dqa = U(f ) 2 c E E 1 (u ◦ g) dqa = (u ◦ g) dqa = U(g) 2 E Ec Let F ∈ Σb and β := qb (F). If fF g ∼ gF f , then βφ(U(f )) + (1 − β)φ(U(g)) = βφ(U(g)) + (1 − β)φ(U(f )) Thus, (2β − 1)φ(U(f )) = (2β − 1)φ(U(g)) If β = 12 , then U(f ) = U(g) and trivially 1 1 1 1 V (fE g) = βφ U(f ) + U(g) + (1 − β)φ U(f ) + U(g) 2 2 2 2 = βφ(U(f )) + (1 − β)φ(U(g)) = V (fF g) If β = 12 , then
1 1 V (fE g) = φ U(f ) + U(g) 2 2 1 1 ≥ φ(U(f )) + φ(U(g)) = V (fF g) 2 2
where the inequality follows from concavity of φ.
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A.2.2. Sufficiency A.2.2.1. Convexity of Dφ . Suppose k l ∈ Dφ and α ∈ (0 1); wlog k < l. Let f g ∈ Fa be such that k = U(f ) and l = U(g). Define A = mins∈S u(f (s)) and B = maxs∈S u(g(s)), and let x y ∈ Z be such that u(x) = A and u(y) = B. By convex-rangedness of qa , there exists E ∈ Σa with qa (E) = (B − [αk + (1−α)l])(B − A)−1 . Verify that U(xE y) = αk+(1−α)l. Hence, Dφ is a convex set. A.2.2.2. Convexity of φ. Suppose k l ∈ Dφ and let f g ∈ Fa be such that k = U(f ) and l = U(g). Define k = mins∈S u(f (s)), k¯ = maxs∈S u(f (s)), ¯ ¯ y y¯ be such that u(x) = l = mins∈S u(g(s)), and l¯ = maxs∈S u(g(s)). Let x x ¯ ¯ ¯ ¯ ¯ ¯ = k, u(y) = l, and u(y) ¯ = l. Also, define ¯ κ := (k¯ − k)(k¯ − k)−1 k, u(x) ¯ ¯ ¯ and λ := (l¯ − l)(¯l¯ − l)−1 . By convex-rangedness of qa , there exist partitions {E1κ E2κ E3κ E4κ } and ¯ {E1λ E2λ E3λ E4λ } of Sa such that E1κ ∪ E2κ = E1λ ∪ E2λ , qa (E1κ ∪ E2κ ) = qa (E1λ ∪ E2λ ) = 12 , qa (E1κ ) = qa (E3κ ) = κ2 , and qa (E1λ ) = qa (E3λ ) = λ . 2 ¯ 2κ xE3κ xE ¯ 4κ and g := yE1λ yE ¯ 2λ yE3λ yE ¯ 4λ . Verify that f Define acts f := xE1κ xE ¯ ¯ and g are symmetric with respect to E = E1κ¯ ∪ E2κ =¯ E1λ ∪ E2λ , and satisfy U(f ) = k and U(g ) = l. By convex-rangedness of qb , there exists F ∈ Σb with qb (F) = 12 . Verify that V (fF g ) = 12 φ(k) + 12 φ(l) = V (gF f ). Hence, by Axiom A5 , 1 1 1 1 φ k + l = V (fE g ) ≥ V (fF g ) = φ(k) + φ(l) 2 2 2 2 As a consequence, 1 1 1 1 φ k + l ≥ φ(k) + φ(l) (16) 2 2 2 2 for all k l ∈ Dφ . By Theorem 3 of Ergin and Gul (2009), the function φ is continuous on Dφ ; hence, by Theorem 86 of Hardy, Littlewood, and Pólya (1952), φ is concave. Q.E.D. A.3. Proof of Theorem 3 By Theorem 3 of Ergin and Gul (2009), Axioms P1 –P6 are equivalent to being represented by (13). Let E ∈ Σa be such that qa (E) = 12 . For any v ∈ Dφ define a preference v on F as follows. Let h ∈ Fa be such that Ec u(h(sa sb )) dqa (sa ) = 12 v and for any f g ∈ F (Z), define f v g iff fE h gE h. (Because of Axiom A2 , the choice of particular h does not mat-
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ter.) Define φv (u) := φ( 12 u + 12 v). From representation (13), it follows that v is represented by v φ 2 u(f (sa sb )) dqa (sa ) dqb (sb ) Sb
E
By Axiom A2 , v =0 for all v ∈ Dφ . Hence, φv and φ0 are equal up to positive affine transformations, that is, φ( 12 u + 12 v) = α(v)φ( 12 u) + β(v) for all u v ∈ Dφ . If Dφ is unbounded, then by Corollary 1 in Aczél (1966, Section 3.1.3), the function φ belongs to the exponential class. If Dφ is bounded, then wlog assume that int Dφ = (0 1). Define the set R := (0 1)2 . Define the functions k : (0 2) → R by k(u) = φ( 12 u) and functions l m n : (0 1) → R by l := β, m := α, and n(u) = φ( 12 u). The functional equation k(u + v) = m(v)n(u) + l(v) holds for all (u v) ∈ R. It follows from the corollary in Aczél (2005) that either k(u) = Cu + B + Pω or k(u) = ωδeCu + B for some arbitrary parameters a B P C ω, and δ with Cωδ = 0. Because n and k coincide on (0 1), it follows that φ is an exponential function up to positive affine transformations in the interior of U . By continuity of φ, this extends to the whole set U . In both cases, φ belongs to the exponential class, that is, it is either linear, strictly concave, or strictly convex. To eliminate the last possibility, observe that Axiom A5 applied to acts f = xF y and g = yF x, and events E and F with qb (F) = 12 and qa (E) = 12 implies that φ( 12 u(x) + 12 u(y)) ≥ 12 φ(u(x)) + 1 φ(u(y)) for all x y ∈ U . 2 It follows from Theorem 1 of Villegas (1964, Section 1) that Axiom A8 delivers countable additivity of qb . An application of the variational formula (8) concludes the proof. Q.E.D. REFERENCES ABDELLAOUI, M., A. BAILLON, L. PLACIDO, AND P. P. WAKKER (2010): “The Rich Domain of Uncertainty: Source Functions and Their Experimental Implementation,” American Economic Review (forthcoming). [52] ACZÉL, J. (1966): Lectures on Functional Equations and Their Applications. New York: Academic Press. [67,71] (2005): “Extension of a Generalized Pexider Equation,” Proceedings of the American Mathematical Society, 133, 3227–3234. [67,71] ANDERSON, E. W., L. P. HANSEN, AND T. J. SARGENT (2003): “A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk, and Model Detection,” Journal of the European Economic Association, 1, 68–123. [60] ANSCOMBE, F., AND R. AUMANN (1963): “A Definition of Subjective Probability,” The Annals of Mathematical Statistics, 34, 199–205. [50] BARILLAS, F., L. P. HANSEN, AND T. J. SARGENT (2009): “Doubts or Variability?” Journal of Economic Theory, 144 (6), 2388–2418. [49] BENIGNO, P., AND S. NISTICÒ (2009): “International Portfolio Allocation Under Model Uncertainty,” Working Paper 14734, NBER. [49]
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MACCHERONI, F., M. MARINACCI, AND A. RUSTICHINI (2004): “Ambiguity Aversion, Malevolent Nature, and the Variational Representation of Preferences,” Mimeo, Collegio Carlo Alberto. [66] (2006a): “Ambiguity Aversion, Robustness, and the Variational Representation of Preferences,” Econometrica, 74, 1447–1498. [47,48,53,54] (2006b): “Dynamic Variational Preferences,” Journal of Economic Theory, 128, 4–44. [57] MACHINA, M., AND D. SCHMEIDLER (1995): “Bayes Without Bernoulli: Simple Conditions for Probabilistically Sophisticated Choice,” Journal of Economic Theory, 67, 108–128. [56] MAENHOUT, P. (2004): “Robust Portfolio Rules and Asset Pricing,” Review of Financial Studies, 17, 951–983. [49] MARKOWITZ, H. M. (1952): “Portfolio Selection,” Journal of Finance, 7, 77–91. [48] NAU, R. (2001): “Uncertainty Aversion With Second-Order Utilities and Probabilities,” in Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, available at http://www.sipta.org/isipta01/proceedings/063.html. [49,53,60] (2006): “Uncertainty Aversion With Second-Order Utilities and Probabilities,” Management Science, 52, 136–145. [49,53,60] NEILSON, W. S. (1993): “Ambiguity Aversion: An Axiomatic Approach Using Second Order Probabilities,” Mimeo. [49,53,57] (2010): “A Simplified Axiomatic Approach to Ambiguity Aversion,” Journal of Risk and Uncertainty, 41, 113–124. [49,53] SAVAGE, L. (1972): The Foundations of Statistics. Dover. [57] SCHMEIDLER, D. (1989): “Subjective Probability and Expected Utility Without Additivity,” Econometrica, 57, 571–587. [53,62] SKIADAS, C. (2003): “Robust Control and Recursive Utility,” Finance and Stochastics, 7, 475–489. [57] TOBIN, J. (1958): “Liquidity Preference as Behavior Toward Risk,” Review of Economic Studies, 25, 65–86. [48] VILLEGAS, C. (1964): “On Qualitative Probability σ-Algebras,” The Annals of Mathematical Statistics, 35, 1787–1796. [67,71] WANG, T. (2003): “A Class of Multi-Prior Preferences,” Mimeo, UBC. [56] WHITTLE, P. (1981): “Risk-Sensitive Linear/Quadratic/Gaussian Control,” Advances in Applied Probability, 13, 764–777. [57] WOODFORD, M. (2006): “An Example of Robustly Optimal Monetary Policy With Near-Rational Expectations,” Journal of the European Economic Association, 4, 386–395. [49]
Dept. of Economics, Harvard University, Littauer Center, 1805 Cambridge Street, Cambridge, MA 02138, U.S.A.;
[email protected]. Manuscript received September, 2008; final revision received August, 2010.
Econometrica, Vol. 79, No. 1 (January, 2011), 75–122
SEARCH AND REST UNEMPLOYMENT BY FERNANDO ALVAREZ AND ROBERT SHIMER1 This paper develops a tractable version of the Lucas and Prescott (1974) search model. Each of a continuum of industries produces a heterogeneous good using a production technology that is continually hit by idiosyncratic shocks. In response to adverse shocks, some workers search for new industries while others are rest unemployed, waiting for their industry’s condition to improve. We obtain closed-form expressions for key aggregate variables and use them to evaluate the model’s quantitative predictions for unemployment and wages. Both search and rest unemployment are important for understanding the behavior of wages at the industry level. KEYWORDS: Search, rest, unemployment, industry wages, persistence.
1. INTRODUCTION THIS PAPER DEVELOPS A TRACTABLE VERSION of Lucas and Prescott (1974) theory of unemployment. We distinguish between “search” and “rest” unemployment. Search unemployment is a costly reallocation activity whereby a worker attempts to move to a better industry. Rest unemployment is a less costly activity whereby a worker waits for her current industry’s condition to improve. We obtain closed-form solutions for the search- and restunemployment rates in terms of reduced-form parameters, which are in turn known functions of preferences and technology. The dynamics of industry wages are determined by the same reduced-form parameters, giving us a tight link between aggregate unemployment and the stochastic process for industrylevel wages. We exploit that link to evaluate quantitatively the role played by search and rest unemployment in our model. Our model economy consists of a continuum of industries, each with many workers and firms. Firms in each industry produce an intermediate good with a constant returns to scale technology using labor only. The growth rate of output per worker is an industry-specific exogenous shock with a constant expected value and a constant variance per unit of time. Intermediate goods are imperfect substitutes in the production of a final consumption good, so industry demand curves slope down, although all workers and firms are price-takers. Workers have standard preferences over consumption and leisure, and markets are complete. At any point in time, workers engage in four mutually exclusive activities, from which they derive different amounts of leisure: work, search unemployment, rest unemployment, and inactivity, that is, out of the labor force. A worker in a given industry can either work, engage in rest unemployment, or leave the industry. A rest-unemployed worker is available to 1 We are grateful for comments by Daron Acemoglu, Gadi Barlevy, Morten Ravn, Marcelo Veracierto, two anonymous referees, the editor, and numerous seminar participants, and for research assistance by Lorenzo Caliendo. This research is supported by a grant from the National Science Foundation.
© 2011 The Econometric Society
DOI: 10.3982/ECTA7686
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return to work in that industry, and that industry only, at no cost. If a worker leaves her industry, she can be either inactive or search unemployed. Searchunemployed workers find a new industry after a random, exponentially distributed amount of time. We consider both the case of directed search, as in Lucas and Prescott (1974), where a successful searcher moves to the industry of her choice, and the case of random search, where the probability that a successful searcher moves to any particular industry is proportional to the number of workers in that industry, in the spirit of the McCall (1970) search model. Finally, workers can costlessly move between search unemployment and inactivity. We perform a quantitative evaluation of the model, focusing on the behavior of industry wages, the key equilibrating force in the Lucas and Prescott (1974) model. Idiosyncratic productivity shocks cause wage dispersion across industries, but we show analytically how this is moderated by workers’ search for better job opportunities. This motivates our empirical approach to evaluating the model: we compare the behavior of wages at the industry level in the model to the U.S. economy, and show a tight theoretical link between the search-unemployment rate and the autocorrelation and variance of wages. We find that our model is consistent with the wage data only if the search-unemployment rate is low, approximately one-quarter of total unemployment; the remaining unemployment is accounted for by rest. We discuss below whether this is empirically reasonable. 1.1. Employment, Unemployment, and Wage Dynamics To better understand the mechanics of our model, let ω denote the log of the wage that would prevail in a particular industry if all its workers were employed, measured in units of the consumption good; the actual log wage exceeds ω if there is rest unemployment in the industry. With our functional forms, ω is a linear function of log productivity, which by assumption follows a Brownian motion. It is also a linearly decreasing function of the log of the number of workers in the industry, reflecting that an increase in output reduces the price of the industry’s good. We determine the stochastic process for ω using two equilibrium conditions. First, the stochastic process is affected by individual workers’ decisions to enter and exit the industry. In particular, some workers may exit to search for another industry, raising ω. Conversely, the arrival of successful searchers puts downward pressure on ω in a way that depends on whether search is directed or random. Second, the decision of a worker to enter or exit an industry depends on the current value of ω and on its stochastic process. Workers exit an industry if current and future wages are expected to be low. In the directed search model, successful searchers enter industries with high current and future wages, while in the random search model they enter all industries. These conditions define a fixed point problem for the stochastic process. We prove
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there is a unique fixed point and that ω follows a regulated Brownian motion with endogenous barriers and drift. We also characterize the barriers and drift as functions of preferences and technology, including whether search is random or directed. Given any equilibrium stochastic process for ω, we can compute the employment rate and the search- and rest-unemployment rates. First we solve for the unique invariant distribution for ω across workers. In an industry at the lower barrier for ω, a fraction of workers leave and become search unemployed, which determines the incidence of search unemployment. Since the average duration of search unemployment is exogenous, this pins down the search-unemployment rate and provides our link between the stochastic process for wages and the search-unemployment rate. We also show that the rest-unemployment rate in an industry is continuously decreasing in ω up to some threshold; above that threshold there is full employment. Finally, we compute the economy-wide employment and rest-unemployment rates by aggregating across industries using the invariant distribution for ω. Search and rest unemployment play distinct roles in our model economy. Search unemployment is a high-cost, irreversible decision that enables workers to move to a new industry. This generates “labor hoarding,” in the sense that the marginal product of labor is not equalized across industries. Rest unemployment is a less costly, reversible decision that does not enable workers to move. Effectively, rest unemployment raises the marginal product of labor in depressed industries by giving workers the option to remain in contact with the industry without incurring the disutility of working. In this sense, the possibility of rest unemployment encourages labor hoarding at the industry level, albeit hoarding that takes the form of unemployment. In deciding whether to search or rest, workers compare the leisure value of the two activities against the promised wage gains. Search may lead to a higher wage in a new industry, while, to the extent that the old industry’s wage can increase in the future, rest may also lead to higher wages.2 The cost and benefit of search unemployment—spending time finding a new industry—are independent of the conditions in the current industry. In contrast, the benefit of rest unemployment—the possibility of high future wages in the current industry— is greater in a less depressed industry since wages are persistent. It follows that workers exit only the most depressed industries to search for new opportunities, while workers in less depressed industries may be rest unemployed, waiting for local conditions to improve. 2 In our model, productivity is a Brownian motion. This assumption is important for our closedform solutions, but in reality there may be transitory productivity shocks or even predictable seasonal cycles. A temporary downturn would create a strong incentive for workers to rest rather than search for a new industry. Our assumption that all productivity shocks are permanent therefore appears to reduce the incidence of rest unemployment, conditional on the other model parameters.
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1.2. Quantitative Evaluation Our analytical results emphasize the separation of optimization—the determination of the stochastic process for ω—and aggregation—the determination of unemployment rates associated with the stochastic process. Indeed, we also show that we can reverse this logic. Starting from observed unemployment rates, we can place restrictions on the barriers and drift in ω, and so say something about the stochastic process for wages. We can thus analyze aggregation without specifying preferences and technology. In addition, starting from a stochastic process for wages, we can uniquely determine the relative disutility of work, search unemployment, and rest unemployment. We can thus analyze optimization without specifying the allocation of workers across industries. We use the separation between optimization and aggregation to make three related points about the quantitative behavior of our model. First, our aggregation result establishes a tight theoretical relationship between the incidence of search unemployment and the autocorrelation of wages at the industry level. Since these results do not depend on the full model, they are robust to the specification of preferences and technology. We then test the theoretical relationship using data for five-digit industries in the United States. We find that the annual average of weekly earnings at the industry level is essentially a random walk. According to our model, this implies that wages rarely hit their regulating barriers, and hence they inherit the persistence of productivity, which we assume is a random walk with drift. But since a worker exits an industry only when ω hits its lower barrier, it follows that the incidence of search unemployment in our model is necessarily small. Given the observed duration of unemployment, we conclude that persistent wages are inconsistent with a high search-unemployment rate in the United States economy. This finding holds both in the directed and random search models. Second, we use our optimization results to show that for our model to match the persistence of wages at the industry level, the disutility of search must be significantly greater than the disutility of work. This result again uses the fact that persistent wages rarely hit their regulating barriers. Combining this with evidence on the variance of wage growth, it follows that the difference between the highest and lowest wage is large, and so are the gains from search for a worker in a low-wage industry. Workers’ optimization then implies that the cost of search is large as well. This finding is closely related to the conclusion in Hornstein, Krusell, and Violante (2009) that search models with plausible search costs cannot generate much wage dispersion. We conjecture that some of the disutility of search actually stands in for different forms of industryspecific human capital, a topic that we have begun to explore in other papers (Alvarez and Shimer (2009a, 2009b)). Third, we use our aggregation results to show that the model can allow for rest unemployment to account for three-quarters of all unemployment, while still having persistent wages. Thus there is no conflict between a high restunemployment rate and persistent wages. Mechanically, given the barriers and
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drift in ω, high rest unemployment simply lowers employment and so raises the wage in some depressed industries, with little effect on the measured autocorrelation of wages. The question remains whether it is reasonable to assume that rest unemployment accounts for three-quarters of all unemployment. To answer this, we need to take a stand on the empirical counterpart of rest unemployment. One feature of rest-unemployed workers is that they eventually return to work in their old industry. Murphy and Topel (1987) and Loungani and Rogerson (1989) documented that this is the case following about threequarters of all unemployment spells in the United States. We view this as suggesting that our model may offer a reasonable description of unemployment and wage dynamics. We discuss the mapping between our model and this type of data further in Section 6.4. 1.3. Relationship to the Literature There are four significant differences between our model and that of Lucas and Prescott (1974). First, we introduce rest unemployment to the framework. Second, we make particular assumptions on the stochastic process for productivity which enable us to obtain closed-form solutions. This leads us to evaluate the model’s properties in novel ways; however, we believe our insights, for example, on the link between search unemployment and the autocorrelation of wages and on the role of rest unemployment, carry over to alternative productivity processes. Third, in Lucas and Prescott (1974), all industries produce a homogeneous good but there are diminishing returns to scale in each industry. In our model, each industry produces a heterogeneous good and has constant returns to scale. We believe our approach is more attractive because the extent of diminishing returns is determined by the elasticity of substitution between goods, which is potentially more easily measurable than the degree of decreasing returns on variable inputs (Atkeson, Khan, and Ohanian (1996)). Finally, our characterization of the random search model is new.3 Our concept of rest unemployment is closely related to the one proposed by Jovanovic (1987), from whom we borrow the term.4 While in both his model and ours, search and rest unemployment coexist, the aims of both papers and hence the setup of the models are different. Jovanovic (1987) focused on the cyclical behavior of unemployment and productivity, and so allowed for both idiosyncratic and aggregate productivity shocks. But to be able to analyze the 3 The closest random search model is the one in Alvarez and Veracierto (1999), which assumed that each searcher is equally likely to find any industry, while we assume here that searchers are more likely to find industries with more workers, as in Burdett and Vishwanath (1988). We view this assumption as no less plausible and it is much more tractable in our current setup. 4 An alternative is “wait unemployment,” but the literature that uses this term emphasizes the behavior of workers waiting for a job in a high-wage primary labor market rather than accepting a readily available job in a low-wage secondary labor market. We study a related concept in our work on unionization (Alvarez and Shimer (2009b)).
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model with aggregate shocks, Jovanovic (1987) assumed that at the end of each period, there is exactly one worker in each location. Hamilton (1988), King (1990), and Gouge and King (1997) all developed models of rest unemployment in the Lucas and Prescott (1974) framework with directed search. King (1990) focused on comparative statics of rest and search unemployment. Hamilton (1988) and Gouge and King (1997) reexamined the business cycle issues in Jovanovic (1987), such as the finding that rest unemployment is likely to be countercyclical, consistent with empirical evidence but not with many models of reallocation. The key difference between those papers and ours is the stochastic process for productivity. While we assume productivity follows a geometric Brownian motion, these earlier papers assumed a two-state Markov process. This coarse parameterization makes their analysis of cyclical fluctuations tractable, but makes it harder to map the model into rich cross-sectional data. For example, wages in their models also follow a twostate Markov process. One of our main contributions is to show that when log productivity follows a Brownian motion, the model is still tractable and the mapping from model to cross-sectional data is direct. Of course, solving our version of the model with aggregate shocks remains a daunting task. In this sense, we view the two approaches as complementary. It is worth stressing that these papers and, to our knowledge, all previous research based on the Lucas and Prescott (1974) search model, assume that productivity is mean-reverting. We conjecture that such an assumption would strengthen our main quantitative finding, that this framework has difficulty generating a near random walk in industry-level wages when the incidence of search unemployment is high. The intuition is straightforward: mean reversion in productivity creates mean reversion in wages even without any endogenous movement of workers in and out of industries. Our paper is also related to a large literature on how labor hoarding affects measured productivity. A number of authors have argued that labor hoarding increases the measured procyclicality of labor productivity (Summers (1986), Burnside, Eichenbaum, and Rebelo (1993)), while others have argued that it may have the opposite effect (Prescott (1986), Horning (1994)). Although our model is not about business cycles, it too suggests that labor hoarding may reduce the response of measured productivity to shocks at the industry level. The key insight is that labor is hoarded by industries rather than by firms and so takes the form of unemployment rather than employment. Following an adverse shock to an industry, some workers become rest unemployed, which raises the productivity—output in units of the final good per worker—of the remaining workers. In Section 2, we describe the economic environment. We analyze in Section 3 a special case where workers can immediately move to the best industry. Without any search cost, there no rest unemployment, since either working in the best industry or dropping out of the labor force dominates this activity. Instead, idiosyncratic productivity shocks lead to a continual reallocation of workers across industries.
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Section 4 characterizes the directed search model. We describe the equilibrium as the solution to a system of two equations in two endogenous variables and various model parameters. We prove that the equilibrium is unique and perform simple comparative statics. In particular, we find that there is rest unemployment only if the cost in terms of foregone leisure is low. We also provide closed-form expressions for the employment, search-unemployment, and rest-unemployment rates, as well as aggregate output. Section 5 gives a parallel characterization of the random search model, now describing the equilibrium as two equations in two different endogenous variables. We again provide closed-form expressions for employment, search unemployment, rest unemployment, and output. The arguments are similar to the directed search case and so we keep our presentation compact. Section 6 develops our measure of wage persistence and quantitatively compares our model with U.S. data. Section 7 concludes. 2. MODEL We consider a continuous time, infinite-horizon model. We focus for simplicity on an aggregate steady state and assume markets are complete. 2.1. Intermediate Goods There is a continuum of intermediate goods indexed by j ∈ [0 1]. Each good is produced in a separate industry with a constant returns to scale technology that uses only labor. In a typical industry j at time t, there is a measure l(j t) workers. Of these, e(j t) are employed, each producing x(j t) units of good j, while the remainder are rest unemployed. x(j t) is an idiosyncratic shock that follows a geometric random walk (1)
d log x(j t) = μx dt + σx dz(j t)
where μx measures the drift of log productivity, σx > 0 measures the standard deviation, and z(j t) is a standard Wiener process, independent across industries. The price of good j, p(j t), and the wage in industry j, w(j t), are determined competitively at each instant t and are expressed in units of the final good. To keep a well behaved distribution of labor productivity, we assume that industry j shuts down according to a Poisson process with arrival rate δ, independent across industries and independent of industry j’s productivity. When this shock hits, all the workers are forced out of the industry. A new industry, also named j, enters with positive initial productivity x0 , keeping the total measure of industries constant. We assume a law of large numbers, so the share of industries experiencing any particular sequence of shocks is deterministic.
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2.2. Final Goods A competitive sector combines the intermediate goods into the final good using a constant returns to scale technology (2)
Y (t) =
θ/(θ−1)
1
y(j t)(θ−1)/θ dj
0
where y(j t) is the input of good j at time t and θ > 0 is the elasticity of substitution across goods. We assume θ = 1 throughout the paper and comment in Section 3 on the role of this assumption. The final goods sector takes the price of the intermediate goods {p(j t)} as given and chooses y(j t) to maximize profits. It follows that (3)
y(j t) =
Y (t) p(j t)θ 2.3. Households
There is a representative household consisting of a measure 1 of members. The large household structure allows for full risk sharing within each household, a standard device for studying complete markets allocations. At each moment in time t, each member of the representative household engages in one of the following mutually exclusive activities: • L(t) household members are located in one of the industries. – E(t) of these workers are employed at the prevailing wage and get leisure 0. – Ur (t) = L(t) − E(t) of these workers are rest unemployed and get leisure br . • Us (t) household members are search unemployed, looking for a new industry and getting leisure bs . • The remaining household members are inactive, getting leisure bi > bs . Household members may costlessly switch between employment and rest unemployment, and between inactivity and searching; however, they cannot switch industries without going through a spell of search unemployment. Workers exit their industry for inactivity or search in three circumstances: first, they may do so endogenously at any time at no cost; second, they must do so when their industry shuts down, which happens at rate δ; and third, they must do so when they are hit by an idiosyncratic shock, according to a Poisson process with arrival rate q, independent across individuals and independent of their industry’s productivity. We introduce the idiosyncratic “quit” shock q to account for separations that are unrelated to the state of the industry. A worker in search unemployment finds a job in the industry of her choice according to a Poisson process with arrival rate α. In Section 5, we consider a
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variant of the model where search-unemployed workers instead find a random industry. We can represent the household’s preferences via the utility function ∞ (4) e−ρt u(C(t)) + bi (1 − E(t) − Ur (t) − Us (t)) 0
+ br Ur (t) + bs Us (t) dt
where ρ > 0 is the discount rate, u is increasing, differentiable, and strictly concave, satisfies the Inada conditions u (0) = ∞ and limC→∞ u (C) = 0, and C(t) is the household’s consumption of the final good. The household finances its consumption using its labor income. 2.4. Equilibrium We look for a competitive equilibrium of this economy. At each instant, each household chooses how much to consume and how to allocate its members between employment in each industry, rest unemployment in each industry, search unemployment, and inactivity to maximize utility subject to technological constraints on reallocating members across industries, taking as given the stochastic process for wages in each industry; each final goods producer maximizes profits by choosing inputs, taking as given the price for all the intermediate goods; and each intermediate goods producer j maximizes profits by choosing how many workers to hire, taking as given the wage in its industry and the price of its good. Moreover, the demand for labor from intermediate goods producers is equal to the supply from households in each industry; the demand for intermediate goods from the final goods producers is equal to the supply from intermediate goods producers; and the demand for final goods from the households is equal to the supply from the final goods producers. Standard arguments imply that for given initial conditions, there is at most one competitive equilibrium of this economy.5 We look for a stationary equilibrium where all aggregate quantities and the joint distribution of wages, productivity, output, employment, and rest unemployment across industries are constant. 3. FRICTIONLESS MODEL To understand the mechanics of the model, we start with a version where nonworkers can instantaneously become workers; formally, this is equivalent to 5 The first welfare theorem implies that any equilibrium is Pareto optimal. Since all households have the same preferences and endowments, if there were multiple equilibria, household utility must be equal in each. But a convex combination of the equilibrium allocations would be feasible and Pareto superior, a contradiction.
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the limit of the model when α → ∞. In this limit, the household does not need to devote any workers to search unemployment. Moreover, assuming bi > br , there is no rest unemployment, since resting is dominated by inactivity. Thus the household divides its time between employment and inactivity. Finally, with costless mobility, all workers must earn a common wage w(t). The household therefore solves ∞ e−ρt u(C(t)) + bi (1 − E(t)) dt max E(t)
0
subject to the budget constraint C(t) = w(t)E(t). Assuming an interior equilibrium, E(t) ∈ (0 1), the first order conditions imply that at each date t, bi = w(t)u (C(t)). Put differently, let ω(t) ≡ log w(t) + log u (C(t)) denote the log wage, measured in units of marginal utility. This is pinned down by preferences alone, ω(t) = log bi for all t. To determine the level of employment and consumption, we aggregate across industries. Consider industry j with productivity x(j t) and l(j t) workers at time t. Output is y(j t) = l(j t)x(j t) and so equation (3) implies the price of the good is p(j t) = (Y (t)/ l(j t)x(j t))1/θ . Since workers are paid their marginal revenue product, p(j t)x(j t), the log wage measured in units of marginal utility is (5)
ω(t) =
log Y (t) + (θ − 1) log x(j t) − log l(j t) + log u (C(t)) θ
We showed in the previous paragraph that ω(t) is pinned down by preferences, while consumption and output are aggregate variables and so are the same in different industries. Thus this equation determines how employment moves with productivity. When θ > 1, more productive industries employ more workers, while if goods are poor substitutes, θ < 1, an increase in labor productivity lowers employment so as to keep output relatively constant. If θ = 1, employment is independent of productivity, an uninteresting case whose analysis we omit from the rest of the paper. To close the model, eliminate l(j t) from y(j t) = x(j t)l(j t) using equation (5). This gives y(j t) = Y (t)(x(j t)u (C(t))e−ω(t) )θ . Substitute this into equation (2) and simplify to obtain 1 1/(θ−1) eω(t) θ−1 = x(j t) dj u (C(t)) 0 With an invariant distribution for x, we can rewrite this equation by integrating across that distribution. Appendix A.1 finds an expression for the invariant distribution. Assuming σx2 (6) δ > (θ − 1) μx + (θ − 1) 2
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we prove that ⎞1/(θ−1)
⎛ (7)
⎜ eω(t) = x0 ⎜ ⎝ u (C(t))
δ 2 x
σ δ − (θ − 1) μx + (θ − 1) 2
⎟ ⎟ ⎠
This equation pins down aggregate consumption C and hence aggregate output Y in terms of model parameters. If condition (6) fails with θ > 1, extremely productive firms would produce an enormous amount of easily substitutable goods, driving consumption to infinity. If it fails with θ < 1, extremely unproductive firms would require a huge amount of labor to produce any of the poorly substitutable goods, driving consumption to zero. Since eω(t) = bi , equation (7) determines the constant level of consumption C = C(t). Finally, employment is also constant and determined from the budget constraint as E = Cu (C)e−ω . For future reference, we note two important properties of the frictionless economy, both of which carry over to the frictional economy. First, if μx + 12 (θ − 1)σx2 = 0, then w = x0 , independent of δ. The condition is equivalent to imposing that xθ−1 is a martingale which, by equation (5), implies employment l is a martingale. This is a reasonable benchmark and ensures that aggregate quantities are well behaved in the limit as δ converges to zero. Second, an increase in the leisure value of inactivity bi raises the marginal utility of consumption u (C) by the same proportion, while employment E decreases in proportion to C. 4. DIRECTED SEARCH MODEL We now return to the model where it takes time to find a new industry, α < ∞. We look for a steady state equilibrium where the household maintains constant consumption, obtains a constant income stream, and keeps a positive and constant fraction of its workers in each of the activities: employment, rest unemployment, search unemployment, and inactivity. Equilibrium is a natural generalization of the frictionless model. Because it is costly to switch industries, different industries pay different wages at each point in time. Workers exit industries when wages fall too low, which puts a lower bound on wages. Searchers enter industries when wages rise too high, providing an upper bound. Between these bounds, there is no endogenous entry or exit of workers, and so wages fluctuate only due to exogenous quits and to productivity shocks. Our approach to characterizing equilibrium mimics our approach in the previous section. Using the household optimization problem alone, we determine
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the stochastic process for the log wage measured in units of marginal utility. We then turn to aggregation to pin down output and employment. 4.1. Household Optimization We assume parameter values are such that some household members work, some search, and some are inactive. The household values its members according to the expected present value of marginal utility that they generate either ¯ To find these bounds, from leisure or from income; this lies in an interval [v v]. ¯ first consider an individual who is permanently inactive. It is immediate from equation (4) that he contributes bi /ρ to the household. Since the household may freely shift workers into inactivity, this must be the lower bound on marginal utility, v = bi /ρ ¯ The household can also freely shift workers into search unemployment, so v is also the expected present value of marginal utility for a searcher. Compared¯ to an inactive worker, a searcher gets low marginal utility today in return for the possibility of moving to an industry where she generates the highest marginal utility. More precisely, a searcher gets flow utility bs and finds the best industry at rate α, giving a capital gain v¯ − v. Since the present value of her utility is v, ¯ ¯ ¯ this implies ρv = bs + α(v¯ − v), pinning down v, ¯ ¯ 1 (9) +κ v¯ = bi ρ (8)
where κ ≡ (bi − bs )/(bi α) is a measure of search costs—the percentage loss in current utility from searching rather than inactivity times the expected duration of search unemployment 1/α. We turn next to the behavior of wages. As in the frictionless model, consider a typical industry j at time t with productivity x(j t) and l(j t) workers. If all the workers were employed, labor demand would pin down the log wage measured in units of marginal utility, (10)
ω(j t) =
log Y + (θ − 1) log x(j t) − log l(j t) + log u (C) θ
We omit the derivation of this equation, which is unchanged from equation (5). If ω(j t) < log br , the household gets more utility from rest unemployment. In equilibrium, employment falls, raising the wage until it is equal to br and households are indifferent between the two activities. To stress this point, we call ω(j t) the log full-employment wage and note that the actual log wage is the maximum of this and log br .
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We claim that in equilibrium, the expected present value of a household member in an industry depends only on the current log full-employment wage
(11)
e−(ρ+q+δ)t max br eω(jt) + (q + δ)v dt ¯ 0 ω(j 0) = ω
v(ω) = E
∞
where expectations are taken with respect to future values of the random variable ω(j t), whose behavior we discuss further below. The discount rate ρ + q + δ accounts for impatience, for the possibility that the worker exits the industry exogenously, and for the possibility that the industry ends exogenously. The time-t payoff is the prevailing wage; this holds whether the worker is employed or rest unemployed because when there is rest unemployment, the worker is indifferent between the two states. In addition, if the worker exogenously leaves the industry, which happens with hazard rate q +δ, the household gets a terminal value v. ¯ In equilibrium, the expected present value of a worker in an industry must lie ¯ If v(ω) > v, ¯ searchers would enter the industry, reducing in the interval [v v]. ¯ ω and with it v(ω). If v(ω) < v, workers would exit the industry, raising ω and v(ω).6 Thus workers’ entry and¯ exit decisions determine two thresholds for the log full-employment wage, ω < ω, ¯ with the value function satisfying ¯ (12)
¯ for all ω v(ω) ¯ v(ω) ∈ [v v] ¯ = v ¯ v(ω) = v if ω > −∞ ¯ ¯ ¯
and
The last equation allows for the possibility that workers never choose to endogenously exit an industry, as will be the case if the leisure from rest unemployment exceeds the leisure from inactivity, br ≥ bi . Finally we turn to the dynamics of the log full-employment wage, a regulated Brownian motion on the interval [ω ω]. ¯ When ω(j t) ∈ (ω ω), ¯ only produc¯ of workers change ω. ¯ Employment and tivity shocks and the deterministic exit wages depend on whether there is rest unemployment. Consider the case with log br > ω, so there is rest unemployment in the most depressed industries. ¯ b ω), log employment falls at rate q, there is no rest unemployIf ω ∈ [log r ¯ ment, and wages change randomly with productivity. If ω ∈ (ω log br ), there ¯ and is rest unemployment, log wages are constant at log br , and employment 6
Note that equation (11) assumes that a worker never endogenously chooses to exit a industry. This is appropriate because a worker never strictly prefers to do so, but rather is always willing to stay if enough other workers exit.
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rest unemployment change randomly with productivity. From equation (10), this implies dω(j t) = μ dt + σω dz(j t), where (13)
μ≡
θ−1 q μx + θ θ
σω ≡
θ−1 σx θ
and
σ ≡ |σω |
that is, in this range ω(j t) has drift μ and instantaneous standard deviation σ. When the thresholds ω and ω ¯ are finite, they act as reflecting barriers, since ¯ would move ω outside the boundaries are offset by the productivity shocks that entry and exit of workers. Expectations in equation (11) are taken with respect to the stochastic process for full-employment wages. These equations uniquely determine the thresholds ω and ω: ¯ ¯ PROPOSITION 1: Equations (11) and (12) uniquely define ω and ω ¯ as functions of model parameters. A proportional increase in bi , br ,¯ and bs raises eω¯ and eω¯ by the same proportion. Moreover, ω < log bi < ω ¯ < ∞ with ω > −∞ if ¯ ¯ and only if br < bi . This generalizes the frictionless equilibrium, where household optimization pins down ω = log bi . With frictions, the log full-employment wage can fluctuate within some boundaries, and changes in the preference for leisure cause proportional changes in those boundaries. There are two ways to prove uniqueness of these boundaries. The proof in Appendix A.2 relies on monotonicity of the value function in the boundaries, showing that an increase in either boundary raises the value function, but an increase in the upper (lower) bound affects v more when ω is close to the upper (lower) bound. Alternatively, in online Appendix B.2 in the Supplemental Material (Alvarez and Shimer (2011)), we prove this result using the solution to an “island planner’s problem” (see also (Leahy 1993)). Notice that the state of an industry in our model is the one dimensional object ω, while in Lucas and Prescott (1974) the state is two dimensional.7 Lucas and Prescott must include productivity x and employment l as separate state variables because they consider a general class of processes for x, in particular allowing for mean reversion. While ω still determines current wages in their setup, it is not a Markov process, so there is no analog of equation (13) in their setup. The assumption that log x is a Brownian motion with drift permits us to reduce the state variable to a single dimension. This simplification is common in the partial equilibrium literature on irreversible investment (Bentolila and Bertola (1990), Abel and Eberly (1996), Caballero and Engel (1999)) and 7 max{br eω } is analogous to R(x l) in Lucas and Prescott (1974) notation. Their production technology implies that Y does not affect R, while risk neutrality ensures that u (C) is constant. Lucas and Prescott (1974) also assumed that Rx > 0—see their equation (1)—which in our setup is equivalent to θ > 1.
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enables us to provide a more complete analytical characterization of the equilibrium than could Lucas and Prescott. Proposition 1 establishes that ω is finite when br < bi . Intuitively, if an industry is hit by sufficiently adverse¯ shocks, workers will leave because rest unemployment is costly and has low expected payoffs. In contrast, when br ≥ bi , rest unemployment is costless and hence workers only leave industries when they shut down. Moreover, if br ≤ bi , there is no rest unemployment in the best industries, ω ¯ > log br . The next proposition addresses whether there is rest unemployment in the worst industries, ω ≷ log br . ¯ PROPOSITION 2: There exists a b¯ r such that in an equilibrium br eω¯ if and only if br b¯ r , with b¯ r = B(κ ρ + q + δ μ σ)bi for some function B that is positive-valued and decreasing in κ with B(0 ρ + q + δ μ σ) = 1. The proof is given in Appendix A.3. Proposition 2 implies that there is rest unemployment if search costs κ are sufficiently high given any br > 0 or, equivalently, if the leisure value of resting br is sufficiently close to the leisure of inactivity bi given any κ > 0. If a searcher finds a job sufficiently fast (so κ is small) or resting gives too little leisure (so br is small), there is no reason to wait for industry conditions to improve and so B is monotone in κ. 4.2. Aggregation Having pinned down the thresholds ω and ω, ¯ we now prove that consump¯ uniquely determined. The approach tion, employment, and unemployment are is again analogous to our analysis of the frictionless model. PROPOSITION 3: There exists a unique equilibrium. The steady state density of workers across log full-employment wages is 2
(14)
f (ω) =
|ηi + θ|eηi (ω−ω) ¯
i=1
2
i=1
¯ ω) eηi (ω− ¯ −1 |ηi + θ| ηi
where η1 < 0 < η2 solve q + δ = −μη +
σ2 2
η2 .
The proof in Appendix A.4 also provides explicit equations for output and consumption (which are equal by market clearing) and the number of workers in industry L.
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We turn next to the rest-unemployment rate. Recall that Ur is the fraction of household members who are rest unemployed. If log br ≤ ω, this is zero. Otherwise, define ω ˆ ≡ log br . In an industry with ω ∈ [ω ω], ˆ the¯ rest-unemployment θ(ω−ω) ˆ ¯ rate is 1 − e . Integrating across such industries using equation (14) gives
(15)
Ur = L
ω ˆ
ω ¯
ˆ ω) ˆ ω) eη1 (ω− eη2 (ω− ¯ −1 ¯ −1 − η η 2 1 ˆ 1 − eθ(ω−ω) f (ω) dω = θ 2 ηi (ω− ¯ ω) e ¯ −1 |θ + ηi | ηi i=1
The remaining household members who are in industries are employed, E = L − Ur . Finally we pin down the search-unemployment rate. Let Ns be the number of workers among L who leave their industry per unit of time, either because conditions are sufficiently bad or because their industry has exogenously shut down. Appendix A.6 takes limits of the discrete time, discrete state-space model to show that this rate is given by (16)
Ns =
θσ 2 f (ω)L + (q + δ)L 2 ¯
The first term gives the fraction of workers who leave their industry to keep ω above ω. The second term is the fraction of workers who exogenously leave ¯ their industry. In steady state, the fraction of workers who leave industries must balance the fraction of workers who arrive in industries. The latter is given by the fraction of workers engaged in search unemployment Us times the rate at which they arrive to the industry α, so αUs = Ns . Solve equation (16) using equation (14) to obtain an expression for the ratio of search unemployment to workers in industries: ⎛ ⎞ (17)
⎜ ⎟ ⎟ 2 Us 1⎜ η2 − η1 ⎜ θσ ⎟ = ⎜ + q + δ⎟ 2 ⎟ ηi (ω− ¯ ω) L α⎜ 2 e ¯ −1 ⎝ ⎠ |θ + ηi | η i i=1
To have an interior equilibrium, we require that Us + Ur + E ≤ 1 so that the labor force is smaller than the total population.8 We deliberately leave the expressions for unemployment as a function of the thresholds ω, ω, ˆ and ω ¯ so as to disentangle optimization—the choice ¯ 8 If this condition fails, all household members participate in the labor market. The equilibrium is equivalent to one with a higher leisure value of inactivity, the value of bi such that Us + Ur + E = 1. In any case, Proposition 4 implies that for br , bs , and bi large enough, the equilibrium has Us + Ur + E < 1.
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of thresholds—from the mechanics of aggregation. This has two advantages. First, we find it useful to exploit this dichotomy in our numerical evaluation of the model in Section 6. Second, the expressions for rest and search unemployment as a function of the thresholds are identical in other variants of the model, including the original Lucas and Prescott (1974) model. For example, suppose the curvature of labor demand comes from diminishing returns at the island level, due to a fixed factor, rather than from imperfect substitutability (see footnote 7). Then the analog of the elasticity of substitution is the reciprocal of the elasticity of revenue with respect to the fixed factor, while the expressions for unemployment are otherwise unchanged. We close this section by noting some homogeneity properties of employment, rest unemployment, search unemployment, and consumption. PROPOSITION 4: Let br = χb¯ r bs = χb¯ s , and bi = χb¯ i for fixed b¯ r , b¯ s , and b¯ i . s +Ur The equilibrium value of the unemployment rate UsU+U and the share of rest r +E Ur unemployed Ur +Us do not depend on χ, the level of productivity x0 , or the utility function u, while u (Y ) is proportional to χ/x0 . PROOF: By inspection, the unemployment rate and the share of rest unemployed are functions of the difference in thresholds ω ¯ − ω and ω ¯ −ω ˆ and ¯ directly or indithe parameters α, δ, q, θ, μ (or μx ), and σ (or σx ), either rectly through the roots ηi . From Proposition 1, the thresholds depend on the same parameters and on the discount rate ρ. Next, Proposition 1 shows that eω¯ and eω¯ are proportional to χ. Then equation (52) implies that u (Y ) inherits the same proportionality. On the other hand, Proposition 1 implies that x0 does not affect any of the thresholds and so equation (52) implies u (Y ) is inversely proportional to x0 . Q.E.D. This proposition shows that the unemployment rate and composition of unemployment is determined by the relative advantage of different leisure activities, while output, and hence consumption and employment, depends on an absolute comparison of leisure versus market production. Indeed, the finding that u (Y ) is proportional to χ/x0 holds in the frictionless benchmark, where an interior solution for the employment rate requires bi = u (Y )w, while the wage is proportional to x0 (see equation (7)). Whether higher productivity lowers or raises equilibrium employment depends on whether income or substitution effects dominate in labor supply. With u(Y ) = log Y , an increase in productivity raises consumption proportionately without affecting employment or labor force participation. 4.3. The Limiting Economy We close this section by discussing a useful limit of the model, when the exogenous shut-down rate of industries δ is zero. To ensure an invariant distribution of productivity and employment, we introduced the assumption that
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industries shut down for technical reasons. Still, with the parameter restriction μx + 12 (θ − 1)σx2 = 0 discussed previously in Section 3, the economy is well behaved even when δ limits to zero. It is clear from Proposition 1 that ω and ω ¯ converge nicely for any value of μx as long as the discount rate ρ is ¯positive. More problematic is whether aggregate employment, unemployment, and output converge. This section verifies that the same parameter restriction yields a well behaved limit of the frictional economy. When μx = − 12 (θ − 1)σx2 and δ → 0, the roots η1 < 0 < η2 in Proposition 3 converge to η1 = −θ and η2 = 2q/θσ 2 . Substituting into equation (14), we find f (ω) =
η2 eη2 (ω−ω) ¯ η2 (ω− ¯ ω) e ¯ −1
If q = 0 as well, this simplifies further to f (ω) = 1/(ω ¯ − ω), that is, f is uni¯ form on its support, while for positive q, the density is increasing in ω. We can also take limits of equations (51) and (52) in Appendix A.4 to prove that the number of workers in industries and output have well-behaved limits. Turning next to unemployment, equations (15) and (17) imply that in the limit, ˆ ω) eη2 (ω− ¯ −1 ˆ ω) + e−θ(ω− ¯ −1 Ur η2 η2 (ω− = e ¯ ω) L ¯ −1 (θ + η2 ) η2
θ
(18)
and
q Us = ¯ ω) L α(1 − e−η2 (ω− ¯ )
Each of these expressions simplifies further when there are no quits, q = 0 and so η2 → 0:9 ˆ ω) θ(ω ˆ − ω) + e−θ(ω− Us θσ 2 Ur ¯ −1 = and = ¯ L θ(ω ¯ − ω) L 2α(ω ¯ − ω) ¯ ¯ The last expression is particularly intuitive. With our parameter restrictions, ¯ − ω)/μ reflects the average duration of an industry μ = − 12 θσ 2 . Then −(ω spell, since wages must fall ¯from ω ¯ to ω and drift down on average at rate μ, while 1/α is the average duration of an¯ unemployment spell. The ratio of durations is L/Us .
(19)
5. RANDOM SEARCH MODEL In this section we analyze an alternative technology for search. Instead of locating the best industry, we assume that agents can only locate industries 9
The order of convergence of δ and q to zero does not affect these results.
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randomly. As in the directed search case, we obtain a separation between optimization and aggregation, with simple reduced-form expressions for the unemployment rates. This shows that our approach is robust to the specification of the mobility technology. Additionally, we find this alternative specification interesting because wages are not regulated from above and hence they can, in principle, have very different statistical behavior. 5.1. Setup As much as possible, our setup parallels that with directed search. We leave our notation unchanged and focus on the differences between the two models. The search unemployed engage in one of two mutually exclusive activities. First, Usr search randomly, finding an industry at rate α. The probability of contacting any particular industry j is proportional to the number of workers in that industry at time t, l(j t), as in Burdett and Vishwanath (1988). The assumption that workers are not more likely to find a high-wage industry is an extreme alternative to our directed search model and is in the spirit of the search problem in McCall (1970). One interpretation is that a worker searching randomly contacts another worker currently in an industry at rate α and is equally likely to contact any worker, regardless of the state of her industry. Second, Usn workers search for a new industry, finding one at the same rate α. This ensures that there are some workers who can get a new industry with productivity x0 off the ground. 5.2. Household Optimization The value of permanent inactivity is given by v as described in equation (8). ¯ In an equilibrium with inactivity and with search, the expected value of a ¯ which is defined in equaworker concluding either search activity must be v, tion (9). An industry j at time t is characterized by the log full-employment wage ω(j t), unchanged from equation (10), with the value of a worker in such an industry still given by equation (11); however, random search affects the stochastic process for wages and thus the value function. As in the directed search model, ω is regulated from below by agents’ willingness to exit an industry with bad prospects, but it is not regulated from above because of the absence of directed search. That is, v(ω) ≥ v for all ω and v(ω) = v if ω > −∞ ¯ ¯ ¯ ¯ The arrival of random searchers causes gross labor force growth at some endogenous rate s, a key variable in what follows. This implies that ω follows a Brownian motion regulated below at ω with dω(j t) = μ dt + σω dz(j t), ¯ where (20)
(21)
μ≡
q−s θ−1 μx + θ θ
σω ≡
θ−1 σx θ
and
σ ≡ |σω |
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Thus the arrival of random searchers puts downward pressure on wages, the opposite effect of exogenous quits. Indeed, it is not surprising that only the difference between these two rates affects the stochastic process for wages. The behavior of wages and employment on (ω ∞) is the same as the behavior in the directed search case on (ω ω), ¯ except¯for the drift of ω. ¯ Turn next to the value of workers who have successfully completed a search spell: ∞ ¯ (22) v(ω)f (ω) dω = v ω ¯
(23)
¯ v(ω0 ) = v
The first equation states that the expected value of a completed random search ¯ where f (ω) is the density of log full-employment wages across spell must be v, industries. The second equation states that the value of a completed search ¯ which pins down ω0 , the initial wage in new for a new industry must be v, industries. We prove in Appendix A.7 that the density of log full-employment wages satisfies ⎧ 2δL¯ ⎪ η1 (ω−ω) ⎪ ⎪ η1 η2 + 2 ((η2 + θ)eη2 (ω−ω) ¯ − (η1 + θ)e ¯ ) ⎪ ⎪ σ L ⎪ ⎪ ⎪ ⎪ θ(η2 − η1 ) ⎪ ⎪ ⎪ ⎪ ⎪ if ω ∈ [ω ω0 ] ⎪ ⎪ ⎪ ¯ ⎪ ⎪ 2δL¯ ⎨ η1 (ω−ω) η1 η2 + 2 ((η2 + θ)eη2 (ω−ω) ¯ − (η1 + θ)e ¯ ) (24) f (ω) = σ L ⎪ ⎪ ⎪ ⎪ θ(η2 − η1 ) ⎪ ⎪ ⎪ ⎪ ⎪ 2δL¯ η1 (ω−ω0 ) ⎪ ⎪ (e − eη2 (ω−ω0 ) ) ⎪ ⎪ 2L ⎪ σ ⎪ + ⎪ ⎪ ⎪ η2 − η1 ⎪ ⎩ if ω > ω0 2 ¯ where η1 < η2 < 0 solves q + δ − s = −μη + σ2 η2 and L/L is the number of workers in a new industry relative to the overall number of workers in industries. Note that there is a kink in the density at ω0 , reflecting the entry of new industries; if δ = 0, the density is smooth. ¯ To close the model, we describe the ratio L/L using the invariant distribution f˜ for the process {ω(t) log l(t)} across industries. Formally, we can first describe {ω(t)} as a one-sided reflected diffusion and jump process on [ω ∞), ¯ be a and then use this process to describe {log l(t)} on the real line. Let Z(t) standard Brownian motion, let N(t) be the counter associated with a homogeneous Poisson process with intensity δ, and let B(t) be an increasing singular
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process: (25)
ω(0) = ω0
¯ log l(0) = log L
dω(t) = μ dt + σ dZ(t) + (ω0 − ω(t)) dN(t) + dB(t) ¯ − log l(t)) dN(t) − θ dB(t) d log l(t) = (s − q) dt + (log(L) T where for all T > 0, 0 I{ω(t)>ω} dB(t) = 0. In this definition we start each in¯ ¯ When the industry dustry at (ω0 L). is destroyed, at rate δ per unit of time, we replace it by a new one with the same initial conditions. Given this recurrence, this process has a unique invariant distribution for all δ > 0. We denote this by f˜(ω l). ¯ The distribution f˜ implies a value for L/L. To see this, notice that equa¯ tion (25) implies that an increase in log L increases all the realizations of each path of {log l(t)} by the same amount. Alternatively, equation (25) can be ¯ which makes no other reference to L. ¯ rewritten for the process log(l(t)/L), ¯ we write f˜(ω l; L). ¯ We have that L is To denote the dependence of f˜ on L, the measure of workers in industries, so that ∞ ∞ ∞ ∞ ¯ dω dl = L¯ lf˜(ω l; L) lf˜(ω l; 1) dω dl L= 0
ω ¯
ω ¯
0
¯ Then we can write a condition for the ratio L/L as (26)
¯ L/L =
∞
0
∞
lf˜(ω l; 1) dω dl
−1
ω ¯
Although we do not explicitly solve for f˜(ω l; 1), equation (25) implies that it depends only on ω, s, and ω0 . ¯ in terms of Equations (20),¯ (22), (23), and (26) determine ω, s, ω0 , and L/L ¯ model parameters. Although we do not have a general existence and uniqueness proof for the random search model, analogous to Proposition 1, later in this section we prove existence in the limit as δ → 0. We also establish uniqueness for a particular case without exogenous quits or rest unemployment. 5.3. Aggregation To close the model, first determine output Y and the number of workers in industries L. We omit the results, which are unchanged from the directed search model in Appendix A.4. We focus instead on the unemployment rates. For the search unemployed, we have (27)
Usr /L = s/α
and
¯ Usn /L = (L/L)(δ/α)
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These equations balance inflows and outflows from each state. For example, random searchers flow into industries at rate sL and find jobs at rate α, so there must be Usr = sL/α such workers. For rest unemployment, we simply have ωˆ ˆ (28) 1 − eθ(ω−ω) f (ω) dω Ur /L = ω ¯
where ω ˆ ≡ min{ω log br }. ¯ 5.4. The Limiting Economy We focus again on the special case with μx = − 12 (θ − 1)σx2 , so we can take the limit as δ converges to zero. In this case, the first block of equations that describe an equilibrium becomes a set of two equations in two unknowns, namely equations (20) and (22) that determine ω and s. This is because neither ω ¯ nor ¯ ¯ L/L affects the density f in this limit. Moreover, the roots ηi satisfy (29)
η1 = −θ
η2 = −
2(s − q) θσ 2
so that the density of workers across industries is exponential: (30)
f (ω) = −η2 eη2 (ω−ω) ¯
Using this expression for f , equation (28) reduces to (31)
Ur η1 η2 ˆ ω) ˆ ω) =1+ eη2 (ω− eη1 (ω− ¯ + ¯ L η2 − η1 η1 − η2
Since no unemployment is accounted for by workers searching for new industries in the limiting economy, we use Us = Usr , unchanged from equation (27), to denote the search unemployment rate. It is straightforward to prove existence of equilibrium in this limiting economy and, under more restrictive conditions, uniqueness of equilibrium: PROPOSITION 5: There exists an equilibrium pair (ω s) for the random search model with μx = − 12 (θ − 1)σx2 and δ → 0. Moreover, s¯ > q + 12 θσ 2 . If q = br = 0, the equilibrium is unique. In that case, an increase in search costs κ reduces the equilibrium values of s and ω, raising the search-unemployment rate. ¯ The proof is given in Appendix A.8. We believe the uniqueness result should hold more generally.
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6. QUANTITATIVE EVALUATION 6.1. Auxiliary Statistical Model We start our quantitative evaluation of the model by describing an auxiliary statistical model for wages in industry j at time t. Suppose that the log wage follows a first order autoregressive process with a common autocorrelation and variance across industries, but possibly with different means for each industry, (32)
log wjt+1 = βw log wjt + log wj· + εjt+1
where εjt is normally distributed with mean zero and standard deviation σw , and is independent across industries and over time. wj· represents an industry fixed effect. We can estimate the parameters of the model using exact maximum likelihood; Appendix A.9 provides details on the estimator. Of course, log wages generated by our model do not follow an AR(1), but rather are a regulated Brownian motion or an even more nonlinear process in the presence of rest unemployment. Still, we regard equation (32) as useful for summarizing properties of log wages in the data and in the model, in the spirit of indirect inference. The following proposition explains why: PROPOSITION 6: Consider a model economy with μx = − 12 (θ − 1)σx2 and δ → 0. Estimate equation (32) using a sample of annual-average wages for J industries during T years. The distribution of the estimated statistics βˆ w and σˆ w /σ depends only on q, θσ, αUs /L, and Ur /L, and on whether search is random or directed. The proof in Appendix A.10 shows that, after an appropriate normalization, the stochastic process for industry wages depends only on these reduced-form parameters. Note that the proposition assumes that μx = − 12 (θ − 1)σx2 , which allows us to focus on the limit without exogenous destruction of industries, δ → 0. Similar equivalence results hold even with a different drift and with exogenous destruction. Two comments are in order. First, if there is no rest unemployment and we measure wages at discrete time intervals (i.e. at the end of every year), the infinite-sample value of βw is a nonmonotonic function of θσ, maximized at an interior value; see the proof of Proposition 6 in Appendix A.10. Our numerical simulations suggest that this result carries over to time-aggregated, finite-sample data with and without rest unemployment. Since one of our main findings is that the model tends to produce too small a value for βˆ w relative to the data, we focus our discussion on the value of θσ that maximizes the expected value (across simulated samples) of βˆ w given the other parameters. Second, in our model economy, there is no reason to include the industry fixed effects wj· in equation (32). We do so because we are concerned that worker heterogeneity across industries may cause permanent differences in wages. In online Appendix B.3 in the Supplemental Material (Alvarez and
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F. ALVAREZ AND R. SHIMER TABLE I MAXIMUM LIKELIHOOD ESTIMATES OF EQUATION (32) USING CES DATA FROM 1990 TO 2008 AT DIFFERENT LEVELS OF AGGREGATIONa Aggregation
2 digit 3 digit 4 digit 5 digit 6 digit
J
βˆ w
σˆ w
15 75 227 297 157
0954 0905 0883 0874 0870
0014 0025 0030 0037 0036
a J indicates the number of industries used in estimation, βˆ is our w estimate of autoregression in wages, and σˆ w is our estimate of the standard deviation of wage innovations.
Shimer (2011)), we offer one justification for this concern—a version of the model with heterogeneity in human capital; most workers are only able to work in a minority of industries, while a few workers can work in all industries. Such scarce labor earns a higher average wage. We find that does not change the distribution of βˆ w or σˆ w , but instead loads entirely onto the fixed effects wj· . On the other hand, if we dropped the industry fixed effects, our estimate βˆ w would be biased toward 1 in this more general environment. 6.2. Persistence of Industry Wages: Data and Model We estimate equation (32) using U.S. data for five-digit North American Industrial Classification System (NAICS) industries at different levels of aggregation. We measure average weekly earnings in industry j and year t, w˜ jt , for J = 297 industries from 1990 to 2008 using data from the Current Employment Statistics (CES) (http://www.bls.gov/ces/), which is all the industries and years with available data. Deflate this by average weekly earnings in the private sector, w¯ t , to obtain that the empirical measure of wages in industry j at time t is wjt ≡ w˜ jt /w¯ t .10 Table I summarizes our empirical findings at different levels of aggregation. Our main focus is on the five-digit level, where our maximum likelihood estimate of the autoregressive coefficient is βˆ w = 087.11 This number is slightly larger in more aggregated data. The table also shows that the standard devi10 In the data, the average real wage fluctuates over time, while according to our model, it is constant. But suppose that labor productivity in industry j at time t is A(t)x(j t), where A(t) is an aggregate shock. With log utility, u(C) ≡ log C, fluctuations in A(t) cause proportional changes in wages measured in units of goods but affect neither wages measured in units of marginal utility nor the various thresholds measured in units of marginal utility nor the search or rest-unemployment rates. According to our model, deflating wages by w¯ t is therefore a perfect control for aggregate productivity shocks. 11 We have also estimated equation (32) using OLS and obtained qualitatively similar results.
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ation of wage changes is about 4 percent per year at the five-digit level and is smaller in more aggregate data, presumably reflecting some correlation in shocks. To get a sense of the magnitude of our persistence estimate, note that βˆ w is a biased estimate of βw due to the finite sample.12 Using Monte Carlo methods, we find that if βw = 099 and we estimate equation (32) using T + 1 = 19 years of data and J = 297 industries, we would obtain βˆ w ≈ 086. We are unlikely to be able reject a unit root using our wage data. Blanchard and Katz (1992) also reported a high persistence in relative wages in manufacturing across U.S. states between 1952 and 1990. They failed to reject a unit root for 47 out of 51 states (including the District of Columbia). Pooling the 51 states and including fixed effects, they estimated an AR(4) using ordinary least squares (OLS). The implied half-life is 11 years and the implied first order autocorrelation is 0.94.13 Like our estimates in Table I, these do not correct for finite-sample biases and so likely understate the true persistence of the wage process. The bottom line is that industry wages in the data are nearly a random walk. We next examine the persistence of wages in our theoretical model, focusing throughout on the case with μx = − 12 (θ − 1)σx2 and δ → 0. Rather than explore different values of the structural parameters, we use Proposition 6 to focus on the reduced-form parameters that affect the persistence of wages. For fixed values of the reduced-form parameters, we simulate a discrete time version of our model to construct a data set of wages. We assume a period length is 1 day and measure the log of annual average wages for J = 297 industries and T + 1 = 19 years, the same as in the data.14 We then estimate equation (32) using the model-generated data. We repeat this many times to obtain an accurate estimate of σˆ w . First suppose there are no exogenous quits, q = 0, and only search unemployment, as is the case if the leisure from rest unemployment is zero, br = 0. Then wage persistence depends only on αUs /L, θσ, and whether search is random or directed. We measure Us /(Us + L) as the unemployment rate, which averaged 5.5 percent in the United States from 1990 to 2008. The mean duration of an in-progress unemployment spell was 031 years, which implies α = 32 per year.15 Putting this together, the rate that unemployed workers 12 See Nickell (1981) for an exact statement of the bias from estimating equation (32) using OLS when J is large and T is finite. We are unaware of a comparable expression using maximum likelihood. 13 Based on the numbers reported in the third column of their Table 1. 14 We draw an initial value of ω from the ergodic distribution of ω across industries, say g(ω). This is not the same as the ergodic distribution of ω across workers, f (ω), because industries that experience negative productivity shocks shed workers but are no more likely to disappear. In the 2 random search model, we find g(ω) = −(2μ/σ 2 )e(2μ/σ )(ω−ω) ¯ , while f is given in equation (30). Both distributions are exponential, but the mean of g is lower. We obtain a similar result in the directed search model.
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find jobs per employed worker is αUs /L ≈ 0186 per year. Note that in our steady state model, this is also the incidence of search unemployment among employed workers. Using this number, we find that in the random search model, βˆ w = 066 when θσ = 060 and is smaller for other values of θσ. In the directed search model, we find that the maximum obtainable persistence is βˆ w = 058, obtained when θσ = 130. Although there is some variation in βˆ w across simulations of the model, its standard deviation is below 002—see the notes in Table II—which is too small to reconcile the model with the data. In summary, the model cannot simultaneously generate a high incidence of search unemployment and persistent wages. Intuitively, this is because a high incidence of search unemployment implies that industry wages are frequently reflecting off the lower bound as workers exit for unemployment, but this implies that persistence must be low. Exogenous quits moderate these results because the model needs a lower endogenous incidence of search unemployment. For example, set q = 012, so about two-thirds of unemployment spells are due to exogenous quits, which are independent of industry conditions. We find that the maximum attainable value of βˆ w increases to 078 in the random search model and 071 in the directed search model, still significantly short of the data. Still higher exogenous quit rates, q = 018, can potentially reconcile the model with the data in Table I. But we do not find this model very interesting. Unemployed workers find jobs at an exogenous rate and lose them mainly due to exogenous quits; the endogenous incidence of search unemployment is αUs /L − q = 0006 per year. Wages are barely regulated by the entry and exit of workers, and so the equilibrating mechanism in Lucas and Prescott (1974) is nearly shut down. Now suppose there is some rest unemployment. In this case, our model generates more persistence in wages for a given quit rate. This happens for two reasons. First, the model does not need to put so many industries near the lower bound ω; instead, unemployment can occur at higher values of the ¯ wage. Second, we measure the actual wage, not the fulllog full-employment employment wage, in the data. Industries with rest unemployment pay a constant wage, which raises the measured persistence βˆ w . To evaluate the quantitative importance of these effects, suppose that search accounts for one-quarter of all unemployment, Us /(Us + L) = 0013, with the remainder accounted for by rest, Ur /(Ur + L) = 0042. This totals the same 55% unemployment rate as before. In Section 6.4 we explain why these numbers may be plausible. We keep the arrival rate of job offers to searchers fixed at α = 32. Table II reports the maximum attainable value of the measured autoregressive coefficient βˆ w , along with the value of θσ that achieves this 15 The empirical duration numbers were constructed by the Bureau of Labor Statistics from the Current Population Survey and may be obtained from http://www.bls.gov/cps/.
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SEARCH AND REST UNEMPLOYMENT TABLE II RESULTS FROM RANDOM AND DIRECTED SEARCH MODELS WITH 13% SEARCH UNEMPLOYMENT, 42% REST UNEMPLOYMENT, AND α = 32a q
θσ
βˆ w
σˆ w /σ
θ
br /bi
κ
Random search
0 001 002 003 004
029 026 021 015 0067
0.842 0.856 0.870 0.885 0.903
0.507 0.517 0.530 0.544 0.561
40 36 30 22 102
0.95 0.95 0.96 0.96 –
20 26 32 50 ∞
Directed search
0 001 002 003 004
060 057 054 049 041
0.822 0.837 0.852 0.868 0.886
0.488 0.492 0.493 0.491 0.429
79 76 72 65 48
0.95 0.96 0.96 0.97 0.99
27 31 38 50 101
a Each row shows a different value of the exogenous quit rate q (second column) and the value of θσ (third column) that maximizes the maximum likelihood estimate of the autoregressive coefficient βˆ w (fourth column). The standard deviation of the estimate of βˆ w across runs of the models ranges from 0008 to 0012, depending on the parameterization. The fifth column shows the maximum likelihood estimate of the standard deviation of the residual in equation (32), which is proportional to σ . The standard deviation of the estimate of σˆ w across runs of the model ranges from 0012 to 0017. The sixth column shows the value of θ such that σˆ w = 0037. The last two columns impose ρ = 005 and show the implied structural parameters br /bi and κ. We assume a period length is 1 day and measure the log of annual earnings for J = 297 industries and T + 1 = 19 years.
maximum, for different values of the exogenous quit rate q, and for both random and directed search models. Note that the structure of our model imposes q ≤ αUs /L ≈ 0042. Two patterns emerge: (i) higher values of q imply more persistence in wages and (ii) for a given value of q, the random search model can generate slightly more persistent wages than the directed search model. The table also shows that rest unemployment significantly raises the measured persistence of wages. In particular, we match the empirical persistence at the five-digit level in the random search model with q = 002 and in the directed search model with q = 003, in both cases leaving substantial room for endogenous search unemployment as workers exit industries at ω. We conclude that rest unemployment ¯ industry wages with high endogenous potentially helps to reconcile persistent unemployment rates. 6.3. Structural Parameters We now return to our structural model and look at how preference parameters affect the measured persistence of wages. We assume throughout that the search-unemployment rate is Us /(Us + L) = 0013 and the rest-unemployment rate is Ur /(Ur + L) = 0042. The search unemployed find jobs at rate α = 32. For a given value of the exogenous quit rate q, we set θσ so as to maximize the autoregressive coefficient in wages, that is, at the values in Table II. The
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remainder of this section explains how we use these facts to uniquely identify all of the structural parameters of our model. To determine θ and σ separately, we use evidence on the estimated standard deviation of the residual in equation (32). The data in Table I indicate that σˆ w = 0037 at the five-digit level. The fifth column of Table II shows σˆ w /σ using model-generated data, which we use to pin down σ. Then the sixth column of Table II shows the implied value of the elasticity of substitution θ, which varies from slightly just over 1 to almost 8, depending on the choice of q and on whether search is random or directed. Evidence in Broda and Weinstein (2006) gives us a sense that the middle of this range may be reasonable. They estimated the elasticity of substitution between goods at the five-digit Standard International Trade Classification (SITC) level using international trade data and reported a median elasticity of 28 (see their Table IV).16 We note that there is no mechanical reason why our model had to deliver an elasticity of substitution in this range. We now determine the values of other structural parameters, starting with the random search model. Equation (27) implies that search-unemployed workers arrive in industries at rate s = αUs /L = 0042. Then equations (29) and (31) determine ω ˆ − ω, which is 004 with q = 0 and 011 with q = 002. ¯ Next, building on our analysis of the value function in Appendix A.8, we use equations (8) and (20) to pin down the relative leisure values of rest unemployment and inactivity. With q = 0, we find br /bi = 095, while q = 002 requires a higher relative value of rest unemployment, br /bi = 096.17 More generally, to generate rest unemployment, br /bi must exceed 090 without exogenous quits and exceed 089 with q = 002. This suggests that while the rest unemployed must pay some cost to remain in contact with their industry, the cost is small. Rest unemployment and inactivity may look quite similar to an outsider who observes individuals’ time use, even though the rest unemployed are much more likely to return to work. Next we consider the search cost κ = (v¯ − v)/bi . We find this by computing ¯ equation (22). With q = 0, we the expected value of finding a new industry from obtain κ = 20, while q = 002 requires κ = 32. In words, the expected cost of moving to a new industry is equal to the utility from 2 or 3 years of inactivity. 16
This elasticity is in line with that used in much of the literature that quantitatively evaluates the Lucas and Prescott (1974) model. Recall that the analog of θ in a model with diminishing returns at the industry level due to a fixed factor is the reciprocal of the elasticity of revenue with respect to the fixed factor. If the fixed factor is capital, then a capital share of 13 is empirically reasonable. Alvarez and Veracierto (1999, 2001) set the elasticity of fixed factor to 036 and 023, respectively, and Kambourov and Manovskii (2007) set it to 032, in line with values of θ between 28 and 43. 17 In the random search model with q = 004, our procedure implies s < q + 12 θσ 2 , which is inconsistent with our structural model, since the benefits from search are infinite (see Proposition 5). We indicate this by setting the search costs to infinity and leaving the value of br /bi blank.
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A strong autocorrelation in wages requires that most industries are far from the lower threshold ω, but this implies that the wage in an average industry is much higher than the¯ wage in the least productive industry. For workers to be willing to endure such an industry, the cost of moving must be large. The determination of the structural parameters in the directed search model is similar. We first use the expression for the search-unemployment rate in equation (18) to pin down ω ¯ − ω, which is 054 with q = 0 and 077 with ¯ for the rest-unemployment rate to find q = 003. We then use the equation ω ˆ − ω, which is 008 without quits and 016 with the high quit rate. Finally, ¯equation (12) gives us the remaining parameters. Without quits, we get br /bi = 095 and κ = 27. With q = 003, both numbers are larger: br /bi = 097 and κ = 56. These search costs are even larger than those in the random search model, and the intuition is similar. We have focused here on the relative leisure values from search unemployment, rest unemployment, and inactivity. To do this, we did not need to take a stand on the utility function. For the model to be consistent with balanced growth, we would additionally require u(C) ≡ log(C). In this case, labor force participation rates pin down the absolute level of the three leisure parameters. Finally, it is worth stressing that these high values of search costs κ are a consequence of persistent wages and are not due to rest unemployment per se. Suppose we shut down rest unemployment but let q → αUs /L so that wages are nearly a random walk, regardless of whether search is random or directed. In either case, the incentive to search for a job is large and so search costs must be large in equilibrium as well, reaching infinity in the limit when wages are a random walk. For example, with a 55 percent search-unemployment rate and no rest unemployment, αUs /L = 0186. Set q = 018 to generate βˆ w = 088 and σˆ w /σ = 074 in the random search model. These reduced-form parameters imply θ = 22, br /bi = 095, and κ = 20, comparable to values in the full model with rest unemployment. Our finding that search costs must be high if wages are persistent is closely related to Hornstein, Krusell, and Violante (2009) finding that when search costs are small, search models cannot generate much wage dispersion. More precisely, in their baseline model, wage offers are random and wages within an employment spell are constant. But they also show that if wages during an employment spell are not persistent, small search costs can be reconciled with a large cross-sectional dispersion in wages. 6.4. Discussion One of our main quantitative findings, established in Section 6.2, is that our model cannot match the empirical persistence of industry wages unless the endogenous incidence of search unemployment is very low. On the other hand, our model can simultaneously have a high rest-unemployment rate and industry wages that are as persistent as in the data. An important, unresolved
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question is the empirical counterpart of search and rest unemployment. According to our model, a spell of search unemployment ends when a worker moves to a new industry, while a spell of rest unemployment may end with the worker returning to her old industry. This suggests using evidence on “stayers” and “switchers” developed by Murphy and Topel (1987) from the March Current Population Survey and by Loungani and Rogerson (1989) from the Panel Study of Income Dynamics (PSID). These papers classify unemployed workers as stayers if they remain in the same two-digit industry after an unemployment spell and as switchers otherwise. Despite differences in their data sources and methodology, both papers find that workers switch industries after about onequarter of all unemployment spells. Moreover, Loungani and Rogerson (1989) reported that switchers account for about one-third of all weeks of unemployment, that is, they stay unemployed slightly longer than stayers. For this reason, we believe our breakdown of 4.2 percent rest unemployment and 1.3 percent search unemployment may be reasonable. Still, this evidence is far from definitive. First, it is not clear that “industries” in the model correspond to industries in the data. Our theoretical notion of industries has two defining characteristics. On the one hand, the goods produced within an industry are homogeneous while the goods produced in different industries are heterogeneous, as captured by the elasticity of substitution θ. This is consistent with the notion of an industry. On the other hand, workers are free to move within an industry but not between industries, perhaps because of some specificity of human capital or because of geographic mobility costs. To the extent that human capital is occupation, not industry, specific (Kambourov and Manovskii (2007)), this suggests that our theoretical construct may be closer to an occupation. In any case, it seems possible that many of the workers classified as stayers in Murphy and Topel (1987) and Loungani and Rogerson (1989) did in fact actively search for a new labor market, for example, by moving to a new geographic area. If the search-unemployment rate is substantially higher than 1.3 percent, then our model will have trouble generating the wage persistence that we observe in the data. Second, one might be concerned that, given how similar the leisure values of rest and inactivity are, many workers in rest unemployment might in practice be misclassified as out of the labor force. To get a sense of the potential importance of this issue, we consider a broader measure of unemployment, including workers who are “marginally attached” or working “part-time for economic reasons.” According to the Bureau of Labor Statistics, this averaged 39 percent higher than the official unemployment rate from 1994 to 2008. We therefore increase the rest-unemployment rate from 42 percent to 81 percent, leave the search-unemployment rate unchanged at 13 percent, and rerun our simulations. Fortunately our results are largely unaffected. In the random search model with q = 002, the maximum autocorrelation βˆ w rises from 0870 to 0873, while the elasticity of substitution θ falls from 30 to 26, the relative value of rest br /bi rises from 096 to 097, and the search cost κ rises from 32
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to 37. In the directed search model with q = 003, the maximum autocorrelation falls from 0868 to 0866, the elasticity of substitution falls from 65 to 54, the relative value of rest rises slightly from 097 to 098, and the search cost increases from 50 to 65. Thus even a substantial change in the incidence of rest unemployment does not much affect our evaluation of our model’s features and shortcomings. The key is that the incidence of endogenous search unemployment must not be too high. Third, our decision to focus on data at the five-digit level was driven in large part by availability issues, but there is no particular reason to think it is costless to move within five-digit industries and costly to move between them. If we defined an industry at a more disaggregated level or perhaps looked at a cross between an industry and a geographic area, we would expect to see more search unemployment, which would reduce the persistence of wages. Fortunately, this may be consistent with the data, since Table I suggests that wages are less persistent in more disaggregated data. Moreover, our approach to backing out the structural parameters would conclude that search costs are lower with a more disaggregated definition of an industry, since wages are less persistent. This too seems reasonable. In this sense, our model does not uniquely pin down the extent of search and rest unemployment, but rather points to the importance of modeling both in a common framework. Our second main quantitative finding, established in Section 6.3, is that for our model to match the empirical persistence and variance of industry wage growth, the search cost has to be very large. Introducing other mobility costs, such as industry-specific human capital, may alleviate this issue. We proved (Alvarez and Shimer (2009a)) an isomorphism between our directed search model and a model with industry-specific human capital. The key difference is quantitative: the time it takes to accumulate industry-specific human capital is likely much longer than the duration of search unemployment. This enables us to calibrate the model without large search costs. Understanding the full richness of the data may require moving beyond our dichotomous treatment of search and rest unemployment. Instead, many different levels of aggregation may be relevant in reality. Moving between “nearby” industries is likely to be cheap, but nearby industries may receive correlated shocks. Workers are therefore sometimes willing to pay a high cost, or give up much of their human capital, to move to distant industry. Such a model would lead to a more nuanced view of search intensity, with our notion of “search” and “rest” as two extremes, but would be less tractable and hence less comprehensible than our approach. Still, we believe that the basic issues we have identified in this paper are likely to be important in that more general framework. 7. CONCLUDING REMARKS This paper characterizes the equilibrium of the Lucas–Prescott search model with directed and random search. Our characterization features a separation
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between optimization and aggregation which keeps our analysis tractable. We illustrate the value of this tractability by examining the link between the incidence of search unemployment and the persistence of wages, and using this to calibrate key parameters. Of course, our model has many implications that we have not yet explored, for example, the link between wages and worker flows. Although this is the first paper to separate optimization from aggregation in the context of the Lucas and Prescott (1974) search model, others have fruitfully used the same idea in different economic environments. Caballero and Engel (1993) evaluated the magnitude and effect of labor adjustment costs by looking at the statistical behavior of employment at the firm level, assuming that in a frictionless model, employment would follow a random walk. Doms and Dunne (1998) performed a similar exercise for capital adjustment costs and irreversibilities. Cole and Rogerson (1999) examined whether the Mortensen and Pissarides (1994) model is consistent with the behavior of job creation and destruction at business cycle frequencies by using a reduced-form version of the original model. Hall (1995) similarly studied the propagation of one-time shocks in an elaboration of this model through a statistical representation of the labor market as a 19-state Markov process. In a companion paper (Alvarez and Shimer (2009b)), we develop a version of our model where unions can keep wages above the market-clearing level and allocate jobs based on seniority. While the optimization problem is more complicated, the same separation between optimization and aggregation applies, and the link between the incidence of search unemployment and wages is unchanged. We find that unions always lead to temporary layoffs, where union members wait for industry conditions to improve, a natural example of rest unemployment. We find that the hazard rate of reentering unemployed is decreasing in duration for union members who are rationed out of jobs. Interestingly, Katz and Meyer (1990) and Starr-McCluer (1993) found that the hazard rates of workers on temporary layoff decrease more strongly with unemployment duration, compared to other unemployed workers. APPENDIX A A.1. Density of Productivity x ˜ denote the steady state density of log productivity, x˜ ≡ log x, across Let fx (x) ˜ = −μx fx (x) ˜ + industries. This solves a Kolmogorov forward equation: δfx (x) σx2 ˜ at all x˜ = x˜ 0 ≡ log x0 . The solution to this equation takes the form f (x) 2 x 1 η˜ x˜ D1 e 1 + D12 eη˜ 2 x˜ if x˜ < log x0 , ˜ = fx (x) D21 eη˜ 1 x˜ + D22 eη˜ 2 x˜ if x˜ > log x0 where η˜ 1 < 0 < η˜ 2 are the two real roots of the characteristic equation (33)
δ = −μx η˜ +
σx2 2 η˜ 2
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For this to be a well defined density, integrating to 1 on (−∞ ∞), we require that D11 = D22 = 0. To pin down the remaining constants, we use two more conditions: the density is continuous at x˜ = log x0 and it integrates to 1. Imposing these boundary conditions delivers ⎧ η˜ 1 η˜ 2 η˜ 2 (x−log ⎪ ⎪ e ˜ x0 ) if x˜ < log x0 , ⎨ η˜ 1 − η˜ 2 ˜ = fx (x) (34) η˜ 1 η˜ 2 η˜ 1 (x−log ⎪ ⎪ ⎩ e ˜ x0 ) if x˜ > log x0 η˜ 1 − η˜ 2 With this notation, we obtain
1/(θ−1)
1 θ−1
x(j t)
(35)
dj
0
=
∞
(θ−1)x˜
e −∞
1/(θ−1) ˜ d x˜ fx (x)
The interior integral converges if η˜ 1 + θ − 1 < 0 < η˜ 2 + θ − 1. The definition of η˜ i in equation (33) implies these inequalities are equivalent to condition (6). With this restriction, we can then simplify equation (35) to obtain equation (7). A.2. Proof of Proposition 1 Throughout this proof we express the value of a worker v as a function not only on the current log wage ω, but also of the lower and upper bound on wages ω and ω, ¯ say v(ω; ω ω). ¯ We start with a preliminary result. ¯ ¯ ¯ It is strictly LEMMA 1: v is continuous and nondecreasing in ω, ω, and ω. increasing in each argument if ω ∈ (ω ω) ¯ and ω ¯ > log br .¯ ¯ PROOF: Define ∞ −(ρ+q+δ)t ¯ ≡E e Iω (ω(j t)) dt ω(j 0) = ω Π(ω ; ω; ω ω) ¯ 0
where Iω (ω(j t)) is an indicator function equal to 1 if ω(j t) < ω and equal to zero otherwise. This discounted occupancy function evaluates to zero at 1 at ω ≥ ω. ¯ We use Πω (ω ; ω; ω ω) ¯ for the density of ω ≤ ω and to ρ+q+δ ¯ ¯ ω or the discounted local time function, where the subscript denotes the partial derivative with respect to the first argument. Then switching the order of integration in equation (11), which is permissible since for −∞ ≤ ω ≤ ω ¯ < ∞ and ¯ we get ρ + q + δ > 0, the function max{br eω } + (q + δ)v is integrable, ¯ ω¯ (36) (max{br eω } + (q + δ)v)Πω (ω ; ω; ω ω) ¯ dω v(ω; ω ω) ¯ = ¯ ¯ ¯ ω ¯
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The value of being in an industry with current log full-employment wage ω is equal to the expected value of future ω weighted by the appropriate discounted local time function. Stokey (2009) proved in Proposition 10.4 that for all ω ∈ [ω ω], ¯ ¯ Πω (ω ; ω; ω ω) (37) ¯ ¯ ⎧ ) ) ¯ 2ω (ζ2 eζ1 ω+ζ2 ω¯ − ζ1 eζ1 ω+ζ )(ζ2 eζ2 (ω−ω − ζ1 eζ1 (ω−ω ) ⎪ ¯ ¯ ⎪ ⎪ ⎪ ζ1 ω+ζ2 ω ¯ ζ1 ω+ζ ¯ 2ω ⎪ (ρ + q + δ)(ζ2 − ζ1 )(e ¯ −e ¯) ⎪ ⎪ ⎨ if ω ≤ ω < ω = ¯ ) ) ¯ ¯ ⎪ 2 ω )(ζ eζ2 (ω−ω (ζ2 eζ1 ω+ζ2 ω¯ − ζ1 eζ1 ω+ζ − ζ1 eζ1 (ω−ω ) ⎪ 2 ¯ ⎪ ⎪ ⎪ ζ1 ω+ζ2 ω ¯ ζ1 ω+ζ ¯ 2ω ⎪ (ρ + q + δ)(ζ2 − ζ1 )(e ¯ −e ⎪ ¯) ⎩ if ω ≤ ω ≤ ω ¯ where ζ1 < 0 < ζ2 are the two roots of the characteristic equation (38)
ρ + q + δ = μζ +
σ2 2 ζ 2
For ω < ω, Πω (ω ; ω; ω ω) ¯ = Πω (ω ; ω; ω ω) ¯ and for ω > ω, ¯ Πω (ω ; ω; ω ¯ ¯ ¯ ¯ ¯ ω) ¯ = Πω (ω ; ω; ¯ ω ω). ¯ That v is continuous follows immediately from equa¯ tions (36) and (37). In particular, the latter equation defines Πω as a continuous function. We next prove that the distribution Π(·; ω; ω ω) ¯ is increasing in each of ω, ¯ ω, and ω ¯ in the sense of first order stochastic dominance. This follows from dif¯ ferentiating equation (37) with respect to each variable and using simple algebra. One can verify that an increase in ω strictly increases Πω (ω ; ω; ω ω) ¯ for ¯ all ω ∈ (ω ω). ¯ This therefore strictly reduces Π(ω ; ω; ω ω) ¯ for ω ∈¯ (ω ω). ¯ ¯¯ for all ω ∈ (ω ¯ ω), Similarly,¯an increase in ω ¯ strictly reduces Πω (ω ; ω; ω ω) ¯ ¯ ¯ for ω ∈ (ω¯ ω). ¯ Finally, an increase which also strictly reduces Π(ω ; ω; ω ω) ¯ ; ω; ω ω) in ω when ω ∈ (ω ω) ¯ reduces Πω (ω ¯ for¯ ω ∈ (ω ω) and raises it ¯ ¯ ¯ for ω ∈ (ω ω). ¯ Once again, this implies a stochastic dominating shift in Π. ω Since the return function max{br e }+(q+δ)v is nondecreasing in ω , weak ¯ monotonicity of v in each argument follows immediately from equation (36). In addition, the return function is strictly increasing when ω > log br , and so we obtain strict monotonicity when the support of the integral includes some ω > log br , that is, when ω ¯ > log br . Q.E.D. Our approach to solving for v may be unfamiliar to some readers, and so it is worth stressing that equations (12) and (36) imply some familiar conditions: (39)
(ρ + q + δ)v(ω; ω ω) ¯ = max{br eω } + (q + δ)v + μvω (ω; ω ω) ¯ ¯ ¯ ¯ σ2 ¯ + vωω (ω; ω ω) 2 ¯
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FIGURE 1.—Illustration of the proof of Proposition 1 when br < bi .
and (40)
¯ ω ω) ¯ = vω (ω; ω ω) ¯ = 0 vω (ω; ¯ ¯ ¯
where subscripts denote partial derivatives with respect to the first argument.18 Together with the “value-matching” conditions v(ω; ¯ ω ω) ¯ = v¯ and ¯ v(ω; ω ω) ¯ = v in equation (12), this is an equivalent representation of the ¯ ¯force participant’s ¯ labor value function. We now return to the proof of Proposition 1. We start by proving the result when br < bi and defer br ≥ bi until the end. ¯ When ω is regulated at the point First, define ω ¯ ∗ to solve v(ω ¯ ∗; ω ¯ ∗ ω ¯ ∗ ) = v. ∗
eω¯ +(q+δ)v
¯ This point is deω ¯ ∗ , it is trivial to solve equation (11) to obtain ρ+q+δ ¯ = v. ◦ picted along the 45 line in Figure 1. Lemma 1 ensures v is continuous and strictly increasing in its first three arguments. Moreover, for any ω < ω ¯ ∗ , we ¯ can make v(ω; ¯ ω ω) ¯ unboundedly large by increasing ω, ¯ while we can make it smaller than v¯¯ by setting ω ¯ =ω ¯ ∗ . Then by the intermediate value theorem, ¯ ω) > ω ¯ ω); ω Ω( ¯ ω)) ≡ v. ¯ Con¯ ∗ solving v(Ω( for any ω < ω ¯ ∗ , there exists a Ω( ¯ ¯ ¯ ¯ ¯ ¯ tinuity of v ensures Ω is continuous, while monotonicity of v ensures it is decreasing. In addition, because the period return function s(ω) ≡ max{br eω } + 18 The first condition, the Hamilton–Jacobi–Bellman equation, can be verified directly by differentiating equation (36) using the definition of Πω in equation (37). The interested reader can consult online Appendix B.1 in the Supplemental Material (Alvarez and Shimer (2011)) for the details of the algebra. The second pair of conditions, “smooth pasting,” follows from equa∂Π (ω ;ω;ωω) ¯ tion (36) because equation (37) implies ω ∂ω ¯ = 0 when ω = ω or ω = ω. ¯ ¯
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¯ ω) is finite. Thus (q + δ)v is bounded below but not above, ω ¯ ∗∗ ≡ limω→−∞ Ω( ¯ ¯ ¯ ∗ ∗∗ ∗ ¯ ω) ∈ (ω Ω( ¯ ω ¯ ) for any ω < ω ¯ . Figure 1 illustrates this function. ∗ ∗ ∗ ∗ ¯ ¯ Similarly, define ω to solve v(ω ; ω ω ) = v. Again solve equation (11) ∗ ¯ ¯ ¯∗ ¯ eω +(q+δ)v ¯ ¯ then ω < ω ¯ ∗ , while equation (8) implies to obtain ¯ ρ+q+δ ¯ = v. Since v < v, ¯ ¯ ¯ r +(q+δ)bi ω∗ = log bi . For any ω ¯ > ω∗ , we can make v(ω; ω ω) ¯ approach ρbρ(ρ+q+δ)
ω , there exists ω = ω . Then by the intermediate value theorem, for any ω ¯¯ < ω∗ that solves v(Ω(ω); ¯ Ω(ω) ¯ ω) ¯ ≡ v. Continuity of ¯v ensures Ω is a¯ Ω(ω) ¯ of ¯v ensures it ¯is decreasing. Thus Ω(ω) ¯ ¯ while monotonicity continuous, ¯ <¯ ω∗ ∗ ¯ ¯ for any ω ¯ >ω . ¯ An equilibrium is simply a fixed point ω ¯ of the composition of the functions ¯ ∗∗ ] Ω¯ ◦ Ω. The preceding argument implies that this composition maps [ω ¯ ∗ ω ¯ into itself and is continuous, and hence has a fixed point. To prove the uniqueness of the fixed point when br < bi , we prove that the composition of the two functions has a slope less than 1, that is, Ω¯ (Ω(ω)) ¯ × ¯ < 1. To start, simple transformations of equation (37) imply ¯that the Ω (ω) ¯ cross partial derivatives of the discounted occupancy function satisfy Πωω¯ (ω ; ω; ω ω) ¯ ¯ −ζ2 (ω −ω) ζ1 ω+ζ2 ω − eζ1 ω+ζ2 ω¯ ) ζ1 ζ2 e(ζ1 +ζ2 )ω¯ (e−ζ1 (ω −ω) ¯ −e ¯ )(e ¯ = ¯ ¯ 2ω 2 2ω (eζ1 ω+ζ − eζ1 ω+ζ ¯ ¯ ) (ρ + q + δ) < 0 ¯ Πωω (ω ; ω; ω ω) ¯ ¯ ) ) ¯ ¯ ¯ 2ω − eζ1 (ω−ω )(eζ1 ω+ζ2 ω¯ − eζ1 ω+ζ ) −ζ1 ζ2 e(ζ1 +ζ2 )ω¯ (eζ2 (ω−ω = ζ1 ω+ζ2 ω ¯ ζ1 ω+ζ ¯ 2ω 2 (e ¯ −e ¯ ) (ρ + q + δ) > 0 where the inequalities use the fact that all the terms in parentheses are positive. Then use integration-by-parts on equation (36) to write ω¯ s(ω) ¯ − s (ω )Π(ω ; ω; ω ω) ¯ dω v(ω; ω ω) ¯ = ρ+q+δ ¯ ¯ ω ¯
where the period return function s(ω ) is nondecreasing and strictly increasing for ω > log br , and Π is the discounted occupancy function. Taking the cross ¯ > 0 > vωω (ω; ω ω). ¯ partial derivatives of this expression gives vωω¯ (ω; ω ω) ¯ ω). ¯ ω ω) ¯ > vω¯ (ω; ω ω) ¯ and vω¯(ω; ω ω) ¯ > vω¯ (ω; ¯ ω ¯ In particular, vω¯ (ω; ¯ ¯ ¯ ¯ 1, these inequalities ¯ ¯ ¯ ¯ > 0 from Lemma imply¯ Now since vω (ω; ω ω) ¯ ¯ ω ω) ¯ vω (ω; vω¯ (ω; ω ω) ¯ ¯ < 1 ¯ ¯ ¯ vω (ω; ¯ ω ω) ¯ + vω¯ (ω; ¯ ω ω) ¯ vω (ω; ω ω) ¯ + vω (ω; ω ω) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
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¯ ω) In particular, this is true when evaluated at any point {ω ω}, ¯ where ω ¯ = Ω( ¯ ¯ and ω = Ω(ω). ¯ Implicit differentiation of the definitions of these functions ¯ ¯ ¯ shows that the first term in the above inequality is −Ω (ω) and the second ¯ term is −Ω (ω), ¯ which proves Ω¯ (Ω(ω)) ¯ Ω (ω) ¯ < 1. ω ¯ ¯ ¯ ¯ Next we prove proportionality of the thresholds e and eω¯ to the leisure values br , bi , and bs . From equations (8) and (9), v and v¯ are homogeneous of degree 1 in the three leisure values. The function¯ max{br eω } + (q + δ)v is also homogeneous of degree 1 in the leisure values and eω . By inspection¯ of equation (37), Πω is unaffected by an equal absolute increase in each of its arguments. Then the integral in equation (36) is homogeneous of degree 1 in the b’s, eω¯ , and eω¯ . The result follows from equation (12). Finally we consider br ≥ bi , so the period return function s(ω) ≥ bi +(q+δ)v for all ω. This implies v(ω; −∞ ω) ¯ ≥ v for all ω and ω. ¯ Then an equilibrium is¯ ¯ ¯ As discussed above, the solution of this equation defined by v(ω; ¯ −∞ ω) ¯ = v. is ω ¯ ∗∗ ∈ (ω ¯ ∗ ∞). A.3. Proof of Proposition 2 First, set br = 0. By Proposition 1, there exists a unique equilibrium char¯ 0 . We now prove that b¯ r ≡ eω¯ 0 . To see why, acterized by thresholds ω0 and ω ¯ ¯ observe that for all br ≤ br , the equations that characterize equilibrium are unchanged from the case of br = 0 because log br ≤ ω0 , and hence the equilibrium ¯ is unchanged. Conversely, for all br > b¯ r , the equations that characterize equilibrium necessarily are changed, and so the equilibrium must have log br > ωbr . ¯ Next we prove that b¯ r /bi = B(κ ρ + q + δ μ σ). Again with br = 0, combine equations (12) and (36), noting the discounted local time function Πω 1 integrates to ρ+q+δ , and use the definitions of v and v¯ in equations (8) and (9): ¯ ω¯ 0 bi = eω Πω (ω; ω0 ; ω0 ω ¯ 0 ) dω ρ+q+δ ¯ ¯ ω0 ¯
and
bi
ω¯ 0 1 +κ = eω Πω (ω; ω ¯ 0 ; ω0 ω ¯ 0 ) dω ρ+q+δ ¯ ω0 ¯
Since Πω is homogeneous of degree 0 in the exponentials of its arguments (see equation (37)), this implies eω¯ 0 and eω¯ 0 are homogeneous of degree 1 in bi . Moreover, ζi depends on ρ + q + δ, μ, and σ by equation (38), and so the density Πω in equation (37) depends on these same parameters. It follows that the solution to these equations can depend only on these parameters and the parameters on the left hand side of the above equations. In particular, this proves eω¯ 0 = bi B(κ ρ + q + δ μ σ). Since b¯ r = eω¯ 0 , the dependence of b¯ r on this limited set of parameters is established.
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Obviously B is positive-valued. By Proposition 1, ω0 < log bi and so B < 1. ¯ We finally prove it is decreasing in κ. Since κ affects ¯ω and ω ¯ only through v, ¯ that the ω and ω to establish that B is decreasing in κ, it suffices to show ¯ that ¯ ¯ ω) ¯ This follows because solve equations (12) and (36) are decreasing in v. Ω( ¯ is increasing in v¯ and Ω(ω) ¯ is unaffected, where these functions are defined ¯ in the proof of Proposition 1. A decrease in v¯ then reduces the composition Ω¯ ◦ Ω. Since the slope of this function is less than 1, it reduces the location of ¯ the fixed point ω ¯ and hence raises ω = Ω(ω). ¯ ¯ ¯ A.4. Proof of Proposition 3 We start by characterizing the stationary distribution of workers across log full-employment wages ω, f , defined on [ω ω]. ¯ Since f is a density, ¯ ω¯ (41) f (ω) dω = 1 ω ¯
By taking the limit of a discrete time, discrete state-space analog of our model, we prove in Appendix A.5 that this density has to satisfy three conditions, equations (42)–(44) below. First, in the interior of its support, the density must solve a Kolmogorov forward equation (42)
(q + δ)f (ω) = −μf (ω) +
σ 2 f (ω) for all ω ∈ (ω ω) ¯ 2 ¯
This captures the requirement that inflows and outflows balance at each point in the support of the density. Workers exit industries because of either quits or shutdowns at rate q + δ, while otherwise ω is a Brownian motion with drift μ and standard deviation σ. Workers whose ω changes leave this point in the density for higher or lower values of ω, while the density picks up mass from points above and below when they are hit by appropriate shocks. In continuous time, this relates the level of f and its derivatives. Second, at the lower bound ω, ¯ σ2 θσ 2 (43) f (ω) − μ + f (ω) = 0 2 2 ¯ ¯ The elasticity of substitution θ appears in this equation because it determines how many workers must exit from depressed industries to regulate ω above ω. ¯ The exogenous separation rate q + δ does not appear in this equation because the ratio of endogenous to exogenous exits is infinite in a short time interval for an industry at the lower bound. Since by definition there are no industries with smaller ω, f (ω) is not fed from below, which explains the difference be¯ and (43). The second order differential equation (42) tween equations (42)
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and the boundary conditions equations (41) and (43) yield equation (14) using standard calculations. We now turn to output, consumption (both of which are equal by the final goods market clearing condition), and employment. Appendix A.5 also establishes that at the upper bound ω, ¯ σ2 L0 θσ 2 (44) f (ω) f (ω) ¯ =δ ¯ − μ+ 2 2 L where L0 is the (endogenous) average number of workers in a new industry. The logic for the left hand side of this equation parallels the logic behind equation (43). There is an extra inflow at ω ¯ coming from newly formed industries, which absorb δL0 workers per unit of time; dividing by L expresses this as a percentage of the workers located in industries. Next, equation (5) implies that the number of workers in a new industry is (45)
L0 = Y u (Y )θ x0θ−1 e−θω¯
which ensure a log full-employment wage ω. ¯ Our last condition relates intermediate and final goods output. Define the productivity of a location x consistent with l workers present in the location, a log full-employment wage ω, and aggregate output and consumption Y . From equation (5), this solves (46)
x = ξ(l ω Y ) ≡
leθω Y u (Y )θ
1/(θ−1)
Use this to compute output in an industry with l workers and log fullemployment wage ω, recognizing that there may be rest unemployment: (47)
Q(l ξ(l ω Y )) = Y
−1/(θ−1)
eω l u (Y )
θ/(θ−1)
eω min 1 br
θ
Using this notation, we can write equation (2) as (48)
1
Y=
Q l(j t) ξ(l(j t) ω(j t) Y )
(θ−1)/θ
θ/(θ−1) dj
0
=
0
1
(θ−1)/θ l(j t) dj Q L ξ(L ω(j t) Y ) L
θ/(θ−1)
Note that these equations have to hold for all t in steady state, so the choice of t is arbitrary. The second equation follows because Q(· ξ(· ω Y ))(θ−1)/θ is linear (equation (47)). To solve this, we change the variable of integration from the name of the industry j to its log full-employment wage ω and number of
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workers l. Let f˜(ω l) be the ergodic density of the joint distribution of workers in industries (ω l). The joint distribution of (ω l) for an individual worker is a strongly convergent Markov process whenever industries shut down at a positive rate, δ > 0, which ensures that f˜ is unique. Without characterizing the dis ω¯ ∞ tribution explicitly, we have Y = ω 0 Q(L ξ(L ω Y ))(θ−1)/θ Ll f˜(ω l) dl dω. ∞ l ¯ solve the inner integral to obtain Since f (ω) = 0 L f˜(ω l) dl, we can (49)
Y=
θ/(θ−1)
ω ¯ (θ−1)/θ
Q(L ξ(L ω Y ))
f (ω) dω
ω ¯
This depends on the known density f rather than the more complicated density f˜. Finally, we solve equations (44), (45), and (49) for L0 , L, and Y . First eliminate L0 between equations (44) and (45), and evaluate f (ω) ¯ using equation (14) to get (50)
L=
−2δY u (Y )θ x0θ−1 e−θω¯ ¯ ω) η1 (ω− ¯ ω) σ 2 (θ + η1 )(θ + η2 )(eη2 (ω− ¯ −e ¯ ) ×
2
|θ + ηi |
i=1
¯ ω) eηi (ω− ¯ −1 ηi
Substitute equation (47) into equation (49) to get (51)
L Y= u (Y ) =
ω ¯
eω min{1 eω /br }θ−1 f (ω) dω
ω ¯ 2
Leω¯ ¯ ω) ˆ |θ + ηi |e−(ω− u (Y ) i=1 ηi (ω− −θ(ω− ˆ ω) (ω− ¯ ω) ˆ ¯ ω) ˆ − e−ηi (ω− e ˆ ω) ¯ −e ¯ ηi (ω− ¯ ω) e × +e ¯ θ + ηi 1 + ηi 2 ¯ ω) eηi (ω− ¯ −1 |θ + ηi | ηi i=1
where we solve the integral using the expression for f in equation (14) and define ω ˆ ≡ max{ω log br }. Eliminating L between these equations and solving for u (Y ) gives an¯ implicit equation for output: (52)
u (Y )1−θ =
¯ ω) ˆ −2δx0θ−1 e−(θω− ¯ ω) η1 (ω− ¯ ω) σ 2 (θ + η1 )(θ + η2 )(eη2 (ω− ¯ −e ¯ )
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2
115
|θ + ηi |
i=1
ˆ ω) −θ(ω− ˆ ω) (ω− ¯ ω) ˆ ¯ ω) ˆ − e−ηi (ω− eηi (ω− ¯ −e ¯ ηi (ω− ¯ ω) e +e × ¯ θ + ηi 1 + ηi
Finally, equation (51) determines the number of workers in labor markets, L. A.5. Derivation of the Density f We use a discrete time, discrete state-space model to obtain the Kolmogorov forward equations and boundary conditions for the density f . Divide [ω ω] ¯ into n intervals of length ω = (ω ¯ − ω)/n. Let the time period be ¯t = (ω/σ)2 and assume that when ω < ω, ¯ it¯decreases with probability 12 (1 + p), 2 where p = μω/σ ; when ω > ω, it increases with probability 12 (1 − p); and otherwise ω stays constant. Note ¯that for ω < ω(t) < ω, ¯ the expected value of ¯ ω(t + t) − ω(t) is μt and the second moment is σ 2 t. As n goes to infinity, this converges to a regulated Brownian motion with drift μ and standard deviation σ. Now let fn (ω t) denote the fraction of workers in industries with log fullemployment wage ω at time t for fixed n. With a slight abuse of notation, let fn (ω) be the stationary distribution. We are interested in characterizing the (ω) . For ω ∈ [ω + ω ω ¯ − ω], the dynamics of ω density f (ω) = limn→∞ fnω ¯ imply (53)
fn (ω t + t) = (1 − (q + δ)t) × 12 (1 + p)fn (ω − ω t) + 12 (1 − p)fn (ω + ω t)
In any period of length t, a fraction (q + δ)t of workers leave due to industry shutdowns and idiosyncratic quits. Thus the workers in industries with ω at t + t are a fraction 1 − (q + δ)t of those who were in industries at ω − ω at t and had a positive shock, plus the same fraction of those who were in industries at ω + ω at t and had a negative shock. Now impose stationarity on fn . Take a second order approximation to fn (ω + ω) and fn (ω − ω) around ω, substituting t and p by the expressions above: ω2 ω2 ω2 fn (ω) = 1 − (q + δ) 2 fn (ω) − μ 2 fn (ω) + f (ω) σ σ 2 n ω2 σ 2 −μfn (ω) + fn (ω) ⇒ (q + δ)fn (ω) = 1 − (q + δ) 2 σ 2
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(ω) Taking the limit as n converges to infinity, fnω → f (ω), solving equation (42). Now consider the behavior of fn at the lower threshold ω. A similar logic ¯ implies
fn (ω t + t) = (1 − (q + δ)t) 12 (1 − p) ¯ ˜ × (fn (ω + ω t) + fn (ω t)(1 − l)) ¯ ¯ The workers at ω at t + t either were at ω + ω or at ω at t; in both cases, ¯ ¯ ¯ they had a negative shock. Moreover, in the latter case, a fraction l˜ ≡ θω of the workers exited the industry to keep ω above ω. Again impose station¯ + ω) at ω; the higher arity, but now take a first order approximation to fn (ω ¯ ¯ order terms will drop out later in any case. Replacing t, p, and l˜ with the expressions described above gives μω ω2 1− fn (ω) = 1 − (q + δ) 2 σ σ2 ¯ ω θω × fn (ω) 1 − f (ω) + 2 2 n ¯ ¯ Again eliminating terms in fn (ω) and taking the limit as n → ∞, we obtain fn (ω) ¯ ¯ → f (ω), solving equation (43). ω ¯ ¯ Now consider the behavior of fn at the upper threshold ω: ¯ t + t) fn (ω ˜ ¯ − ω t) + fn (ω ¯ t)(1 + l)) = (1 − (q + δ)t) 12 (1 + p)(fn (ω + δtL0 /L Compared to the equation at the lower threshold, the only significant change is the last term, which reflects the fact that on average a fraction L0 /L of workers enter at the upper threshold when a new industry is created. Recall also that industries are destroyed at rate δ per unit of time and hence δtL0 /L is the fraction of workers added to the upper threshold due to newly created industries. Impose stationarity and take limits to get μω ω2 fn (ω) 1+ ¯ = 1 − (q + δ) 2 σ σ2 ω θω ω2 L0 − fn (ω) × fn (ω) ¯ 1+ ¯ +δ 2 2 2 σ L Eliminate terms in fn (ω) ¯ and take the limit as n → ∞ to obtain solving equation (44).
fn (ω) ¯ ω
→ f (ω), ¯
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A.6. Exit Rates From Industries A worker exits her industry if the log full-employment wage is ω and the ¯ In the industry is hit by an adverse shock, if the industry closes, or if she quits. discrete time, discrete state-space model, the first event hits a fraction 12 (1 − p)l˜ of the workers who survive in an industry with ω = ω, so Ns t ≡ (1 − (q + ¯˜ ˜ n (ω)L + (q + δ)tL. Reexpress ω, l, and p in terms of δ)t) 12 (1 − p)lf ¯ fn (ω) t, take the limit as n → ∞, and use ω → f (ω), to get equation (16). A.7. Density f : Random Search Since f is a density, ∞ (54) f (ω) dω = 1 ω ¯
By taking the limit of a discrete time, discrete state-space analog of our model, we find that this density has to satisfy three additional conditions. First, at ω > ω, it must solve a Kolmogorov forward equation, unchanged from equa¯ except for the effect of random search: tion (42) (55)
(q + δ − s)f (ω) = −μf (ω) +
σ 2 f (ω) for all ω > ω 2 ¯
Second, at the lower bound ω, we have the same condition as in the directed ¯ search case, equation (43). Finally, in contrast to the directed search case, there is a kink in f at the point where new industries are created, (56)
f− (ω) ¯ − f+ (ω) ¯ =
2δL¯ σ 2L
where L¯ is the average number of workers in a new industry. The derivation of this condition, which reflects the addition of directed searchers into new industries, is standard and hence is omitted. Solving these equations and emphasizing the dependence of f on the boundary ω and the arrival rate of new ¯ workers s, we obtain equation (24). A.8. Proof of Proposition 5 Express the value of a worker v as a function not only of the current log wage ω, but also of the lower bound on wages ω and the arrival rate of search ¯ unemployed workers s, v(ω; ω s). We start by proving a preliminary result. ¯ LEMMA 2: v is continuous and nondecreasing in ω, and continuous and non¯ if ω > ω. increasing in s. It is strictly monotone in each argument ¯
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PROOF: As in the proof of Lemma 1, we can write the value function as ∞ v(ω; ω s) = (57) (max{br eω } + (q + δ)v)Πω (ω ; ω; ω s) dω ¯ ¯ ¯ ω ¯
where again Πω is the discounted local time function. Taking limits of equation (37), we obtain ⎧ ) ζ2 (ω−ω ) ζ2 eζ1 (ω−ω) − ζ1 eζ1 (ω−ω ) ⎪ ¯ (ζ2 e ¯ ¯ ⎪ ⎪ ⎪ ⎪ (ρ + q + δ)(ζ − ζ ) 2 1 ⎪ ⎪ ⎨ if ω ≤ ω < ω Πω (ω ; ω; ω s) = (58) ¯ ) ζ2 (ω−ω) ⎪ ¯ ζ2 eζ2 (ω−ω (ζ2 eζ1 (ω−ω) ⎪ ¯ ¯ − ζ1 e ¯ ) ⎪ ⎪ ⎪ ⎪ (ρ + q + δ)(ζ2 − ζ1 ) ⎪ ⎩ if ω ≤ ω ≤ ∞ where ζ1 < 0 < ζ2 are the two roots of the characteristic equation (59)
ρ + q + δ = μ(s)ζ +
σ2 2 ζ 2
where μ(s) is a decreasing function from equation (21). The remainder of the proof follows the structure of the proof of Lemma 1 and so is omitted. Q.E.D. Now rewrite equations (20) and (22) to define two implicit functions of s, (60) (61)
v(ω1 (s); ω1 (s) s) = v ¯ ¯ ¯ ∞ ¯ v(ω; ω2 (s) s)f (ω; ω2 (s) s) dω = v ¯ ¯ ω2 (s) ¯
where our notation also emphasizes the dependence of the cross-sectional density of ω on ω and s. An equilibrium is a solution s to ¯ (62) ω1 (s) = ω2 (s) ¯ ¯ Lemma 2 and equation (30) imply that these are continuous and increasing functions of s. We then argue that (63) ω2 q + 12 θσ 2 = −∞ < ω1 q + 12 θσ 2 and lim ω2 (s) > lim ω1 (s) s→∞ ¯ s→∞ ¯ ¯ ¯ If s ≤ q + 12 θσ 2 , the integral on the left hand side of equation (61) does not converge for all ω2 (s), while as s → q + 12 θσ 2 , ω2 → −∞. On the other hand, ¯ s. At the other end of the ¯ range, as s → ∞, the denω1 is finite for any ¯ sity f puts almost all its weight at ω. This implies that the left hand side of ¯ ω2 (∞) ∞). Since v¯ > v, we must have equation (61) converges to v(ω2 (∞); ¯ ¯ ¯
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ω2 (∞) > ω1 (∞). The existence of ω solving equation (62) then follows from ¯ ¯ ¯ intermediate the value theorem. For the case where br = 0, so there is no rest unemployment, and q = 0, equations (60) and (61) simplify further. In this case, the left hand sides of both equations are proportional to eω¯ , hence taking the ratio gives ρ η2 (s)(1 + ζ1 (s) + η2 (s)) v¯ ≡1+ κ= (1 + η2 (s))(ζ1 (s) + η2 (s)) v v ¯ ¯ where the notation emphasizes that the roots ηi for f and ζi given by equation (59) are functions of μ, which depends on s as in equation (21). In a Mathematica file available on request, we prove that the right hand side of equation (64) converges to ∞ as s goes to its lower bound and it converges to 1 as s goes to ∞. Moreover, a lengthy algebraic argument shows that this ¯ v ≥ 1, there is a unique soratio is decreasing in s everywhere. Thus, since v/ ¯ lution s∗ , and hence a unique equilibrium. It is immediate that the equilibrium ∗ value of s is decreasing in the search cost κ, and since ω1 is increasing, the ¯ value of ω decreases too. ¯ (64)
A.9. Maximum Likelihood Estimator We estimate equation (32) using exact maximum likelihood. For each industry j, we assume the first observation is drawn from the ergodic distribution, that is, log wj0 is normally distributed with mean log wj· and variance σw2 /(1 − β2w ). Subsequently, for t = {0 1 T − 1}, log wjt+1 is normally distributed with mean βw log wjt + (1 − βw ) log wj· and variance σw2 , where, in our data, T = 18. The log likelihood function is, therefore, J T 1 (log wjt − βw log wjt−1 − (1 − βw ) log wj· )2 − 2 2σw j=1 t=1 2 2 + (1 − βw )(log wj0 − log wj· )
1 1 + J log(1 − β2w ) − J(T + 1) log σw − J(T + 1) log(2π) 2 2 Note that OLS treats wj0 as exogenous, whereas maximum likelihood recognizes that it is drawn from the ergodic distribution and so contains information. It is straightforward to maximize the likelihood function numerically. We start with an initial guess for βˆ w and use that to compute the industry fixed effects wˆ j· using each of those first order conditions. We then compute the standard deviation of the innovations σˆ w from its first order condition. Finally, we use the first order condition for βˆ w to update our initial guess and iterate
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until convergence. The last step of the algorithm involves a cubic equation in βˆ w , but it always has a unique solution on the interval (−1 1). A.10. Proof of Proposition 6 We start with the random search model. Wages are a Brownian motion dω(j t) = μ dt + σ dz(j t), regulated on [ω ∞). Using equation (21), μx = ¯ Equation (27) implies that un− 12 (θ − 1)σx2 implies μ = − 12 θσ 2 + (q − s)/θ. employed workers enter industries at rate s = αUs /L. Now define ω(j ˜ t) = (ω(j t) − ω)/σ. The stochastic process for ω ˜ is ¯ αUs /L − q 1 dt + dz(j t) d ω(j ˜ t) = − θσ + 2 θσ for ω(j ˜ t) ≥ 0. This depends only on θσ, αUs /L, and q. Next, equation (31) implies (ω ˆ − ω)/σ depends only on Ur /L and the same three reduced-form ¯ parameters. This implies that the stochastic process for the maximum of (ω(j t) − ω)/σ and (ω ˆ − ω)/σ depends on those four parameters, and so ¯ ¯ ˆ must estimates of βw and σˆ w using this type of data. Since this is a linear transformation of max{ω(j t) ω}, ˆ it follows that βˆ w and σˆ w /σ must also depend only on those four parameters. Note that the theoretical infinite sample correlation between ω(j ˜ t) and ω(j ˜ t + 1) decreases when the drift in ω(j ˜ t) becomes more negative, since ω ˜ spends more time bouncing off its lower bound 0. Since the drift is maxi mized at θσ = 2(αUs /L − q), it follows that the infinite-sample, non-timeaggregated, no rest-unemployment estimate of βˆ w is maximized at this value of θσ for given αUs /L − q. In the directed search model, wages are a Brownian motion dω(j t) = μ dt + σ dz(j t), regulated on [ω ω]. ¯ Using equation (13), μx = − 12 (θ − 1)σx2 ¯ 1 2 implies μ = − 2 θσ + q/θ. Again let ω(j ˜ t) = (ω(j t) − ω)/σ. This has the ¯ ˆ same autoregressive coefficient βw as ω, but satisfies a simpler law of motion, q 1 dt + dz(j t) d ω(j ˜ t) = − θσ + 2 θσ for ω(j ˜ t) ∈ [0 (ω ¯ − ω)/σ]. Invert equation (18) to express this interval as ¯ θσ qL ω(j ˜ t) ∈ 0 − log 1 − 2q αUs Once again, this depends only on θσ, αUs /L, and q. The remainder of the proof is unchanged, except that there is no longer a closed-form solution for the value of θσ that maximizes βˆ w . When θσ is close to zero, ω ˜ lies in a small
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interval and so is nearly serially uncorrelated. When θσ is large, ω ˜ has a strong negative drift, which eliminates the serial correlation as in the random search model. REFERENCES ABEL, A. B., AND J. C. EBERLY (1996): “Optimal Investment With Costly Reversibility,” Review of Economic Studies, 63 (4), 581–593. [88] ALVAREZ, F., AND R. SHIMER (2009a): “Unemployment and Human Capital,” Mimeo, University of Chicago. [78,105] (2009b): “Unions and Unemployment,” Mimeo, University of Chicago. [78,79,106] (2011): “Supplement to ‘Search and Rest Unemployment’,” Econometrica Supplemental Material, 79, http://www.econometricsociety.org/ecta/Supmat/7686_extensions.pdf. [88,98,109] ALVAREZ, F., AND M. VERACIERTO (1999): “Labor-Market Policies in an Equilibrium Search Model,” NBER Macroeconomics Annual, 14, 265–304. [79,102] (2001): “Severance Payments in an Economy With Frictions,” Journal of Monetary Economics, 47 (3), 477–498. [102] ATKESON, A., A. KHAN, AND L. OHANIAN (1996): “Are Data on Industry Evolution and Gross Job Turnover Relevant for Macroeconomics?” Carnegie–Rochester Conference Series on Public Policy, 44, 216–250. [79] BENTOLILA, S., AND G. BERTOLA (1990): “Firing Costs and Labour Demand: How Bad Is Eurosclerosis?” Review of Economic Studies, 47 (3), 381–402. [88] BLANCHARD, O. J., AND L. F. KATZ (1992): “Regional Evolutions,” Brookings Papers on Economic Activity, 23 (1992-1), 1–76. [99] BRODA, C., AND D. E. WEINSTEIN (2006): “Globalization and the Gains From Variety,” Quarterly Journal of Economics, 121 (2), 541–585. [102] BURDETT, K., AND T. VISHWANATH (1988): “Balanced Matching and Labor Market Equilibrium,” Journal of Political Economy, 96 (5), 1048–1065. [79,93] BURNSIDE, C., M. EICHENBAUM, AND S. REBELO (1993): “Labor Hoarding and the Business Cycle,” Journal of Political Economy, 101 (2), 245–273. [80] CABALLERO, R. J., AND E. M. R. A. ENGEL (1993): “Microeconomic Adjustment Hazards and Aggregate Dynamics,” Quarterly Journal of Economics, 108 (2), 359–383. [106] (1999): “Explaining Investment Dynamics in US Manufacturing: A Generalized (S s) Approach,” Econometrica, 67 (4), 783–826. [88] COLE, H. L., AND R. ROGERSON (1999): “Can the Mortensen–Pissarides Matching Model Match the Business-Cycle Facts?” International Economic Review, 40 (4), 933–959. [106] DOMS, M. E., AND T. DUNNE (1998): “Capital Adjustment Patterns in Manufacturing Plants,” Review of Economic Dynamics, 1 (2), 409–429. [106] GOUGE, R., AND I. KING (1997): “A Competitive Theory of Employment Dynamics,” Review of Economic Studies, 64 (1), 1–22. [80] HALL, R. (1995): “Lost Jobs,” Brookings Papers on Economic Activity, 1, 221–273. [106] HAMILTON, J. D. (1988): “A Neoclassical Model of Unemployment and the Business Cycle,” Journal of Political Economy, 96 (3), 593–617. [80] HORNING, B. C. (1994): “Labor Hoarding and the Business Cycle,” International Economic Review, 35 (1), 87–100. [80] HORNSTEIN, A., P. KRUSELL, AND G. L. VIOLANTE (2009): “Frictional Wage Dispersion in Search Models: A Quantitative Assessment,” Mimeo. [78,103] JOVANOVIC, B. (1987): “Work, Rest, and Search: Unemployment, Turnover, and the Cycle,” Journal of Labor Economics, 5 (2), 131–148. [79,80] KAMBOUROV, G., AND I. MANOVSKII (2007): “Occupational Specificity of Human Capital,” Mimeo. [102,104]
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KATZ, L. F., AND B. D. MEYER (1990): “Unemployment Insurance, Recall Expectations, and Unemployment Outcomes,” Quarterly Journal of Economics, 105 (4), 973–1002. [106] KING, I. P. (1990): “A Natural Rate Model of Frictional and Long-Term Unemployment,” Canadian Journal of Economics, 23 (3), 523–545. [80] LEAHY, J. V. (1993): “Investment in Competitive Equilibrium: The Optimality of Myopic Behavior,” Quarterly Journal of Economics, 108 (4), 1105–1133. [88] LOUNGANI, P., AND R. ROGERSON (1989): “Cyclical Fluctuations and Sectoral Reallocation: Evidence From the PSID,” Journal of Monetary Economics, 23 (2), 259–273. [79,104] LUCAS, R. E. JR., AND E. C. PRESCOTT (1974): “Equilibrium Search and Unemployment,” Journal of Economic Theory, 7, 188–209. [75,76,79,80,88,91,100,102,106] MCCALL, J. (1970): “Economics of Information and Job Search,” Quarterly Journal of Economics, 84 (1), 113–126. [76,93] MORTENSEN, D., AND C. PISSARIDES (1994): “Job Creation and Job Destruction in the Theory of Unemployment,” Review of Economic Studies, 61, 397–415. [106] MURPHY, K. M., AND R. TOPEL (1987): “The Evolution of Unemployment in the United States: 1968–85,” in NBER Macroeconomics Annual, ed. by S. Fischer. Cambridge, MA: MIT Press, 11–58. [79,104] NICKELL, S. J. (1981): “Biases in Dynamic Models With Fixed Effects,” Econometrica, 49 (6), 1417–1426. [99] PRESCOTT, E. C. (1986): “Response to a Skeptic,” Federal Reserve Bank of Minneapolis Quarterly Review, 10 (4). [80] STARR-MCCLUER, M. (1993): “Cyclical Fluctuations and Sectoral Reallocation: A Reexamination,” Journal of Monetary Economics, 31 (3), 417–425. [106] STOKEY, N. L. (2009): The Economics of Inaction: Stochastic Control Models With Fixed Costs. Princeton, NJ: Princeton Unversity Press. [108] SUMMERS, L. H. (1986): “Some Skeptical Observations on Real Business Cycle Theory,” Federal Reserve Bank of Minneapolis Quarterly Review, 10 (4). [80]
Dept. of Economics, University of Chicago, Chicago, IL 60637, U.S.A.; [email protected] and Dept. of Economics, University of Chicago, Chicago, IL 60637, U.S.A.; robert. [email protected]. Manuscript received January, 2008; final revision received June, 2010.
Econometrica, Vol. 79, No. 1 (January, 2011), 123–157
THE DISTRIBUTION OF WEALTH AND FISCAL POLICY IN ECONOMIES WITH FINITELY LIVED AGENTS BY JESS BENHABIB, ALBERTO BISIN, AND SHENGHAO ZHU1 We study the dynamics of the distribution of wealth in an overlapping generation economy with finitely lived agents and intergenerational transmission of wealth. Financial markets are incomplete, exposing agents to both labor and capital income risk. We show that the stationary wealth distribution is a Pareto distribution in the right tail and that it is capital income risk, rather than labor income, that drives the properties of the right tail of the wealth distribution. We also study analytically the dependence of the distribution of wealth—of wealth inequality in particular—on various fiscal policy instruments like capital income taxes and estate taxes, and on different degrees of social mobility. We show that capital income and estate taxes can significantly reduce wealth inequality, as do institutions favoring social mobility. Finally, we calibrate the economy to match the Lorenz curve of the wealth distribution of the U.S. economy. KEYWORDS: Wealth distribution, Pareto, fat tails, capital income risk.
1. INTRODUCTION RATHER INVARIABLY ACROSS A LARGE CROSS SECTION of countries and time periods income and wealth distributions are skewed to the right2 and display heavy upper tails,3 that is, slowly declining top wealth shares. The top 1% of the richest households in the United States hold over 33% of wealth4 and the 1 We gratefully acknowledge Daron Acemoglu’s extensive comments on an earlier paper on the same subject, which led us to the formulation in this paper. We also acknowledge the ideas and suggestions of Xavier Gabaix and five referees that we incorporated into the paper, as well as conversations with Marco Bassetto, Gerard Ben Arous, Alberto Bressan, Bei Cao, In-Koo Cho, Gianluca Clementi, Isabel Correia, Mariacristina De Nardi, Raquel Fernandez, Leslie Greengard, Frank Hoppensteadt, Boyan Jovanovic, Stefan Krasa, Nobu Kiyotaki, Guy Laroque, John Leahy, Omar Licandro, Andrea Moro, Jun Nie, Chris Phelan, Alexander Roitershtein, Hamid Sabourian, Benoite de Saporta, Tom Sargent, Ennio Stacchetti, Pedro Teles, Viktor Tsyrennikov, Gianluca Violante, Ivan Werning, Ed Wolff, and Zheng Yang. Thanks to Nicola Scalzo and Eleonora Patacchini for help with “impossible” Pareto references in dusty libraries. We also gratefully acknowledge Viktor Tsyrennikov’s expert research assistance. This paper is part of the Polarization and Conflict Project CIT-2-CT-2004-506084 funded by the European Commission-DG Research Sixth Framework Programme. 2 Atkinson (2002), Moriguchi and Saez (2005), Piketty (2003), Piketty and Saez (2003), and Saez and Veall (2003) documented skewed distributions of income with relatively large top shares consistently over the last century, respectively, in the United Kingdom, Japan, France, the United States, and Canada. Large top wealth shares in the United States since the 1960s were also documented, for example, by Wolff (1987, 2004). 3 Heavy upper tails (power law behavior) for the distributions of income and wealth are also well documented, for example, by Nirei and Souma (2004) for income in the United States and Japan from 1960 to 1999, by Clementi and Gallegati (2005) for Italy from 1977 to 2002, and by Dagsvik and Vatne (1999) for Norway in 1998. 4 See Wolff (2004). While income and wealth are correlated, and have qualitatively similar distributions, wealth tends to be more concentrated than income. For instance, the Gini coefficient
© 2011 The Econometric Society
DOI: 10.3982/ECTA8416
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top end of the wealth distribution obeys a Pareto law, the standard statistical model for heavy upper tails.5 Which characteristics of the wealth accumulation process are responsible for these stylized facts? To answer this question, we study the relationship between wealth inequality and the structural parameters in an economy in which households choose optimally their life-cycle consumption and savings paths. We aim at understanding first of all heavy upper tails, as they represent one of the main empirical features of wealth inequality.6 Stochastic labor endowments can, in principle, generate some skewness in the distribution of wealth, especially if the labor endowment process is itself skewed and persistent. A large literature indeed studies models in which households face uninsurable idiosyncratic labor income risk (typically referred to as Bewley models). Yet the standard Bewley models of Aiyagari (1994) and Huggett (1993) produce low Gini coefficients and cannot generate heavy tails in wealth. The reason, as discussed by Carroll (1997) and by Quadrini (1999), is that at higher wealth levels, the incentives for further precautionary savings taper off and the tails of wealth distribution remain thin. To generate skewness with heavy tails in wealth distribution, a number of authors have, therefore, successfully introduced new features, like, for example, preferences for bequests, entrepreneurial talent that generates stochastic returns (Quadrini (1999, 2000), De Nardi (2004), Cagetti and De Nardi (2006)),7 and heterogeneous discount rates that follow an exogenous stochastic process (Krusell and Smith (1998)). Our model is related to these papers. We study an overlapping generations economy where households are finitely lived and have a “joy of giving” bequest motive. Furthermore, to capture entrepreneurial risk, we assume households of the distribution of wealth in the United States in 1992 is 78, while it is only 57 for the distribution of income (Diaz-Gimenez, Quadrini, and Rios-Rull (1997)); see also Feenberg and Poterba (2000). 5 Using the richest sample of the United States, the Forbes 400—during 1988–2003, Klass, Biham, Levy, Malcai, and Solomon (2007) found, for example, that the top end of the wealth distribution obeys a Pareto law with an average exponent of 149. 6 A related question in the mathematics of stochastic processes and in statistical physics asks which stochastic difference equations produce stationary distributions which are Pareto; see, for example, Sornette (2000) for a survey. For early applications to the distribution of wealth, see, for example, Champernowne (1953), Rutherford (1955), and Wold and Whittle (1957). For the recent econophysics literature on the subject, see, for example, Mantegna and Stanley (2000). The stochastic processes which generate Pareto distributions in this whole literature are exogenous, that is, they are not the result of agents’ optimal consumption-savings decisions. This is problematic, as, for example, the dependence of the distribution of wealth on fiscal policy in the context of these models would necessarily disregard the effects of policy on the agents’ consumption–savings decisions. 7 In Quadrini (2000), the entrepreneurs receive stochastic idiosyncratic returns from projects that become available through an exogenous Markov process in the “noncorporate” sector, while there is also a corporate sector that offers nonstochastic returns.
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face stochastic stationary processes for both labor and capital income. In particular, we assume (i) (the log of) labor income has an uninsurable idiosyncratic component and a trend-stationary component across generations,8 and (ii) capital income also is governed by stationary idiosyncratic shocks, possibly persistent across generations. This specification of labor and capital income requires justification. The combination of idiosyncratic and trend-stationary components of labor income finds some support in the data; see Guvenen (2007). Most studies of labor income require some form of stationarity of the income process, although persistent income shocks are often allowed to explain the cross-sectional distribution of consumption; see, for example, Storesletten, Telmer, and Yaron (2004). While some authors (e.g., Primiceri and van Rens (2006)), adopted a nonstationary specification for individual income, it seems hardly the case that such a specification is suggested by income and consumption data; see for example, the discussion of Primiceri and van Rens (2006) by Heathcote (2008).9 The assumption that capital income contains a relevant idiosyncratic component is not standard in macroeconomics, although Angeletos and Calvet (2006) and Angeletos (2007) introduced it to study aggregate savings and growth.10 Idiosyncratic capital income risk appears, however, to be a significant element of the lifetime income uncertainty of individuals and households. Two components of capital income are particularly subject to idiosyncratic risk: ownership of principal residence and private business equity, which account for, respectively, 282% and 27% of household wealth in the United States according to the 2001 Survey of Consumer Finances (SCF; Wolff (2004) and Bertaut and Starr-McCluer (2002)).11 Case and Shiller (1989) documented a large standard deviation, on the order of 15%, of yearly capital gains or losses on owner-occupied housing. Similarly, Flavin and Yamashita (2002) measured the standard deviation of the return on housing, at the level of individual houses, from the 1968–1992 waves of the Panel Study of Income Dynamics (see http://psidonline.isr.umich.edu/), obtaining a similar number, 14%. Returns on private equity have an even higher idiosyncratic dispersion across households, a consequence of the fact that private equity is highly concentrated: 75% of all private equity is owned by households for which it constitutes at least 50% of their total net worth (Moskowitz and Vissing-Jorgensen (2002)). In the 1989 SCF studied by Moskowitz and Vissing-Jorgensen (2002), both the capital gains and earnings on private equity exhibit very substantial variation, as does excess returns to private over public equity investment, even conditional 8 In fact, trend-stationarity of income is assumed mostly for simplicity. More general stationary processes can be accounted for. 9 See Heathcote, Storesletten, and Violante (2008) for an extensive survey. 10 See also Angeletos and Calvet (2005) and Panousi (2008). 11 From a different angle, 67.7% of households own a principal residence (16.8% own other real estate) and 11.9% of households own unincorporated business equity.
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on survival.12 Evidently, the presence of moral hazard and other frictions renders complete risk diversification or concentration of each household’s wealth under the best investment technology hardly feasible.13 Under these assumptions on labor and capital income risk,14 the stationary wealth distribution is a Pareto distribution in the right tail. The economics of this result is straightforward. When labor income is stationary, it accumulates additively into wealth. The multiplicative process of wealth accumulation then tends to dominate the distribution of wealth in the tail (for high wealth). This is why Bewley models, calibrated to earnings shocks with no capital income shocks, have difficulties producing the observed skewness of the wealth distribution. The heavy tails in the wealth distribution in our model are populated by the dynasties of households which have realized a long streak of high rates of return on capital income. We analytically show that it is capital income risk rather than stochastic labor income that drives the properties of the right tail of the wealth distribution.15 An overview of our analysis is useful to navigate over technical details. If wn+1 is the initial wealth of an nth generation household, we show that the dynamics of wealth follows wn+1 = αn+1 wn + βn+1 where αn+1 and βn+1 are stochastic processes representing, respectively, the effective rate of return on wealth across generations and the permanent income of a generation. If αn+1 and βn+1 are independent and identically distributed (i.i.d.) processes, this dynamics of wealth converges to a stationary distribution with a Pareto law Pr(wn > w) ∼ kw−μ with an explicit expression for μ in terms of the process for αn+1 (μ turns out to be independent of βn+1 ).16 12 See Angeletos (2007) and Benhabib and Zhu (2008) for more evidence on the macroeconomic relevance of idiosyncratic capital income risk. Quadrini (2000) also extensively documented the role of idiosyncratic returns and entrepreneurial talent in explaining the heavy tails of wealth distribution. 13 See Bitler, Moskowitz, and Vissing-Jorgensen (2005). 14 Although we emphasize the interpretation with stochastic returns, our model also accommodates a reduced form interpretation of stochastic discounting as in Krusell and Smith (1998). 15 An alternative approach to generate fat tails without stochastic returns or discounting is to introduce a “perpetual youth” model with bequests, where the probability of death (and/or retirement) is independent of age. In these models, the stochastic component is not stochastic returns or discount rates but the length of life. For models that embody such features, see Wold and Whittle (1957), Castaneda, Diaz-Gimenez, and Rios-Rull (2003), and Benhabib and Bisin (2006). 16 See Kesten (1973) and Goldie (1991).
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But αn+1 and βn+1 are endogenously determined by the life-cycle savings and bequest behavior of households. Only by studying the life-cycle choices of households can we characterize the dependence of the distribution of wealth— and of wealth inequality in particular—on the various structural parameters of the economy, for example, technology, preferences and fiscal policy instruments like capital income taxes and estate taxes. We show that capital income and estate taxes reduce the concentration of wealth in the top tail of the distribution. Capital and estate taxes have an effect on the top tail of wealth distribution because they dampen the accumulation choices of households experiencing lucky streaks of persistent high realizations in the stochastic rates of return. We show by means of simulations that this effect is potentially very strong. Furthermore, once αn+1 and βn+1 are obtained from households’ savings and bequest decisions, it becomes apparent that the i.i.d. assumption is very restrictive. Positive autocorrelations in αn+1 and βn+1 capture variations in social mobility in the economy, for example, economies in which returns on wealth and labor earning abilities are in part transmitted across generations. Similarly, it is important to allow for the possibility of a correlation between αn+1 and βn+1 to capture institutional environments where households with high labor income have better opportunities for higher returns on wealth in financial markets. By using some new results in the mathematics of stochastic processes (due to Saporta (2004, 2005) and to Roitershtein (2007)), we are able to show that even in this case the stationary wealth distribution has a Pareto tail, and we can compute the effects of social mobility on the tail analytically.17 Finally, we calibrate and simulate our model to obtain the full wealth distribution, rather than just the tail. The model performs well in matching the (Lorenz curve of the) empirical distribution of wealth in the United States.18 Section 2 introduces the household’s life-cycle consumption and savings decisions. Section 3 gives the characterization of the stationary wealth distribution with power tails and a discussion of the assumptions underlying the result. In Section 4, our results for the effects of capital income and estate taxes on the tail index are stated. Section 4, reports on comparative statics for the bequest motive, the volatility of returns, and the degree of social mobility as measured by the correlation of rates of return on capital across generations. In Section 5, we do a simple calibration exercise to match the Lorenz curve and the fat tail of the wealth distribution in the United States, and to study the effects of capital 17 Champernowne (1953) authored the first paper to explore the role of stochastic returns on wealth that follow a Markov chain to generate an asymptotic Pareto distribution of wealth. Recently, Levy (2005), in the same tradition, studied a stochastic multiplicative process for returns and characterized the resulting stationary distribution; see also Levy and Solomon (1996) for more formal arguments and Fiaschi and Marsili (2009). These papers, however, do not provide the microfoundations necessary for consistent comparative static exercises. Furthermore, they all assume i.i.d. processes for αn+1 and βn+1 , and an exogenous lower barrier on wealth. 18 We also explore the differential effects of capital and estate taxes, of social mobility on the tail index for top wealth shares, and of the Gini coefficient for the whole wealth distribution.
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income tax and estate tax on wealth inequality. Most proofs and several technical details are buried in Appendices A and B. Replication files are posted as Supplemental Material (Benhabib, Bisin, and Zhu (2011)). 2. SAVING AND BEQUESTS Consider an economy populated by households who live for T periods. At each time t, households of any age from 0 to T are alive. Any household born at time s has a single child entering the economy at time s + T , that is, at his parents’ deaths. Generations of households are overlapping but are linked to form dynasties. A household born at time s belongs to the n = Ts th generation of its dynasty. It solves a savings problem which determines its wealth at any time t in its lifetime, leaving its wealth at death to its child. The household faces idiosyncratic rates of return on wealth and earnings at birth, which remain, however, constant in its lifetime. Generation n is, therefore, associated to a rate of return on wealth rn and to earnings yn .19 Consumption and wealth at t of a household born at s depend on the generation of the household n through rn and yn , and on its age τ = t − s We adopt the notation c(s t) = cn (t − s) and w(s t) = wn (t − s) respectively, for consumption and wealth for a household of generation n = Ts at time t. Such household inherits wealth w(s s) = wn (0) at s from its previous generation. If b < 1 denotes the estate tax, then wn (0) = (1 − b)w(s − T s) = (1 − b)wn−1 (T ). Each household’s momentary utility function is denoted u(cn (τ)). Households also have a preference for leaving bequests to their children. In particular, we assume “joy of giving” preferences for bequests: generation n’s parents’ utility from bequests is φ(wn+1 (0)), where φ denotes an increasing bequest function.20 A household of generation n born at time s chooses a lifetime consumption path cn (t − s) to maximize
T
e−ρτ u(cn (τ)) dτ + e−ρT φ(wn+1 (0))
0
19 Without loss of generality, we can add a deterministic growth component g > 0 to lifetime earnings: y(s t) = y(s s)eg(t−s) where y(s t) denotes the earnings at time t of an agent born at time s (in generation n) with yn = y(s s). In fact, this is the notation we use in Appendix A. Importantly, the aggregate growth rate of the economy is independent of g. We can also easily allow for general trend-stationary earning processes across generations (with trend g not necessarily equal to gT ). In this case, our results hold for the appropriately discounted measure of wealth (or, equivalently, for the ratio of individual and aggregate wealth); see the NBER version of this paper (Benhabib and Bisin (2009)). Finally, Zhu (2010) allowed for stochastic returns of wealth inside each generation. 20 Note that we assume that the argument of the parents’ preferences for bequests is after-tax bequests. We also assume that parents correctly anticipate that bequests are taxed and that this accordingly reduces their joy of giving.
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subject to w˙ n (τ) = rn wn (τ) + yn − cn (τ) wn+1 (0) = (1 − b)wn (T ) where ρ > 0 is the discount rate, and rn and yn are constant from the point of view of the household. In the interest of closed form solutions, we make the following assumption. ASSUMPTION 1: Preferences satisfy u(c) =
c 1−σ 1−σ
φ(w) = χ
w1−σ 1−σ
with elasticity σ ≥ 1 Furthermore, we require rn ≥ ρ and χ > 021 The dynamics of individual wealth is easily solved for; see Appendix A. 3. THE DISTRIBUTION OF WEALTH In our economy, after-tax bequests from parents are initial wealth of children. We can construct then a discrete time map for each dynasty’s wealth accumulation process. Let wn = wn (0) denote the initial wealth of the n’th dynasty. Since wn is inherited from generation n − 1, wn = (1 − b)wn−1 (T ) The rates of return of wealth and earnings are stochastic across generations. We assume they are also idiosyncratic across individuals. Let (rn )n and (yn )n denote, respectively, the stochastic processes for the rates of return of wealth and earnings over generations n.22 We obtain a difference equation for the initial wealth of dynasties, mapping wn into wn+1 : (1)
wn+1 = αn wn + βn
21 The condition rn ≥ ρ (on the whole support of the random variable rn ) is sufficient to guarantee that agents will not want to borrow during their lifetime. The condition σ ≥ 1 guarantees that rn is larger than the endogenous rate of growth of consumption, rnσ−ρ . It is required to produce a stationary nondegenerate wealth distribution and could be relaxed if we allowed the elasticities of substitution for consumption and bequest to differ, at a notational cost. Finally, χ > 0 guarantees positive bequests. 22 We avoid as much as possible the notation required for formal definitions on probability spaces and stochastic processes. The costs in terms of precision seem overwhelmed by the gain of simplicity. Given a random variable xn for instance, we simply denote the associated stochastic process as (xn )n
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where (αn βn )n = (α(rn ) β(rn yn ))n are stochastic processes induced by (rn yn )n . They are obtained as solutions of the households’ savings problem and hence they endogenously depend on the deep parameters of our economy; see Appendix A, equations (5) and (6), for closed form solutions of α(rn ) and β(rn yn ). The multiplicative term αn can be interpreted as the effective lifetime rate of return on initial wealth from one generation to the next, after subtracting the fraction of lifetime wealth consumed and before adding effective lifetime earnings, netted for the affine component of lifetime consumption.23 It can be shown that α(rn ) is increasing in rn . The additive component βn can, in turn, be interpreted as a measure of effective lifetime labor income, again after subtracting the affine part of consumption. 3.1. The Stationary Distribution of Initial Wealth In this section, we study conditions on the stochastic process (rn yn )n which guarantee that the initial wealth process defined by (1) is ergodic. We then apply a theorem from Saporta (2004, 2005) to characterize the tail of the stationary distribution of initial wealth. While the tail of the stationary distribution of initial wealth is easily characterized in the special case in which (rn )n and (yn )n are i.i.d.,24 we study more general stochastic processes which naturally arise when studying the distribution of wealth. A positive autocorrelation in rn and yn , in particular, can capture variations in social mobility in the economy, for example, economies in which returns on wealth and labor earning abilities are in part transmitted across generations. Similarly, correlation between rn and yn allows, for example, for households with high labor income to have better opportunities for higher returns on wealth in financial markets.25 To induce a limit stationary distribution of (wn )n , it is required that the contractive and expansive components of the effective rate of return tend to balance, that is, that the distribution of αn display enough mass on αn < 1 as well as some on αn > 1, and that effective earnings βn be positive, hence acting as a reflecting barrier. We impose assumptions on (rn yn )n which are sufficient to guarantee the existence and uniqueness of a limit stationary distribution of (wn )n ; see Assumptions 2 and 3 in Appendix B. In terms of (αn βn )n , these assumptions guarantee that (αn βn )n > 0, that E(αn |αn−1 ) < 1 for any αn−1 , and finally that 23 A realization of αn = α(rn ) < 1 should not, however, be interpreted as a negative return in the conventional sense. At any instant, the rate of return on wealth for an agent is a realization of rn > 0 that is, is positive. Also, note that because bequests are positive under our assumptions, αn is also positive; see the Proof of Proposition 1. 24 The characterization is an application of the well known Kesten–Goldie theorem in this case, as αn and βn are i.i.d. if rn and yn are. 25 See Arrow (1987) and McKay (2008) for models in which such correlations arise endogenously from non-homogeneous portfolio choices in financial markets.
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αn > 1 with positive probability; see Lemma A.1 in Appendix B.26 In terms of fundamentals, these assumptions require an upper bound on the (log of the) mean of rn as well as that rn be large enough with positive probability.27 Under these assumptions we can prove the following theorem, based on a theorem in Saporta (2005). THEOREM 1: Consider wn+1 = α(rn )wn + β(rn yn )
w0 > 0
Let (rn yn )n satisfy Assumption 2 and 3 as well as a regularity assumption.28 Then the tail of the stationary distribution of wn , Pr(wn > w), is asymptotic to a Pareto law Pr(wn > w) ∼ kw−μ where μ > 1 satisfies N−1 1/N μ (2) (α−n ) = 1 lim E N→∞
n=0
When (αn )n is i.i.d., condition (2) reduces to E(α)μ = 1, a result established by Kesten (1973) and Goldie (1991).29 We now turn to the characterization of the stationary wealth distribution of the economy, aggregating over households of different ages. 26 We also assume that βn is bounded, although the assumption is stronger than necessary. In Proposition 1, we also show that the state space of (αn βn )n is well defined. Furthermore, by Assumption 2, (rn )n converges to a stationary distribution and hence (α(rn ))n also converges to a stationary distribution. 27 Suppose preferences are logarithmic. Then it is required that
eρT + ρχ − 1 (1 − b)ρχ ρT e + ρχ − 1 1 rn > log T (1 − b)ρχ E(ern T ) <
with positive probability.
We thank an anonymous referee for pointing this out. As an example of parameters that satisfy these conditions for the log utility case, suppose that ρ = 04, χ = 25, T = 45, b = 2, ζ = 15, and that the rate of return on wealth is i.i.d. with four states (see Section 5 for details regarding the model’s calibration along these lines). The probabilities with these four states are 8, 12, 07, and 01. The first three states of before-tax rate of return are 08, 12, and 15. The above two inequalities imply that the fourth state of before-tax rate of return could belong to the open interval (169 286). 28 See Appendix B, Proof of Theorem 1, for details. 29 The term N−1 n=0 α−n in (2) arises from using repeated substitutions for wn See Brandt (1986) for general conditions to obtain an ergodic solution for stationary stochastic processes satisfying (1).
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3.2. The Stationary Distribution of Wealth in the Population We have shown that the stationary distribution of initial wealth in our economy has a power tail. The stationary wealth distribution of the economy can be constructed by aggregating over the wealth of households of all ages τ from 0 to T . The wealth of a household of generation n and age τ born with wealth wn = wn (0), return rn , and income yn is a deterministic map, as the realizations of rn and yn are fixed for any household during its lifetime. In Appendix B, we show that, under our assumptions, the process (wn rn )n is ergodic and has a unique stationary distribution. Let ν denote the product measure of the stationary distribution of (wn rn )n . In Appendix A, we derive the closed form for wn (τ) the wealth of household of generation n and age τ (equation (4)): wn (τ) = σw (rn τ)wn + σy (rn τ)yn We can then define F(w; τ) = 1 − Pr(wn (τ) > w), the cumulative distribution function of the stationary distribution of wn (τ), as F(w; τ) =
l
Pr(yj )
I{σw (rn τ)wn +σy (rn τ)yj ≤w} dν
j=1
where I is an indicator function. The cumulative distribution function of wealth w in the population is then defined as T 1 F(w; τ) dτ F(w) = T 0 We can now show that the power tail of the initial wealth distribution implies that the distribution of wealth w in the population displays a tail with exponent μ in the following sense: THEOREM 2: Suppose the tail of the stationary distribution of initial wealth wn = wn (0) is asymptotic to a Pareto law, Pr(wn > w) ∼ kw−μ . Then the stationary distribution of wealth in the population has a power tail with the same exponent μ. Note that this result is independent of the demographic characteristics of the economy, that is, of the stationary distribution of the households by age. The intuition is that the power tail of the stationary distribution of wealth in the population is as thick as the thickest tail across wealth distributions by age. Since under our assumptions each wealth distribution by age has a power tail with the same exponent μ, this exponent is inherited by the distribution of wealth in the population as well.30 30 The tail of the stationary wealth distribution of the population is independent of any deterministic growth component g > 0 to lifetime earning as introduced in Appendix A.
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4. WEALTH INEQUALITY: SOME COMPARATIVE STATICS We study in this section the tail of the stationary wealth distribution as a function of preference parameters and fiscal policies. In particular, we study stationary wealth inequality as measured by the tail index of the distribution of wealth, μ, which is analytically characterized in Theorem 1. The tail index μ is inversely related to wealth inequality, as a small index μ implies a heavier top tail of the wealth distribution (the distribution declines more slowly with wealth in the tail). In fact, the exponent μ is inversely linked 1 to the Gini coefficient G = 2μ−1 , the classic statistical measure of inequality31 First, we study how different compositions of capital and labor income risk affect the tail index μ. Second, we study the effects of preferences, in particular the intensity of the bequest motive. Third, we characterize the effects of both capital income and estate taxes on μ Finally, we address the relationship between social mobility and μ. 4.1. Capital and Labor Income Risk If follows from Theorem 1 that the stochastic properties of labor income risk, (βn )n have no effect on the tail of the stationary wealth distribution. In fact, heavy tails in the stationary distribution require that the economy has sufficient capital income risk, with αn > 1 with positive probability. Consider instead an economy with limited capital income risk, in which αn < 1 with probability 1 and β¯ is the upper bound of βn In this case, it is straightforward to show that β , where α the stationary distribution of wealth would be bounded above by 1−α 32 is the upper bound of αn More generally, we can also show that wealth inequality increases with the capital income risk households face in the economy. PROPOSITION 1: Consider two distinct i.i.d. processes for the rate of return on wealth, (rn )n and (rn )n . Suppose α(rn ) is a convex function of rn .33 If rn second order stochastically dominates rn , the tail index μ of the wealth distribution under (rn )n is smaller than under (rn )n . We conclude that it is capital income risk (idiosyncratic risk on return on capital), and not labor income risk, that determines the heaviness of the tail of the stationary distribution given by the tail index: the higher is capital income risk, the more unequal is wealth. 31
See, for example, Chipman (1976). Since the distribution of wealth in our economy is typi1 as the Gini of the tail. cally Pareto only in the tail, we refer G = 2μ−1 32 Of course, this is true a fortiori in the case where there is no capital risk and αn = α < 1 33 This is typically the case in our economy if constant relative risk aversion parameter σ is √
T
T 2 A(r )t not too high. A sufficient condition is 2( 2 − 1)T 0 teA(rn )t dt − σ−1 t e n dt > 0 where √σ 0 −1 A(rn ) = (rn (σ − 1) + ρ)σ , which holds since T ≥ t if σ < (1 − 2( 2 − 1))−1 = 48284.
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4.2. The Bequest Motive Wealth inequality depends on the bequest motive, as measured by the preference parameter χ. PROPOSITION 2: The tail index μ decreases with the bequest motive χ A household with a higher preference for bequests will save more and accumulate wealth faster. This saving behavior induces an higher effective rate of return of wealth across generations αn , on average, which in turn leads to higher wealth inequality. 4.3. Fiscal Policy To study the effects of fiscal policy, first we redefine the random rate of return rn as the pre-tax rate and introduce a capital income tax, ζ so that the post-tax return on capital is (1 − ζ)rn Fiscal policies in our economy are then captured by the parameters b and ζ representing, respectively, the estate tax and the capital income tax. PROPOSITION 3: The tail index μ increases with the estate tax b and with the capital income tax ζ. Furthermore, let ζ(rn ) denote a nonlinear tax on capital, such that the net n rate of return of wealth for generation n becomes rn (1 − ζ(rn )) Since ∂α > 0 ∂rn the corollary below follows immediately from Proposition 3. COROLLARY 1: The tail index μ increases with the imposition of a nonlinear tax on capital ζ(rn ). Taxes have, therefore, a dampening effect on the tail of the wealth distribution in our economy: the higher are taxes, the lower is wealth inequality. The calibration exercise in Section 2 documents that, in fact, the tail of the stationary wealth distribution is quite sensitive to variations in both capital income taxes and estate taxes. Becker and Tomes (1979), on the contrary, found that taxes have ambiguous effects on wealth inequality at the stationary distribution. In their model, bequests are chosen by parents to essentially offset the effects of fiscal policy, limiting any wealth equalizing aspects of these policies. This compensating effect of bequests is present in our economy as well, although it is not sufficient to offset the effects of estate and capital income taxes on the stochastic returns on capital. In other words, the power of Becker and Tomes’ (1979) compensating effect is due to the fact that their economy has no capital income risk. The main mechanism through which estate taxes and capital income taxes have an equalizing effect on the wealth distribution in our economy is by reducing the capital income risk, along the lines of Proposition 1, not its average return.
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4.4. Social Mobility We turn now to the study of the effects of different degrees of social mobility on the tail of the wealth distribution. Social mobility is higher when (rn )n and (yn )n (and hence when (αn )n and (βn )n ) are less autocorrelated over time. We provide here expressions for the tail index of the wealth distribution as a function of the autocorrelation of (αn )n in the following two distinct cases,34 where 0 < θ < 1 and (ηn )n is an i.i.d. process with bounded support.35 MA(1)
ln αn = ηn + θηn−1
AR(1)
ln αn = θ ln αn−1 + ηn
PROPOSITION 436 : Suppose that ln αn satisfies MA(1). The tail of the limiting distribution of initial wealth wn is then asymptotic to a Pareto law with tail exponent μMA which satisfies EeμMA (1+θ)ηn = 1 If instead ln αn satisfies AR(1), the tail exponent μAR satisfies Ee(μAR /(1−θ))ηn = 1 In either the MA(1) or the AR(1) case, the higher is θ, the lower is the tail exponent. That is, the more persistent is the process for the rate of return on wealth (the higher are frictions to social mobility), the fatter is the tail of the wealth distribution.37 5. A SIMPLE CALIBRATION EXERCISE As we have already discussed in the Introduction, it has proven hard for standard macroeconomic models, when calibrated to the U.S. economy, to produce wealth distributions with tails as heavy as those observed in the data. The analytical results in the previous sections suggest that capital income risk should prove very helpful in matching the heavy tails. Our theoretical results are, however, limited to a characterization of the tail of the wealth distribution, and questions remain about the ability of our model to match the entire wealth distribution. To this end, we report on a simulation exercise which illustrates 34 The stochastic properties of (yn )n , and hence of (βn )n as we have seen, do not affect the tail index. 35 We thank Zheng Yang for pointing out that boundedness of ηn guarantees boundedness of αn under our assumptions. 36 We thank Xavier Gabaix for suggesting the statement of this proposition and outlining an argument for its proof. 37 The results easily extend to MA(k) and AR(k) processes for ln αn
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the ability of the model to match the Lorenz curve of the wealth distribution in the United States.38 We calibrate the parameters of the models as follows. First of all, we set the fundamental preference parameters in line with the macroeconomic literature: σ = 2, ρ = 004. We also set the preference for bequest parameter χ = 025 and working life span T = 45. The labor earnings process, yn is set to match mean earnings in $10,000 units, 4239 We pick a standard deviation of yn equal to 95 and we also assume that earnings grow at a yearly rate g equal to 1% over each household lifetime.40 The calibration of the cross-sectional distribution of the rate of return on wealth, rn is rather delicate, as capital income risk typically does not appear in calibrated macroeconomic models. We proceed as follows. First of all, we map the model to the data by distinguishing two components of rn : a common economy-wide rate of return r E and an idiosyncratic component rnI The common component of returns, r E represents the value-weighted returns on the market portfolio, including, for example, cash, bonds, and public equity. The idiosyncratic component of returns, rnI is composed for the most part of returns on the ownership of a principal residence and on private business equity. According to the Survey of Consumer Finances, ownership of a principal residence and private business equity account for about 50% of household wealth portfolios in the United States. We then map rn into data according to 1 1 rn = r E + rnI 2 2 For the common economy-wide rate of return r E , which is assumed to be constant over time in the model, we choose a range of values between 7 and 9 percent before taxes, about 1–3 percentage points below the rate of return on public equity. Unfortunately, no precise estimate exists for the distribution of the idiosyncratic component of capital income risk to calibrate the distribution of rnI . Flavin and Yamashita (2002) studied the after-tax return on housing at the level of individual houses from the 1968–1992 waves of the Panel Study of Income Dynamics. They obtained a mean after-tax return of 66% with a standard deviation of 14%. Returns on private equity were estimated by Moskowitz and Vissing-Jorgensen (2002) from the 1989-1998 Survey of Consumer Finances data. They found mean returns comparable to those on public equity, but they lacked enough time series variation to estimate their standard deviation, which 38 For the data on the U.S. economy, the tail index is from Klass et al. (2007), who used the Forbes 400 data. The rest of data for the United States economy are from Diaz-Gimenez, Quadrini, Ríos-Rull, and Rodríguez (2002), who used the 1998 Survey of Consumer Finances. 39 More specifically, we choose a discrete distribution for yn , taking values 75, 251, 501, 1254, 2507, and 7522 with probability 14 , 36 , 11 , 1 , 1 , and 641 , respectively. 64 64 64 64 64 40 This requires straightforwardly extending the model along the lines delineated in footnote 19.
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they end up proxying with the standard deviation of an individual publicly traded stock. Based on these data, Angeletos (2007) adopted a baseline calibration for capital income risk with an implied mean return around 7% and a standard deviation of 20% Allowing for a private equity risk premium, we choose mean values for rnI between 7 and 9 percent. With regard to the standard deviation, in our model rnI is constant over an agent’s lifetime. Interpreting rnI as a mean over the yearly rates of return estimated in the data and assuming independence, a 3% standard deviation of rnI corresponds to a standard deviation of yearly returns on the order of 20% as in Angeletos (2007). We then choose a range of standard deviations of rnI between 2 and 3 percent. With regard to social mobility, we present results for the case in which rn is i.i.d. across generations (perfect social mobility), as well as for different degrees of autocorrelation of rn (imperfect social mobility). The capital income risk process rn is formally modelled as a discrete Markov chain. In the case in which rn is i.i.d., the Markov transition matrix for rn has identical rows.41 We then introduce frictions to social mobility by moving a mass εlow of probability from the off-diagonal terms to the diagonal term in the first row of the Markov transition matrix for rn , that is, the row corresponding to the probability distribution of rn+1 conditional on rn being lowest. We do the same shift of a mass εhigh of probability in the last row of the Markov transition matrix for rn , that is, the row corresponding to the probability distribution of rn+1 conditional on rn being highest. This introduces persistence of low and high rates of return of wealth across generations. For our baseline simulation, in Table I we report the relevant statistics of the rn process at the stationary distribution for εlow = 0 01, and εhigh = 0 01 02 05 respectively. Finally, we set the estate tax rate b = 2 (which is the average tax rate on bequests), and the capital income tax ζ = 15 in the baseline, but in Section 5.2 we study various combinations of fiscal policy. With this calibration we simulate the stationary distribution of the economy.42 We then calculate the top percentiles of the simulated wealth distribution, the Gini coefficient of the whole distribution (not just the Gini of the tail), the quintiles, and the tail index μ. While we are mostly concerned with the wealth distribution, we also report the capital income to labor income ratio implied in the simulation as an extra check. We aim at a ratio not too distant from 5 the value implied by the standard calibration of macroeconomic production models (with a constant return to scale Cobb–Douglas production 41 We choose two discrete Markov processes for rn , the first with mean (at the stationary distribution) on the order of 9 percent and the second on the order of 7 percent. More specifically, the first process takes values [08 12 15 32] with probability rows (in the i.i.d. case) of the transition equal to [8 12 07 01]; the second process has support [065 12 15 27] with probability rows (in the i.i.d. case) equal to [93 01 01 05] 42 We note that under these calibrations for rn and other parameters, we check that the conditions of Assumptions 2 and 3 are satisfied and, therefore, that the restrictions on α hold.
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J. BENHABIB, A. BISIN, AND S. ZHU TABLE I BASELINE CALIBRATION OF rn a Economy
E(rn )
σ(rn )
corr(rn rn−1 )
εlow = 0, εhigh = 0 εhigh = 01 εhigh = 02 εhigh = 05
0921 0922 0922 0925
0311 0313 0316 0325
0 0148 0342 0812
εlow = 01, εhigh = 0 εhigh = 01 εhigh = 02 εhigh = 05
0892 0892 0892 0893
0223 0224 0224 0227
0571 0613 0619 0952
a All the statistics are obtained from the simulated stationary distribution of rn except the auto-correlation corr(rn rn−1 ) when εlow = εhigh = 0 which is 0 analytically.
function with capital share equal to 13 ) We report first, as a baseline, the case with εlow = 01 and various values for εhigh First of all, note that the wealth distributions which we obtain in the various simulations in Table II match quite successfully the top percentiles of the United States. Furthermore, note that the tail of the simulated wealth distribution economy gets thicker by increasing εhigh that is, by increasing corr(rn rn−1 ) In particular, the better fit is obtained with substantial imperfections in social mobility (εhigh = 02) in which case the 99th–100th percentile of wealth in the U.S. economy is matched almost exactly. More surprisingly, perhaps, the Lorenz curve (in quintiles) of the simulated wealth distributions, Table III, matches reasonably well that of the United States; and so does the Gini coefficient. Once again, εhigh = 02 appears to represent the better fit in terms of the Lorenz curve and the Gini coefficient
TABLE II PERCENTILES OF THE TOP TAIL; εlow = 01 Percentiles Economy
United States εhigh = 0 εhigh = 01 εhigh = 02 εhigh = 05
90th–95th
95th–99th
99th–100th
113 118 116 105 087
231 204 202 182 151
347 261 275 341 457
139
THE DISTRIBUTION OF WEALTH TABLE III TAIL INDEX, GINI, AND QUINTILES; εlow = 01 Quintiles Economy
United States εhigh = 0 εhigh = 01 εhigh = 02 εhigh = 05
Tail Index μ
Gini
First
Second
Third
Fourth
Fifth
149 1796 1256 1038 716
803 646 655 685 742
−003 033 032 029 024
013 058 056 051 042
05 08 078 071 058
122 123 12 11 09
817 707 714 739 786
(even though the tail index of this calibration is lower than the U.S. economy’s, but the tail index is imprecisely estimated with wealth data).43 Furthermore, the capital income to labor income ratio implied by the simulations takes on reasonable values: it goes from 3 for εhigh = 0 to 6 for εhigh = 05. In the εhigh = 02 calibration, the capital–labor ratio is almost exactly 5 5.1. Robustness As a robustness check, we report the calibration with εlow = 0 In this case, the simulated wealth distributions also have Gini coefficients close to that of the U.S. economy and Lorenz curves which also match that of the United States rather well. Table IV reports the top percentiles of the U.S. economy and of the simulated wealth distribution. Table V reports instead the tail index, the Gini coefficient, and the Lorenz curve of the U.S. economy and of the simulated wealth distribution.44 Note that the calibration with i.i.d. capital income risk rn (εlow = εhigh = 0) does particularly well. TABLE IV PERCENTILES OF THE TOP TAIL; εlow = 0 Percentiles Economy
United States εhigh = 0 εhigh = 01 εhigh = 02 εhigh = 05
90th–95th
95th–99th
99th–100th
113 1 082 073 026
231 207 173 154 06
347 38 49 544 836
43 The calibration with εhigh = 05 with even more frictions to social mobility, also fares well, although in this case the tail index is < 1 which implies that the tails are so thick that the theoretical distribution has no mean. In this case, Assumption 3(ii) in Appendix B is violated. 44 Again, for εhigh = 05 we have μ < 1 See footnote 43.
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J. BENHABIB, A. BISIN, AND S. ZHU TABLE V TAIL INDEX, GINI, AND QUINTILES; εlow = 0 Quintiles
Economy
United States εhigh = 0 εhigh = 01 εhigh = 02 εhigh = 05
Tail Index μ
Gini
First
Second
Third
Fourth
Fifth
149 1795 1254 1036 713
803 738 786 808 933
−003 023 018 017 006
013 041 033 003 01
05 057 046 042 014
122 092 074 067 023
817 788 827 844 947
We also report the simulation for the economy with a different Markov process for rn with pre-tax mean of 7% Table VI reports the relevant statistics of the rn process at the stationary distribution, in this case, for εlow = 0 1 and εhigh = 2 respectively.45 Tables VII and VIII collect the results regarding the simulated wealth distribution for this process of capital income risk. While still in the ballpark of the U.S. economy, these calibrations match it much more poorly than the previous ones with a higher mean of rn . Interestingly, they induce a higher Gini coefficient than in the U.S. distribution, suggesting that our model, in general, does not share the difficulties experienced by standard calibrated macroeconomic models to produce wealth distributions with tails as heavy as those observed in the data. 5.2. Tax Experiments The tables below illustrate the effects of taxes on the tail index and the Gini coefficient. We calibrate the parameters of the economy, other than b and ζ, as before, with rn as in Table I, εhigh = 02, and εlow = 01 and we vary b and ζ. Table IX reports the effects of capital income taxes and estate taxes on the tail index μ Taxes have a significant effect on the inequality of the wealth distribution as measured by the tail index. This is especially the case for the capital income TABLE VI CALIBRATION OF rn WITH MEAN 7%
45
Economy
E(rn )
σ(rn )
corr(rn rn−1 )
εlow = 0, εhigh = 02 εlow = 01, εhigh = 02
772 0738
467 0415
0356 0542
A more extensive set of results is available from the authors upon request.
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THE DISTRIBUTION OF WEALTH TABLE VII PERCENTILES OF THE TOP TAIL Percentiles Economy
90th–95th
95th–99th
99th–100th
113 066 076
231 232 236
347 675 646
United States εlow = 01 εhigh = 02 εlow = 0 εhigh = 02
tax, which directly affects the stochastic returns on wealth. The implied Gini of the tail46 is very high with no (or low) taxes,47 while it is reduced to 66 with a 30% estate tax and a 15% capital income tax. We now turn to the Gini coefficient of the whole distribution. The results are in Table X. We see that the Gini coefficient consistently declines as the capital income tax increases, but the decline is quite moderate and the estate taxes can even have ambiguous effects. A tax increase has the effect of reducing the concentration of wealth in the tail of the distribution. This effect is, however, partly offset by greater inequality at lower wealth levels. In general, a decrease in the rate of return on wealth (e.g., due to a tax increase) has the effect of increasing the permanent labor income of households, because future labor earnings are discounted at a lower rate. For rich households, whose wealth consists mainly of physical wealth rather than labor earnings, a lower capital income tax rate generates an approximately proportional wealth effect on consumption and savings. On the other hand, the positive wealth effect of a tax reduction has a relatively large effect for households whose physical wealth is relatively low. These households will smooth their consumption based on their lifetime labor earnings and will hence react to a tax reduction by decumulating TABLE VIII TAIL INDEX, GINI, AND QUINTILES Quintiles Economy
United States εlow = 01 εhigh = 02 εlow = 0 εhigh = 02
Tail Index μ
Gini
First
Second
Third
Fourth
Fifth
149 1514 1514
803 993 978
−003 −022 −016
013 003 003
05 009 008
122 016 015
817 994 991
1 As before, the tail Gini is G = 2μ−1 When the tail index μ is less than 1, the wealth distribution has no mean, so that again, Assumption 3(ii) in Appendix B is violated. In this case, theoretically the Gini coefficient is not defined. In Table X, however, we report the simulated value, computed from the simulated wealth distribution. 46 47
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J. BENHABIB, A. BISIN, AND S. ZHU TABLE IX TAX EXPERIMENTS—TAIL INDEX μ b\ζ
0 1 2 25
0
05
15
2
68 689 7 706
76 772 785 793
994 1014 1038 1051
1177 1205 1238 1257
physical wealth proportionately faster than households that are relatively rich in physical wealth. As a result of this effect, wealth inequality between rich and poor households as measured by physical wealth tends to increase. Of course, the effects of a tax increase on relatively poor households would be moderated (perhaps eliminated) if tax revenues were to be redistributed toward the less wealthy. Nonetheless the results of Table X suggests a word of caution in evaluating the effects on wealth inequality of proposed fiscal policies like the abolition of estate taxes or the reduction of capital taxes. For instance, Castaneda, DiazGimenez, and Rios-Rull (2003) and Cagetti and De Nardi (2007) found very small (or even perverse) effects of eliminating bequest taxes in their calibrations in models with a skewed distribution of earnings but no capital income risk.48 If the capital income risk component is a substantial fraction of idiosyncratic risk, such fiscal policies could have sizeable effects in increasing wealth inequality in the top tail of the distribution of wealth which may not show up in measurements of the Gini coefficient.49 TABLE X TAX EXPERIMENTS—GINI b\ζ
0 1 2 3
48
0
05
15
2
.779 .768 .778 .754
.769 .730 .724 .726
.695 .693 .679 .680
.674 .677 .674 .677
See also our discussion of the results of Becker and Tomes (1979) previously in this section. Empirical studies also indicate that higher and more progressive taxes did, in fact, significantly reduce income and wealth inequality in the historical context; notably, for example, Lampman (1962) and Kuznets (1955). Most recently, Piketty (2003) and Piketty and Saez (2003) argued that redistributive capital and estate taxation may have prevented holders of very large fortunes from recovering from the shocks that they experienced during the Great Depression and World War II because of the dynamic effects of progressive taxation on capital accumulation and pre-tax income inequality. This line of argument has been extended to the United States and 49
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143
6. CONCLUSION The main conclusion of this paper is that capital income risk, that is, idiosyncratic returns on wealth, has a fundamental role in affecting the distribution of wealth. Capital income risk appears to be crucial in generating the heavy tails observed in wealth distributions across a large cross section of countries and time periods. Furthermore, when the wealth distribution is shaped by capital income risk, the top tail of wealth distribution is very sensitive to fiscal policies, a result which is often documented empirically but is hard to generate in many classes of models without capital income risk. Higher taxes in effect dampen the multiplicative stochastic return on wealth, which is critical to generate the heavy tails. Interestingly, this role of capital income risk as a determinant of the distribution of wealth seems to have been lost by Vilfredo Pareto. He explicitly noted that an identical stochastic process for wealth across households will not induce the skewed wealth distribution that we observe in the data (see Pareto (1897), note 1 to No. 962, p. 315–316). He therefore introduced skewness into the distribution of talents or labor earnings of households (1897, notes to No. 962, p. 416). Left with the distribution of talents and earnings as the main determinant of the wealth distribution, he was perhaps lead to his Pareto law, enunciated by Samuelson (1965) as follows: In all places and all times, the distribution of income remains the same. Neither institutional change nor egalitarian taxation can alter this fundamental constant of social sciences.50
APPENDIX A: CLOSED FORM SOLUTIONS We report here only the closed form solutions for the dynamics of wealth in the paper. Let the age at time t of a household born at time s ≤ t be denoted τ = t − s An agent born at time s belongs to generation n = Ts Let the human capital at time t of a household born at s h(s t) = hn (t − s) = hn (τ) be defined T as hn (τ) = 0 yn e−(rn −g)τ dτ.51 We adopt the notation wn (0) = wn The optimal consumption path satisfies cn (τ) = m(τ)(wn (τ) + hn (τ)) Japan, and the United States and Canada, respectively, by Moriguchi and Saez (2005) and Saez and Veall (2003). 50 See Chipman (1976) for a discussion on the controversy between Pareto and Pigou regarding the interpretation of the law. To be fair to Pareto, he also had a “political economy” theory of fiscal policy (determined by the controlling elites) which could also explain the Pareto law; see Pareto (1901, 1909). 51 To save on notation in the text, we restrict to the case in which g = 0
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The propensity to consume out of financial and human wealth, m(τ) is independent of wn (τ) and hn (τ), and is decreasing in age τ in the estate tax b and in capital income tax ζ: (3)
m(τ) =
1 −(rn −(rn −ρ)/σ)(T −τ) 1 − e rn − ρ rn − σ (1−σ)/σ −(rn −(rn −ρ)/σ)(T −τ)
(1 − b)
+χ
1/σ
e
−1
The dynamics of individual wealth as a function of age τ satisfies (4)
wn (τ) = σw (rn τ)wn + σy (rn τ)yn
with σw (rn τ) = ern τ
eA(rn )(T −τ) + A(rn )B(b) − 1 eA(rn )T + A(rn )B(b) − 1
σy (rn τ) = ern τ
e(g−rn )T − 1 g − rn
×
eA(rn )(T −τ) + A(rn )B(b) − 1 e(rn −g)(T −τ) − 1 − eA(rn )T + A(rn )B(b) − 1 e(rn −g)T − 1
and A(rn ) = rn −
rn − ρ σ
B(b) = χ1/σ (1 − b)(1−σ)/σ
The dynamics of wealth across generation is then wn+1 = αn wn + βn with (5)
α(rn ) = (1 − b)ern T
A(rn )T
e
A(rn )B(b) + A(rn )B(b) − 1
and (6)
β(rn yn ) = (1 − b)yn
e(g−rn )T − 1 rn T A(rn )B(b) e A(r )T n g − rn e + A(rn )B(b) − 1
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APPENDIX B: PROOFS The stochastic processes for (rn yn ) and the induced processes for (αn βn )n = (α(rn ) β(rn yn ))n are required to satisfy the following assumptions. ASSUMPTION 2: The stochastic process (rn yn )n is a real, irreducible, aperiodic, stationary Markov chain with finite state space r¯ × y¯ := {r 1 r m } × {y 1 y l }. Furthermore, it satisfies Pr(rn yn | rn−1 yn−1 ) = Pr(rn yn | rn−1 ) where Pr(rn yn | rn−1 yn−1 ) denotes the conditional probability of (rn yn ) given (rn−1 yn−1 )52 A stochastic process (rn yn )n which satisfies Assumption 2 is a Markov modulated chain. This assumption would be satisfied, for instance, if a single Markov chain, corresponding, for example, to productivity shocks, drove returns on capital (rn )n as well as labor income (yn )n 53 ASSUMPTION 3: Let P denote the transition matrix of (rn )n : Pii = Pr(ri |ri ). Let α(¯r) denote the state space of (αn )n as induced by the map α(rn ) Then r¯ , y¯ , and P are such that (i) r¯ × y¯ 0 (ii) Pα(¯r) < 1 (iii) ∃r i such that α(r i ) > 1, and (iv) Pii > 0 for any i. We are now ready to show the following lemma. LEMMA A.1: Assumption 2 on (rn yn )n implies that (αn βn )n is a Markov modulated chain. Furthermore, Assumption 3 implies that (αn βn )n is reflective, that is, it satisfies (i) (αn βn )n > 0 (ii) E(αn |αn−1 ) < 1 for any αn−1 54 (iii) αi > 1 for some i = 1 m, and (iv) the diagonal elements of the transition matrix P of αn are positive. PROOF: Let A be the diagonal matrix with elements Aii = αi and Aij = 0, j = i. Note that E(αn |αn−1 ) for any αn−1 can be written as Pα(¯r) < 1. Let r = {r 1 r m } denote the state space of rn Similarly, let y = {y 1 y l } denote the state space of yn Let α = {α1 αm } and β = {β1 βl } denote the state spaces of, respectively, αn and βn as they are induced through the 52 While Assumption 2 requires rn to be independent of (yn−1 yn−2 ), it leaves the autocorrelation of (rn )n unrestricted in the space of Markov chains. Also, Assumption 2 allows for (a restricted form of) autocorrelation of (yn )n as well as correlation of yn and rn 53 For the use of Markov modulated chains, see Saporta (2005) in her remarks following Theorem 2 or Saporta (2004, Section 2.9, p. 80). See instead Roitersthein (2007) for general Markov modulated processes. 54 We could only require that the mean of the unconditional distribution of α be less than 1 that is, if E(α) < 1, but in this case, the stationary distribution of wealth may not have a mean.
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maps (5) and (6). We shall show that the maps (5) and (6) are bounded in rn and yn Therefore, the state spaces of αn and βn are well defined. It immediately follows that if (rn yn )n is a Markov modulated chain (Assumption 2), so is (αn βn )n . We now show that under Assumption 3(i), (αn βn )n is greater than 0 and bounded with probability 1 in rn and yn Recall that B(b) = χ1/σ (1 − b)(1−σ)/σ > 0. Note that α(rn ) = (1 − b)
B(b)
T
−(rn −ρ)/σT
e
−A(rn )(T −t)
e
−rn T
dt + e
B(b)
0
Therefore, αn > 0 and bounded. Furthermore, note that T β(rn yn ) = α(rn )yn e(g−rn )t dt 0
and the support of yn is bounded by Assumption 2. Thus (βn )n ≥ 0 and is bounded. Therefore, (αn βn ) is a Markov modulated process provided (βn )n is positive and bounded. Furthermore, Assumption 3(ii) implies directly that (ii) P α ¯ < 1 Assumption 3(iii) also directly implies αi > 1 for some i = 1 m. Finally P is the transition matrix of rn as well as αn Therefore, Assumption 3(iv) implies that the elements of the trace of the transition matrix of αn are positive. Q.E.D. PROOF OF THEOREM 1: We first define rigorously the regularity of the Markov modulated process (αn βn )n . In singular cases, particular correlations between αn and βn can create degenerate distributions that eliminate the randomness of wealth. We rule this out by means of the following technical regularity conditions55 : CONDITIONS: The Markov modulated process (αn βn )n is regular, that is, Pr(α0 x + β0 = x|α0 ) < 1
for any x ∈ R+
and the elements of the vector α¯ = {ln α1 · · · ln αm } ⊂ Rm + are not integral multiples of the same number.56 55 We formulate these regularity conditions on (αn βn )n , but they can be immediately mapped back into conditions on the stochastic process (rn yn )n . 56 Theorems which characterize the tails of distributions generated by equations with random multiplicative coefficients rely on this type of nonlattice assumption from renewal theory; see for example Saporta (2005). Versions of these assumption are standard in this literature; see Feller (1966).
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147
Saporta (2005, Proposition 1, Section 4.1) established that, for finite Markov N−1 chains, limN→∞ (E n=0 (α−n )μ )1/N = λ(Aμ P ), where λ(Aμ P ) is the dominant root of Aμ P .57 Condition (2) can then be expressed as λ(Aμ P ) = 1 The theorem then follows directly from Saporta (2005, Theorem 1), if we show (i) that there exists a μ that solves λ(Aμ P ) = 1 and (ii) that such μ > 1. Saporta showed that μ = 0 is a solution to λ(Aμ P ) = 1 or, equivalently, to ln(λ(Aμ P )) = 0 This follows from A0 = I and P being a stochastic matrix. Let Eα(r) denote the expected value of αn at its stationary distribution (which exists as it is implied by the ergodicity of (rn )n , in turn a consequence of AssumpμP) tion 2). Saporta, under the assumption Eα(r) < 1 showed that d ln λ(A <0 δμ at μ = 0 and that ln(λ(Aμ P )) is a convex function of μ.58 Therefore, if there exists another solution μ > 0 for ln(λ(Aμ P )) = 0 it is positive and unique. To assure that μ > 1, we replace the condition Eα(r) < 1 with Proposition 3(ii), P α ¯ < 1 This implies that the column sums of AP are < 1. Since AP is positive and irreducible, its dominant root is smaller than the maximum column sum. Therefore, for μ = 1 λ(Aμ P ) = λ(AP ) < 1. Now note that if (αn βn )n is reflective, by Proposition 1, Pii > 0 and αi > 1 for some i. This implies that the trace of Aμ P goes to infinity if μ does (see also Saporta (2004, Proposition 2.7)). But the trace is the sum of the roots, so the dominant root of Aμ P λ(Aμ P ) goes to infinity with μ. It follows that for the solution of Q.E.D. ln(λ(Aμ P )) = 0, we must have μ > 1 This proves (ii). PROOF OF THEOREM 2: We first show by Lemma A.2 that the process (wn rn−1 )n is ergodic59 and thus has a unique stationary distribution. If we denote with φ the product measure of the stationary distribution of (wn rn−1 )n , and we denote with ν the product measure of the stationary distribution of (wn rn )n , the relationship between φ and v is v(dw rn ) = (Pr(rn |rn−1 )φ(dw rn−1 )) rn−1
Ergodicity of (wn rn−1 )n then implies ergodicity of (wn rn )n which then also has a unique stationary distribution. Actually, Lemma A.1 shows that (wn rn−1 )n is V -uniformly ergodic, which is stronger than ergodicity. For the Recall that the matrix AP has the property that the ith column sum equals the expected value of αn conditional on αn−1 = αi . When (αn )n is i.i.d., P has identical rows, so transition probabilities do not depend on the state αi In this case, Aμ P has identical column sums given by Eαμ and equal to λ(Aμ P ) 58 This follows because limn→∞ n1 ln E(α0 α−1 · · · αn−1 )μ = ln(λ(Aμ P )) and because the moments of nonnegative random variables are log convex (in μ); see Loeve (1977, p. 158). 59 Actually Lemma A.2 shows that (wn rn−1 )n is V -uniformly ergodic, which is stronger than ergodicity. For the mathematical concepts such as V -uniform ergodicity, ψ-irreducibility, and petite sets, see Meyn and Tweedie (2009). 57
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J. BENHABIB, A. BISIN, AND S. ZHU
mathematical concepts such as V -uniform ergodicity, ψ-irreducibility, and petite sets which we use in the proof, see Meyn and Tweedie (2009). LEMMA A.2: The process (wn rn−1 )n is V -uniformly ergodic. PROOF: As in Theorem 1, wn+1 = α(rn )wn + β(rn yn ) As assumed in Theorem 1, the process (rn yn )n satisfies Assumptions 2 and 3. Let αL = mini=12m {α(ri )} and βL = mini=12m;j=12l {β(rn yn )}. Thus βL βL is the lower bound of the state space of wn . Let X = [ 1−α L +∞) × 1−αL {¯r1 r¯m }. Assumptions 2 and 3, and the regularity assumption of (αn βn )n guarantee that the process visits, with positive probability in finite time, a dense subset of its support; see Brandt (1986) and Saporta (2005, Theorem 2, p. 1956). The stochastic process (wn rn−1 )n is then ψ-irreducible and aperiodic.60 Let α˜ = maxi=12m {E(α(rn )|α(ri ))}. From Lemma 3(ii), we know U E(αn |αn−1 ) < 1 for any αn−1 . Thus α˜ < 1. Let wˆ = β1−+1 , where βU = α˜ βL ˆ × {¯r1 r¯m }. Pick a function maxi=12m;j=12l {β(rn yn )}. Let C = [ 1−αL w] V (wn rn−1 ) = wn so that E(V (wn+1 rn )|(wn rn−1 )) = E(wn+1 |(wn rn−1 )) = E(α(rn )|rn−1 )wn + E(β(rn yn )|rn−1 ) ≤ wn − 1 + (βU + 1)IC (wn rn−1 ) = V (wn rn−1 ) − 1 + (βU + 1)IC (wn rn−1 ) Thus (wn rn−1 )n satisfies the drift condition of Tweedie (2001). For a sequence of measurable set Bn with Bn ↓ ∅, there are two cases: (i) Bn is contained in a compact set in X and (ii) Bn has forms of (xn +∞) × r¯i or of the union of such sets. In both cases it is easy to show that lim sup P((w r) Bn ) = 0
n→∞ (wr)∈C
where P(· ·) is the one-step transition probability of the stochastic process (wn rn−1 )n . Thus (wn rn−1 )n satisfies the uniform countable additivity condition of Tweedie (2001). 60 Alternatively to the regularity assumption, we could assume a continuous distribution for yn (and hence for βn ) Irreducibility would then easily follow; see Meyn and Tweedie (2009, p. 76).
149
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As a consequence, (wn rn−1 )n satisfies condition A of Tweedie (2001): V (wn rn−1 ) = wn is everywhere finite and (wn rn−1 )n is ψ-irreducible. By Theorem 3 of Tweedie (2001), we know that the set C is petite.61 Also we have E(V (wn+1 rn )|(wn rn−1 )) − V (wn rn−1 ) = E(wn+1 |(wn rn−1 )) − wn = E(α(rn )|rn−1 )wn + E(β(rn yn )|rn−1 ) − wn ≤ −(1 − α)w ˆ n + βU IC (wn rn−1 ) = −(1 − α)V ˆ (wn rn−1 ) + βU IC (wn rn−1 ) We then have that (wn rn−1 )n is ψ-irreducible and aperiodic, V (wn rn−1 ) = wn is everywhere finite, and the set C is petite. By Theorem 16.1.2 of Meyn and Tweedie (2009), we then obtain that (wn rn−1 )n is V -uniformly ergodic. Q.E.D. The wealth of a household of age τ wn (τ), is given by (4). Recall that we use the notational shorthand wn = wn (0) The cumulative distribution function of the stationary distribution of wealth of a household of age τ, F(w; τ), is then given by F(w; τ) =
l
Pr(yj )
I{σw (rn τ)wn +σy (rn τ)yj ≤w} dν
j=1
where I is an indicator function and ν is the product measure of the stationary distribution of (wn rn ), which exists and is unique as a direct consequence of Lemma A.2. The cumulative distribution function of wealth w in the population is then T 1 F(w) = Fτ (w) dτ T 0 Note that P(wn (τ) > w) =
l
Pr(yj )
I{σw (rn τ)wn +σy (rn τ)yj >w} dν
j=1
61
Note that (i) every subset of a petite set is petite and (ii) when we pick any w, such that U
ˆ the proof goes through. By these two facts we could show that every w > β1−+1 , to replace w, α˜ compact set of X is petite. Thus by Theorem 6.2.5 of Meyn and Tweedie (2009) we know that (wn rn−1 )n is a T -chain. For another example of stochastic process in economics with the property that every compact set is petite, see Nishimura and Stachurski (2005).
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and σw (rn τ) and σy (rn τ) are continuous functions of rn and τ. Since the number of states of rn is finite and τ ∈ [0 T ], there exist σwL , σwU , and σyU such that 0 < σwL ≤ σw (rn τ) ≤ σwU and σy (rn τ) ≤ σyU . Let y U = max{y¯1 y¯l }. We have I{σw (rn τ)wn +σy (rn τ)yj >w} ≥ I{σwL wn >w} and I{σw (rn τ)wn +σy (rn τ)yj >w} ≤ I{σwU wn +σyU y U >w} Hence w − σyU y U w P wn > L ≤ P(wn (τ) > w) ≤ P wn > σw σwU We then have
T
1 − F(w) =
P(wn (τ) > w) 0
1 dτ T
Thus w − σyU y U w P wn > L ≤ 1 − F(w) ≤ P wn > σw σwU and 1 − F(w) 1 − F(w) ≤ lim sup ≤ (σwU )μ k −μ w→+∞ w w−μ w→+∞
(σwL )μ k ≤ lim inf
since limw→+∞ P(wwn−μ>w) = k. We conclude that the wealth distribution in the population has a power tail with the same exponent μ, that is, 0 < k1 ≤ lim inf
w→+∞
1 − F(w) 1 − F(w) ≤ lim sup ≤ k2 w−μ w−μ w→+∞
Q.E.D.
We can also show the following claim: CLAIM 1: When (rn )n is i.i.d., the asymptotic power law property with the same power μ is preserved for each age cohort and the whole economy: ∃k˜ > 0 such that 1 − F(w) ˜ = k w→+∞ w−μ lim
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151
PROOF: When (rn )n is i.i.d., 1 − F(w) T 1 = P(wn (τ) > w) dτ T 0 T l m w − σy (ri τ)yj 1 dτ P wn > Pr(ri ) Pr(yj ) = σw (ri τ) T 0 i=1 j=1 Since σw (ri τ) and σy (ri τ) are continuous functions of τ on [0 T ], there exist τ˜ i , τˆ i ∈ [0 T ] such that for ∀t ∈ [0 T ], σw (ri τ) ≤ σw (ri τ˜ i ) and σy (ri τ) ≤ σy (ri τˆ i ). Thus w − σy (ri τ)yj w − σy (ri τˆ i )yj ≤ P wn > P wn > σw (ri τ) σw (ri τ˜ i ) When w is sufficiently large, w − σy (ri τˆ i )yj P wn > σw (ri τ˜ i ) w−μ since limw→+∞
P(wn >w) w−μ
is bounded
= c. Thus by the bounded convergence theorem, we have
w − σy (ri τ)yj P w > n T 1 σw (ri τ) dτ lim −μ w→+∞ 0 w T w − σy (ri τ)yj T P wn > 1 σw (ri τ) dτ = lim −μ w→+∞ w T 0
Thus 1 − F(w) w→+∞ w−μ lim
w − σy (ri τ)yj T P wn > m l 1 σw (ri τ) dτ Pr(ri ) Pr(yj ) = lim w−μ T 0 w→+∞ i=1 j=1 T m μ =k Pr(ri ) (σw (ri τ)) dτ 0 Q.E.D. i=1
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PROOF OF PROPOSITION 1: Since μ > 1, (αn )μ is an increasing convex function in αn . If α(rn ) is a convex function of rn , then α(rn )μ is also a convex function of rn ; hence, −α(rn )μ is a concave function of rn . By second order stochastic dominance, we have E(−α(rn )μ ) ≥ E(−α(rn )μ ) so Eα(rn )μ ≤ Eα(rn )μ and 1 = Eα(rn )μ ≤ Eα(rn )μ . Let μ solve Eα(rn )μ = 1. Suppose μ > μ. By Holder’s inequality, we have Eα(rn )μ < (Eα(rn )μ˜ )μ/μ = 1. This is a contradiction. Thus we have μ ≤ μ. Q.E.D. PROOF OF PROPOSITION 2: From the definition of αn , we have α(rn ) =
(1 − b)ern T T −1/σ (σ−1)/σ χ (1 − b) eA(rn )t dt + 1 0
n > 0Thus an infinitesimal increase in χ shifts and it is easy to show that ∂α ∂χ the state space a to the right. Therefore, elements of the nonnegative matrix [Aμ P ] increase, which implies that the dominant root λ(Aμ P ) increases. However, we know from Saporta (2005) that ln(λ(Aμ P )) is a convex function of μ.62 At μ = 0, it is equal to zero, since A0 is the identity matrix and P is a stochastic matrix with dominant root equal to unity. At μ = 0, the function ln(λ(Aμ P )) is also decreasing. (See Saporta (2005, Proposition 2, p. 1962).) Then ln(λ(Aμ P )) must be increasing at the positive value of μ which solves ln λ(Aμ P ) = 0 Therefore, to preserve ln(λ(Aμ P )) = 0, μ must decline. Q.E.D.
PROOF OF PROPOSITION 3: From (5), we have α(rn ) =
ern T T
χ−1/σ (1 − b)−1/σ
eA(rn )t dt + (1 − b)−1
0
Thus
∂αn ∂b
< 0. To see
∂αn ∂ζ
< 0, we rewrite the expression of α(rn ) as
α(rn ) = (1 − b)
B(b)
T
−(rn −ρ)/σT
e
−A(rn )(T −t)
e
−rn T
dt + e
B(b)
0 n) = σ−1 ≥ Note that A(rn ) = rn − rnσ−ρ and B(b) = χ1/σ (1 − b)(1−σ)/σ . Then ∂A(r ∂rn σ ∂αn 0, since σ ≥ 1 by Assumption 1, and also B(b) > 0. Thus ∂rn > 0. Higher ζ n means lower rn . We have ∂α < 0. Now the proof is identical to the proof ∂ζ
62
See Loeve (1977, p. 158).
153
THE DISTRIBUTION OF WEALTH
of Proposition 2 in the reverse direction since ∂αn > 0. ∂χ
∂αn ∂b
< 0 and
∂αn ∂ζ
< 0, whereas Q.E.D.
PROOF OF PROPOSITION 4: We apply the results of Roitershtein (2007) on exponents of the tails of the limiting distribution. In the MA(1) case where ln αn = ηn + θηn−1 we have n
ln αt = θη0 + ηn +
t=1
Thus
n−1 (1 + θ)ηt t=1
n μ 1 lim ln E αt n→+∞ n t=1
n−1 1 μ n ln αt 1 ln Ee t=1 = lim ln Eeμ t=1 (1+θ)ηt n→+∞ n n→+∞ n n−1 n−1 1 1 μ(1+θ)ηt = lim ln Ee = lim ln Eeμ(1+θ)ηt = ln Eeμ(1+θ)ηt n→+∞ n n→+∞ n t=1 t=1
= lim
n
Thus limn→+∞ n1 ln(E(
t=1
αt )μ ) = 0 implies
Eeμ(1+θ)ηt = 1 Consider, in turn, the AR(1) case ln αn = θ ln αn−1 + ηn We have n t=1
Thus
1 − θn−t+1 θ(1 − θn ) ln α0 + ηt 1−θ 1 − θ t=1 n
ln αt =
n μ 1 lim ln E αt n→+∞ n t=1
n 1 μ n ln αt 1 n−t+1 )/(1−θ))η t ln Ee t=1 = lim ln Eeμ t=1 ((1−θ n→+∞ n n→+∞ n n 1 n−t+1 )/(1−θ))μη t = lim ln(Ee((1−θ ) = ln(Ee(1/(1−θ))μηt ) n→+∞ n t=1
= lim
154
J. BENHABIB, A. BISIN, AND S. ZHU
n
Thus limn→+∞ n1 ln(E(
t=1
αt )μ ) = 0 implies
Ee(μ/(1−θ))ηt = 1
Q.E.D. REFERENCES
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Dept. of Economics, New York University, 19 West 4th Street, 6th Floor, New York, NY 10012, U.S.A. and NBER; [email protected],
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Dept. of Economics, New York University, 19 West Street, 6th Floor, New York, NY 10012, U.S.A. and NBER; [email protected], and Dept. of Economics, National University of Singapore, Faculty of Arts & Social Sciences, AS2 Level 6, 1 Arts Link, Singapore 117570; [email protected]. Manuscript received February, 2009; final revision received June, 2010.
Econometrica, Vol. 79, No. 1 (January, 2011), 159–209
SELECTION AND COMPARATIVE ADVANTAGE IN TECHNOLOGY ADOPTION BY TAVNEET SURI1 This paper investigates an empirical puzzle in technology adoption for developing countries: the low adoption rates of technologies like hybrid maize that increase average farm profits dramatically. I offer a simple explanation for this: benefits and costs of technologies are heterogeneous, so that farmers with low net returns do not adopt the technology. I examine this hypothesis by estimating a correlated random coefficient model of yields and the corresponding distribution of returns to hybrid maize. This distribution indicates that the group of farmers with the highest estimated gross returns does not use hybrid, but their returns are correlated with high costs of acquiring the technology (due to poor infrastructure). Another group of farmers has lower returns and adopts, while the marginal farmers have zero returns and switch in and out of use over the sample period. Overall, adoption decisions appear to be rational and well explained by (observed and unobserved) variation in heterogeneous net benefits to the technology. KEYWORDS: Technology, heterogeneity, comparative advantage.
1. INTRODUCTION FOOD SECURITY is a major social and economic issue across many sub-Saharan African economies. Agricultural yields (output per acre) have fallen in the last 1 I would like to thank three anonymous referees for extensive comments that have substantially improved the paper. I am extremely grateful to Michael Boozer for many discussions on this project. A special thanks to Joe Doyle, Billy Jack, Gustav Ranis, Paul Schultz, Tom Stoker, and Christopher Udry. I would also like to thank Ken Chay, Robert Evenson, Michael Greenstone, Penny Goldberg, Koichi Hamada, Thomas Jayne, Fabian Lange, Ashley Lester, Anandi Mani, Sharun Mukand, Ben Polak, Steve Pischke, Roberto Rigobon, James Scott, T. N. Srinivasan, and seminar audiences at Berkeley Economics Department, Haas School of Business, Harvard/MIT, Hunter College, London School of Economics, the NBER Productivity Lunch, NEUDC, Oxford, Princeton, Sloan School of Management, Stanford, Stanford Graduate School of Business, University of California at San Diego, University of Virginia, and Yale for all their comments. I would like to acknowledge the financial support of the Yale Center for International and Area Studies Dissertation Fellowship, the Lindsay Fellowship, and the Agrarian Studies Program at Yale. The data come from the Tegemeo Agricultural Monitoring and Policy Analysis (TAMPA) Project, which is between Tegemeo Institute at Egerton University, Kenya and Michigan State University, and is funded by USAID. A special thanks to Thomas Jayne and Margaret Beaver at Michigan State University for all their help. I am sincerely grateful to the Tegemeo Institute and Thomas Jayne for including me in the 2004 survey and to Tegemeo for their hospitality during the field work. I would like to thank James Nyoro, Director of Tegemeo, as well as Tegemeo Research Fellows Miltone Ayieko, Joshua Ariga, Paul Gamba, and Milu Muyanga, Senior Research Assistant Frances Karin, and Research Assistants Bridget Ochieng, Mary Bundi, Raphael Gitau, Sam Mburu, Mercy Mutua, and Daniel Kariuki, and all the field enumeration teams. An additional thanks to Margaret Beaver and Daniel Kariuki for their work on cleaning the data. The rainfall data come from the Climate Prediction Center, part of the USAID/FEWS project: a special thanks to Tim Love for his help with these data.
© 2011 The Econometric Society
DOI: 10.3982/ECTA7749
160
TAVNEET SURI TABLE I TRENDS IN YIELDS OF STAPLES: AVERAGE ANNUAL % CHANGES IN YIELDS (HG/HA) BY DECADEa 1961–1970
1971–1980
1981–1990
1991–2004
Kenya Maize Wheat
0362 5646
2373 2333
1169 −3078
−1198 0984
India Maize Wheat Rice
1502 4876 0954
0842 2514 1714
1900 3343 3310
2572 1235 0838
Mexico Maize Wheat
2057 4586
4267 3204
−0548 −0255
1447 1664
Zambia Maize
−0267
10403
1571
−1707
a Surprisingly, in the 1960’s the levels of yields of maize in Kenya, India, and Mexico were comparable and very similar (all ranging between 10,000 and 12,000 hg/ha). The levels of yields in Zambia were slightly lower (about 8000 hg/ha), whereas those in Uganda and Malawi were comparable to those in Kenya. Source: FAOSTAT Online Database.
decade across many of these economies, despite the widespread availability of technologies that increase yields. For instance, Table I shows the falling yields of staple crops over the last decade in Kenya, compared with increasing yields in India and Mexico. Kenya has clearly not been able to take the same advantage of improvements in agricultural technologies as have India and Mexico during their green revolutions. This is not an isolated example, and is very worrisome for economic policies intended to enhance food production and agricultural incomes. The leading edge of agricultural technology has improved over the past few decades. Field trials at experiment stations across Kenya have shown that hybrid maize and fertilizers can increase yields and profits of maize significantly, with increases ranging from 40% to 100% (see Gerhart (1975), Kenya Agricultural Research Institute (1993a, 1993b), Karanja (1996)).2 Despite this, the aggregate adoption rates of hybrid maize and fertilizer remain far below 100%, with no sustained increases in adoption over the past decade. This contrasts with the experience in more developed countries; for example, adoption rates 2
The Fertilizer Use Recommendations Project (FURP) in Kenya also shows high returns to hybrid maize and fertilizer (see Hassan, Njoroge, Njore, Otsyula, and Laboso (1998b)). These results are confirmed for fertilizer in the small sample of Duflo, Kremer, and Robinson (2008a) for western Kenya. In fact, Duflo, Kremer, and Robinson (2008b) analyzed the rates of return to fertilizer and show high, but variable, returns to fertilizer.
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of hybrid in the United States are close to 100% (Griliches (1980), Dixon (1980)). The facts that improved agricultural technologies are not universally adopted and that adoption rates have remained persistently low over a long period of time illustrate the empirical puzzle that motivates this paper. I find that if farmer heterogeneity is properly taken into account, there is no puzzle. The farmers with high net returns to the technology adopt it and the farmers with low returns do not. Persistent lack of adoption is a reflection of the distribution of (observable and unobservable) costs and benefits of the technology. The approach here models households’ adoption decisions in an environment with household-specific heterogeneity in the costs and benefits, and hence profits, to the technology. I estimate how the returns to the technology vary across farmers and then compare these returns to the adoption decisions of farmers. I find that farmers with low (or zero) returns to the technology are precisely the farmers who do not adopt it. In particular, I use a generalized Roy model along the lines of Heckman and Vytlacil (1998), where the expected profits from hybrid maize are heterogeneous and drive adoption decisions. This theoretical model implies a yield function for maize with a correlated random coefficient structure, which is estimated using a generalization of Chamberlain’s (1982, 1984) method for fixed effects models, applied here to a correlated random coefficients model. The estimation results imply a distribution of returns that can endogenously affect adoption decisions. I therefore provide an empirical resolution of the puzzle that, despite high average returns to hybrid use, the marginal returns are low, and, given the infrastructure constraints faced by farmers, adoption decisions are on the whole rational. The estimation strategy in this paper allows for two different forms of household-specific heterogeneity in maize production: absolute advantage (independent of the technology used) and comparative advantage, which measures the relative productivity of a farmer in hybrid over nonhybrid. The novel econometric contribution here is to analyze the role of comparative advantage in adoption decisions, to empirically estimate its importance, and to estimate its distribution. I also discuss why this approach is preferable in development settings to related approaches used to study firms in developed countries, where the evolution of firm efficiencies is more likely to be important in consistently estimating output and investment equations. I use an extensive panel data set representative of maize-growing Kenya for the period 1996–2004. I first document the adoption patterns of households for both hybrid maize and fertilizer. Aggregate adoption is stable over the period, but, surprisingly, at least 30% of households switch into and out of hybrid use from period to period.3 The panel structure of the data allows me to identify the distribution of returns to hybrid maize and to estimate the correlated 3 This lack of a trend with underlying switching of use from period to period is seen in other parts of Africa. Dercon and Christiaensen (2005) found identical patterns for fertilizer use in
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random coefficient structure of the underlying yield functions. The panel also allows me to construct counterfactual distributions of returns for all the farmers in the sample, including those who do not use hybrid. It is important for identification that some farmers switch in and out of use of the technology, and also that there are farmers in the sample who use hybrid maize in every period. I discuss how the identification assumptions used are supported by the timing of maize production, other characteristics of the production process, and the fact that shocks to yields like rainfall are observed in the data. I find strong evidence of heterogeneity in the returns to hybrid maize, especially in the sample excluding the 2 districts (out of a total of 22) with high adult mortality. I estimate the distribution of returns, and compare the mean of this distribution to estimates from standard approaches that fail to take full account of farmer heterogeneity, such as ordinary least squares (OLS), instrumental variable (IV), and fixed effects models.4 OLS estimates of the average return are in the range of 50–100%, models with fixed effects suggest returns of close to 0%, and IV estimates (using supply constraints as instruments) are on the order of 150–200%. The estimated mean gross return from my approach is 60%, but some farmers have returns as high as 150%, while there are many who have returns either close to zero or (in some cases) negative. These estimated returns control for input use, but do not account for other costs (such as the costs of accessing the technology) and are therefore gross, rather than fully net returns. The joint distribution of estimated returns and adoption decisions displays some remarkable features. There are three subgroups of farmers in the sample. A small group of farmers has extremely high counterfactual returns to hybrid (about 150%), yet they choose not to adopt. This is rather striking and seems to deepen the initial puzzle, but is well explained by supply and infrastructure constraints, such as long distances to seed and fertilizer distributors. These farmers have higher costs to using hybrid as they have poor access to input suppliers. Their overall net counterfactual returns are therefore rather low. A comparatively larger group of farmers has lower, though still high, returns and they adopt hybrid in every period. Finally, a third group of farmers has essentially zero net returns and they switch in and out of use of hybrid from period to period. These are marginal farmers who switch easily when subject to shocks to the cost of and access to hybrid seed and fertilizer. The heterogeneity in returns to this technology has important implications for policy. For one, encouraging complete adoption of a technology that has a large average return among existing adopters may be very inefficient due to Ethiopia. Duflo, Kremer, and Robinson (2008b) also found a lot of switching behavior in fertilizer use in western Kenya. 4 When there is heterogeneity in returns, IV estimates a local average treatment effect, that is, the returns for only the subpopulation that is affected by the instrument. This could explain why OLS and IV are so different here.
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the much lower returns for the nonadopters. In addition, knowing the distribution of returns to a technology allows for focused policy interventions that can be cost effective. For example, for farmers who would have high returns but are constrained on the supply side, alleviating their constraints by targeted distribution of inputs and infrastructure improvements could improve yields dramatically. Similarly, the unconstrained farmers would benefit from the development of new hybrid strains. This research also illustrates the importance of the distributional consequences of policy and sheds light on the “scaling up” of policy (see Attanasio, Meghir, and Szekely (2003)).5 The literature provides a number of explanations for low adoption, ranging from learning models (Foster and Rosenzweig (1995) and Conley and Udry (2010)), informational barriers, credit constraints, taste preferences, differences in agroecological conditions, and local costs and benefits to time inconsistency and the lack of effective commitment devices (see Duflo, Kremer, and Robinson (2008a)). The dominant approach has been to frame adoption decisions in a learning environment where benefits and costs to the technologies are homogeneous but unknown, and are learned over time. The approach in this paper is in contrast to much of the literature. Since these technologies have been available for many years, and are well known and understood6 (as in Duflo, Kremer, and Robinson (2008a, 2008b)), the approach here assumes the benefits and costs are known ex ante, but are spatially heterogeneous across farmers. The rest of this paper is structured as follows. Section 2 outlines some relevant empirical literature. Section 3 describes the institutional context and the data. Section 4 discusses a theoretical framework for adoption decisions, clarifying the role of comparative advantage and the identification assumptions needed to estimate the yield function. It also describes the empirical model and its estimation.7 Section 5 discusses some descriptive baseline regressions, such as OLS, fixed effects, IV, various treatment effects, and preliminary evidence that heterogeneity in returns is relevant. Section 6 presents the results from the correlated random coefficient model and the associated distribution of returns. Section 7 discusses implications of the results and a host of alternative models. Section 8 concludes. 5
The questions posed here cannot be answered with an experiment which randomizes the technology across farmers without specific assumptions. With experimental data, one can test for the presence of heterogeneous returns, but estimating the distribution of returns requires assumptions about the underlying selection process, which is randomized away in such an experiment (see Heckman, Smith, and Clements (1997)). 6 For example, in my sample, about 90% of households have used hybrid maize at some point in the past. The mean number of years since first use for hybrid maize is 19.4 years and for fertilizer is 20.5 years. 7 Estimation programs are provided in the Supplemental Material (Suri (2011)).
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2. LITERATURE REVIEW This section briefly summarizes some of the empirical literature on technology adoption, focusing on a few studies in sub-Saharan Africa (see Suri (2006) for a more detailed review).8 I describe the literature that has looked at the roles of heterogeneity, credit constraints, learning externalities, and, finally, the more recent experimental research on similar technologies in Kenya. The seminal empirical paper on technology adoption is by Griliches (1957), who looked at heterogeneity across local conditions in the adoption speeds of hybrid corn in the midwestern United States and emphasized the role of expected profits and scale. He noted how the speed of adoption across geographical space depended on the suppliers of the technology and when the seed was adapted to local conditions. For Kenya, Gerhart (1975) tracked the adoption of hybrid maize in western Kenya in the early 1970’s. He highlighted the fast diffusion of hybrid and identified risk, education, credit availability, extension services, and use of fertilizer as constraints. There is a vast literature that describes the observable heterogeneity that drives adoption decisions. For example, Schultz (1963) and Weir and Knight (2000) emphasized education. Various CIMMYT (The International Wheat and Maize Improvement Center) studies9 across Kenya highlighted the unavailability and untimely delivery of the technologies, labor and input use, costs, and unfavorable climactic conditions.10 There are a number of papers that focus on credit constraints, mostly using self-reports (see Croppenstedt, Demeke, and Meschi (2003) for Ethiopia, Salasya, Mwangi, Verkuijl, Odendo, and Odenya (1998) for western Kenya). Dercon and Christiaensen (2005) showed that the possibility of a poor harvest (and hence very low consumption) can account for the low use of fertilizer in Ethiopia. Their data showed similar adoption patterns to Kenya, with a lot of switching of technology use from period to period. 8 The literature on technology adoption is too vast to review here: excellent reviews are Sunding and Zilberman (2001), Sanders, Shapiro, and Ramaswamy (1996), Rogers (1995), Feder, Just, and Zilberman (1985), David (2003), and Hall (2004). For the theoretical literature, see Besley and Case (1993), Banerjee (1992), and Just and Zilberman (1983). For studies of land management practices, see Mugo, Place, Swallow, and Lwayo (2000); for agricultural extension, see Evenson and Mwabu (1998); for property rights, see Place and Swallow (2000). 9 See Doss (2003) and De Groote, Doss, Lyimo, and Mwangi (2002) for a review of all the CIMMYT surveys in Kenya. 10 For example, Makokha, Kimani, Mwangi, Verkuijl, and Musembi (2001) looked at fertilizer and manure use in Kiambu district. The main (self-reported) constraints to use were unavailability and untimely delivery, high labor costs, and high prices of inputs. Ouma, Murithi, Mwangi, and Verkuijl (2002) found that, in Embu district, gender, agroclimatic zone, manure use, hiring of labor, and extension services were significant determinants of the adoption of improved seed and fertilizer. Wekesa, Mwangi, Verkuijl, Danda, and De Groote (2003) looked at the adoption of various hybrids and fertilizer in the coastal lowlands where the nonavailability and high cost of seed, unfavorable climatic conditions, perceptions of sufficient soil fertility, and lack of money were reasons for low use.
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Much of the academic literature has focused on the learning externality described by Besley and Case (1993), which I find little evidence for. These papers mostly studied the green revolution in India, where the learning externality was certainly key in the rapid growth of new hybrid varieties developed for India. Foster and Rosenzweig (1995) considered the adoption of high yielding varieties in India and found that farmers with more experienced neighbors are more profitable than those without. Munshi (2003) found these impacts to be heterogeneous across crops. Conley and Udry (2010) studied the adoption of fertilizer in the small-scale pineapple industry in Ghana and found evidence of social learning within information neighborhoods (defined by farmers as the people they discuss farming with).11 More recently, there has been a growing experimental literature. A number of field trials at Kenya Agricultural Research Institute (KARI) experiment stations have shown large increases in yields from hybrid seed and fertilizer. One of the early experimental studies was the Fertilizer Use Recommendations Project (FURP) in the early 1990’s to understand optimal rates of fertilizer use (see Corbett (1998)). FURP recorded yields about half of those found at experiment stations (KARI (1993a, 1993b)). Hassan et al. (1998b) showed higher adoption and faster diffusion of hybrid in high potential areas, blaming poor extension, infrastructure, and seed distribution in the marginal areas. Hassan, Murithi, and Kamau (1998a) found that farmers apply less fertilizer than optimal, leading to an estimated 30% yield gap. De Groote, Overholt, Ouma, and Mugo (2003) considered an ex ante impact assessment of the Insect Resistant Maize for Africa project that develops Genetically Modified (GM) maize varieties that are more resistant to stem borers. They experimentally estimated a 13.5% crop loss, valued at about $80 million, due to these insects.12 Duflo, Kremer, and Robinson (2008a) ran experiments to understand the returns to fertilizer and the low use of fertilizer in western Kenya. They found that the average rate of return for investing in top-dressing fertilizer is between 52% and 85%, and that learning effects were extremely small. Their most significant contribution was to implement the Savings and Fertilizer Initiative (SAFI) as a commitment device for farmers. They showed that offering SAFI at harvest time (vs. planting) increases adoption by between 11 and 14 percentage points, and they showed that this effect on increasing adoption is about equivalent to a 50% subsidy to the fertilizer price. They concluded that behavioral biases prevent farmers from making profitable investments on their farms and, hence, that small subsidies at the right time (i.e., at harvest) or offering farmers a commitment device to investing in fertilizer at this time can improve adoption rates. Duflo, Kremer, and Robinson 11 Other studies of learning in sub-Saharan Africa include Bandiera and Rasul (2003) for sunflowers in Mozambique and Moser and Barrett (2003) for a rice production method in Madagascar. 12 See http://apps.cimmyt.org/english/wpp/gen_res/irma.htm and Smale and De Groote (2003) for more information on the CIMMYT IRMA and GM projects in Kenya.
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(2008b) looked at the heterogeneity in the returns to fertilizer. They found an average annual return of about 70%, but zero returns to a combination of fertilizer and hybrid, and some evidence of heterogeneity, stating that “the returns to fertilizer are smaller when the control plot does better.” The empirical findings of Duflo, Kremer, and Robinson (2008a, 2008b) on adoption patterns are consistent with the evidence here, although, as the discussion in Section 7 indicates, I find different reasons for the differential returns and, therefore, for the lack of adoption. 3. INSTITUTIONAL CONTEXT AND DATA I now describe the relevant institutional detail and the data, which help motivate the model and empirical approach used. Maize13 is the main staple in Kenya, with 90% of the population depending on it for income (Nyameino, Kagira, and Njukia (2003)). Hybrid maize increases yields and can be more resistant to agricultural stress (Hassan et al. (1998b)). Both fertilizer and hybrid have been available since the 1960’s: more than 20 modern varieties of seed have been released since 1955, although later releases have not shown large yield improvements.14 In the data, about 70% of plots are planted with a hybrid variety released in 1986. Government recommendations for the types and quantities of hybrid seed and fertilizer vary across the country,15 and both must be purchased each season. Hybrid seed replanted from the previous year’s harvest (i.e., recycled hybrid seed) has little yield advantage over nonhybrid. Most farmers (about 80% in the sample) use either both hybrid and fertilizer or neither. From 1965 to 1980, hybrid variety 611 diffused in western Kenya “at rates as fast as or faster than among farmers in the U.S. corn belt during the 1930’s– 1940’s” (Gerhart (1975)), but this changed in the 1990’s.16 The most important 13
McCann (2005) described the fascinating history of maize in Africa. Smale and Jayne (2003) provided an excellent review of maize policy in Kenya. 14 Karanja (1996) stated “newly released varieties in 1989 had smaller yield advantages over their predecessors than the previously released ones. . . research yields were exhibiting a ‘plateau effect’.” Examples he gives are KSII, which was followed in time by H611 (with a 40% yield advantage), then H622 (16%), and then H611C (12%). H626, which had a 1% yield advantage over H625, was released 8 years later. 15 See Ouma et al. (2002), Hassan et al. (1998b), Salasya et al. (1998), Wekesa et al. (2003), and Karanja (1996). 16 Reform of the cereal sector began in 1988, followed by some liberalization in 1994. Smale and Jayne (2003) and Karanja (1996) attributed early successes to good germplasm, effective research, good distribution, and coordinated marketing of inputs and outputs. This changed in the 1990’s as earlier policies of large subsidies, price supports, pan-territorial seed/output pricing, and restrictions on cross-district trade resulted in large fiscal deficits. The National Cereals and Produce Board accrued losses of 5% of gross domestic product (GDP) in the 1980’s. For more on the reforms, see Jayne, Yamano, Nyoro, and Awuor (2001), Karanja, Jayne, and Strasberg (1998), Jayne, Myers, and Nyoro (2005), Nyoro, Kiiru, and Jayne (1999), and Wanzala, Jayne, Staatz, Mugera, Kirimi, and Owuor (2001).
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government policy for this sector has been pan-territorial seed pricing. The hybrid seed was all produced by KARI on research stations and then distributed by Kenya Seed Company. The price of seed was not driven by market forces, shows little variance over time, and was fixed across the country. In the data, only 1% of seed purchases are not from Kenya Seed Company. This panterritorial seed pricing created poor incentives for suppliers to locate in distant areas or far from markets.17 3.1. Data The data set comes from the Tegemeo Agricultural Monitoring and Policy Analysis Project, a joint project between Tegemeo Institute at Egerton University, Kenya and Michigan State University. It is a household level panel survey of Kenya, representative of rural maize-growing areas. Figure 1 shows a map of the survey villages and the population density across the country (darker shades represent greater density). Although data are available for the years 1997, 1998, 2000, 2002, and 2004, for most of the analysis in this paper, I use the
FIGURE 1.—Population density and location of sample villages. 17
The seed market was liberalized post 2004; in late 2008, fertilizer was subsidized.
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FIGURE 2.—Hybrid maize adoption patterns by province.
1997 and 2004 data. The 1997 and 2004 surveys collect detailed agricultural input and output data, consumption, income, demographics, infrastructure, and credit information.18 The panel sample covers just over 1200 households. Figures 2 and 3 show the heterogeneity in hybrid and fertilizer use across provinces and over time. Figure 2 shows the stability in aggregate hybrid maize adoption over time and the persistence of cross-sectional differences.19 In principle, hybrid use could be a continuous variable as farmers plant quantities of hybrid, but only 2% of farmers in the sample plant both hybrid and nonhybrid in a given season. I therefore take hybrid use to be binary throughout the paper. Figure 3A shows the trends across provinces in the fraction of households that use inorganic fertilizer on maize and Figure 3B shows the total value (in constant Kenyan shillings) of inorganic fertilizer used, both showing similar persistent cross-sectional differences.20 Figures 4A and 4B show the distribution of yields for 1997 and 2004 by technology, illustrating that mean yields are much higher and the variance of yields lower in the hybrid sector, although 18 The 2000 and 1998 surveys are similar, but data on family labor were not collected in 2000, and 1998 covers only a subsample of 612 households. The 2002 survey was a short proxy survey. 19 The Coast province looks rather different in 2004. All the results in the paper are similar if the households in the Coast province are dropped from the sample (59 households). In addition, looking across wealth/asset or acreage quintiles, the pattern is identical, with no systematic temporal variation in average adoption. 20 Aggregate yields are not stable over time. There is a sharp drop in yields around 1997–1998, the result of El Nino. However, there are no dynamics in any of the main inputs. For example, the acreage farmers plant to maize is constant over this period. For more on these trends, see Suri (2006).
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FIGURE 3A.—Fraction of households using inorganic fertilizer by province.
both these summary measures could be artifacts of selection (as I discuss in Section 4). Table IIA shows summary statistics for my sample of households for 1997 and 2004. Of the 26 different types of fertilizer used, Table IIA shows the three most popular (diammonium phosphate (DAP), calcium ammonium nitrate (CAN), and monoammonium phosphate (MAP)). Table IIB breaks out
FIGURE 3B.—Real expenditure on inorganic fertilizer by province.
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FIGURE 4A.—Marginal distribution of yields by sector, 1997.
some of these variables by hybrid/nonhybrid use for 1997 and 2004. Yields are significantly lower across the board in the nonhybrid sector. Maize acreage and total seed planted are not different across sectors in 1997 and only just in 2004. Fertilizer, land preparation costs, and main season rainfall are different across the two sectors. Finally, hired labor is only barely different in 2004 and family labor is only barely different in 1997.
FIGURE 4B.—Marginal distribution of yields by sector, 2004.
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TABLE IIA SUMMARY STATISTICS BY SAMPLE YEARa
Yield (log maize harvest per acre) Acres planted Total seed planted (kg per acre) Total purchased hybrid planted (kg per acre) Hybrid (dummy) Fertilizer (kg DAP (diammonium phosphate) per acre) Fertilizer (kg MAP (monoammonium phosphate) per acre) Fertilizer (kg CAN (calcium ammonium nitrate) per acre) Total fertilizer expenditure (KShs per acre) Land preparation costs (KShs per acre) Family labor (hours per acre) Hired labor (KShs per acre) Main season rainfall (mm) Distance to closest fertilizer seller (km) Household size
1997 Sample
2004 Sample
5907 (1153) 1903 (3217) 9575 (7801) 6273 (6926) 0658 (0475) 20300 (38444) 1566 (10165) 6473 (24727) 13617 (22463) 96088 (12371) 29325 (34749) 17660 (33464) 62083 (25643) 6288 (9774) 7109 (2671)
6350 (0977) 1957 (2685) 9072 (6863) 5080 (5260) 0604 (0489) 24610 (34001) 0308 (4538) 8957 (21702) 13546 (18312) 54143 (10228) 35427 (35268) 14274 (21303) 72811 (29329) 3469 (5964) 8409 (3521)
a Standard deviations are given in parentheses. KShs is Kenyan shillings (exchange rate over this period was KShs 75 = $ 1). All monetary variables are in real terms.
The land preparation costs and labor variables merit some discussion. The land preparation costs variable reported in the summary statistics covers costs that are incurred either before or right at the time of the seed choice and that do not include labor costs. The survey collects the labor data for each component of the production process, separately for hired and family labor. On average, about 75% of the total labor used for maize by these households is family labor, which is difficult to value (it should be valued at the shadow wage, which is not observable).21 The main production activities that use labor are land preparation (about 23% of total labor), planting (10%), weeding (36%), and harvesting (30%). Harvesting includes all the postharvest activities (threshing, winnowing, bagging, storage), which put the maize into a form fit for sale or consumption. The survey records the harvest of maize in bags, that is, after all postharvest activities. Table IIC summarizes the labor variables by sector. Planting labor tends to be significantly different across the sectors since it is positively correlated with fertilizer use. Harvest and postharvest labor are also higher in the hybrid sector, unsurprising as the yields of hybrid are higher. In terms of weeding, hired labor is only different across the two sectors in 2004, and family weeding differs only in 1997. Total weeding labor is not different across the sectors in either 21
When I impute the shadow wage for family labor as the district level hired labor wage, I find that about half my sample has negative profits, implying this is not the correct shadow wage, and that these households tend to undervalue family labor relative to the district level wage rate.
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No. of households Yield (log maize harvest per acre) Total maize acres cultivated Total seed planted (kg per acre) Fertilizer (kg DAP per acre) Fertilizer (kg CAN per acre) Land preparation costs (KShs/acre) Expenditure on fertilizer (KShs/acre) Inorganic fertilizer use (dummy) Main season rainfall (mm) Hired labor (KShs/acre) Family labor (hours/acre) Distance to closest fertilizer seller (km) Household size
2004 Sample
Hybrid
Nonhybrid
Hybrid
Nonhybrid
791 6296 (0934)
411 5158 (1167)
726 6751 (0692)
476 5738 (1030)
1982 (3557) 9669 (6569)
1753 (2428) 9394 (9750)
2087 (3029) 8746 (4156)
1758 (2042) 9569 (9608)
28755 (44115) 4028 (13266) 37148 (37294) 5488 (13909) 9087 (29715) 1442 (7152) 12708 (24961) 3235 (13622) 10439 (12427) 80108 (12117) 65983 (10797) 36083 (9010) 19223 (25429) 28264 (74053) 18933 (19647) 53309 (12114) 07421 (04378) 02311 (04221) 08994 (03009) 04055 (04915) 65170 (22882) 56144 (29388) 82541 (21520) 57969 (33205) 18643 (26806) 15767 (43478) 16165 (21974) 11390 (19916) 26035 (26413) 35657 (46171) 3436 (3361) 37058 (37633) 4684 (7993) 9374 (1193) 2419 (2420) 5069 (8760) 7162 (2616)
7007 (2773)
8457 (3340)
8336 (3783)
a Standard deviations are given in parentheses. The mean of yields is higher and the standard deviation of yields is lower in the hybrid sector.
year. If I value family labor at the district hired labor wage, the dominant cost differences between hybrid and nonhybrid production come from seed and fertilizer costs, and not from labor cost differences. Finally, Table IID shows the transitions of hybrid use in the data, with 30% of households switching in and out of use over 1997, 2000, and 2004. Including 2002, this fraction becomes 39%. In the raw data, households that always plant hybrid have the highest average yields, followed by households that switch in and out of use. Those that never use hybrid have the lowest average yields. 4. THEORY AND EMPIRICAL FRAMEWORK This section first describes the theoretical foundations underlying my empirical model and then the empirical framework. I am interested in estimating a distribution of returns to the hybrid technology to assess if the farmers who do not adopt hybrid are those who do not benefit from it. I do this by estimating the yield (production) function underlying farmers’ adoption decisions. The exact structure of the adoption decisions is never estimated, so the adoption model below is for illustrative purposes. I base the model on a comparison of
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SELECTION AND COMPARATIVE ADVANTAGE TABLE IIC BREAKDOWN OF LABOR COSTS BY HYBRID/NONHYBRID USEa 1997 Sample
2004 Sample
Hybrid
Nonhybrid
Hybrid
Nonhybrid
Hired labor (KShs/acre) Land preparation Planting Weeding Harvest Postharvest activities Fertilizer application Other
4084 (1171) 1598 (3189) 6788 (1149) 3651 (6425) 2365 (4726) 1521 (8467) 0501 (9727)
5142 (3178) 1054 (2935) 6518 (1484) 1966 (7661) 9865 (3086) 1002 (1564) 0 (0)
4082 (8691) 1434 (3541) 6357 (1088) 1511 (3035) 2411 (5767) 1324 (7932) 2384 (2816)
5018 (1096) 7667 (2169) 4031 (9155) 9642 (3405) 4737 (1821) 2374 (2804) 1126 (2275)
Family labor (hours/acre) Land preparation Planting Weeding Harvest Postharvest activities Fertilizer application Other
4795 (1135) 2912 (3181) 9375 (1071) 4245 (4604) 4407 (5577) 3171 (8778) 0047 (1313)
1024 (2146) 3523 (4519) 1275 (1516) 4954 (1430) 3989 (7000) 1935 (1509) 0 (0)
5193 (9822) 3827 (5613) 1201 (1513) 5877 (7236) 6854 (9176) 3928 (9976) 2041 (1362)
9779 (1801) 3955 (4901) 1346 (1602) 4960 (6280) 4469 (5998) 1510 (6706) 2814 (2473)
a Standard deviations are given in parentheses. On average, family labor accounts for 75% of total labor use (the rest being hired labor). The main production activities that use labor are listed above. Land preparation accounts for 23% of total labor use on average, planting accounts for 10%, weeding accounts for 36%, and harvesting and postharvesting activities account for 30%. Postharvesting activities include threshing, winnowing, bagging, and storing the maize.
TABLE IID TRANSITIONS ACROSS HYBRID/NONHYBRID SECTORS FOR THE SAMPLE PERIODS 1997, 2000, AND 2004a Transition in Terms of Technology Used (1997 2000 2004)
N N N N H N N H H N H N H H H H
N H H H N H N H
Fraction of Sample (%) (N = 1202 Households)
2038 283 607 491 599 316 715 4950
a This table shows all the possible three period transitions in my sample of farmers and the fraction of my sample that experiences each of these transitions. The three periods correspond to 1997, 2000, and 2004. In the first column, the three letters represent the transition history with respect to technology, where “H” represents the use of hybrid and “N” represents the use of nonhybrid. For example, the transition “N N N” stands for farmers who used nonhybrid maize in all three periods; they make up about 20.4% of my sample. The survey instrument asks about hybrid use in multiple sections of the questionnaire (since it is a rather large part of household decisions). We check and confirm the coherency of these responses in the field, which greatly reduces the likelihood that the observed switching behavior is appreciably affected by measurement error.
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profits under hybrid and nonhybrid, and it clearly illustrates the role of the fixed costs of acquiring hybrid seed and fertilizer in the adoption decision, which is important for identification. The model shows how the unobserved heterogeneity in the yield function is a key determinant in the profit comparison underlying the hybrid decision. The aim of this section is, therefore, to derive the fundamental of the yield function that is of interest: the distribution of farmers’ comparative advantage in the production of hybrid maize, which is the correlated random coefficient in the yield function. I then discuss the underlying identification assumptions, their validity, and the estimation of this yield function.22 4.1. Adoption Decisions Under Profit Maximization I start with a simplified technology choice model where the farmer decides between hybrid and nonhybrid seed. In reality, the technology choice is a joint decision of using both hybrid and fertilizer or neither, which I elaborate on in Section 4.5. For ease of exposition, I use the hybrid/nonhybrid choice as a parsimonious representation of this joint choice. The timing of the farmer’s decisions is as follows. Each year, the farmer decides on his seed technology at the beginning of the growing season, just before the rains begin. The farmer makes his decision based on all his current information, his forward looking expectations as to the coming year’s growing conditions, and the relative costs and benefits (differing productivities) of the two types of seed, which are assumed to be known to him. The farmer is assumed to be risk-neutral and chooses a seed type to maxiH N mize profits per area of land. He compares πit∗ (pit ait bit wit ) and πit∗ (pit cit wit ), the maximized profit functions under hybrid and nonhybrid, respectively, where pit is the expected output price of maize for both hybrid and nonhybrid maize,23 ait represents the (fixed) cost of obtaining hybrid seed (due to availability differences), bit is per-unit cost of hybrid seed, cit is the (very low, if not zero) per-unit costs of replanting nonhybrid seed from the previous year’s harvest, and wit ≡ (w1it w2it wJit ) represents the vector of input prices for k the inputs Xjit , j = 1 J and k ∈ H N. The profit functions are (1)
πitH = pit YitH − (bt sit + ait ) −
J
H wjit Xjit
j=1
22 Note that profit functions themselves are difficult to estimate here due to the widespread use of family labor, which is hard to value. Households do not value family labor in the same way as hired labor. Below, I show that the yield function here is similar to a gross revenue function (as in Levinsohn and Petrin (2003)). As a robustness check, I estimate a model using the value of yields and valuing all the inputs where possible using their prices. The results are extremely similar. Finally, if I assume that family labor is valued at hired labor wages, I can estimate profit functions, which also show qualitatively similar results. 23 There is no distinction in the output market between hybrid and nonhybrid maize.
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(2)
πitN = pit YitN − cit sit −
J
175
N wjit Xjit
j=1
where YitH and YitN are the yields of hybrid and nonhybrid maize, respectively, and sit is the quantity of seed used. In addition, I have imposed the following restrictions based on the institutional context in Section 3. First, the quantity of seed used for a given area of land is the same whether it is hybrid or nonhybrid seed.24 Second, since the seed price is fixed across space, bt is indexed only by t and not i. H N Therefore, the farmer chooses to plant hybrid when πit∗ > πit∗ or when J J ait w w (bt − cit ) ∗ H H N N jit jit ∗ ∗ Yit∗ − − Yit∗ − > (3) Xjit Xjit + sit pit pit pit pit j=1 j=1 I can rewrite the second term on the right-hand side of the inequality in (3) as the normalized (by the price of maize) cost difference between hybrid and nonhybrid seed. In fact, it is the case that sit∗ ≈ s∗ , as evidenced in Tables IIA and IIB, where both the quantity of seed used for a given area of land and the land area planted to maize are similar over time. This comes from standard seeding rates for maize and since land markets barely exist, there is little change in land ownership and land cropped to maize over time (see Suri (2006)).25 From (3), the optimized profits from using hybrid over nonhybrid are greater when J J wjit ∗H wjit ∗N ait δsit ∗H ∗N Yit − (4) Xjit − Yit − Xjit > + ≡ Ait +Δsit p p p p it it it it j=1 j=1 where δsit = (bt − cit )s∗ , Ait = ait /pit , and Δsit = δsit /pit . In reality, the value of cit , the cost of nonhybrid, is close to zero (relative to the hybrid cost) and the output price pit does not vary much across space (time by province dummies explain 60% of the variation in pit ). From the evidence on input use in Table IIB, the optimized quantities of inputs apart from fertilizer tend to be about equal under hybrid and nonhybrid. If, in addition, I assume that fertilizer is only used with hybrid (as in the data) and that a fixed amount is used per land allocated to hybrid in each year, then 24 There are standard seeding rates for maize that do not vary by seed type. This is borne out by the empirical seeding rates not varying much over time and not varying across hybrid and nonhybrid sectors; see Tables IIA and IIB. 25 Note that the second term on the right-hand side of equation (3) can be ignored if the cost of obtaining hybrid seed, ait , is much greater than the straight cost difference between hybrid and nonhybrid seed, which seems to be borne out in reality.
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fertilizer costs are subsumed in bt in equation (3) and hence in Δsit .26 If this is the case, then the farmer chooses hybrid if (5)
H
N
(Yit∗ − Yit∗ ) > Ait + Δsit
It is clear that adoption decisions based on profits depend fundamentally on yield comparisons. The above assumptions simply isolate the components of the adoption decision that are important to the identification and estimation of the yield function, which I now describe. 4.2. The Underlying Yield Functions Underlying the profit functions, assume Cobb–Douglas production functions of the form k γjH H H βH (6) Xijt euit Yit = e t (7)
βN t
YitN = e
j=1 k
γjN
Xijt
N
euit
j=1
These production functions for hybrid and nonhybrid maize have different parameters on the inputs, γ H and γ N , to allow for differential complementarity between the seed variety and the inputs (although the same set of potential inN puts are used). uH it and uit are sector-specific errors that may be the composite of time-invariant farm characteristics and time-varying shocks to production, and the β’s are sector-specific aggregate returns to yields. Taking logs, (8)
H H yitH = βH t + xit γ + uit
(9)
yitN = βNt + xit γ N + uNit
I now place additional structure on the unobserved productivities and impose the following factor structure, as in Lemieux (1993, 1998) and Carneiro, 26
This is purely for expositional purposes, since the decision rule it leads to in equation (25) can be amended to include other input costs. Even if I allow the optimized inputs to differ by hybrid and nonhybrid for a given farmer (such as for harvest labor), this will create no bias in the empirical work since the estimated yield function does not depend on the counterfactual input use and conditions on inputs for the observed hybrid/nonhybrid choice. The other reason for taking the approach in the text is that for all inputs other than harvest/postharvest labor, the assumptions are empirically true, and valuing labor is problematic in this context. Therefore, departures from the strict assumptions in the text should be thought of as captured in the error term ϑit in the decision rule in equation (25).
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Hansen, and Heckman (2003): (10)
H H uH it = θi + ξit
(11)
uNit = θiN + ξitN
Farmers are assumed to know θiH and θiN , but not ξitH and ξitN , when making their seed choice. The transitory errors, ξitH and ξitN , are assumed to be uncorrelated with each other and with the Xit ’s, and do not affect the hybrid decision (as per Heckman and Honore (1990)). It is clear that, in expected terms, N H N E(uH it − uit ) = (θi − θi ) plays a role in the hybrid choice. I discuss this further in Section 4.4. Following Lemieux (1998), since the relative magnitudes of the unobserved θiH and θiN are not identified, I define the relative productivity of a farmer in hybrid over nonhybrid as (θiH − θiN ), using the decomposition of θiH and θiN , (12)
θiH = bH (θiH − θiN ) + τi
(13)
θiN = bN (θiH − θiN ) + τi
where the projection coefficients are bH = (σH2 − σHN )/(σH2 + σN2 − 2σHN ), bN = (σHN − σN2 )/(σH2 + σN2 − 2σHN ), and σHN ≡ Cov(θiH θiN ), σH2 ≡ Var(θiH ), and σN2 ≡ Var(θiN )27 The τi is farmer i’s absolute advantage: its effect on yields does not vary by technology and it is (by construction of the linear projection) orthogonal to (θiH − θiN ). The gain, (θiH − θiN ) can be redefined to be farmer-specific comparative advantage, θi , as (14)
θi ≡ bN (θiH − θiN )
Defining φ ≡ bH /bN − 1, equations (12) and (13) become (15)
θiH = (φ + 1)θi + τi
(16)
θiN = θi + τi
I am interested in the structural parameter φ and the distribution of θi , both fundamentals of the production function. The θi ’s are the key unobservables that determine selection into hybrid. 27 The τi ’s in equations (12) and (13) are the same. To see this, subtracting equation (13) from equation (12) gives θiH −θiN = (bH −bN )(θiH −θiN ). For the τi ’s to be equal across sectors, bH − bN must be equal to 1, which is easily shown: bH − bN = (σH2 − σHN − σHN + σN2 )/(σH2 + σN2 − 2σHN ) = 1.
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N Using this decomposition of uH it and uit , I can rewrite equations (8) and (9)
28
(17)
H H yitH = βH t + τi + (φ + 1)θi + Xit γ + ξit
(18)
yitN = βNt + τi + θi + Xit γ N + ξitN
I use a generalized yield equation of the form (19)
yit = hit yitH + (1 − hit )yitN
and substituting in equations (17) and (18), I get (20)
N N yit = βNt + θi + (βH t − βt )hit + Xit γ
+ φθi hit + Xit (γ H − γ N )hit + τi + εit Equation (20) is my basic empirical specification; Section 4.5 describes the estimation in detail. Since, the coefficient on hit , φθi (the fifth term in the equation), depends on the unobserved θi , this is a correlated random coefficient (CRC) model where the θi ’s are correlated with the adoption decision.29 I estimate two components of this model: the coefficient φ that describes how important differences in comparative advantage are in this economy, and, second, the distribution of θi and hence the corresponding distribution of the heterogeneous returns to hybrid. Intuitively, in a specification like equation (20), θi measures farmer i’s relative productivity in hybrid over nonhybrid, that is, his comparative advantage in hybrid. The θi ’s are a fundamental of the production function, but also play a role in the adoption decision. If the farmers with high θi ’s have lower gains to switching to hybrid from nonhybrid, then φ < 0. In that case, relative to the mean θi in the population, a farmer would need a small θi to meet the adoption criterion. This is the notion of selection on the basis of comparative advantage, in that those with lower baseline productivities have larger gains to switching to the new technology. The coefficient φ therefore describes the sorting in the economy. If φ < 0, there is less inequality in yields in this economy as compared to an economy where individuals are randomly allocated to a technology. On the other hand, if φ > 0, then the self-selection process leads to greater inequality in yields. To see this, from equation (20), if we let μi = θi + τi , then μi is a household-specific intercept (average yield) and φθi is the householdspecific return to hybrid. Since θi and τi are uncorrelated by construction, the N Note that this decomposition of uH it and uit implies a more complex Cobb–Douglas proγH φ+1 H H k H τ i θi eβt ( j=1 Xijtj )eξit , and similarly for nonhybrid duction function for hybrid maize, Yit = e e maize. 29 Also note that equation (20) is a generalization of the household fixed effects model (see Suri (2006)). 28
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covariance between μi and φθi is φσθ2 . The sign of φ is therefore the sign of the covariance between a household’s overall average yield and its return to hybrid. A negative φ implies that farmers who do better on average, do worse at hybrid and vice versa for a positive φ. 4.3. The Role of Fixed Costs in the Adoption Decision In this section, I discuss the implications of profit maximization to illustrate how changes in infrastructure and access to seed and fertilizer distributors affect adoption decisions. This profit maximization approach is a generalization of the strict Roy model where adoption decisions are based purely on the outcome (yields). I combine the yield function decomposition of Section 4.2 with the profit maximization problem of Section 4.1 to discuss how the farm-specific comparative advantage parameters combine with the costs of acquiring seed and fertilizer to determine the hybrid adoption decision. Rewriting (4) in log output and using (8) and (9), a farmer chooses hybrid if (21)
N s H N E(uH it − uit ) > Ait + Δit − (βt − βt ) +
J N H (γjN x∗jit − γjH x∗jit ) j=1
In the data, the revenue in hybrid is about double that in nonhybrid. 30% of this is due to differential seed and fertilizer costs, 4% is land preparation cost differences, 7% is hired labor cost differences, family labor goes the other way, and the rest is profit differences and costs of acquiring seed over and above H N the seed price. Apart from fertilizer, the optimized inputs x∗jit and x∗jit are not substantially different, so if γjH γjN for all inputs j, then the last term in this expression is zero for all inputs except fertilizer.30 If for fertilizer, I assume as above that fertilizer is used only with hybrid and in fixed proportions per area of land, then fertilizer is subsumed in the Δsit . In this case, the adoption rule reduces to (22)
N s H N E(uH it − uit ) > Ait + Δit − (βt − βt )
Substituting in equations (10) and (11), and using equations (15) and (16), (23)
N (θiH − θiN ) > Ait + Δsit − (βH t − βt )
(24)
N φθi > Ait + Δsit − (βH t − βt )
30 It is important to emphasize again that the assumption that the term involving the optimized inputs is zero is purely for expositional convenience, in addition to it being roughly true in the data. The estimated yield equation will condition on the observed input intensities. So even without this simplifying assumption, conditional on the observed input intensities, the counterfactual input intensities do not affect the observed outputs, so their role in the adoption decisions does not create bias in the empirical work.
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Equation (24) implies that the technology choice will depend on (i) unobserved, farmer-specific, time-invariant comparative advantage θi that comes from the underlying production function, (ii) pure macroeconomic factors afN fecting the differential productivity of hybrid and nonhybrid seed (βH t − βt ), (iii) potentially time-varying costs of obtaining hybrid, Ait , and (iv) the real relative purchase costs of hybrid seed, Δsit . The key aspect of the Ait costs is that they affect the demand for hybrid seed, but not production and yields directly.31 This framework illustrates how to empirically relate my comparative advantage estimates to observables. Let αi ≡ Et [Ait ] be the population time mean of the real fixed costs of acquiring hybrid seed for each farmer, and let ϑit ≡ Ait − αi . Rewriting equation (24), (25)
φθi − αi > Δsit − (βNt − βH t ) + ϑit
where αi is the permanent component of the fixed costs (that is, the average real fixed costs for a farmer over time) and ϑit denotes the period to period fluctuations in these costs. Clearly, φθi cannot be separately distinguished from the permanent component of fixed costs, αi . In the empirics, I relate the estimated comparative advantage to permanent aspects of infrastructure, since both act in an equivalent way to drive adoption decisions across farmers. By contrast, the changes in the fixed costs, ϑit , drive adoption decisions for a given farmer over time. Section 4.4.3 discusses the empirical evidence on these costs. 4.4. Identification of the Generalized Yield Function The basic equation I estimate is equation (20): N N yit = βNt + θi + (βH t − βt )hit + Xit γ
+ φθi hit + Xit (γ H − γ N )hit + τi + εit In this section, I discuss the identification assumptions needed to estimate this and the justifications for them. While the projections in equations (15) and (16) only impose uncorrelatedness of the absolute advantage, τi , and the comparative advantage, θi , my empirical work does not require this level of generality. So, as in Lemieux (1993), I use the stronger sufficient assumption of mean independence of the composite error (τi + εit ) and the comparative advantage component (θi ), and the histories of the regressors, that is, that (26)
E(τi + εit |θi hi1 hiT Xi1 XiT ) = 0
Δsit , the relative per-unit costs of hybrid seed normalized by the output price also play a role, as equation (24) makes clear. But, as this varies mostly across time and not by individual, it is absorbed by time dummies in the empirical specifications. 31
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With respect to the absolute advantage component, τi , the assumption in equation (26) is not restrictive for two reasons. First, by definition, τi does not affect the differential return to growing hybrid (it is differenced out of θiH − θiN as per equations (15) and (16) above). Second, Heckman and Honore (1990) discussed the identification of the Roy model and showed that if the absolute advantage plays a role in adoption decisions, then the Roy model has no empirical content. This assumption is essentially, therefore, the standard strict exogeneity requirement used in panel data models.32 The strict exogeneity assumption on the transitory part of the composite error, εit , is more restrictive, but since the data include measures of shocks to yields, this assumption is more plausible than if I did not observe such shocks. In terms of the primitives of the model, (27)
εit ≡ hit ξitH + (1 − hit )ξitN
The mean independence assumption regarding εit implies that ξitH and ξitN from equations (17) and (18) do not affect the farmer’s decision to use hybrid and, crucially, the farmer’s switching behavior. I discuss the various types of possible shocks and how they relate to yields and hybrid decisions in the following two subsections. Given the long lag between planting and harvest (an average of j 4 or more months), I consider two “types” of ξit (j ∈ H N) transitory shocks to yields: those that occur after the seed choice has been made and those that occur before. I discuss each separately below and then discuss the implications for the switching behavior. 4.4.1. Shocks Realized After the Technology Choice Central to identification is the fact that the hybrid seed choice is made before the farmer experiences most of the agricultural shocks to yields. However, the shocks postplanting may affect optimal input use differentially for hybrid and nonhybrid, and hence pose a problem for identification. So it is key that the most important transitory shock postplanting—rainfall—is observable in the data and can be controlled for. There may be other unobservable shocks to yields that are correlated with inputs, such as weeds, pests, and disease. Hassan, Onyango, and Rutto (1998c) used survey data to rank farmers’ perceptions of 32 The assumption in equation (26) implies that E(τi |θi ) = 0 by the law of iterated expectations. As shown in Section 4.5.1, θi can be written as the linear projection θi = λ0 + λ1 hi1 + λ2 hi2 + λ3 hi1 hi2 + vi . Given the mean independence of τi from the adoption history discussed above, the additional assumption that E(τi |θi ) = 0 only implies E(τi |vi ) = 0. This last assumption is not actually needed for the empirical work, as the relationship between these two error components is left unspecified—they can be correlated, for example, and my empirical approach is still consistent. The mean independence assumption in (26) is therefore “overly strong,” but is useful for expositional convenience concerning the adoption behavior and the assumptions on the composite error term. However, I do make the strict exogeneity assumption regarding τi , as described.
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the most important maize production shocks and what options are available to them to deal with these shocks. They found that inadequate and/or erratic rainfall is one of the most widespread and one of the two worst shocks to yields across all zones in Kenya.33 In addition, given the timing of production, the inputs that could be adjusted after the technology choice are only a subsample of the labor inputs, since the empirical specification will allow for the joint hybrid–fertilizer decision. The main labor activities are land preparation, planting, weeding, harvest, and postharvest. Land preparation and planting are costs incurred before the technology choice, and harvest and postharvest after all shocks are realized. The only input that could therefore be an issue is weeding labor. Table IIC shows that the use of weeding labor in the sample is not very different across hybrid and nonhybrid farmers. Duflo, Kremer, and Robinson (2008b) compared plots that use hybrid and fertilizer, those that use just fertilizer, and those that use nonhybrid and no fertilizer. Other than the cost of the seed and fertilizer, they stated that farmers reported no differences in the time spent weeding across these plots and their field officers observed no differences in weeding. Overall, it seems that shocks realized after the hybrid choice are well captured by the variables in my data, since rainfall is observed and postplanting input use is unlikely to be differential across technologies. 4.4.2. Shocks Realized Before the Technology Choice Even though the identification assumptions seem robust to shocks realized after the technology choice, there could be an issue with shocks that happen between 1997 and 2004 and affect the use of hybrid as well as yields. An example may be changes in household structure, for example, the death of adults in the household due to HIV, that affect the quality of labor as well as the decision to use hybrid. In all the empirical results reported, I control directly for household structure (excluding these controls does not affect the results; see Suri (2006)). In addition, all the results in this paper are reported for the 33 I have also investigated other potential shocks. For example, Hassan, Onyango, and Rutto (1998c) also identified Striga (a weed) as an important issue in three agroecological zones (pests and most diseases do not rank high). Hassan and Ransom (1998) carefully analyzed Striga and stated there is “no conclusive evidence that local vs. improved maize, time of planting, and cropping pattern either encourage or discourage Striga.” The occurrence of Striga therefore seems to be uncorrelated with the hybrid decision and should not pose a problem for identification. As a robustness check, where I can match my households to these three zones, I drop these households and the results are extremely similar. Another potential shock is temperature: maize grows best in a temperature range of 24–30◦ C and has trouble germinating at temperatures above 38◦ C (see Pingali and Pandey (1999) and McCann (2005)). All the maize areas in Kenya fit within this range of temperatures. Looking at data on monthly temperatures for 1956–2006 by latitude and longitude, the maximum observed temperature is 23◦ . The warmest part of the country in my sample is probably the Coast province. As a robustness check, I drop the households in the Coast province and the results are similar.
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full sample as well as for the sample that excludes two districts where much of the adult mortality in the sample was concentrated over this period and where HIV is prevalent. The results are stronger and more consistent in the latter case, implying that my description of adoption decisions is most relevant for the sample that excludes the two high mortality districts, what I refer to as the non-HIV sample.34 4.4.3. The Switching Behavior in Practice The identification assumptions just described translate directly into assumptions on what drives the hybrid switching behavior in the data. Essentially, the unobserved time-varying variables that drive the switching should not be correlated with yields. I argue that the switching behavior is driven by exogenous changes in the availability of seed and fertilizer. If hybrid seed and/or fertilizer are not easily available at the time of planting, this would manifest itself in a supply constraint or a higher cost of using hybrid, and the farmer would switch to nonhybrid.35 There is a lot of qualitative evidence that this happens often. Farmers get their seed and fertilizer from very small local distributors, who get their supplies from slightly larger distributors and so on. In the survey, farmers who use nonhybrid were asked why they do not use improved seed varieties: 14% blamed unavailability of seed as one of the top two reasons for this; 65% put high costs as one of the top two reasons. The 2000 survey (data not shown; see Suri (2006)) showed that one of the common fertilizers (MAP) was not used at all in the sample due to a macrolevel unavailability. Several of the CIMMYT and FURP studies mentioned earlier cited unavailability of inputs, issues that were also at the forefront of the Africa Fertilizer Summit in 2006. In addition, there are certainly changes in the pricing and availability of seed and fertilizer from period to period, especially since the Kenyan government plays such an active role in these markets.36 34 The survey does not necessarily identify the cause of death accurately, so these are not necessarily all HIV related deaths. However, these two districts are well known to be the areas with extremely high HIV prevalence rates, so it is likely that the high mortality rates in these two districts are a consequence. Also see Morons, Mutunga, Munene, and Cheluget (2003). 35 The framework in Dercon and Christiaensen (2005), for example, shows the switching to be related to households’ ability to smooth downside consumption risks. This would be compatible with my framework. 36 The price of fertilizer varies tremendously over time, since its supply and availability vary. It is crucial that the fertilizer be available right before planting. Since the early 2000’s, the National Cereals and Produce Board (NCPB) has been importing fertilizer in bulk and sometimes selling it at lower prices to farmers. A Lexis Nexis search covered only one Kenyan newspaper and returned 55 articles between 1999 and 2004 documenting the role of the government in fertilizer, seed, and maize markets. The articles discuss changes in prices of the inputs in different parts of the country, often due to shortages (DAP in 1999 and 2004), delays in the imports by the NCPB (as happened in 2004), a stop to MAP fertilizer aid by Japan (late 1990’s on), and the dramatic changes in the structure of the institutions that help farmers get access to inputs.
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Finally, Tables IIA and IIB show the changes in infrastructure between 1997 and 2004. These have been due to private investment in fertilizer distribution networks; now there are more than 10 importers, 500 wholesalers, and 7000 retailers in the country (see Ariga, Jayne, and Nyoro (2006)). For example, the distance to the closest fertilizer seller fell from 6.3 km in 1997 to 3.5 km in 2004 (a fall from 4.9 km to 2.4 km for hybrid farmers and from 9.3 km to 5.1 km for the nonhybrid). 4.5. Estimating a Model With Heterogeneous Returns I now describe how to estimate the CRC model in equation (20).37 An alternative estimation framework would be to use contemporary industrial organization methods. However, in the face of the correlated heterogeneity in equation (20), methods such as those of Olley and Pakes (1996) and Levinsohn and Petrin (2003) that focus on unobserved time-varying productivity are inconsistent, since the returns to hybrid are heterogeneous and correlated with the decision to use hybrid. Also, in the contexts those papers examined, productivity shifts for existing firms and the productivities for the firms leaving and entering the industry are of key importance in estimating responses to regulatory or trade environments. In my scenario, given the static but cross sectionally heterogeneous nature of the returns to hybrid and the underlying productivities of Kenyan farmers, it is of less importance to deal with time-varying productivities, since the approach controls for the dominant time-varying shocks to output effectively. 4.5.1. Empirical Identification of the CRC Model The estimation strategy I use is a generalization of the Chamberlain (1982, 1984) correlated random effects approach. It parallels Chamberlain (1984) in how the model is identified and how the parameters of the model are estimated, most importantly φ. Later, I describe how to use these estimates to derive a distribution of the predicted θi ’s. For simplicity, I first describe how to estimate the model without covariates, (28)
yit = δ + βhit + θi + φθi hit + uit
N where uit ≡ τi + εit and assuming βH t − βt ≡ β ∀t. Relaxing this assumption empirically does not change the results. To estimate equation (28), I eliminate the dependence of the observed θi ’s on the endogenous input (hit ) by following Chamberlain to exploit the linear
37 This empirical model is similar to models of individual-specific heterogeneity in Heckman and Vytlacil (1998), Card (2000, 1998), Deschênes (2001), Carneiro, Hansen, and Heckman (2003), Carneiro and Heckman (2002), and Wooldridge (1997). Lemieux (1998) used the same model to look at whether the return to union membership varies along observable and unobservable dimensions.
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projection of θi on the full history of the inputs, although the projection I use is more general. The θi ’s are projected onto not just the history of the hybrid decisions, but also their interactions, so that, in the two period case, the projection error is orthogonal to hi1 and hi2 individually as well as to their product, hi1 hi2 by construction.38 The generalized linear projection used is θi = λ0 + λ1 hi1 + λ2 hi2 + λ3 hi1 hi2 + vi In addition, the θi ’s are normalized so that θi = 0.39 This normalization implies that λ0 can be written as a function of λ1 , λ2 , and λ3 from equation (29). Here, the coefficient λ3 is of crucial importance to the empirical identification and estimation of the model, and is where this is a generalization of Chamberlain’s approach. The hi1 hi2 interaction term is necessary in the projection to ensure that vi is orthogonal to every possible history of hybrid use. Since all the hit variables describing the use of hybrid are dummies, to estimate λ3 it is necessary to have farmers in the sample that have planted hybrid in both periods. Also note that the projection does not have a behavioral interpretation, but is used purely to purge θi of its dependence on the full histories of the inputs. Substituting the projection into the yield equation for each of the two time periods gives (29)
(30)
yi1 = (δ + λ0 ) + [λ1 (1 + φ) + β + φλ0 ]hi1 + λ2 hi2 + [λ3 (1 + φ) + φλ2 ]hi1 hi2 + (vi + φvi hi1 + ui1 )
(31)
yi2 = (δ + λ0 ) + λ1 hi1 + [λ2 (1 + φ) + β + φλ0 ]hi2 + [λ3 (1 + φ) + φλ1 ]hi1 hi2 + (vi + φvi hi2 + ui2 )
The corresponding reduced forms are (32)
yi1 = δ1 + γ1 hi1 + γ2 hi2 + γ3 hi1 hi2 + ςi1
(33)
yi2 = δ2 + γ4 hi1 + γ5 hi2 + γ6 hi1 hi2 + ςi2
Equations (32) and (33) give six reduced form coefficients (γ1 γ2 γ3 γ4 γ5 γ6 ), from which the five structural parameters (λ1 λ2 λ3 β φ) can be estimated using minimum distance. The structural parameters are overidentified 38 Chamberlain’s correlated random effects (CRE) model uses the projection θi = λ1 hi1 + λ2 hi2 + vi , which, when substituted into the yield function, gives yit = λ0 + λ1 hi1 + λ2 hi2 + βhit + φλ0 + φλ1 hi1 hit + φλ2 hi2 hit + vi + φvi hit + uit . Even though vi is uncorrelated with hi1 and hi2 individually (by the nature of the projection), it could be correlated with their product, hi1 hi2 , so that E[vi hi1 hi2 ] = 0. The CRC projection must, therefore, also include all the interactions of the hybrid histories. 39 This normalization is used so that β in equation (28) corresponds to the average return to hybrid as in a standard fixed effects model. This normalization fixes the average θi in the sample and does not change the ordering of the θi ’s across farmers.
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given these minimum distance restrictions: (34)
γ1 = (1 + φ)λ1 + β + φλ0 γ2 = λ2 γ3 = (1 + φ)λ3 + φλ2 γ4 = λ1 γ5 = (1 + φ)λ2 + β + φλ0 γ6 = (1 + φ)λ3 + φλ1
There may seem to be six structural parameters, since λ0 is in equations (29) and (34). However, given the normalization θi = 0, then λ0 = −λ1 hi1 − λ2 hi2 − λ3 hi1 hi2 , where hi1 and hi2 are the averages of the adoption decisions in periods one and two, and hi1 hi2 is the average of their interaction. From the restrictions above, for example, φ could be (inefficiently) estimated as a combination of the reduced form parameters in two ways: φ = (γ6 − γ3 )/(γ4 − γ2 ) and 1 + φ = (γ1 − γ5 )/(γ4 − γ2 ). Note that if γ2 and γ4 are equal, then φ is not identified. Requiring γ2 and γ4 to be different means that the reduced form effect of the hybrid decision in period two on yields in period one has to be different from the reduced form effect of the hybrid decision in period one on yields in period two. Since these are reduced form coefficients, this could structurally arise from a number of mechanisms. One example directly from the theory would be differential shocks to the costs in each period, that is, different ϑit (see equation (25)) in each period. In Table VII, the null that γ2 = γ4 can clearly be rejected. In addition, in the extension discussed in Section 4.5.2, where the use of fertilizer is endogenized (the preferred model), the problem becomes heavily overidentified and φ is a more complicated function of the underlying reduced form parameters. 4.5.2. Extensions to the Basic Model I consider the following extensions to the basic model:40 (i) Covariates: All the identification arguments presented above generalize when covariates are included. Covariates can be either exogenous or endogenous. Exogenous covariates are uncorrelated with the θi ’s and, therefore, enter only the reduced form equations (32) and (33). Endogenous covariates are correlated with the θi ’s and hence also enter the projection in equation (29). In 40 An extension to three periods is straightforward. The problem is heavily overidentified with 9 structural parameters to estimate from the 21 reduced form coefficients for the basic case. Models with three time periods are not presented here as the 2000 survey does not collect data on family labor inputs. See Suri (2006) for these results.
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the case where fertilizer is the other endogenous covariate, the CRC projection generalizes to (35)
θi = λ0 + λ1 hi1 + λ2 hi2 + λ3 hi1 hi2 + λ4 hi1 fi1 + λ5 hi2 fi1 + λ6 hi1 hi2 fi1 + λ7 hi1 fi2 + λ8 hi2 fi2 + λ9 hi1 hi2 fi2 + λ10 fi1 + λ11 fi2 + vi
where fit for t = 1 2 represents the use of fertilizer in each period. Adding more endogenous covariates does complicate the CRC model and it can become cumbersome.41 In this case, there are 22 reduced form coefficients and only 14 structural coefficients to be estimated, so the problem is heavily overidentified. (ii) Joint choice variables: The two-sector model presented above (hybrid/nonhybrid) can be extended to multiple sectors. Since the use of hybrid seed and fertilizer is clearly a joint decision in the data, as a further check, the use of fertilizer is incorporated into the technology sector. I therefore also estimate a model that looks at the heterogeneity in returns in a technology sector, where a farmer is in the technology sector when he uses both hybrid and fertilizer, else he is not in the technology sector. 5. DESCRIPTIVE REGRESSIONS This section provides estimates of the returns to hybrid from models that assume homogeneity in these returns. I also present descriptive evidence that illustrates the importance of selection and heterogeneity in returns, and that there may be selection on these heterogeneous returns. 5.1. OLS and Fixed Effects Estimates The OLS estimates of the yield functions, controlling for various inputs, are in the first data columns of Table IIIA. The estimates of the average return to hybrid are extremely large: households that plant hybrid have 54–100% higher yields. The last two columns of Table IIIA report the household fixed effects results. The coefficient on hybrid decreases to about 9%, indicating a substantial role for heterogeneity in the production function that is fixed across households.42 The simple household fixed effects estimates are consistent under the assumption of strict exogeneity. Chamberlain’s correlated random effects (CRE) approach illustrates how the fixed effects model is overidentified 41 There is some justification for treating only fertilizer as endogenous with respect to θi . First, the summary statistics by sector in Table IIB show that there are not big differences across sectors in the use of other inputs. Second, using the estimates from the hybrid problem in Table VIIIA, I correlate the predicted θi ’s with the inputs. Only the correlations between the θi ’s and fertilizer are important in magnitude and/or significance. 42 The fixed effects framework imposes restrictive assumptions on the underlying adoption process. Apart from the fixed effect, the adoption decision cannot depend on observed outcomes except under restrictive assumptions on the transitory component of yields (see Ashenfelter and
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BASIC OLS AND FIXED EFFECTS (FE) SPECIFICATIONS: DEPENDENT VARIABLE IS YIELDS (LOG MAIZE HARVEST PER ACRE)a OLS, Pooled
OLS, Pooled
Hybrid 1074 (0040) 0695 (0039) Acres (× 1000) — − Seed kg per acre (× 10) — − Land preparation costs — − per acre (× 1000) Fertilizer per acre — − (× 1000) Hired labor per acre — − (× 1000) Family labor per acre — − (× 1000) Year = 2004 Constant Province dummies R-squared
OLS, Pooled
FE
FE
0541 (0041) 0017 (0070) 0090 (0065) 0035 (5749) — −0509 (0140) 0184 (0024) — 0179 (0032) 0066 (0016) — 0075 (0023) 0075 (0009)
—
0054 (0012)
0037 (0006)
—
0027 (0008)
0374 (0050)
—
0467 (0072)
0501 (0038) 0480 (0035) 0566 (0041) 0444 (0032) 0587 (0044) 5200 (0038) 4636 (0080) 3954 (0113) 5896 (0051) −2383 (5582) No Yes Yes — — 0.266 0.400 0.502 0.049 0.089
a Standard errors are given in parentheses. All regressions have 2404 observations (two periods). Covariates not reported include household size, controls for the age–sex composition of the household (henceforth this includes variables for the number of boys (aged <16 years), the number of girls, the number of men (aged 17–39), the number of women, and the number of older men (aged >40 years)), the main season rainfall, and the average long term main season rainfall. Results are almost identical if the sample is for three periods without family labor as a covariate. Hired labor is measured in KShs per acre and family labor is measured in hours per acre. The OLS and household fixed effects specifications run are yit = δ + βhit + Xit γ + εit and yit = δ + ai + βhit + Xit γ + εit .
and testable with panel data.43 Intuitively, the CRE model tests the fact that if the fixed effects model is valid, then the only way the history of hit affects the current outcome is through a fixed effect that is the same in every period. Table IIIB shows these tests. It shows both the reduced form and structural estimates for the CRE model. Covariates can be treated as either exogenous or endogenous (the latter are assumed to be correlated with the fixed effects). I report estimates where all the covariates are allowed to be endogenous.44 The Card (1985)). Such assumptions can be motivated by myopia or ignorance of the potential gains from planting hybrid. Both of these seem unrealistic here, since hybrid maize was introduced in the 1960’s with widespread use of extension services to promote the technology (Evenson and Mwabu (1998)) and data in the survey instrument support this as 90% of farmers have used hybrid seed at some point in the past. 43 With a data generating process of yit = δ + βhit + αi + uit , the CRE model is based on the assumption of strict exogeneity, that is, E(uit |hi1 hiT αi ) = 0. For the CRE model, the minimum distance estimator is the minimum χ2 estimator if the weight matrix used is the inverse of the variance–covariance matrix of the reduced form coefficients. See Suri (2006) for more detail on this approach for the current setting. 44 If I treat only hybrid as endogenous and all the other covariates as exogenous, the results are similar.
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SELECTION AND COMPARATIVE ADVANTAGE TABLE IIIB
CRE MODEL REDUCED FORMS AND STRUCTURAL ESTIMATES DEPENDENT VARIABLE IS YIELDS (LOG MAIZE HARVEST PER ACRE)a Reduced Form Estimates
Without Covariates Yields, 1997
Yields, 2004
With Covariates Yields, 1997
Yields, 2004
With Covariates and Interactions of Covariates with Hybrid Yields, 1997
Yields, 2004
Hybrid, 0.674 (0075) 0.538 (0065) 0.579 (0064) 0.415 (0060) 0.467 (0242) 0.501 (0228) 1997 Hybrid, 0.809 (0072) 0.723 (0062) 0.411 (0065) 0.563 (0063) 1.214 (0259) 0.630 (0230) 2004 Optimal Minimum Distance (OMD) Structural Estimates
β λ1 λ2 χ21
Without Covariates
With Covariates
With Covariates and Interactions of Covariates With Hybrid
00322 (00701) 05795 (00621) 07332 (00684) 44.63
01588 (00653) 04166 (00570) 04062 (00622) 0.193
−03039 (02522) 05683 (02103) 10447 (02351) 460.5
a Standard errors are given in parentheses. Reduced forms are estimated without covariates, with covariates (acreage, real fertilizer expenditure, real land preparation costs, seed, labor, household size, household age–sex composition variables, and rainfall variables) and with all covariates interacted with hybrid. The covariates are all treated as endogenous. Results are the same if three periods without family labor are used. Results are similar if all covariates except hybrid are treated as exogenous. The reduced form for each t , the projection used to estimate the structural model by minimum distance, and the structural model are, respectively, yit = δt +γ1t hit +γ2t hi2 +γ3t hi3 +Xi πt +εit , ai = λ0 + λ1 hi1 + λ2 hi2 + vi , and yit = δ + βhit + ai + uit . Structural coefficients β and projection coefficients (λ’s) are reported. OMD are optimal (weighted by inverted reduced form variance–covariance matrix) minimum distance estimates. Equally weighted (using the identity matrix) and diagonally weighted (using only the diagonal elements from the OMD weight matrix) minimum distance results are similar.
CRE estimates of the return to hybrid are close to the household fixed effects estimates in Table IIIA, but the χ2 values on the overidentification test allow me to reject the fixed effects model.45 The last column of Table IIIB shows estimates for the case where hybrid is interacted with all the covariates. Although the coefficient on hybrid is negative, when evaluated at the mean input levels, the mean return to hybrid is positive. 5.2. IV and Treatment Effect Estimates This section presents IV and control function estimates of the returns to hybrid, using the Heckman two-step estimator (also see Garen (1984)). In par45 This overidentification test is an omnibus test and has low power against any specific alternative. It is, therefore, not that surprising that I am able to reject the overidentifying restrictions.
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ticular, I present estimates of the average treatment effects (ATE), the treatment on the treated (TT), marginal treatment effects (MTE), and local average treatment effects (LATE) under nonrandom assignment (see Björklund and Moffitt (1987) and Heckman, Tobias, and Vytlacil (2001)).46 These approaches do not fully address the issues at hand and do not exploit the panel nature of the data, but they provide useful benchmarks. With assumptions of normality, it is straightforward to estimate the selection corrected treatment effects using a two-step control function procedure. The first stage is a probit describing the hybrid adoption decision, from which selection correction terms are computed and used as controls in second stage sector-specific yield functions. The ATE uses estimates from this second step. The TT adjusts the estimated ATE for the sample of those who actually plant hybrid. The slope of the MTE describes whether people who are more likely to use hybrid for unobservable reasons (i.e., have a higher error in the selection equation) have higher or lower returns from planting hybrid. The exclusion restriction I use is the distance to the closest fertilizer store (not to where fertilizer is actually purchased), which proxies for the availability of the technologies. The treatment effects are shown in Table IV separately for 1997 and 2004. The ATE’s are all large and positive, ranging from 1.3 to 2.4, with smaller TT estimates. The MTE slope is consistently negative and significant: −099 in 2004 and −251 in 1997. A nonzero MTE slope implies heterogeneity in returns and the sign of the MTE slope provides information on the underlying selection process. The negative MTE slopes imply that the farmers who are more likely to use hybrid are those who have the lower relative returns to using hybrid, that is, there is negative selection on returns in hybrid. I also estimate the IV (LATE) specification. An issue with IV estimates is that the results are often different if different instruments are used. This is attributed to underlying heterogeneity in the population where the separate instruments affect a different subset of the population, resulting in the different LATE estimates (i.e., an average of heterogeneous returns that is instrument dependent). However, the instrument itself does not always highlight which subset of the population it affects (Heckman (1997)). The lower panel of Table IV reports two versions of IV. The first data column reports an estimate of over 200%, which uses the distance to the closest stock of fertilizer as the excluded instrument. The second data column uses the interactions of this distance measure with dummies for the household’s asset quintile as excluded instruments (the distance and asset quintile main effects are included in both 46 The treatment effects are defined as follows. The ATE is given by E((y H − y N )|X = x) = x(γ H − γ N ), the TT is given by TT(x z h(Z) = 1) = x(γ H − γ N ) + (ρH σH − ρN σN )φ(zπ)/(zπ), and the MTE is given by MTE(x usi ) = x(γ H − γ N ) + (ρH σH − ρN σN )usi , where y H − y N is the yield gain from hybrid, uH and uN are as defined earlier, σH2 = i i N s s 2 Var(uH ), σ = Var(u ), u is the error in the selection equation, ρH = Corr(uH i N i ui ) and i i N s ρN = Corr(ui ui ); φ(·) and (·) represent the normal probability and cumulative distribution functions, respectively, and Var(usi ) = 1.
TABLE IV HECKIT AND TREATMENT EFFECT ESTIMATES UNDER NONRANDOM ASSIGNMENT (ATE, TT, MTE, LATE)a Implied Treatment Effects
Year
Hybrid Sector
Nonhybrid Sector
ATE
TT
MTE Slope
1997 2004
−0854 (0170) −0957 (0181)
1.659 (0864) 0.028 (0152)
2.391 1.279
0.917 0.921
−2512 (0880) −0985 (0237)
IV (LATE) Estimates (Conditional on Covariates)
First stage: Effect of distance First stage: Effect of distance interacted with wealth quintile (× 100) Second wealth quintile (coefficient on interaction) Third wealth quintile (coefficient on interaction) Fourth wealth quintile (coefficient on interaction) Fifth wealth quintile (coefficient on interaction) F test p-value on excluded instruments Second stage: Effect of predicted hybrid on yields
−0288 (0108)
—
— — — — 0.008 2.768 (1123)
−0221 (0302) −0057 (0032) 0329 (0288) 0507 (0273) 0.108 1536 (0816)
a Standard errors are given in parentheses. All regressions control for covariates. The first two upper panel data columns show (for each year) the two-step selection corrected ˆ , and y N = X γ N + λN [φ(Z π)/(1 estimates of coefficients on the inverse Mills ratio for hybrid and nonhybrid sectors from hi = Zi π + usi , yiH = Xi γ H + λH [φ(Zi π)/(Z ˆ − i i π)] i iˆ ˆ . The third data column reports the average treatment effect accounting for selection and the fourth data column reports the treatment on the treated. Finally, the MTE (Zi π))] slope is just the difference in the λ coefficients for the hybrid and nonhybrid selection terms (the difference between coefficients reported in the first two data columns). ATE, TT, and MTE are all evaluated at the mean Xi ’s. The lower panel reports two set of IV estimates. The first data column uses the distance to closest fertilizer supplier (not where fertilizer is purchased) as the excluded instrument. The second data column uses the distance interacted with wealth quintiles as excluded instruments (controlling for the asset quintile dummies and the distance main effect in both stages). All regressions control for the full set of covariates as per earlier tables (including household size and controls for the age–sex composition of the household as above).
SELECTION AND COMPARATIVE ADVANTAGE
Heckman Two-Step Estimates: Selection Correction λ
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stages). This strategy allows distance to be more of a constraint for poorer households. These estimates are on the order of 150%, still very large when compared with the earlier OLS and household fixed effects estimates, and indicate high returns to hybrid for the supply (of seed and fertilizer) constrained farmers. Overall, the preliminary evidence from these approaches indicates large positive returns for at least some subpopulations of farmers, but also evidence of substantial heterogeneity in returns with negative selection into the use of hybrid. 5.3. Motivation for Heterogeneity in Returns This section presents some preliminary regressions that further motivate heterogeneity in returns. Previous research has developed tests for heterogeneity in returns for experimental data (see Heckman, Smith, and Clements (1997)). Since the data here are not experimental, the results are not reported, but using such tests I do reject the null of no heterogeneity.47 Table V presents some evidence of selection on the part of farmers in their adoption decisions. The adoption history of a farmer is split into a set of dummy variables to analyze whether farmers with different histories have different returns (see Jakubson (1991) and Card and Sullivan (1988)). A “joiner” is a farmer who does not plant hybrid the first period, but does the next, and a “leaver” is a farmer who plants hybrid the first period, but not the next. Similarly, “hybrid stayers” always use hybrid and “nonhybrid stayers” always use nonhybrid. Table V compares the yields for each of these groups in 1997 and 2004 separately. If there was no selection at all, we would expect the leavers in 1997 to be no different from the hybrid stayers in 1997, and no different from the joiners in 2004. Similarly, the joiners in 1997 and the leavers in 2004 should be no different from the omitted group (the nonhybrid stayers). I can reject most of these restrictions, implying that there is selection, and leavers and joiners are distinct (unlike in a fixed effects model). Finally, in Table VI, I look for heterogeneity in the returns to hybrid along observables, estimating OLS and fixed effects yield functions separately for hybrid and nonhybrid farmers. In the OLS case, the returns to some of the covariates are different, although not family or hired labor. In the case of the fixed effects specifications, it is only the return to fertilizer that is different across hybrid and nonhybrid sectors. The last row reports estimates of the return to hybrid (evaluated at the mean inputs), still showing a significant return to hybrid. 47
These tests include looking at the Frechet–Hoeffding bounds, bounding the variance in the percentiles of the returns distribution, and testing whether this bound is different from zero (see Suri (2006)).
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SELECTION AND COMPARATIVE ADVANTAGE TABLE V
SELECTION RETURNS BY HYBRID HISTORY (JOINERS, LEAVERS, AND STAYERS): DEPENDENT VARIABLE IS YIELDS (LOG MAIZE HARVEST PER ACRE)a Without Covariates Variable
1997 Yield
Hybrid stayers 1505 (0066) Leavers 0809 (0094) Joiners 1007 (0114) Acres (× 100) — Seed kg per acre (× 10) — Land preparation costs — per acre (× 1000) Fertilizer per acre (× 1000) — Hired labor per acre (× 1000) — Family labor per acre (× 1000) —
With Covariates
2004 Yield
1997 Yield
2004 Yield
1280 (0056) 0648 (0079) 0883 (0096) — — —
0869 (0073) 0537 (0084) 0469 (0101) 0561 (0782) 0218 (0035) 0066 (0023)
0683 (0063) 0370 (0069) 0498 (0084) −0744 (0802) 0197 (0032) 0058 (0021)
— — —
0063 (0012) 0028 (0008) 0415 (0075)
0061 (0012) 0057 (0010) 0318 (0064)
a Standard errors are in parentheses. In each year, the regression y = δ + μ h i 1 i11 + μ2 hi10 + μ3 hi01 + Xi π + ui is run both with and without the covariates, where hi11 is the dummy indicating that farmer i is a hybrid stayer (plants hybrid in both periods), hi10 indicates he is a leaver (plants hybrid the first year, not the second), and hi01 indicates he is a joiner (plants hybrid the second year, not the first). The coefficients reported are μ1 , μ2 , and μ3 . Covariates included that are not reported above are province dummies, household size, variables for the age–sex composition of the household (includes variables for the number of boys (aged <16 years), the number of girls, the number of men (aged 17–39), the number of women, and the number of older men (aged >40 years)), the main season rainfall, and the average long term main season rainfall. Note that hired labor is measured in KShs per acre and family labor is measured in hours per acre.
6. CRC ESTIMATES This section describes the results for the CRC model. I report estimates for the pure hybrid model described in detail above (with and without covariates). In addition, I report results for the endogenous covariates and joint hybrid– fertilizer technology sector models. Tables VII, VIIIA, VIIIB, and VIIIC present the CRC model reduced form and structural estimates. These tables report only the optimal minimum distance (OMD) estimates where the weight matrix used in the minimum distance problem is the inverse of the variance–covariance matrix of the reduced form coefficients. If the minimum distance problem is overidentified, the χ2 test statistic on the overidentification test is the value of the minimand in the OMD problem.48 48
Equally weighted minimum distance (EWMD) estimates use the identity matrix as the weight matrix and diagonally weighted minimum distance (DWMD) estimates use the OMD matrix with the off-diagonal elements set to zero (see Pischke (1995)). The OMD estimates are efficient, but may be biased in small samples and can, therefore, be outperformed by EWMD (see Altonji and Segal (1996)). For all the results in the paper, the EWMD and DWMD estimates are extremely similar to OMD, but the OMD estimates are asymptotically efficient, so only they are reported. See Suri (2006) for the EWMD and DWMD estimates.
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TAVNEET SURI TABLE VI HETEROGENEITY BY OBSERVABLE RETURNS IN THE HYBRID/NONHYBRID SECTOR: DEPENDENT VARIABLE IS YIELDS (LOG MAIZE HARVEST PER ACRE)a OLS With Covariates
Variable
Acres (× 10) Seed kg per acre (× 10) Land preparation costs per acre (× 1000) Fertilizer per acre (× 1000) Hired labor per acre (× 1000) Family labor per acre (× 1000) Year = 2004 Average return (β) when returns vary by observables (evaluated at mean X’s) Number of observations
Hybrid
FE With Covariates
Nonhybrid
Hybrid
Nonhybrid
0144 (0056) −0053 (0149) −0381 (0153) −0941 (0379) 0281 (0035) 0129 (0036) 0219 (0047) 0147 (0063) 0056 (0018) 0135 (0031) 0033 (0024) 0097 (0060) 0064 (0008) 0143 (0032) 0047 (0007) 0026 (0010) 0297 (0064) 0435 (0081) 0568 (0050) 0595 (0068) 0.480 (0048) 1517
0040 (0011) 0081 (0086) 0054 (0011) 0035 (0029) 0497 (0094) 0581 (0177) 0467 (0058) 0689 (0096) 0.091 (0076)
887
1517
887
j
a Standard errors are given in parentheses. The specification y = δj +X γ j + ε , j ∈ H N , is estimated separately ti it it for the sample of farmers who use hybrid and nonhybrid maize. The covariates included that are not reported are province dummies, household size, age–sex composition of the household (includes variables for the number of boys (aged <16 years), the number of girls, the number of men (aged 17–39), the number of women, and the number of older men (aged >40 years)), the main season rainfall, and the average long term main season rainfall. The results are similar for the three period sample without family labor as a covariate. Note that hired labor is measured in KShs per acre and family labor is measured in hours per acre.
Throughout, the results show that the selection into hybrid is negative, with the farmers having the lowest yields in nonhybrid having the highest returns to planting hybrid. The estimated φ is consistently negative, which illustrates that the households that do better on average, do relatively worse at hybrid as the sign of φ describes the sorting process. The CRC model results are, therefore, consistent with the earlier MTE results. Table VII presents the two period reduced forms for the CRC model. The first data column shows the reduced form estimates without covariates as a benchmark. The second data column controls for all the inputs and the third for all the inputs interacted with hybrid. The reduced forms are estimated via seemingly unrelated regressions with the most general variance–covariance estimation. The estimates in Table VII are for the basic specification described above, where hybrid is the only endogenous variable and the projection used is given by equation (29). Table VIIIA reports the structural OMD results and the χ2 statistics for all three specifications. The estimates of φ are consistently negative. I also report the structural estimates for the case where the sample excludes two districts with very high adult mortality between 1997 and 2004,49 49 Thanks to an anonymous referee for pointing this out. Jayne and Yamano (2004) documented how this mortality affects the value of high value crops produced, but not the value or productivity of maize.
TABLE VII
Without Covariates Yields, 1997
Hybrid, 1997 0833 (0121) Hybrid, 2004 1139 (0122) Hybrid 1997 × hybrid 2004 −0458 (0156) Acres (× 10) — Seed kg per acre (× 10) — Land preparation cost per acre (× 1000) — Fertilizer per acre (× 1000) — Hired labor per acre (× 1000) — Family labor per acre (× 1000) — R-squared 0.285
With Endogenous Covariates
With Interactions With Hybrid
Yields, 2004
Yields, 1997
Yields, 2004
Yields, 1997
Yields, 2004
0471 (0099) 0766 (0103) −0194 (0132) — — — — — — 0.232
0719 (0103) 0702 (0110) −0358 (0132) 0106 (0067) 0230 (0052) 0124 (0025) 0079 (0018) 0025 (0014) 0399 (0115) 0.454
0316 (0088) 0508 (0092) −0098 (0110) −0006 (0098) 0433 (0060) 0133 (0039) 0042 (0014) 0053 (0010) 0186 (0071) 0.441
0926 (0252) 0474 (0122) −0084 (0147) −0220 (0302) 0211 (0062) 0033 (0051) 0281 (0073) 0008 (0014) 0676 (0187) 0.486
0139 (0092) 0520 (0222) −0115 (0117) −0799 (0300) 0200 (0086) 0353 (0077) 0106 (0035) 0049 (0019) 0198 (0128) 0.489
a Standard errors are given in parentheses. Reduced forms are estimated with standard covariates and then with interactions of all the covariates with hybrid. Covariates not reported include main season rainfall, household size, and age–sex composition of the household (includes variables for the number of boys (aged <16 years), the number of girls, the number of men (aged 17–39), the number of women, and the number of older men (aged >40 years)). Note that hired labor is measured in KShs per acre and family labor is measured in hours per acre. Where the covariates are interacted with hybrid, only main effects of the covariates are reported. Reduced forms are for the case where all covariates are exogenous: only hybrid is correlated with the comparative advantage, θi . See Table VIII for the projection and structural estimates. The reduced form equations run are yi1 = δ1 + γ1 hi1 + γ2 hi2 + γ3 hi1 hi2 + ξi1 and yi2 = δ2 + γ4 hi1 + γ5 hi2 + γ6 hi1 hi2 + ξi2 for 1997 and 2004, respectively.
SELECTION AND COMPARATIVE ADVANTAGE
TWO PERIOD BASIC COMPARATIVE ADVANTAGE CRC MODEL REDUCED FORM ESTIMATES: DEPENDENT VARIABLE IS YIELDS (LOG MAIZE HARVEST PER ACRE)a
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196
TABLE VIIIA TWO PERIOD BASIC COMPARATIVE ADVANTAGE CRC MODEL OMD STRUCTURAL ESTIMATESa With Only Hybrid Endogenous Full Sample Without Covariates
With Covariates
With Interactions With Hybrid
Without Covariates
With Covariates
With Interactions With Hybrid
0648 (0093) 1007 (0112) 1636 (4854) −0543 (1874) −0794 (0411) 40.089
0565 (0087) 0665 (0104) −1690 (4316) 1023 (1480) −1317 (1262) 11.25
0456 (0090) 0473 (0116) −0485 (0199) 3534 (2405) −1782 (1374) 139.5
0471 (0099) 1139 (0122) −4800 (9173) 2287 (4222) −1010 (0228) 175.5
0305 (0089) 0710 (0112) −0936 (0308) 0623 (0100) −1518 (0310) 114.1
0139 (0092) 0466 (0123) −0497 (0257) 0790 (0169) −2196 (1142) 305.2
a Standard errors are given in parentheses. The reduced forms for these estimates are reported in Table VII (for the case where only the hybrid decision is endogenous, i.e., correlated with the θi ’s). The projection used in this model is θi = λ0 + λ1 hi1 + λ2 hi2 + λ3 hi1 hi2 + vi . The structural coefficients reported are the average return to hybrid (β), the comparative advantage coefficient (φ), and the projection coefficients (λ’s). OMD is optimal weighted (the weight matrix is the inverted reduced form variance–covariance matrix) minimum distance. Results from diagonally weighted (the weight matrix is the OMD weight matrix with the off-diagonal elements set to zero) and equally weighted minimum distance are similar. The χ2 statistic on the overidentification test is the value of the OMD minimand. Results are reported for two samples: the full sample and the sample without two districts where HIV is prevalent. In addition, minimum distance results for three different specifications are reported: without covariates, with covariates, and with covariates and interactions of the covariates with the hybrid decision (reported in Table VII). All the specifications with covariates assume that all covariates are exogenous: only hybrid is correlated with the comparative advantage, θi . Covariates include acreage, land preparation costs, fertilizer, hired labor, family labor, main season rainfall, household size and age–sex composition of the household (includes variables for the number of boys (aged <16 years), the number of girls, the number of men (aged 17–39), the number of women, and the number of older men (aged >40 years)).
TAVNEET SURI
λ1 λ2 λ3 β φ χ21
Without HIV Districts
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JOINT SECTOR COMPARATIVE ADVANTAGE CRC MODEL OMD STRUCTURAL ESTIMATESa With Joint Sector Fertilizer-Hybrid Decision Full Sample
β φ
Without HIV Districts
With Covariates
With Interactions With Hybrid
With Covariates
With Interactions With Hybrid
0639 (0095) −1602 (1684)
1148 (0813) −3133 (4003)
0420 (0051) −1687 (0554)
0901 (0175) −2051 (1282)
a Standard errors are given in parentheses. OMD is optimal weighted (the weight matrix is the inverted reduced form variance–covariance matrix) minimum distance. Results from diagonally weighted (the weight matrix is the OMD weight matrix with the off-diagonal elements set to zero) and equally weighted minimum distance are similar. The structural coefficients reported are average return to hybrid (β) and the comparative advantage coefficient (φ). Results are reported for two samples: the full sample and the sample without two HIV districts and are for two periods of data. In addition, minimum distance results for two different specifications are reported: with covariates and with covariates and interactions of the covariates with the hybrid decision. All the specifications with covariates assume that all covariates other than hybrid and/or fertilizer are exogenous. Covariates include acreage, land preparation costs, fertilizer, hired labor, family labor, main season rainfall, household size, and age–sex composition of the household (includes variables for the number of boys (aged <16 years), the number of girls, the number of men (aged 17–39), the number of women, and the number of older men (aged >40 years)).
what I call the non-HIV sample. The results are much more precise and stable across specifications for this sample. Tables VIIIB and VIIIC present structural estimates of φ for the case where fertilizer endogenously enters the farmer’s decisions. In Table VIIIB, I report results for the case where there is a joint hybrid–fertilizer decision on the part of the farmer so that he is in the technology sector if he uses both hybrid and fertilizer;50 otherwise, he is not. Results for φ are again consistently negative, and significant mostly in the specifications that exclude the two high mortalTABLE VIIIC COMPARATIVE ADVANTAGE CRC MODEL OMD ESTIMATES: BOTH HYBRID AND FERTILIZER ENDOGENOUSa With Both Fertilizer and Hybrid as Endogenous Projection: θi = λ0 + λ1 hi1 + λ2 hi2 + λ3 hi1 hi2 + λ4 hi1 fi1 + λ5 hi2 fi1 + λ6 hi1 hi2 fi1 + λ7 hi1 fi2 + λ8 hi2 fi2 + λ9 hi1 hi2 fi2 + λ10 fi1 + λ11 fi2 + vi Full Sample
β φ
Without HIV Districts
With Covariates
With Interactions With Hybrid
With Covariates
With Interactions With Hybrid
0088 (0096) −0449 (0176)
0915 (0417) −3772 (2707)
0603 (0060) −1788 (0277)
0686 (0174) −2118 (0641)
a The notes for Table VIIIB apply here also.
50 The fertilizer decision I use involves chemical fertilizers. The results are similar if I include the use of manure as part of the fertilizer decision.
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ity districts. Finally, in Table VIIIC, I allow for fertilizer to be endogenous and correlated with the θi ’s, using the projection in equation (35). Again, the estimates of φ in Table VIIIC are consistently negative across all OMD specifications, again more so for the sample without the two high mortality districts. Overall, to summarize these results, the null hypothesis that φ is equal to zero is often rejected and estimates of φ are consistently negative.51 These results hold predominantly for the non-HIV districts in the sample, that is, after leaving out 2 of the overall 22 districts in the sample. For these households, the selection into hybrid is negative, with the farmers having the lowest yields in nonhybrid having the highest returns to planting hybrid. i 6.1. Recovering the Distribution of θ Given the CRC structural estimates of the λ’s and the form of the projection given by either equations (29) or (35), the θi ’s can be predicted for a given history of hybrid (and fertilizer) use. However, to do so, I must assume that the projections describe the true conditional expectation of the θi ’s. This is essentially an assumption only for the case of equation (35). For equation (29), since hit is binary and each history is accounted for, the projection is essentially saturated (for example, polynomials in adoption would be redundant since adoption is a dummy variable). Using the predicted distribution of θi , the predicted τi ’s can be derived via the yield function. Intuitively, the process of recovering the distribution of θi is building a counterfactual set of returns for the nonadopters using a weighted average of the experience of every type of farmer, where “type” refers to the farmer’s hybrid history. For the simple two period model (Table VIIIA), since the predicted θi ’s are obtained from the projection in equation (29), the distribution of the predicted θi ’s has just four mass points. There are only four possible hybrid histories: nonhybrid stayers, hybrid stayers, leavers, and joiners. The results indicate that the nonhybrid stayers have the most negative θi , followed by the hybrid stayers, then the leavers and joiners. While the nonhybrid stayers have the lowest predicted θi ’s, they have the highest returns to hybrid as φ < 0.52 Using the projection in equation (35), I can similarly estimate the distribution of θi using the estimates in Table VIIIC. Here, the distribution of the θi ’s come from the projection predicted θi ’s is continuous, since the predicted in equation (35), which includes the amount of fertilizer used. To plot the distribution of the θi ’s and the overall return to hybrid, I use the estimates from 51 The results without covariates and across all EWMD and DWMD specifications are similar. In addition, the following robustness checks give similar results: dropping the Coast province, allowing for depreciation of assets used in land preparation (for example, a tractor, which only 2% of households own), valuing yields and all the inputs using prices where possible, and using estimated profits instead of yields. 52 I do not report these estimates as they do not account for the endogeneity of fertilizer (see Suri (2006)).
SELECTION AND COMPARATIVE ADVANTAGE
199
FIGURE 5A.—Distribution of comparative advantage.
FIGURE 5B.—Distribution of returns.
the third data column of Table VIIIC, which excludes the two HIV districts and controls only for covariates. This is a simpler model to compute the overall return to hybrid, since hybrid is not interacted with all the covariates. The results are not different across the last two columns in Table VIIIC, so I opt to use this simpler model for the purposes of plotting the distribution of the comparative advantage and the returns. Figure 5A shows the means of the θi by transition: again, the nonhybrid stayers have the most negative θi , followed by the hybrid stayers, then the leavers and joiners. Figure 5B shows the distribution of returns across the sample by transition. The distribution of returns is given by β + φθi , where β is the structural coefficient on hit in equation (20), that is, the average return to hybrid. Since φ < 0, the ordering of returns is the reverse of the ordering of θi ’s, that is, the nonhybrid stayers have the most negative θi ’s, but the highest positive returns. The joiners and leavers have close to zero returns (they are the marginal farmers) and the hybrid stayers have lower positive returns. Note that the sign of φ determines the ordering of the magnitude of the returns across transitions; this ordering is by no means mechanical. Figure 5C shows the distribution of the predicted θi ’s by transition. Finally, as a check, Figure 5D shows
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FIGURE 5C.—Endogenous hybrid and fertilizer use.
FIGURE 5D.—Distribution of tau, by adoption decision in 1997.
the corresponding distribution of the predicted τi ’s, which were constructed to be orthogonal to the hybrid choice. 7. DISCUSSION Figure 5B resolves one puzzle about low adoption rates: the hybrid joiners and leavers are marginal farmers in the sense of having very low returns as compared to the average return. However, while the empirical results resolve this one puzzle of why some farmers move in and out of hybrid despite high returns for the average farmer, the results indicate a further puzzle: the very large counterfactual returns to growing hybrid for the nonhybrid stayers. To examine this latter puzzle, I return to the decision rule in equation (25) and relate the estimated gross returns to observable supply and demographic characteristics of the farmers (just for the sample excluding the two HIV districts). Table IX illustrates that the households with the lower predicted θi ’s have much higher cost determinants. Recall that throughout the sample period, the price of hybrid seed was fixed across the country. This meant that seed suppliers had no
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incentives to locate far away from markets or roads as there was no compensating price increase. The regressions in Table IX therefore correlate the predicted θi ’s with observable cost measures in the data.53 The observables considered are the distance from the household to the closest fertilizer seller, distance to the closest matatu (public transport) stop, distance to the closest motorable road, distance to the closest extension services, a set of dummies for education of the household head, a dummy for whether the household tried to get credit, a dummy for whether the household tried to get credit but did not receive any, a dummy for whether the household received any credit, and province dummies. Villages are large and heterogeneous, and these measures of costs and infrastructure vary even within a village. The infrastructure and education variables correlate strongly with the predicted θi ’s, but the credit variables do not. The last column shows similar results without province dummies.54 These results indicate that while there are high potential returns to the farmers who never grow hybrid during the sample period, they face higher costs and supply constraints. The IV/LATE estimates in Table IV also suggested that distance and infrastructure are constraining factors. From Table IV, the IV estimates are about 150%, which implies that those affected by the infrastructure instrument are the nonhybrid stayers, since these IV estimates are of the same magnitude as the estimated returns for the nonhybrid stayers. Thus, given existing seed supply locations and infrastructure constraints, the adoption decisions appear to be quite rational. Liberalizing the seed market and seed prices may have large benefits. In addition, while I do not have the data for a social return calculation to expanding supply, encouraging greater seed supply to the more remote areas of Kenya could have large benefits, but will also have large costs of expanding infrastructure. 7.1. Competing Explanations Finally, I discuss some alternative explanations of adoption behavior that are popular in the literature. The first is a model of learning, where households are uninformed about the benefits of a technology and they experiment to learn what the returns are. Learning models give rise to the familiar S-shaped curves of adoption rates over time (see Griliches (1957)). In my data, adoption rates show no such dynamics. Aggregate adoption remains constant over the span 53 As pointed out by an anonymous referee, given how the θi ’s are estimated, it could induce a mechanical positive correlation between the estimated θi ’s and the costs even if no such correlation exists. Simulating the model for a range of true values of φ showed that the bias is small and in the opposite direction to the sign of the correlations reported in Table IX. Whenever φ < 0 (as is the case for my estimates), for example, the bias would always mean a positive coefficient between costs and the estimated θi ’s, whereas the correlations in Table IX are all negative. 54 These results are all for the model with endogenous hybrid and fertilizer (estimates from Table VIIIC).
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TABLE IX CORRELATES OF ESTIMATED θi ’S: DEPENDENT VARIABLE IS THE ESTIMATED θi ’Sa (1)
(2)
(3)
(4)
(5)
−0301 (0122) −0904 (0503) 0032 (0298) −0130 (0155) — — — No
−0289 (0121) −0887 (0501) −0034 (0298) −0063 (0155) −0138 (0153) — — Yes (0002) Yes
−0285 (0121) −0901 (0502) −0016 (0298) −0063 (0155) — 0027 (0347) — Yes (0002) Yes
−0285 (0121) −0898 (0501) −0028 (0299) −0061 (0155) — — −0047 (0154) Yes (0000) Yes
−0315 (0063) −0978 (0285) −0021 (0165) 0002 (0091) — — −0164 (0144) Yes (0000) No
Yes
a Standard errors are given in parentheses. This table reports the correlations of the predicted θ ’s with various observables. The predicted θ ’s used are from the two period i i
case with endogenous hybrid and fertilizer use (Table VIIIC). The observables are the average of the 1997 and 2004 values. Columns (1)–(4) show the results while controlling for province dummies. Since a lot of the variation in infrastructure may be aggregate, column (5) also shows the results without province dummies, but just for the “Received credit” variable on the credit side (the results are no different for the other credit variables).
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Distance to closest fertilizer seller (× 100) Distance to motorable road (× 100) Distance to matatu stop (× 100) Distance to extension services (× 100) Tried to get credit (× 10) Tried but did not receive credit (× 10) Received credit (× 10) Dummies for household head education (p-value on joint significance) Province dummies
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of almost 10 years. In fact, there are no dynamics in any of the inputs, even after a large rainfall shock due to El Nino in 1998. Any conventional model of learning does not appear to be borne out by these aggregates, although there may have been learning effects closer to the introduction of hybrid in Kenya: Gerhart (1975) showed S-shaped curves in the 1970’s, curves that flattened out at low aggregate adoption rates prior to 1997. The switching between hybrid and nonhybrid from period to period may look like experimentation, but recall that hybrid and fertilizer have been available for a few decades now. In addition, 90% and 83% of households have used hybrid and fertilizer before, respectively. Table IID does not show any systematic persistence in adoption; the cycling behavior is about as prevalent as the noncycling. In the qualitative parts of the survey, farmers who were using traditional varieties were asked why they were not using hybrid and only about 0.3% cited experimenting or on trial. In addition, Duflo, Kremer, and Robinson (2008a) found little evidence of learning in western Kenya. Learning therefore seems to be unimportant in this setting.55 There are different varieties of hybrid, so it may be possible that households are learning about these types and are switching to newer varieties over time. In this case, there should still be aggregate increases in adoption over time. For two periods of the survey (2002 and 2004), the data include exactly what type of hybrid variety was used. In 2002, 61.5% of maize plots were planted with hybrid 614, 6% each of 625 and 627, and 4% of 511. The release dates of these varieties were 1986, 1981, 1996, and the 1960’s, respectively. In 2004, 69.5% of plots were planted with 614, 5% with 627, and 4% each with 625 and 511. It seems that most households use rather old varieties of hybrid seed. As a further check, I categorize the type of hybrid used in 2004 as new or not, where anything released after 1990 is defined as new. I cannot look at leavers as I do not know what hybrid type they used in 1997, and in 2004 they do not use hybrid. Of the hybrid stayers, about 11.5% use new hybrids and 12.9% of the joiners use new hybrids. These two means are not statistically significantly different from each other (the p-value is 0.702). A second set of alternative explanations deals with the role of credit constraints. Credit constraints do not seem to be of first order importance for several reasons. First, a strikingly large number of households are able to get access to credit, especially for agricultural purposes, and of those that try to get credit, most get it. In 2004, 41% of all households tried to get credit and 83% of them got it. In 1997, these figures were collected for agricultural credit, where 33.7% of households tried to get agricultural credit and 90% got it. There 55
On the methodological side, a learning model with heterogeneity in returns would imply a random coefficient model with feedback. Chamberlain (1993) showed that such models suffer from identification problems. Gibbons, Katz, Lemieux, and Parent (2002) estimated a random coefficient model with learning, but they relied on the structure of the learning process and an extremely long panel (the National Longitudinal Survey of Youth) to identify the model.
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are no significant differences between the hybrid stayers, nonhybrid stayers, leavers, and joiners in the three credit variables (whether the farmer tried to get credit, whether he received it, and whether he tried to get credit but did not receive it). The p-values on these F -tests are all above 0.20. Also, none of these credit variables correlates strongly with comparative advantage in Table IX. A caveat here is that these variables are not necessarily good measures of credit constraints, since households that may be constrained may not even try to get credit. The patterns in Table V also do not fit a pure liquidity constraints story. Say the variation in adoption is completely explained by liquidity constraints. In this case, if a nonhybrid farmer does better than average in terms of yields, it will lead him to use hybrid in the next period, but in that second period, he should have the average hybrid yield, such that nonhybrid households that have higher yields than average in the first period should have no different than average hybrid yields in the second period. So, for 1997–2004, the coefficient on joiners would be greater than zero in 1997 and equal to the coefficient on the hybrid stayers in 2004. Similarly, the leavers should do worse than the hybrid stayers in 1997 and no different than the nonhybrid stayers in 2004. These joint restrictions are rejected in Table V. Liquidity constraints alone do not, therefore, seem to explain the patterns in the data. The final alternative hypothesis is differences in tastes between hybrid and nonhybrid (as in Latin America). In Kenya, hybrid and nonhybrid output are indistinguishable and sold at the same price. The survey asked farmers why they chose the variety they plant, and an equal fraction of hybrid and nonhybrid farmers said it was because they preferred the taste. The differential tastes hypothesis does not seem to fit the Kenyan case. 8. CONCLUSION This paper examines the adoption decisions and benefits of hybrid maize in the context of Kenya, where adoption trends have been relatively constant for the past decade, yet vary considerably across space. To examine this setting, in contrast to much of the literature on technology adoption, I use a framework that abandons learning about a homogeneous technology, and that instead considers a model and empirical approach that allows for heterogeneous returns to hybrid that correlate with the hybrid adoption decision. My framework emphasizes the large disparities in farming and input supply characteristics across the maize growing areas of Kenya. Within this framework, I find that for the non-HIV districts in the sample, returns to hybrid maize vary greatly. Furthermore, in these districts, farmers who are on the margin of adopting and disadopting (and who do so during my sample period) experience little change in yields. The experimental evidence for Kenya, along with simple OLS and IV estimates, has indicated average high, positive returns to these agricultural technologies. This has indicated a puzzle as to why adoption rates had flattened
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out well below 100% despite the seemingly large gains that could be had by adopting hybrid. The experimental and IV evidence appeared to indicate that maize growers in Kenya were leaving “money on the table” by not adopting hybrid seed. However, the experiments and IV results reveal little else about the returns. The framework of heterogeneous returns in this paper allows the estimation of not just average returns, but also the distribution of returns across the sample of farmers. I find strong evidence of heterogeneity in returns to hybrid maize, with comparative advantage playing an important role in the determination of yields and adoption decisions. My findings regarding the heterogeneity in returns have important implications for policy. Looking at a distribution of returns across the sample of farmers allows me to separate out farmers with low returns from those with high returns into a group that could be targeted by policy interventions. For a small group of farmers in the sample (only 20% of the sample), returns to growing hybrid (as opposed to their observed choice of nonhybrid) would be extremely high, yet they do not adopt hybrid maize. I show that these farmers have high fixed costs that prevent them from adopting hybrid, as they have poor access to seed and fertilizer distributors. In terms of policy, alleviating these constraints would increase yields for these farmers although this may not be a socially optimal intervention. Furthermore, liberalization of the seed supply market and of hybrid seed prices might encourage seed suppliers to locate in the more distant areas. For a large fraction of my sample, the returns to hybrid maize are smaller and these farmers choose to adopt. While I do not build risk into the choice framework used in this paper, farmers may use hybrid maize, even when the mean returns to hybrid are low, as it helps insure them against bad outcomes. For these farmers, since they do not seem to be constrained, they would benefit from improved research and development efforts in developing new hybrid strains and a biotechnology effort similar to that of, say, India, where releases of newer hybrids occur often and lead to continual yield improvements. Finally, the heterogeneity in returns to the hybrid technology, on the whole, suggests quite rational and relatively unconstrained adoption of existing hybrid strains, in contrast to the evidence from the experimental and IV literatures. REFERENCES ALTONJI, J., AND L. SEGAL (1996): “Small Sample Bias in GMM Estimation of Covariance Structures,” Journal of Business and Economic Statistics, 14, 353–366. [193] ARIGA, J., T. S. JAYNE, AND J. NYORO (2006): “Factors Driving the Growth in Fertilizer Consumption in Kenya, 1990–2005: Sustaining the Momentum in Kenya and Lessons for Broader Replicability in Sub-Saharan Africa,” Working Paper 24/2006, Tegemeo Institute. [184] ASHENFELTER, O., AND D. CARD (1985): “Using the Longitudinal Structure of Earnings to Estimate the Effects of Training Programs,” Review of Economics and Statistics, 67 (4), 648–666. [187,188] ATTANASIO, O., C. MEGHIR, AND M. SZEKELY (2003): “Using Randomized Experiments and Structural Models for Scaling Up: Evidence From the PROGRESA Evaluation,” Mimeo, University College London. [163]
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The Sloan School of Management, Massachusetts Institute of Technology, 100 Main Street, E62-517, Cambridge, MA 02142, U.S.A.; [email protected]. Manuscript received February, 2008; final revision received September, 2009.
Econometrica, Vol. 79, No. 1 (January, 2011), 211–251
THE EFFECT OF EXPECTED INCOME ON INDIVIDUAL MIGRATION DECISIONS BY JOHN KENNAN AND JAMES R. WALKER1 This paper develops a tractable econometric model of optimal migration, focusing on expected income as the main economic influence on migration. The model improves on previous work in two respects: it covers optimal sequences of location decisions (rather than a single once-for-all choice) and it allows for many alternative location choices. The model is estimated using panel data from the National Longitudinal Survey of Youth on white males with a high-school education. Our main conclusion is that interstate migration decisions are influenced to a substantial extent by income prospects. The results suggest that the link between income and migration decisions is driven both by geographic differences in mean wages and by a tendency to move in search of a better locational match when the income realization in the current location is unfavorable. KEYWORDS: Migration, dynamic discrete choice models, job search, human capital.
1. INTRODUCTION THERE IS AN EXTENSIVE LITERATURE on migration.2 Most of this work describes patterns in the data: for example, younger and more educated people are more likely to move; repeat and especially return migration account for a large part of the observed migration flows. Although informal theories explaining these patterns are plentiful, fully specified behavioral models of migration decisions are scarce, and these models generally consider each migration event in isolation, without attempting to explain why most migration decisions are subsequently reversed through onward or return migration. This paper develops a model of optimal sequences of migration decisions, focusing on expected income as the main economic influence on migration. The model is estimated using panel data from the National Longitudinal Survey of Youth (NLSY) on white males with a high-school education. We emphasize that migration decisions are reversible and that many alternative locations must be considered. Indeed (as we show in Section 2), repeat migration is a prominent feature of the data, and in many cases people choose to return to 1 The National Science Foundation and the NICHD provided research support. We thank Taisuke Otsu for outstanding research assistance. We also thank the editor and referees for very detailed constructive criticism of several earlier versions of the paper. We are grateful to Joe Altonji, Kate Antonovics, Peter Arcidiacono, Gadi Barlevy. Philip Haile, Bruce Hansen, Igal Hendel, Yannis Ioannides, Mike Keane, Derek Neal, John Pencavel, Karl Scholz, Robert Shimer, Chris Taber, Marcelo Veracierto, Ken Wolpin, Jim Ziliak, and seminar and conference participants at the Chicago Federal Reserve Bank, Carnegie–Mellon, Duke, Iowa, IZA, Ohio State, Penn State, Rochester, SITE, the Upjohn Institute, Virginia, Wisconsin, and Yale for helpful comments. 2 See Greenwood (1997) and Lucas (1997) for surveys.
© 2011 The Econometric Society
DOI: 10.3982/ECTA4657
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a location that they had previously chosen to leave, even though many unexplored alternative locations are available. A dynamic model is clearly necessary to understand this behavior. Structural dynamic models of migration over many locations have not been estimated before, presumably because the required computations have not been feasible.3 A structural representation of the decision process is of interest for the usual reasons: we are ultimately interested in quantifying responses to income shocks or policy interventions not seen in the data, such as local labor demand shocks or changes in welfare benefits. Our basic empirical question is the extent to which people move for the purpose of improving their income prospects. Work by Keane and Wolpin (1997) and by Neal (1999) presumes that individuals make surprisingly sophisticated calculations regarding schooling and occupational choices. Given the magnitude of geographical wage differentials, and given the findings of Topel (1986) and Blanchard and Katz (1992) regarding the responsiveness of migration flows to local labor market conditions, one might expect to find that income differentials play an important role in migration decisions. We model individual decisions to migrate as a job search problem. A worker can draw a wage only by visiting a location, thereby incurring a moving cost. Locations are distinguished by known differences in wage distributions and amenity values. We also allow for a location match component of preferences that is revealed to the individual for each location that is visited. The decision problem is too complicated to be solved analytically, so we use a discrete approximation that can be solved numerically, following Rust (1994). The model is sparsely parameterized. In addition to expected income, migration decisions are influenced by moving costs (including a fixed cost, a reduced cost of moving to a previous location, and a cost that depends on distance), by differences in climate, and by differences in location size (measured by the population in each location). We also allow for a bias in favor of the home location (measured as the state of residence at age 14). Age is included as a state variable, entering through the moving cost, with the idea that if the simplest human capital explanation of the relationship between age and migration rates is correct, there should be no need to include a moving cost that increases with age. 3
Holt (1996) estimated a dynamic discrete choice model of migration, but his framework modeled the move–stay decision and not the location-specific flows. Similarly, Tunali (2000) gave a detailed econometric analysis of the move–stay decision using microdata for Turkey, but his model does not distinguish between alternative destinations. Dahl (2002) allowed for many alternative destinations (the set of states in the United States), but he considered only a single lifetime migration decision. Gallin (2004) modeled net migration in a given location as a response to expected future wages in that location, but he did not model the individual decision problem. Gemici (2008) extended our framework and considered family migration decisions, but defined locations as census regions.
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Our main substantive conclusion is that interstate migration decisions are indeed influenced to a substantial extent by income prospects. There is evidence of a response to geographic differences in mean wages, as well as a tendency to move in search of a better locational match when the income realization in the current location is unfavorable. More generally, this paper demonstrates that a fully specified econometric model of optimal dynamic migration decisions is feasible and that it is capable of matching the main features of the data, including repeat and return migration. Although this paper focuses on the relationship between income prospects and migration decisions at the start of the life cycle, suitably modified versions of the model can potentially be applied to a range of issues, such as the migration effects of interstate differences in welfare benefits, the effects of joint career concerns on household migration decisions, and the effects on retirement migration of interstate differences in tax laws.4 2. MIGRATION DYNAMICS The need for a dynamic analysis of migration is illustrated in Table I, which summarizes 10-year interstate migration histories for the cross-section sample of the NLSY, beginning at age 18. Two features of the data are noteworthy. First, a large fraction of the flow of migrants involves people who have already moved at least once. Second, a large fraction of these repeat moves involves people returning to their original location. Simple models of isolated move– stay decisions cannot address these features of the data. In particular, a model of return migration is incomplete unless it includes the decision to leave the TABLE I INTERSTATE MIGRATION, NLSY 1979–1994a Less Than High School
High School
Some College
College
Total
322 80 248% 210
919 223 243% 195
758 224 296% 190
685 341 498% 202
2,684 868 323% 198
Repeat moves (% of all moves) Return migration (% of all moves) Home Not home
524%
487%
474%
505%
495%
327% 155%
331% 71%
291% 68%
232% 86%
281% 84%
Movers who return home
613%
565%
513%
428%
502%
Number of people Movers (age 18–27) Movers (%) Moves per mover
a The sample includes respondents from the cross-section sample of the NLSY79 who were continuously interviewed from ages 18 to 28 and who never served in the military. The home location is the state of residence at age 14.
4
See, for example, Kennan and Walker (2010) and Gemici (2008).
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initial location as well as the decision to return. Moreover, unless the model allows for many alternative locations, it cannot give a complete analysis of return migration. For example, a repeat move in a two-location model is necessarily a return move, and this misses the point that people frequently decide to return to a location that they had previously decided to leave, even though many alternative locations are available. 3. AN OPTIMAL SEARCH MODEL OF MIGRATION We model migration as an optimal search process. The basic assumption is that wages are local prices of individual skill bundles. We assume that individuals know the wage in their current location, but to determine the wage in another location, it is necessary to move there, at some cost. This reflects the idea that people may be more productive in some locations than in others, depending on working conditions, residential conditions, local amenities, and so forth. Although some information about these things can of course be collected from a distance, we view the whole package as an experience good. The model aims to describe the migration decisions of young workers in a stationary environment. The wage offer in each location may be interpreted as the best offer available in that location.5 Although there are transient fluctuations in wages, the only chance of getting a permanent wage gain is to move to a new location. One interpretation is that wage differentials across locations equalize amenity differences, but a stationary equilibrium with heterogeneous worker preferences and skills still requires migration to redistribute workers from where they happen to be born to their equilibrium location. Alternatively, it may be that wage differentials are slow to adjust to location-specific shocks, because gradual adjustment is less costly for workers and employers.6 In that case, our model can be viewed as an approximation in which workers take current wage levels as an estimate of the wages they will face for the foreseeable future. In any case, the model is intended to describe the partial equilibrium response of labor supply to wage differences across locations; from the worker’s point of view, the source of these differences is immaterial, provided that they are permanent. A complete equilibrium analysis would of course be much more difficult, but our model can be viewed as a building-block toward such an analysis. 5 This means that we are treating local match effects as relatively unimportant: search within the current location quickly reveals the best available match. 6 Blanchard and Katz (1992), using average hourly earnings of production workers in manufacturing, by state, from the Bureau of Labor Statistics (BLS) establishment survey, described a pattern of “strong but quite gradual convergence of state relative wages over the last 40 years.” For example, using a univariate AR(4) model with annual data, they found that the half-life of a unit shock to the relative wage is more than 10 years. Similar findings were reported by Barro and Sala-i Martin (1991) and by Topel (1986).
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Suppose there are J locations, and individual i’s income yij in location j is a random variable with a known distribution. Migration decisions are made so as to maximize the expected discounted value of lifetime utility. In general, the level of assets is an important state variable for this problem, but we focus on a special case in which assets do not affect migration decisions: we assume that the marginal utility of income is constant, and that individuals can borrow and lend without restriction at a given interest rate. Then expected utility maximization reduces to maximization of expected lifetime income, net of moving costs, with the understanding that the value of amenities is included in income, and that both amenity values and moving costs are measured in consumption units. This is a natural benchmark model, although of course it imposes strong assumptions. There is little hope of solving this expected income maximization problem analytically. In particular, the Gittins index solution of the multiarmed bandit problem cannot be applied because there is a cost of moving.7 But by using a discrete approximation of the wage and preference distributions, we can compute the value function and the optimal decision rule by standard dynamic programming methods, following Rust (1994). 3.1. The Value Function Let x be the state vector (which includes wage and preference information, current location, and age, as discussed below). The utility flow for someone who chooses location j is specified as u(x j) + ζj , where ζj is a random variable that is assumed to be independent and identically distributed (i.i.d.) across locations and across periods, and independent of the state vector. Let p(x |x j) be the transition probability from state x to state x if location j is chosen. The decision problem can be written in recursive form as V (x ζ) = max(v(x j) + ζj ) j
where v(x j) = u(x j) + β
¯ ) p(x |x j)v(x
x
and ¯ v(x) = Eζ V (x ζ) and where β is the discount factor and Eζ denotes the expectation with respect to the distribution of the J-vector ζ with components ζj . We assume that 7 See Banks and Sundaram (1994) for an analysis of the Gittins index in the presence of moving costs.
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ζj is drawn from the type I extreme value distribution. In this case, following McFadden (1974) and Rust (1987), we have ¯ exp(v(x)) = exp(γ) ¯
J
exp(v(x k))
k=1
where γ¯ is the Euler constant. Let ρ(x j) be the probability of choosing location j, when the state is x. Then ¯ ρ(x j) = exp(γ¯ + v(x j) − v(x)) We compute v by value function iteration, assuming a finite horizon T . We include age as a state variable, with v ≡ 0 at age T + 1, so that successive iterations yield the value functions for a person who is getting younger and younger. 4. EMPIRICAL IMPLEMENTATION A serious limitation of the discrete dynamic programming method is that the number of states is typically large, even if the decision problem is relatively simple. Our model, with J locations and n points of support for the wage distribution, has J(n+1)J states for each person at each age. Ideally, locations would be defined as local labor markets, but we obviously cannot let J be the number of labor markets; for example, there are over 3,100 counties in the United States. Indeed, even if J is the number of States, the model is computationally infeasible,8 but by restricting the information available to each individual, an approximate version of the model can be estimated; this is explained below. 4.1. A Limited History Approximation To reduce the state space to a reasonable size, it seems natural in our context to use an approximation that takes advantage of the timing of migration decisions. We have assumed that information on the value of human capital in alternative locations is permanent, so if a location has been visited previously, the wage in that location is known. This means that the number of possible states increases geometrically with the number of locations. In practice, however, the number of people seen in many distinct locations is small. Thus by restricting the information set to include only wages seen in recent locations, it is possible to drastically shrink the state space while retaining most of the information actually seen in the data. Specifically, we suppose that the number 8 And it will remain so: for example, if there are 50 locations, and the wage distribution has 5 support points, then the number of dynamic programming states is
40414063873238203032156980022826814668800
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of wage observations cannot exceed M, with M < J, so that it is not possible to be fully informed about wages at all locations. Then if the distributions of location match wage and preference components in each of J locations have n points of support, the number of states for someone seen in M locations is J(Jn2 )M , the number of possible M-period histories describing the locations visited most recently, and the wage and preference components found there. For example, if J = 50, n = 3, and M = 2, the number of states at each age is 10,125,000, which is manageable. This approximation reduces the number of states in the most obvious way: we simply delete most of them.9 Someone who has “too much” wage information in the big state space is reassigned to a less-informed state. Individuals make the same calculations as before when deciding what to do next, and the econometrician uses the same procedure to recover the parameters governing the individual’s decisions. There is just a shorter list of states, so people with different histories may be in different states in the big model, but they are considered to be in the same state in the reduced model. In particular, people who have the same recent history are in the same state, even if their previous histories were different. Decision problems with large state spaces can alternatively be analyzed by computing the value function at a finite set of points, and interpolating the function for points outside this set, as suggested by Keane and Wolpin (1994).10 In our context this would not be feasible without some simplification of the state space, because of the spatial structure of the states. Since each location has its own unique characteristics, interpolation can be done only within locations, and this means that the set of points used to anchor the interpolation must include several alternative realizations of the location match components for each location; allowing for n alternatives yields a set of nJ points, which is too big when J = 50 (even if n is small). On the other hand, it is worth noting that our limited history approximation works only because we have discretized the state space. If the location match components are drawn from continuous distributions, the state space is still infinite, even when the history is limited (although interpolation methods could be used in that case).
9 Note that it is not enough to keep track of the best wage found so far: the payoff shocks may favor a location that has previously been abandoned, and it is necessary to know the wage at that location so as to decide whether to go back there (even if it is known that there is a higher wage at another location). 10 For example, this method was used by Erdem and Keane (1996) to analyze the demand for liquid laundry detergent, and by Crawford and Shum (2005) to analyze the demand for pharmaceuticals. In these applications, the agents in the model do not know the flow payoffs from the various available choices until they have tried them, just as our agents do not know the location match components until they have visited the location.
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4.2. Wages The wage of individual i in location j at age a in year t is specified as wij (a) = μj + υij + G(Xi a t) + ηi + εij (a) where μj is the mean wage in location j, υ is a permanent location match effect, G(X a t) represents a (linear) time effect and the effects of observed individual characteristics, η is an individual effect that is fixed across locations, and ε is a transient effect. We assume that η, υ, and ε are independent random variables that are identically distributed across individuals and locations. We also assume that the realizations of η and υ are seen by the individual.11 The relationship between wages and migration decisions is governed by the difference between the quality of the match in the current location, measured by μj + υij , and the prospect of obtaining a better match in another location k, measured by μk + υik . The other components of wages have no bearing on migration decisions, since they are added to the wage in the same way no matter what decisions are made. The individual knows the realization of the match quality in the current location and in the previous location (if there is one), but the prospects in other locations are random. Migration decisions are made by comparing the expected continuation value of staying, given the current match quality, with the expected continuation values associated with moving. 4.3. State Variables and Flow Payoffs Let = ( M−1 ) be an M vector containing the sequence of recent locations (beginning with the current location), and let ω be an M vector recording wage and utility information at these locations. The state vector x consists of , ω, and age. The flow payoff for someone whose “home” location is h is specified as 0
1
u˜ h (x j) = uh (x j) + ζj where uh (x j) = α0 w( 0 ω) +
K
αk Yk ( 0 ) + αH χ( 0 = h)
k=1
+ ξ( 0 ω) − Δτ (x j) 11 An interesting extension of the model would allow for learning, by relaxing the assumption that agents know the realizations of η and υ. In particular, such an extension might help explain return migration, because moving reveals information about the wage components. Pessino (1991) analyzed a two-period Bayesian learning model along these lines and applied it to migration data for Peru.
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Here the first term refers to wage income in the current location. This is augmented by the nonpecuniary variables Yk ( 0 ), representing amenity values. The parameter αH is a premium that allows each individual to have a preference for their native location (χA denotes an indicator meaning that A is true). The flow payoff in each location has a random permanent component ξ; the realization of this component is learned only when the location is visited. This location match component of preferences is analogous to the match component of wages (υ), except that ξ can only be inferred from observed migration choices, whereas both migration choices and wages are informative about υ. The cost of moving from 0 to j for a person of type τ is represented by Δτ (x j). The unexplained part of the utility flow, ζj , may be viewed as either a preference shock or a shock to the cost of moving, with no way to distinguish between the two. 4.4. Moving Costs 0
Let D( j) be the distance from the current location to location j and let A( 0 ) be the set of locations adjacent to 0 (where states are adjacent if they share a border). The moving cost is specified as Δτ (x j) = γ0τ + γ1 D( 0 j) − γ2 χ(j ∈ A( 0 )) − γ3 χ(j = 1 ) + γ4 a − γ5 nj χ(j = 0 ) We allow for unobserved heterogeneity in the cost of moving: there are several types, indexed by τ, with differing values of the intercept γ0 . In particular, there may be a “stayer” type, meaning that there may be people who regard the cost of moving as prohibitive, in all states. The moving cost is an affine function of distance (which we measure as the great circle distance between population centroids). Moves to an adjacent location may be less costly (because it is possible to change states while remaining in the same general area). A move to a previous location may also be less costly, relative to moving to a new location. In addition, the cost of moving is allowed to depend on age, a. Finally, we allow for the possibility that it is cheaper to move to a large location, as measured by population size nj . It has long been recognized that location size matters in migration models (see, e.g., Schultz (1982)). California and Wyoming cannot reasonably be regarded as just two alternative places, to be treated symmetrically as origin and destination locations. For example, a person who moves to be close to a friend or relative is more likely to have friends or relatives in California than in Wyoming. One way to model this in our framework is to allow for more than one draw from the distribution of payoff shocks in each location.12 Alternatively, location size may affect moving costs; for example, 12 Suppose that the number of draws per location is an affine function of the number of people already in that location and that migration decisions are controlled by the maximal draw for
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friends or relatives might help reduce the cost of the move. In practice, both versions give similar results. 4.5. Transition Probabilities ˜ a), where x˜ = ( 0 1 x0υ x1υ x0ξ x1ξ ) The state vector can be written as x = (x 0 and where xυ indexes the realization of the location match component of wages in the current location, and similarly for the other components. The transition probabilities are ⎧ ˜ a = a + 1, 1 if j = 0 , x˜ = x, ⎪ ⎪ ⎪ ⎪ ⎪ 1 if j = 1 , x˜ = ( 1 0 x1υ x0υ x1ξ x0ξ ), a = a + 1, ⎪ ⎨ p(x | x j) = 1 if j ∈ / { 0 1 }, x˜ = (j 0 sυ x0υ sξ x0ξ ), ⎪ 2 ⎪ n ⎪ ⎪ ⎪ (1 1) ≤ (sυ sξ ) ≤ (nυ nξ ), a = a + 1, ⎪ ⎩ 0 otherwise. This covers several cases. First, if no migration occurs this period, then the state remains the same except for the age component. If there is a move to a previous location, the current and previous locations are interchanged, and if there is a move to a new location, the current location becomes the previous location and the new location match components are drawn at random. In all cases, age is incremented by one period. 4.6. Data Our primary data source is the National Longitudinal Survey of Youth 1979 Cohort (NLSY79); we also use data from the 1990 Census. The NLSY79 conducted annual interviews from 1979 through 1994, and subsequently changed to a biennial schedule. The location of each respondent is recorded at the date of each interview and we measure migration by the change in location from one interview to the next. We use information from 1979 to 1994 so as to avoid the complications arising from the change in the frequency of interviews. To obtain a relatively homogeneous sample, we consider only white nonHispanic high-school graduates with no post-secondary education, using only each location. This leads to the following modification of the logit function describing choice probabilities: ρ(x j) =
ξj ; J ξk
ξk = (1 + ψnk ) exp(υk ( ω))
k=1
Here nj is the population in location j, and ψ can be interpreted as the number of additional draws per person.
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the years after schooling is completed.13 Appendix A describes our selection procedures. The NLSY oversamples people whose parents were poor, and one might expect that the income process for such people is atypical and that the effect of income on migration decisions might also be atypical. Thus we use only the “cross-section” subsample, with the poverty subsample excluded. The sample includes only people who completed high school by age 20 and who never enrolled in college. We exclude those who ever served in the military and also those who report being out of the labor force for more than 1 year after age 20. We follow each respondent from age 20 to the 1994 interview or the first year in which some relevant information is missing or inconsistent. Our analysis sample contains 432 people, with continuous histories from age 20 comprising 4,274 person-years. There are 124 interstate moves (2.9 percent per annum). In each round of the NLSY79, respondents report income for the most recent calendar year. Wages are measured as total wage and salary income, plus farm and business income, adjusted for cost of living differences across states (using the American Chamber of Commerce Research Association (ACCRA) Cost of Living Index). We exclude observations with positive hours or weeks worked and zero income. We use information from the Public Use Micro Sample (PUMS) from the 1990 Census to estimate state mean effects (μj ), since the NLSY does not have enough observations for this purpose. From the PUMS we select white highschool men aged 19–20 (so as to avoid selection effects due to migration).14 We estimate state mean wage effects using a median regression with age and state dummies.15 We condition on these estimated state means in the maximum likelihood procedure that jointly estimates the remaining parameters of the wage process, and the utility and cost parameters governing migration decisions.16 13 Attrition in panel data is an obvious problem for migration studies, and one reason for using NLSY data is that it minimizes this problem. Reagan and Olsen (2000, p. 339) reported that “Attrition rates in the NLSY79 are relatively low. . . The primary reason for attrition are death and refusal to continue participating in the project, not the inability to locate respondents at home or abroad.” 14 The parameters governing migration decisions and the parameters of the wage process are estimated jointly to account for selection effects due to migration (although in practice these effects are empirically negligible). The state mean effects are specified as age-invariant and are estimated using wages observed at the beginning of the work life to minimize the potential effects of selection. We include observations for 19-year-olds from the PUMS to increase the precision of the estimated state means. 15 We measure wages as annual earnings and exclude individuals with retirement income, social security income, or public assistance; we also exclude observations if earnings are zero despite positive hours or weeks worked. 16 See Kennan and Walker (2011) for a detailed description of the estimation procedure.
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5. ESTIMATION In this section we discuss the specification and computation of the likelihood function. 5.1. Discrete Approximation of the Distribution of Location Match Effects We approximate the decision problem by using discrete distributions to represent the distributions of the location match components, and computing continuation values at the support points of these distributions. We first describe this approximation and then describe the specification of the other components of wages. For given support points, the best discrete approximation Fˆ for any distribution F assigns probabilities so as to equate Fˆ with the average value of F over each interval where Fˆ is constant. If the support points are variable, they are chosen so that Fˆ assigns equal probability to each point.17 Thus if the distribution of the location match component υ were known, the wage prospects associated with a move to state k could be represented by an n-point distribution with equally weighted support points μˆ k + υ(q ˆ r ) 1 ≤ r ≤ n, where υ(q ˆ r ) is the qr quantile of the distribution of υ, with qr =
2r − 1 2n
for 1 ≤ r ≤ n. The distribution of υ is, in fact, not known, but we assume that it is symmetric around zero. Thus, for example, with n = 3, the distribution of μj + υij in each state is approximated by a distribution that puts mass 13 on μj (the median of the distribution of μj + υij ), with mass 13 on μj ± τυ , where τυ is a parameter to be estimated. The location match component of preferences is handled in a similar way. 5.2. Fixed Effects and Transient Wage Components Even though our sample is quite homogeneous, measured earnings in the NLSY are highly variable, both across people and over time. Moreover, the variability of earnings over time is itself quite variable across individuals. Our aim is to specify a wage components model that is flexible enough to fit these data, so that we can draw reasonable inferences about the relationship between measured earnings and the realized values of the location match component. For the fixed effect η, we use a (uniform) discrete distribution that is symmetric around zero, with seven points of support, so that there are three parameters 17
See Kennan (2006).
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to be estimated. For the transient component ε, we need a continuous distribution that is flexible enough to account for the observed variability of earnings. We assume that ε is drawn from a normal distribution with zero mean for each person, but we allow the variance to vary across people. Specifically, person i initially draws σε (i) from some distribution, and subsequently draws εit from a normal distribution with mean zero and standard deviation σε (i), with εit drawn independently in each period. The distribution from which σε is drawn is specified as a (uniform) discrete distribution with four support points, where these support points are parameters to be estimated. 5.3. The Likelihood Function The likelihood of the observed history for each individual is a mixture over heterogeneous types. Let Li (θτ ) be the likelihood for individual i, where θτ is the parameter vector, for someone of type τ and let πτ be the probability of type τ. The sample log likelihood is K
N Λ(θ) = log πτ Li (θτ ) i=1
τ=1
For each period of an individual history, two pieces of information contribute to the likelihood: the observed income and the location choice. Each piece involves a mixture over the possible realizations of the various unobserved components. In each location there is a draw from the distribution of location match wage components, which is modeled as a uniform distribution over the finite set Υ = {υ(1) υ(2) υ(nυ )}. We index this set by ωυ , with ωυ (j) representing the match component in location j, where 1 ≤ ωυ (j) ≤ nυ . Similarly, in each location there is a draw from the location match preference distribution, which is modeled as a uniform distribution over the finite set Ξ = {ξ(1) ξ(2) ξ(nξ )}, indexed by ωξ . Each individual also draws from the distribution of fixed effects, which is modeled as a uniform distribution over the finite set H = {η(1) η(2) η(nη )}, and we use ωη to represent the outcome of this. Additionally, each individual draws a transient variance from a uniform distribution over the set ς = {σε (1) σε (2) σε (nε )}, with the outcome indexed by ωε . The unobserved components of wages and preferences for individual i are then represented by a vector ωi with Ni + 3 elements, ωi = {ωiξ ωiη ωiε ωiυ (1) ωiυ (2) ωiυ (nη )}, where Ni is the number of locations visited by this individual. The set of possible realizations of ωi is denoted by Ω(Ni ); there are nξ nη nε (nυ )Ni points in this set and our discrete approximation implies that they are equally likely. We index the locations visited by individual i in the order in which they appear, and we use the notation κ0it and κ1it to represent the position of the current and previous locations in this index. Thus κit = (κ0it κ1it ) is a pair of integers between 1 and Ni . For example, in the case of someone who never
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moves, κ0it is always 1 and κ1it is 0 (by convention), while for someone who has just moved for the first time, κit = (2 1). The likelihood is obtained by first conditioning on the realizations of ωi and then integrating over these realizations. Let ψit (ωi θ) be the likelihood of the observed income for person i in period t. Given ωi , the transient income component in period t is given by εit (ωi ) = wit − μ 0 (it) − G(Xi ait θτ ) − υ(ωiυ (κ0it )) − η(ωiη ) Thus ψit (ωi θτ )
wit − μ 0 (it) − G(Xi ait θτ ) − υ(ωiυ (κ0it )) − η(ωiη ) 1 φ = σε (ωiε ) σε (ωiε )
where φ is the standard normal density function. Let λit (ωi θτ ) be the likelihood of the destination chosen by person i in period t. Recall that ρ(x j) is the probability of choosing location j when the state is x. Then λit (ωi θτ ) = ρh(i) (i t) ωiυ (κ0it ) ωiυ (κ1it ) ωiξ (κ0it ) ωiξ (κ1it ) ait 0 (i t + 1) θτ Here the probability that i chooses the next observed location, 0 (i t + 1), depends on the current and previous locations, the values of the location match components at those locations, the individual’s home location h(i), and the individual’s current age. The parameter vector θτ includes the unknown coefficients in the flow payoff function and the support points in the sets Υ , H, Ξ, and ς. Finally, the likelihood of an individual history for a person of type τ is Li (θτ ) =
1 nη nε nξ (nυ )Ni T
i × ψit (ωi θτ )λit (ωi θτ ) ωi ∈Ω(Ni )
t=1
5.4. Identification The relationship between income and migration decisions in our model can be identified using the variation in mean wages across locations or by using the variation in the location match component of wages. We assume that the
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wage components (ηi υij εijt ) and the location match component of preferences ξij are all independently and identically distributed across individuals and states, and that εijt is i.i.d. over time. Alternatively, we can allow for unobserved amenities, represented by a component of ξ that is common to all individuals and that may be correlated with the state mean wages (μj ). In this case identification relies on variation in υ. We also assume that the unobserved heterogeneity in moving costs is i.i.d. across individuals, and that it is independent of the wage components and of the preference components. Our basic empirical results use variation in both μ and υ to identify the effect of income differences. In the context of an equilibrium model of wage determination, this can be justified by assuming constant returns to labor in each location, so that wage differences across locations are determined entirely by productivity differences and are thus independent of differences in amenity values. Clearly, this is a strong assumption. Accordingly (in Section 6.6 below), we also present estimates that control for regional differences in unobserved amenity values. 5.4.1. Nonparametric Identification of the Choice Probabilities If the match component of wages could be observed directly, it would be relatively straightforward to identify the effect of wages on migration decisions. But the observed wages include individual fixed effects and transient effects, so that the match component is observed only with error. In addition, there is selection bias in the match component of the observed wages, since an unfavorable draw from the υ distribution is more likely to be discarded (because it increases the probability of migration). The maximum likelihood procedure deals with the measurement error problem by integrating over the distributions of η and ε, and it deals with the selection problem by maximizing the joint likelihood of the wage components and the migration decisions. But this of course rests on specific parametric assumptions, and even then it does not give a transparent description of how identification is achieved. Thus it is useful to consider identification in a broader context.18 The basic identification argument can be well illustrated in a simplified situation in which there are just two observations for each person. Define the wage residual for individual i in period t in location j(t) as yit = wit − μj − G(Xi a t) = ηi + υij(t) + εit Recall that the wage components (η υ ε) are assumed to be independent, with zero means. The probability of moving (in the first period) depends on the 18
Identification of dynamic discrete choice models was analyzed by Magnac and Thesmar (2002), and by Abbring and Heckman (2007); identification of static equilibrium discrete choice models was analyzed by Berry and Haile (2010).
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location match component: denote this probability by ρ(υ). The process that generates the wage and migration data can then be represented as y2 = η + υ + ε2 ρ(υ) move
y1 = η + υ1 + ε1 1−ρ(υ) stay
y2 = η + υ˜ s + ε2 The question is whether it is possible to recover the function ρ(υ) from these data. For movers, we have two observations on the fixed effect η, contaminated by errors drawn from distinct distributions. Let υ˜ m denote the censored random variable derived from υ by discarding the realizations of υ for those who choose to stay and let υ˜ s denote the corresponding censored random variable derived by discarding the realizations of υ for those who choose to move. The observed wages for movers are y1 = η + υ˜ m + ε1 and y2 = η + υ + ε2 , where υ is a random draw from the υ distribution (which is independent of υ˜ m ). Then, under the regularity condition that the characteristic function of the random vector (y1 y2 ) is nonvanishing, Lemma 1 of Kotlarski (1967) implies that the (observed) distribution of (y1 y2 ) for movers identifies the distributions of η, υ˜ m + ε1 , and υ + ε2 . For stayers, we have two observations on η + υ˜ s , with measurement errors ε1 and ε2 . Thus Kotlarski’s lemma implies that the distribution of (y1 y2 ) for stayers identifies the distributions of η + υ˜ s , ε1 , and ε2 . This means that the distributions of η, υ, ε1 , ε2 , υ˜ m , and υ˜ s are all identified (either directly or by deconvolution). The choice probabilities ρ(υ) are then identified by Bayes theorem, fυ˜ m (υ) =
ρ(υ)fυ (υ) Prob(j(2) = j(1))
where fυ˜ 1 and fυ are the conditional and unconditional density functions, and j(2) = j(1) indicates a move. This argument shows that the effect of income on migration decisions is generically identified under our assumptions. In particular, under the null hypothesis that migration has nothing to do with income, the choice probability function ρ(υ) is a constant, while the expected income maximization model
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predicts that ρ(υ) is an increasing function. Since the shape of this function is identified in the data, the effect of income is identified nonparametrically. This is true even if there is unobserved heterogeneity with respect to moving costs and the location match component of preferences. Moreover, the identification argument does not rely on a particular distribution of the payoff shocks. What is identified is the average relationship between income and migration decisions, after integrating over the distributions of preference shocks and moving costs. 5.4.2. Identification of the Income Coefficient We have shown that the relationship between wages and migration choice probabilities is nonparametrically identified, given panel data on wage and migration outcomes, under the assumption that the wage is a sum of independent components. Although in general this relationship might be quite complicated, in our parametric model it is encapsulated in the income coefficient α0 . We now illustrate how this parameter is identified in our model, using an argument along the lines of Hotz and Miller (1993). Fix home location and age, with no previous location. Assume that there is no unobserved heterogeneity in moving costs and that there is no location match component of preferences. Then the state consists of the current location and the location match component of wages, and the choice probabilities corresponding to the function ρ(υ) in the nonparametric argument above are given by
ρ( υs j) =
⎧ exp(−Δ j + βV¯0 (j)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ exp(βV¯s ( )) + exp(−Δ k + βV¯0 (k)) ⎪ ⎪ ⎪ ⎪ k= ⎪ ⎨ j = ⎪ exp(βV¯s ( )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ exp(βV¯s ( )) + exp(−Δ k + βV¯0 (k)) ⎪ ⎪ ⎪ ⎪ k= ⎩ j =
where Δ j is the cost of moving from location to location j, V¯s (j) is the expected continuation value in j, given the location match component υs before knowing the realization of ζ, and V¯0 (j) is the expected continuation value before knowing the realization of υ: 1 ¯ V¯0 (j) = Vs (j) n s=1 n
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The probability of moving from to j, relative to the probability of staying, is ρ( υs j) = exp −Δ j + β(V¯0 (j) − V¯s ( )) ρ( υs ) Thus n ρ( υs j) 1 = −Δ j + β(V¯0 (j) − V¯0 ( )) log n s=1 ρ( υs ) and n 1 ρ( υs j) ρ(j υs ) = −Δ j − Δj log n s=1 ρ( υs ) ρ(j υs j) This identifies the round-trip moving cost between and j. The one-way moving costs are identified under weak assumptions on the moving cost function; for example, symmetry is obviously sufficient. In the model, the round-trip moving cost between two nonadjacent locations (for someone aged a with no previous location) is given by Δ j + Δj = 2(γ0 + γ4 a + γ1 D(j )) − γ5 (nj + n ) Since distance and population vary independently, one can choose three distinct location pairs, such that the three moving cost equations are linearly independent; these equations identify γ1 , γ5 , and γ0 + γ4 a. Then by choosing two different ages, γ0 and γ4 are identified, and by comparing adjacent and nonadjacent pairs, γ2 is identified. If the continuation value in all states is increased by the same amount, then the choice probabilities are unaffected, so one of the values can be normalized to zero.19 We assume V¯0 (J) = 0. Then n 1 ρ( υs J) = −Δ j − βV¯0 ( ) log n s=1 ρ( υs ) This identifies V¯0 ( ), since we assume that β is known, and Δ j has already been identified. Once V¯0 is identified, V¯s ( ) is identified by the equation ρ( υs j) = −Δ j + β(V¯0 (j) − V¯s ( )) log ρ( υs ) 19 It might seem that the choice probabilities are also invariant to a rescaling of the continuation values, but we have already normalized the scale by assuming additive payoff shocks drawn from the extreme value distribution.
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Given that the expected continuation values in all states are identified, the flow payoffs are identified by V¯s ( ) = γ¯ + α0 υs + A ¯ ¯ + log exp(βVs ( )) + exp(Δ k + βV0 (k)) k=
where A represents amenity values and other fixed characteristics of location (both observed and unobserved), and where υs represents the location match component of wages. The income coefficient α0 is identified by differencing this equation with respect to s (thereby eliminating A ), and A is then identified as the only remaining unknown in the equation. 5.5. Computation Since the parameters are embedded in the value function, computation of the gradient and hessian of the log-likelihood function is not a simple matter (although, in principle, these derivatives can be computed using the same iterative procedure that computes the value function itself). We maximize the likelihood using a version of Newton’s algorithm with numerical derivatives. We also use the downhill simplex method of Nelder and Mead, mainly to check for local maxima. This method does not use derivatives, but it is slow.20 6. EMPIRICAL RESULTS Our basic results are shown in Table II. We set β = 095 T = 40, and M = 2; we show below that our main results are not very sensitive to changes in the discount factor or the horizon length.21 The table gives estimated coefficients and standard errors for four versions of the model that highlight both the effect of income on migration decisions and the relevance of the location match component of preferences. Unobserved heterogeneity in moving costs is introduced by allowing for two types, one of which is a pure stayer type (representing peo20 Given reasonable starting values (for example, 50% type probabilities and a fixed cost for the mover type that roughly matches the average migration rate, with variances near 1 for the wage components and all other parameters set to zero), the maximal likelihood is reached, using Nelder–Mead iterations followed by a switch to Newton’s method, in less than a day on a cluster of parallel CPUs, with one CPU per home location; each likelihood evaluation requires about 9 seconds. We found the Newton procedure to be well behaved in the sense that it almost always reached the same answer, no matter what starting values were used: we have estimated hundreds of different versions of the model and found very few local maxima; even in these cases, the likelihood and the parameter values were very close to the “true” maximum. 21 The validity of the estimates is checked in Appendix B: the estimated coefficients were used to simulate 100 replicas of each person in the data and the maximum likelihood procedure was applied to the simulated data. The null hypothesis that the data were generated by the true data generating process (DGP) is accepted by a likelihood ratio test.
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J. KENNAN AND J. R. WALKER TABLE II INTERSTATE MIGRATION, YOUNG WHITE MENa θˆ
Utility and cost Disutility of moving (γ0 ) Distance (γ1 ) (1000 miles) Adjacent location (γ2 ) Home premium (αH ) Previous location (γ3 ) Age (γ4 ) Population (γ5 ) (millions) Stayer probability Cooling (α1 ) (1000 degree-days) Income (α0 ) Location match preference (τξ ) Wages Wage intercept Time trend Age effect (linear) Age effect (quadratic) Ability (AFQT) Interaction (Age, AFQT) Transient s.d. 1 Transient s.d. 2 Transient s.d. 3 Transient s.d. 4 Fixed effect 1 Fixed effect 2 Fixed effect 3 Wage match (τυ ) Log likelihood Exclude income: χ2 (1) Exclude match preference: χ2 (1)
σˆ θ
4790 0265 0808 0331 2757 0055 0653 0510 0055 0312 – −5165 −0035 7865 −2364 0014 0147 0217 0375 0546 1306 0113 0298 0936 0384
0.565 0.182 0.214 0.041 0.356 0.020 0.179 0.078 0.019 0.100
θˆ
4514 0280 0787 0267 2544 0062 0653 0520 0036 – –
σˆ θ
0.523 0.178 0.211 0.031 0.300 0.019 0.178 0.079 0.019
θˆ
4864 0311 0773 0332 3082 0060 0635 0495 0048 – 0168
σˆ θ
0.601 0.187 0.220 0.047 0.449 0.020 0.177 0.087 0.018 0.049
θˆ
4851 0270 0804 0337 2818 0054 0652 0508 0056 0297 0070
σˆ θ
0.604 0.184 0.216 0.045 0.416 0.020 0.179 0.082 0.019 0.116 0.099
0.244 −5175 0.246 −5175 0.246 −5168 0.244 0.008 −0033 0.008 −0033 0.008 −0035 0.008 0.354 7876 0.356 7877 0.356 7870 0.355 0.129 −2381 0.130 −2381 0.130 −2367 0.129 0.065 0015 0.066 0014 0.066 0014 0.065 0.040 0152 0.040 0152 0.040 0147 0.040 0.007 0218 0.007 0218 0.007 0217 0.007 0.015 0375 0.015 0375 0.015 0375 0.015 0.017 0547 0.017 0547 0.017 0546 0.017 0.028 1307 0.028 1307 0.028 1306 0.028 0.035 0112 0.035 0112 0.035 0113 0.035 0.035 0296 0.035 0296 0.035 0298 0.035 0.017 0934 0.017 0934 0.017 0936 0.017 0.017 0387 0.018 0387 0.018 0384 0.018
−4,214.880 0.09
−4,221.426 13.09 5.25
−4,218.800 7.93
−4,214.834
a There are 4,274 (person-year) observations, 432 individuals, and 124 moves.
ple with prohibitive moving costs); little is gained by introducing additional types or by replacing the stayer type with a type with a high moving cost. We find that distance, home and previous locations, and population size all have highly significant effects on migration. Age and local climate (represented by the annual number of cooling degree-days) are also significant.22 Our main 22 The “cooling” variable is the population-weighted annual average number of cooling degree days (in thousands) for 1931–2000, taken from Historical Climatography Series 5-2 (Cooling Degree Days); see United States National Climatic Data Center (2002). For example, the cooling degree-day variable for Florida is 3.356, meaning that the difference between 65◦ and the mean daily temperature in Florida, summed over the days when the mean was above 65◦ , averaged 3,356 degree-days per year (over the years 1931–2000). We explored various alternative specifi-
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INDIVIDUAL MIGRATION DECISIONS TABLE III WAGE PARAMETER ESTIMATES (IN 2010 DOLLARS) AFQT Percentile
Average wages Age 20 in 1979 Age 20 in 1989 Age 30 in 1989
25
50
75
25,827 18,472 40,360
27,522 20,166 42,850
29,216 21,861 45,340
Low
Location match Fixed effect support State means
Middle
High
−8,366 0 8,366 −20,411 −6,498 −2,454 0 2,454 6,498 20,411 Low (WV) Rank 5 (OK) Median (MO) Rank 45 (RI) High (MD) 12,698 14,530 16,978 19,276 22,229
finding is that, controlling for these effects, migration decisions are significantly affected by expected income changes. This holds regardless of whether the location match component of preferences is included in the specification. Since the estimated effect of this component is negligible and it enlarges the state space by a factor of about 100, we treat the specification that excludes this component as the base model in the subsequent discussion. 6.1. Wages The estimated parameters of the wage process are summarized in Table III, showing the magnitudes of the various components in 2010 dollars. As was mentioned above, there is a great deal of unexplained variation in wages across people and over time for the same person; moreover, there are big differences in the variability of earnings over time from one individual to the next.23 The wage components that are relevant for migration decisions in the model are also quite variable, suggesting that migration incentives are strong. For example, the 90–10 differential across state means is about $4,700 a year, and the value of replacing a bad location match draw with a good draw is almost $17,000 a year. cations of the climate amenity variables. Including heating degree-days had little effect on the results (see Table X below). The number of states that are adjacent to an ocean is 23. We considered this as an additional amenity variable, and also estimated models including annual rainfall and the annual number of sunny days, but found that these variables had virtually no effect. 23 As indicated in Table II, the individual characteristics affecting wages include age, AFQT score, and an interaction between the two. The interaction effect is included to allow for the possibility that the relationship between AFQT scores and wages is stronger for older workers, either because ability and experience are complementary or because employers gradually learn about ability, as argued by Altonji and Pierret (2001).
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J. KENNAN AND J. R. WALKER TABLE IV MOVING COST EXAMPLES γ0
α0
θ 4.790 0.312 Young mover Average mover
Age
Distance
Adjacent
Population
Previous
Cost
0055 20 24355
0265 1 0664
0808 0 0427
0653 1 0728
2757 0 0371
$384,743 $312,146
6.2. Moving Costs and Payoff Shocks Since utility is linear in income, the estimated moving cost can be converted to a dollar equivalent. Some examples are given in Table IV. For the average mover, the cost is about $312,000 (in 2010 dollars) if the payoff shocks are ignored. One might wonder why anyone would ever move in the face of such a cost and, in particular, whether a move motivated by expected income gains could ever pay for itself. According to the estimates in Table III, a move away from a bad location match would increase income by $8,366, on average, and a move from the bottom to the top of the distribution of state means would increase income by $9,531. A move that makes both of these changes would mean a permanent wage increase of $17,897, or $311,939 in present value (assuming a remaining work life of 40 years, with β = 095). The home premium is equivalent to a wage increase of $23,106 and the cost of moving to a previous location is relatively low. Thus in some cases the expected income gains would be more than enough to pay for the estimated moving cost. Of course in most cases this would not be true, but then most people never move. More importantly, the estimates in Table IV do not refer to the costs of moves that are actually made, but rather to the costs of hypothetical moves to arbitrary locations. In the model, people choose to move only when the payoff shocks are favorable, and the net cost of the move is, therefore, much less than the amounts in Table IV. Consider, for example, a case in which someone is forced to move, but allowed to choose the best alternative location. The expected value of the maximum of J − 1 draws from the extreme value distribution is γ¯ + log(J − 1) (where γ¯ is the Euler constant), so if the location with the most favorable payoff shock is chosen, the expected net cost of the move is reduced by log(J − 1)/α0 . Using the estimated income coefficient, this is a reduction of $271,959. Moreover, this calculation refers to a move made in an arbitrary period; in the model, the individual can move later if the current payoff shocks are unfavorable, so the net cost is further reduced. Of course people actually move only if there is, in fact, a net gain from moving; the point of the argument is just that this can quite easily happen, despite the large moving cost estimates in Table IV. In Section 6.3 below, we analyze the average costs of moves that are actually made, allowing for the effects of the payoff shocks. Another way to interpret the moving cost is to consider the effect of a $10,000 migration subsidy, payable for every move, with no obligation to stay
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in the new location for more than one period. This can be analyzed by simulating the model with a reduction in γ0 such that γ0 /α0 falls by $10,000, with the other parameters held fixed. We estimate that such a subsidy would lead to a substantial increase in the interstate migration rate: from 2.9% to about 4.9%. 6.2.1. Moving Costs and Payoff Shocks: An Example To understand the relationship between moving costs and prospective income gains, it is helpful to consider an example in which these are the only influences on migration decisions. Suppose that income in each location is either high or low, the difference being y, and suppose that the realization of income in each location is known. Then the odds of moving are given by (1)
1 − λL = exp(−γ0 )[JL − 1 + JH eβ V ] λL
(2)
1 − λH = exp(−γ0 )[JH − 1 + JL e−β V ] λH
where λL is the probability of staying in one of JL low-income locations (and similarly for λH and JH ), and where V is the difference in expected continuation values between the low-income and high-income locations. This difference is determined by the equation (3)
e V =
eα0 y (JL + (JH − 1 + eγ0 )eβ V ) JL − 1 + eγ0 + JH eβ V
For example, if β = 0, then V = α0 y, while if moving costs are prohibitive 0 y (e−γ0 = 0), then V = α1−β . These equations uniquely identify α0 and γ0 (these parameters are in fact overidentified, because there is also information in the probabilities of moving to the same income level).24 If γ0 < β V , then the odds of moving from a low-income location are greater than JH to 1, and this is contrary to what is seen in the data (for any plausible value of JH ). By making γ0 a little larger than β V and letting both of these be large in relation to the payoff shocks, the probability of moving from the low-income location can be made small. But then the probability of moving from the high-income location is almost zero, which is not true in the data. In other words, if the probability of moving from a high-income location is not negligible, then the payoff shocks cannot be negligible, since a payoff shock is the only reason for making such a move. 24 It is assumed that λL λH JL JH y, and β are given. Dividing (1) by (2) and rearranging terms yields a quadratic equation in eβ V that has one positive root and one negative root. Since eβ V must be positive, this gives a unique solution for V . Equation (1) then gives a unique solution for γ0 , and inserting these solutions into equation (3) gives a unique solution for α0 y.
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The net cost of moving from a low-income location to a high-income location is γ0 − β V , while the net cost of the reverse move is γ0 − β V . The difference is 2β V , and equations (1) and (2) show that β V determines the relative odds of moving from low-income and high-income locations. Thus β V is identified by the difference between λL and λH ; this difference is small in the data, so β V must be small. The magnitude of γ0 is then determined by the level of λL and λH , and since these are close to 1 in the data, the implication is that γ0 is large and that it is much larger than β V . Since β V is roughly the present value of the difference in income levels, the upshot is that the moving cost must be large in relation to income. For example, suppose JL = JH = 25, with β = 095. In our data, the migration probability for someone in the bottom quartile of the distribution of state mean wages is 5.5% (53 moves in 964 person-years); for someone in the top quartile, it is 2.1% (16 moves in 754 person-years). If 1 − λL = 53/964 and 1 − λH 16/754, then γ0 = 734, and V = 102, and the implied moving cost is γ0 /α0 = 853 y. Taking y to be the difference in the mean wages for states in the top and bottom quartiles gives γ0 /α0 = $304,670 (in 2008 dollars). On the other hand, if λL = 07, the implied moving cost is only 144 y, or $51,449. We conclude that the moving cost estimate is large mainly because the empirical relationship between income levels and migration probabilities is relatively weak. 6.3. Average Costs of Actual Moves Our estimates of the deterministic components of moving costs are large because moves are rare in the data. But moves do occur and, in many cases, there is no observable reason for a move, so the observed choice must be attributed to unobserved payoff shocks, including random variations in moving costs. Given this heterogeneity in moving costs, both across individuals and over time for the same individual, the question arises as to how large the actual moving costs are, conditional on a move being made.25 Because the payoff shocks are drawn from the type I extreme value distribution, this question has a relatively simple answer. The cost of a move may be defined as the difference in the flow payoff for the current period due to the move. Since a move to location j exchanges ζ 0 for ζj , the average cost of a move from 0 to j, given state x, is ¯ j) = Δ(x j) − E(ζj − ζ 0 | dj = 1) Δ(x where dj is an indicator variable for the choice of location j. Thus, for example, if a move from 0 to j is caused by a large payoff shock in location j, the cost 25 See Sweeting (2007) for a similar analysis of switching costs in the context of an empirical analysis of format switching by radio stations.
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of the move may be much less than the amount given by the deterministic cost Δ(x j). In logit models, the expected gain from the optimal choice, relative to an arbitrary alternative that is not chosen, is a simple function of the probability of choosing the alternative (see Anas and Feng (1988) and Kennan (2008)). In the present context, this result means that the average increase in the gross continuation value for someone who chooses to move from 0 to j is given by ˜ ˜ E(v(x j) − v(x 0 ) | dj = 1) = −
log(ρ(x 0 )) 1 − ρ(x 0 )
˜ where v(x j) is the continuation value when the state is x and location j is chosen, which includes the current flow payoff and the discounted expected continuation value in location j: ˜ v(x j) = v(x j) + ζj = u(x j) + β
p(x | x j)V (x ) + ζj
x
The deterministic part of the moving cost is Δ(x j) = u(x 0 ) − u(x j) = v(x 0 ) − v(x j) + β
(p(x | x j) − p(x | x 0 ))V (x ) x
˜ ˜ = v(x 0 ) − ζ 0 − v(x j) + ζj +β (p(x | x j) − p(x | x 0 ))V (x ) x
This implies that the average moving cost, net of the difference in payoff shocks, is ¯ j) ≡ Δ(x j) − E(ζj − ζ 0 | dj = 1) Δ(x =
log(ρ(x 0 )) + β (p(x | x j) − p(x | x 0 ))v(x ) 1 − ρ(x 0 ) x
Since some of the components of the state vector x are unobserved, we compute expected moving costs using the conditional distribution over the unobservables, given the observed wage and migration history. Recall that the likelihood of an individual history, for a person of type τ, is T
i 1 i i Li (θτ ) = ψit (ω θτ )λit (ω θτ ) nη nε nξ (nυ )Ni i t=1 ω ∈Ω(Ni )
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Thus the conditional probability of ωi is Ti
t=1
Q(ωi ) =
ω∈Ω(Ni ) Ti
=
ψit (ωi θτ )λit (ωi θτ ) Ti
ψit (ω θτ )λit (ω θτ )
t=1
ψit (ωi θτ )λit (ωi θτ )
t=1
nη nε nξ (nυ )Ni Li (θτ )
The unobserved part of the state variable consists of the location match components of wages and preferences. Since the distributions of these components have finite support, there is a finite set Ω(Ni ) of possible realizations corresponding to the observed history for individual i; this set is indexed by ωi . Let x(ωi ) be the state implied by ωi (including the location match components in the current location, and in the previous location, if any). Then if individual i moves to location j in period t, the moving cost is estimated as i ¯ Q(ωi )Δ(x(ω ) j) Δˆit = ωi ∈Ω(Ni )
The estimated average moving costs are given in Table V. There is considerable variation in these costs, but for a typical move the cost is negative. The interpretation of this is that the typical move is not motivated by the prospect of a higher future utility flow in the destination location, but rather by unobserved TABLE V AVERAGE MOVING COSTSa Previous Location
None Home Other Total
Move Origin and Destination From Home
To Home
Other
Total
−$147,619 [56] –
$138,095 [1] $18,686 [40] $113,447 [2] $25,871 [43]
−$39,677 [2] −$124,360 [10] −$67,443 [5] −$97,656 [17]
−$139,118 [59] −$9,924 [50] −$87,413 [15] −$80,768 [124]
−$150,110 [8] −$147,930 [64]
a The number of moves in each category is given in brackets.
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factors yielding a higher current payoff in the destination location, compared with the current location. That is, the most important part of the estimated moving cost is ζ 0 − ζj , the difference in the payoff shocks. In the case of moves to the home location, on the other hand, the estimated cost is positive; most of these moves are return moves, but where the home location is not the previous location, the cost is large, reflecting a large gain in expected future payoffs due to the move. 6.4. Goodness of Fit To keep the state space manageable, our model severely restricts the set of variables that are allowed to affect migration decisions. Examples of omitted observable variables include duration in the current location and the number of moves made previously. In addition, there are, of course, unobserved characteristics that might make some people more likely to move than others. Thus it is important to check how well the model fits the data. In particular, since the model pays little attention to individual histories, one might expect that it would have trouble fitting panel data. One simple test of goodness of fit can be made by comparing the number of moves per person in the data with the number predicted by the model. As a benchmark, we consider a binomial distribution with a migration probability of 2.9% (the number of moves per person-year in the data). Table VI shows the predictions from this model: about 75% of the people never move; of those who do move, about 14% move more than once.26 The NLSY data are quite different: about 84% never move and about 56% of movers move more than once. An obvious interpretation of this is mover–stayer heterogeneity: some people are more likely to move than others, and these people account for more than their share of the observed moves. We simulated the corresponding statistics for the model by starting 100 replicas of the NLSY individuals in the observed initial locations and using the model (with the estimated parameters TABLE VI GOODNESS OF FIT Moves
None One More Movers with more than one move Total observations
Binomial
NLSY
Model
3251 753% 915 212% 154 36% 14.4% 432
361 836% 31 72% 40 93% 56.3% 432
36,257 839% 2,466 57% 4,478 104% 64.5% 43,201
26 Since we have an unbalanced panel, the binomial probabilities are weighted by the distribution of years per person.
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J. KENNAN AND J. R. WALKER TABLE VII RETURN MIGRATION STATISTICS Movers
NLSY
Model
Proportion who Return home Return elsewhere Move on
347% 48% 605%
375% 62% 619%
Proportion who ever Leave home Move from not-home Return from not-home
144% 400% 257%
137% 425% 323%
shown in Table II) to generate a history for each replica, covering the number of periods observed for this individual. The results show that the model does a good job of accounting for the heterogeneous migration probabilities in the data. The proportion of people who never move in the simulated data matches the proportion in the NLSY data almost exactly, and although the proportion of movers who move more than once is a bit high in the simulated data, the estimated model comes much closer to this statistic than the binomial model does. 6.4.1. Return Migration Table VII summarizes the extent to which the model can reproduce the return migration patterns in the data (the statistics in the “Model” column refer to the simulated data set used in Table VI). The model attaches a premium to the home location and this helps explain why people return home. For example, in a model with no home premium, one would expect that the proportion of movers going to any particular location would be roughly 1/50, and this obviously does not match the observed return rate of 35%. The home premium also reduces the chance of initially leaving home, although this effect is offset by the substantial discount on the cost of returning to a previous location (including the home location): leaving home is less costly if a return move is relatively cheap. The simulated return migration rates match the data reasonably well. The main discrepancy is that the model overpredicts the proportion who ever return home from an initial location that is not their home location. That is, the model has trouble explaining why people seem so attached to an initial location that is not their “home.” One potential explanation for this is that our assignment of home locations (the state of residence at age 14) is too crude; in some cases, the location at age 20 may be more like a home location than the location at age 14. More generally, people are no doubt more likely to put down roots the longer they stay in a location, and our model does not capture this kind of duration dependence.
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INDIVIDUAL MIGRATION DECISIONS TABLE VIII ANNUAL MIGRATION RATES BY AGE AND CURRENT LOCATIONa All
Age
20–25 26–34 All
Not at Home
At Home
N
Moves
Migration Rate
N
Moves
Migration Rate
N
Moves
Migration Rate
2,359 1,915 4,274
84 40 124
3.6% 2.1% 2.9%
244 228 472
40 20 60
16.4% 8.8% 13.4%
2,115 1,687 3,802
44 20 64
2.1% 1.2% 1.7%
a At home means living now in the state of residence at age 14.
6.5. Why Are Younger People More Likely to Move? It is well known that the propensity to migrate falls with age (at least after age 25 or so). Table VIII replicates this finding for our sample of high-school men. A standard human capital explanation for this age effect is that migration is an investment: if a higher income stream is available elsewhere, then the sooner a move is made, the sooner the gain is realized. Moreover, since the work life is finite, a move that is worthwhile for a young worker might not be worthwhile for an older worker, since there is less time for the higher income stream to offset the moving cost (Sjaastad (1962)). In other words, migrants are more likely to be young for the same reason that students are more likely to be young. Our model encompasses this simple human capital explanation of the age effect on migration.27 There are two effects here. First, consider two locations paying different wages and suppose that workers are randomly assigned to these locations at birth. Then, even if the horizon is infinite, the model predicts that the probability of moving from the low-wage to the high-wage location is higher than the probability of a move in the other direction, so that eventually there will be more workers in the high-wage location. This implies that the (unconditional) migration rate is higher when workers are young.28 Second, the human capital explanation says that migration rates decline with age because the horizon gets closer as workers get older. This is surely an important reason for the difference in migration rates between young adult workers and those 27
Investments in location-specific human capital might also help explain why older workers are less likely to move. Marriage might be included under this heading, for example, as in Gemici (2008). It is worth noting that if we take marital status as given, it has essentially no effect on migration in our sample in simple logit models of the move–stay decision that include age as an explanatory variable. 28 One way to see this is to consider the extreme case in which there are no payoff shocks. In this case, all workers born in the low-wage location will move to the high-wage location at the first opportunity (if the wage difference exceeds the moving cost), and the migration rate will be zero from then on.
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within sight of retirement. But the workers in our sample are all in their 20s or early 30s, and the prospect of retirement seems unimportant for such workers. We find that the simple human capital model does not fully explain the relationship between age and migration in the data. Our model includes age as a state variable to capture the effects just discussed. The model also allows for the possibility that age has a direct effect on the cost of migration; this can be regarded as a catchall for whatever is missing from the simple human capital explanation. The results in Table II show that this direct effect is significant. 6.6. Decomposing the Effects of Income on Migration Decisions Migration is motivated by two distinct wage components in our model: differences in mean wages (μj ) across locations and individual draws from the location match distribution (υij ). The relevance of these components can be considered separately, first by suppressing the dispersion in υ, so that wages affect migration decisions only because of differences in mean wages across locations, and alternatively by specifying the wage distribution at the national level, so that migration is motivated only by the prospect of getting a better draw from the same wage distribution (given our assumption that location match effects are permanent). Consider an economy in which everyone has the same preferences over locations and also the same productivity in each location. In a steady state equilibrium, everyone is indifferent between locations: there are wage differences, but these just equalize the amenity differences. People move for other reasons, but there are just as many people coming into each location as there are going out. There should be no correlation between wages and mobility in the steady state. Nevertheless, if moving costs are high, at any given time one would expect to see flows of workers toward locations with higher wages as part of a dynamic equilibrium driven by local labor demand shocks. As was mentioned above (in footnote 6), there is some evidence that local labor market shocks have long-lasting effects. So in a specification that uses only mean wages in each location (with no location match effects), we should find a relationship between mean wages and migration decisions. This is, in fact, what we find in Table IX (in the “State Means” column). But we also find that the exclusion of location match wage effects is strongly rejected by a likelihood ratio test. Even if differences in mean wages merely equalize the amenity differences between locations, the model predicts a relationship between wage realizations and migration decisions, because of location match effects: if the location match component is bad, the worker has an incentive to leave. This motivates the “National Wages” column of Table IX, where it is assumed that mean wages are the same in all locations (as they would be if measured wage differences merely reflect unmeasured amenities). We find that workers who have unusually low wages in their current location are indeed more likely to move. Finally, the “Regional Amenities” column shows that the results are robust to the inclusion of regional amenity differences. Section 5.4.2 shows that the
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INDIVIDUAL MIGRATION DECISIONS TABLE IX ALTERNATIVE INCOME SPECIFICATIONS Base Model θˆ
State Means σˆ θ
θˆ
National Wages σˆ θ
θˆ
Regional Amenitiesa σˆ θ
θˆ
σˆ θ
Disutility of moving 4790 0.565 4567 0.533 4755 0.568 4761 0.611 Distance 0265 0.182 0253 0.183 0269 0.183 0353 0.236 Adjacent location 0808 0.214 0810 0.213 0805 0.213 0777 0.234 Home premium 0331 0.041 0273 0.032 0328 0.040 0373 0.051 Previous location 2757 0.356 2558 0.299 2728 0.345 2730 0.423 Age 0055 0.020 0061 0.019 0055 0.020 0052 0.020 Population 0653 0.179 0661 0.181 0649 0.179 0682 0.234 Stayer probability 0510 0.078 0517 0.079 0513 0.078 0495 0.093 Cooling 0055 0.019 0040 0.019 0055 0.019 0031 0.044 Income 0312 0.100 0318 0.181 0315 0.099 0290 0.130 Wage intercept −5165 0.244 −5440 0.238 −4024 0.270 −5133 0.249 Time trend −0035 0.008 −0051 0.005 −0011 0.009 −0033 0.008 Age effect (linear) 7865 0.354 8112 0.366 7445 0.381 7842 0.360 Age effect (quadratic) −2364 0.129 −2321 0.134 −2386 0.128 −2364 0.131 Ability (AFQT) 0014 0.065 0062 0.060 0024 0.064 0011 0.066 Interaction (age, AFQT) 0147 0.040 0158 0.041 0140 0.039 0144 0.041 Transient s.d. 1 0217 0.007 0231 0.007 0217 0.007 0217 0.007 Transient s.d. 2 0375 0.015 0385 0.016 0372 0.015 0375 0.015 Transient s.d. 3 0546 0.017 0559 0.018 0544 0.017 0546 0.017 Transient s.d. 4 1306 0.028 1331 0.028 1305 0.027 1306 0.028 Fixed effect 1 0113 0.035 0253 0.013 0357 0.039 0113 0.036 Fixed effect 2 0298 0.035 0547 0.011 0167 0.041 0297 0.036 Fixed effect 3 0936 0.017 1028 0.014 0905 0.023 0933 0.017 Wage match 0384 0.017 – 0363 0.023 0384 0.019 Log likelihood −4,214.88 −4,267.88 −4,215.83 −4,203.58 a The “Regional Amenities” model includes 12 regional dummy variables (coefficients not shown).
model is identified even if each location has an unobserved amenity value that is common to all individuals. In practice, we do not have enough data to estimate the complete model with a full set of fixed effects for all 50 locations. As a compromise, we divide the states into 13 regions and present estimates for a model with fixed amenity values for each region.29 This has little effect on
29 The regions are as follows: (i) Northeast (NE, ME, VT, NH, MA, RI, CT); (ii) Atlantic (DE, MD, NJ, NY, PA); (iii) Southeast (SE, VA, NC, SC, GA, FL); (iv) North Central (NC, MN, MI, WI, SD, ND); (v) Midwest (OH, IN, IL, IA, KS, NB, MO); (vi) South (LA, MS, AL, AR); (vii) South Central (OK, TX); (viii) Appalachia (TN, KY, WV); (ix) Southwest (AZ, NM, NV); (x) Mountain (ID, MT, WY, UT, CO); (xi) West (CA, HI); (xii) Alaska and (xiii) Northwest (OR, WA).
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the estimated income coefficient; moreover, a likelihood ratio test accepts the hypothesis that there are no regional amenity differences.30 6.7. Sensitivity Analysis Our empirical results are inevitably based on some more or less arbitrary model specification choices. Table X explores the robustness of the results with respect to some of these choices. The general conclusion is that the parameter estimates are robust. In particular, the income coefficient estimate remains positive and significant in all of our alternative specifications. The results presented so far are based on wages that are adjusted for cost of living differences across locations. If these cost of living differences merely compensate for amenity differences, then unadjusted wages should be used to measure the incentive to migrate. This specification yields a slightly lower estimate of the income coefficient without much effect on the other coefficients and the likelihood is lower (mainly because there is more unexplained variation in the unadjusted wages). Thus, in practice, the theoretical ambiguity as to whether wages should be adjusted for cost of living differences does not change the qualitative empirical results: either way, income significantly affects migration decisions. The other specifications in Table X are concerned with sensitivity of the estimates to the discount factor (β), the horizon length (T ), heterogeneity in moving costs and the inclusion of a second climate variable (heating degreedays).31 Again, the effect of income is quite stable across these alternative specifications. 7. MIGRATION AND WAGES 7.1. Spatial Labor Supply Elasticities We use the estimated model to analyze labor supply responses to changes in mean wages for selected states. We are interested in the magnitudes of the migration flows in response to local wage changes and in the timing of these responses. Since our model assumes that the wage components relevant to migration decisions are permanent, it cannot be used to predict responses to wage innovations in an environment in which wages are generated by a stochastic 30 Alaska is the only region that has a significant (positive) coefficient; this is perhaps not surprising given that the model specifies the utility flow as a linear function of average temperature and Alaska is an outlier in this respect. 31 Table X is a sample of many alternative specifications that were tried. As was mentioned earlier, size (as measured by population) may affect migration either as a scaling factor on the payoff shocks or as a variable affecting the cost of migration. We experimented with these alternatives and also expanded the moving cost specification to allow quadratic effects of distance, location size, and climate variables; none of these experiments changed the results much.
TABLE X ALTERNATIVE SPECIFICATIONS Base Model θˆ
σˆ θ
4790 0.565 0265 0.182 0808 0.214 0331 0.041 2757 0.356 0055 0.020 0653 0.179 0510 0.078 0055 0.019 – – 0312 0.100 −5165 0.244 −0035 0.008 7865 0.354 −2364 0.129 0014 0.065 0147 0.040 0217 0.007 0375 0.015 0546 0.017 1306 0.028 0113 0.035 0298 0.035 0936 0.017 0384 0.017 −4,214.880
θˆ
σˆ θ
4702 0.556 0282 0.182 0797 0.214 0324 0.040 2711 0.366 0056 0.019 0639 0.181 0512 0.079 0057 0.019 – – 0263 0.095 −5125 0.270 −0030 0.010 7826 0.385 0.128 −2379 0053 0.067 0133 0.040 0220 0.007 0380 0.016 0553 0.016 1322 0.029 0138 0.034 0311 0.034 0966 0.020 0400 0.019 −4,282.473
θˆ
T = 20 σˆ θ
4694 0.578 0297 0.198 0840 0.232 0465 0.052 2809 0.347 0060 0.020 0696 0.186 0486 0.080 0069 0.027 – – 0451 0.139 −5172 0.244 −0035 0.008 7875 0.354 −2366 0.129 0016 0.065 0144 0.040 0217 0.007 0375 0.015 0546 0.017 1306 0.028 0.035 0112 0298 0.035 0937 0.017 0384 0.017 −4,214.038
θˆ
1 Cost Type σˆ θ
4487 0.613 0266 0.186 0811 0.224 0382 0.041 2806 0.332 0068 0.022 0650 0.179 0494 0.079 0056 0.022 – – 0359 0.110 −5168 0.244 −0035 0.008 7869 0.355 −2366 0.129 0017 0.065 0144 0.040 0217 0.007 0375 0.015 0546 0.017 1306 0.028 0112 0.035 0298 0.035 0936 0.017 0384 0.017 −4,214.034
θˆ
σˆ θ
5287 0.560 0262 0.182 0781 0.214 0185 0.022 3373 0.322 0074 0.020 0645 0.163 0 – 0020 0.014 – – 0148 0.069 −5199 0.243 −0035 0.008 7906 0.353 −2375 0.129 0012 0.065 0153 0.040 0218 0.007 0375 0.015 0546 0.017 1306 0.028 0111 0.035 0298 0.035 0938 0.017 0382 0.017 −4,231.700
Heating θˆ
σˆ θ
4764 0.556 0274 0.193 0794 0.220 0323 0.039 2770 0.363 0058 0.019 0683 0.190 0.079 0505 0099 0.031 0020 0.012 0304 0.098 −5166 0.244 −0035 0.008 7867 0.355 −2365 0.129 0014 0.065 0147 0.040 0217 0.007 0375 0.015 0546 0.017 1306 0.028 0113 0.035 0298 0.035 0936 0.017 0384 0.017 −4,214.082
INDIVIDUAL MIGRATION DECISIONS
Disutility of moving Distance Adjacent location Home premium Previous location Age Population Stayer probability Cooling Heating Income Wage intercept Time trend Age effect (linear) Age effect (quadratic) Ability (AFQT) Interaction (age, AFQT) Transient s.d. 1 Transient s.d. 2 Transient s.d. 3 Transient s.d. 4 Fixed effect 1 Fixed effect 2 Fixed effect 3 Wage match Log likelihood
β = 9
No COLA
243
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J. KENNAN AND J. R. WALKER
process. Instead, it is used to answer comparative dynamics questions: we use the estimated parameters to predict responses in a different environment. First we do a baseline calculation, starting people in given locations and allowing them to make migration decisions in response to the wage distributions estimated from the Census data. Then we do counterfactual simulations, starting people in the same locations, but facing different wage distributions. We take a set of people who are distributed over states as in the 1990 Census data for white male high-school graduates aged 20–34. We assume that each person is initially in the home state at age 20 and we allow the population distribution to evolve over 15 years by iterating the estimated transition probability matrix. We consider responses to wage increases and decreases that represent a 10% change in the mean wage of an average 30-year-old for selected states. First, we compute baseline transition probabilities using the wages that generated the parameter estimates. Then we increase or decrease the mean wage in a single state and compare the migration decisions induced by these wage changes with the baseline. Supply elasticities are measured relative to the supply of labor in the baseline calculation. For example, the elasticity of L w , the response to a wage increase in California after 5 years is computed as w L where L is the number of people in California after 5 years in the baseline calculation, and L is the difference between this and the number of people in California after 5 years in the counterfactual calculation. Figure 1 shows the results for three large states that are near the middle of the one-period utility flow distribution. The supply elasticities are above 05. Adjustment is gradual, but is largely completed in 10 years. Our conclusion from this exercise is that despite the low migration rate in the data, the supply of labor responds quite strongly to spatial wage differences. 7.2. Migration and Wage Growth Our model is primarily designed to quantify the extent to which migration is motivated by expected income gains. Interstate migration is a relatively rare event and our results indicate that many of the moves that do occur are motivated by something other than income gains. This raises the question of whether the income gains due to migration are large enough to be interesting. One way to answer this question is to compare the wages of the mover and the stayer types as time goes by, using simulated data. Table XI shows results for a simulation that starts 1,000 people at home in each of the 50 states at age 20 and measures accumulated income and utility gains at age 34 (the oldest age in our NLSY sample).32 Migration increases the total utility flow by a modest but nontrivial amount. Most of the gain comes from improved location 32 The results are weighted by the state distribution of white male high-school graduates aged 19–20 from the 1990 Census.
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INDIVIDUAL MIGRATION DECISIONS
FIGURE 1.—Responses to wage changes.
matches; even though there is considerable dispersion in mean wages across states, the estimated dispersion in the location match component of wages is much larger and, therefore, a much more important source of income gains due to migration. The dollar value of the nonpecuniary gains due to (climate) amenities is also larger than the gains from moving toward high-wage states. The importance of the home location can be seen by simulating migration decisions with the home premium parameter set to zero. The results are shown in Table XII. With no attachment to a home location, the annual migration rate increases to 7.3% and the mover type moves about once every 6 years. By age 34, the accumulated gains due to migration exceed 20% of the base utility level. Given that people are willing to forgo gains of this magnitude to stay in their home location, it follows that the costs of forced displacements (due to natural disasters such as hurricane Katrina, for example) are very high. TABLE XI MIGRATION GAINSa
Migration Rate
Mover type Stayer type Gain Percentage gain Standard deviation
523% 0
Mean μ
Match υ
17,222 17,192 31 0.1% 1,460
966 −58 1024 4.8% 6,835
a Migration gains are measured in 2010 dollars.
Amenity α 1 Y1 α0
Total
4,677 4,374 303 1.4% 3,018
22,865 21,508 1,357 6.3% 7,572
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J. KENNAN AND J. R. WALKER TABLE XII MIGRATION GAINS WITH NO HOME LOCATION
Mover type Stayer type Gain Percentage gain
Migration Rate
Mean
Match
Amenity
Total
13.58% 0
17,223 17,192 31 0.1%
2,876 −21 2,897 13.5%
5,847 4,350 1,497 7.0%
25,946 21,521 4,424 20.6%
8. CONCLUSION We have developed a tractable econometric model of optimal migration in response to income differentials across locations. The model improves on previous work in two respects: it covers optimal sequences of location decisions (rather than a single once-for-all choice) and it allows for many alternative location choices. Migration decisions are made so as to maximize the expected present value of lifetime income, but these decisions are modified by the influence of unobserved location-specific payoff shocks. Because the number of locations is too large to allow the complete dynamic programming problem to be modeled, we adopt an approximation that truncates the amount of information available to the decision-maker. The practical effect of this is that the decisions of a relatively small set of people who have made an unusually large number of moves are modeled less accurately than they would be in the (computationally infeasible) complete model. Our empirical results show a significant effect of expected income differences on interstate migration for white male high-school graduates in the NLSY. Simulations of hypothetical local wage changes show that the elasticity of the relationship between wages and migration is roughly 0.5. Our results can be interpreted in terms of optimal search for the best geographic match. In particular, we find that the relationship between income and migration is partly driven by a negative effect of income in the current location on the probability of out-migration: workers who get a good draw in their current location tend to stay, while those who get a bad draw tend to leave. The main limitations of our model are those imposed by the discrete dynamic programming structure: given the large number of alternative location choices, the number of dynamic programming states must be severely restricted for computational reasons. Goodness of fit tests indicate that the model nevertheless fits the data reasonably well. From an economic point of view, the most important limitation of the model is that it imposes restrictions on the wage process, implying that individual fixed effects and movements along the age–earnings profile do not affect migration decisions. A less restrictive specification of the wage process would be highly desirable.
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APPENDIX A: THE SAMPLE In this appendix, we describe the selection rules use to construct the analysis sample of 432 respondents with 4,274 person-years. As noted in the text, we applied strict sample inclusion criteria to obtain a relatively homogeneous sample. In Table XIII we report the selection rules and the number of respondents deleted by each rule. The NLSY79 contains three subsamples, a nationTABLE XIII SAMPLE SELECTION Respondents
White non-Hispanic males (cross-section sample) Restrictions applied to respondents Ever in military High school dropouts and college graduates Attended college Older than age 20 at start of sample period Missing AFQT score Attend or graduate from high school at age 20 Not in labor force for more than 1 year after age 19 Location at age 20 not reported Income information inconsistent Died before age 30 Residence at age 14 not reported In jail in 1993 Subtotal Restrictions applied to periods Delete periods after first gap in history Delete periods before age 20 Analysis sample Years per person 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total
−246 −1,290 −130 −134 −41 −87 −44 −20 −1 −4 −2 −1 −2,000
Person-Years
2,439
39,024
439
6,585
−1 −6
−1,104 −1,207 432
4,274
14 16 19 14 14 14 13 9 34 61 53 44 45 44 38 432
14 32 57 56 70 84 91 72 306 610 583 528 585 616 570 4,274
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ally representative cross-section sample, a supplemental sample of minorities and economically disadvantaged youth, and a sample of individuals in the military in 1979. We start with the 2,439 white non-Hispanic males in the crosssection sample. We exclude respondents who ever served in the military and we include only those with exactly a high-school education. We assume that permanent labor force attachment begins at age 20; thus we exclude respondents who were born in 1957 and who were, therefore, not interviewed until they were already more than 20 years old. We drop those who are in school or report graduating from high school at age 20. Since we use the AFQT (conducted in 1980) to help explain wages, we drop individuals with missing AFQT scores. Respondents who report being out of the labor force for more than 1 year after age 19, due to disability, tending house, or “other,” are dropped on the grounds that they are not typical of this population. We use residence at age 14 as the home location, so we drop people for whom this variable is missing; we also drop people whose location at age 20 is unknown. We dropped one person who never reported income after age 19. We also dropped four people who died in their 30s, again on the grounds that they are atypical. Finally, we dropped one individual who was incarcerated in 1993 (after reporting remarkably high incomes in earlier years). Application of these criteria produced a sample of 439 individuals and 6,585 person years. We apply two period-level restrictions. The first is that the histories must be continuous: we follow individuals from age 20 to their first noninterview or the 1994 interview. Since a missed interview means that location is unknown, we discard all data for each respondent after the first missed interview. Finally, we delete observations before age 20 from the analysis sample. Seven respondents have information only during their teenage years. Our final sample contains 4,274 periods for 432 men. There are 124 interstate moves, with an annual migration rate of 2.9 percent. More than a onethird of the moves (43) were returns to the home location. There are 361 people who never moved, 31 who moved once, 33 who moved twice, and 7 who moved three times or more. The median age is 25, reflecting the continuoushistory restriction. APPENDIX B: VALIDATION OF ML ESTIMATES The parameter estimates from Table II were used to generate 100 replicas of each NLSY observation, starting from the actual value in the NLSY data and allowing the model to choose the sequence of locations. Table XIV gives maximum likelihood estimates using the simulated data. The last column reports the t-value that tests the difference between the estimates and the individual parameters of the data generating process; the last row reports a likelihood ratio test of the hypothesis that the data were generated by the process that did, in fact, generate them (assuming that the simulation program works). The estimated coefficients are close to the true values and the χ2 test accepts the
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INDIVIDUAL MIGRATION DECISIONS TABLE XIV ESTIMATES FROM SIMULATED MIGRATION HISTORIES Base Model θˆ
Disutility of moving Distance Adjacent location Home premium Previous location Age Population Stayer probability Cooling Income Wage intercept Time trend Age effect (linear) Age effect (quadratic) Ability (AFQT) Interaction (age, AFQT) Transient s.d. 1 Transient s.d. 2 Transient s.d. 3 Transient s.d. 4 Fixed effect 1 Fixed effect 2 Fixed effect 3 Wage match Log likelihood, χ2 (24)
100 Reps σˆ θ
4790 0.565 0265 0.182 0808 0.214 0331 0.041 2757 0.356 0055 0.020 0653 0.179 0510 0.078 0055 0.019 0312 0.100 −5165 0.244 −0035 0.008 7865 0.354 −2364 0.129 0014 0.065 0147 0.040 0217 0.007 0375 0.015 0546 0.017 1306 0.028 0113 0.035 0298 0.035 0936 0.017 0384 0.017 −4,214.88
θˆ
σˆ θ
4732 0.058 0272 0.015 0808 0.017 0333 0.004 2709 0.031 0056 0.002 0648 0.017 0517 0.008 0055 0.002 0307 0.008 −5158 0.033 −0035 0.001 7853 0.050 −2359 0.018 0017 0.010 0142 0.007 0217 0.001 0376 0.002 0545 0.002 1306 0.004 0114 0.003 0300 0.003 0938 0.002 0383 0.002 −473,830.3
t
−100 042 004 042 −152 025 −030 100 −007 −056 021 040 −023 028 035 −067 007 037 −084 014 040 068 137 −017 995
truth. We take this as evidence that our estimation and simulation programs work. REFERENCES ABBRING, J. H., AND J. J. HECKMAN (2007): “Econometric Evaluation of Social Programs, Part III: Distributional Treatment Effects, Dynamic Treatment Effects, Dynamic Discrete Choice, and General Equilibrium Policy Evaluation,” in Handbook of Econometrics, Vol. 6B, ed. by J. J. Heckman and E. Leamer. New York: Elsevier. [225] ALTONJI, J. G., AND C. R. PIERRET (2001): “Employer Learning and Statistical Discrimination,” Quarterly Review of Economics, 116, 313–350. [231] ANAS, A., AND C. M. FENG (1988): “Invariance of Expected Utilities in Logit Models,” Economics Letters, 27, 41–45. [235] BANKS, J. S., AND R. K. SUNDARAM (1994): “Switching Costs and the Gittins Index,” Econometrica, 62, 687–694. [215] BARRO, R. J., AND X. SALA-I MARTIN (1991): “Convergence Across States and Regions,” Brookings Papers on Economic Activity, 1, 107–158. [214]
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BERRY, S. T., AND P. A. HAILE (2010): “Nonparametric Identification of Multinomial Choice Demand Models With Heterogeneous Consumers,” Unpublished Manuscript, Yale University. [225] BLANCHARD, O. J., AND L. F. KATZ (1992): “Regional Evolutions,” Brookings Papers on Economic Activity, 1, 1–37. [212,214] CRAWFORD, G. S., AND M. SHUM (2005): “Uncertainty and Learning in Pharmaceutical Demand,” Econometrica, 73, 1137–1173. [217] DAHL, G. B. (2002): “Mobility and the Return to Education: Testing a Roy Model With Multiple Markets,” Econometrica, 70, 2367–2420. [212] ERDEM, T., AND M. P. KEANE (1996): “Decision-Making Under Uncertainty: Capturing Dynamic Brand Choice Processes in Turbulent Consumer Goods Markets,” Marketing Science, 15, 1–20. [217] GALLIN, J. H. (2004): “Net Migration and State Labor Market Dynamics,” Journal of Labor Economics, 22, 1–21. [212] GEMICI, A. (2008): “Family Migration and Labor Market Outcomes,” Unpublished Manuscript, New York University. [212,213,239] GREENWOOD, M. J. (1997): “Internal Migration in Developed Countries,” in Handbook of Population and Family Economics, Vol. 1B, ed. by M. R. Rosenzweig and O. Stark. North Holland. [211] HOLT, F. (1996): “Family Migration Decisions: A Dynamic Analysis,” Unpublished Manuscript, University of Virginia. [212] HOTZ, V., AND R. MILLER (1993): “Conditional Choice Probabilities and the Estimation of Dynamic Models,” The Review of Economic Studies, 60, 497–529. [227] KEANE, M. P., AND K. I. WOLPIN (1994): “The Solution and Estimation of Discrete Choice Dynamic Programming Models by Simulation and Interpolation: Monte Carlo Evidence,” Review of Economics and Statistics, 76, 648–672. [217] KEANE, M. P., AND K. I. WOLPIN (1997): “The Career Decisions of Young Men,” Journal of Political Economy, 105, 473–522. [212] KENNAN, J. (2006): “A Note on Discrete Approximations of Continuous Distributions,” Unpublished Manuscript, University of Wisconsin–Madison. [222] (2008): “Average Switching Costs in Dynamic Logit Models,” Unpublished Manuscript, University of Wisconsin–Madison. [235] KENNAN, J., AND J. R. WALKER (2010): “Wages, Welfare Benefits and Migration,” Journal of Econometrics, 156, 229–238. [213] (2011): “Supplement to ‘The Effect of Expected Income on Individual Migration Decisions’,” Econometrica Supplementary Material, 79, http://www.econometricsociety.org/ecta/ Supmat/4657_data and programs.zip. [221] KOTLARSKI, I. I. (1967): “On Characterizing the Gamma and Normal Distribution,” Pacific Journal of Mathematics, 20, 69–76. [226] LUCAS, R. E. B. (1997): “Internal Migration in Developing Countries,” in Handbook of Population and Family Economics, Vol. 1B, ed. by M. R. Rosenzweig and O. Stark. North Holland. [211] MAGNAC, T., AND D. THESMAR (2002): “Identifying Dynamic Discrete Decision Processes,” Econometrica, 70, 801–816. [225] MCFADDEN, D. D. (1974), “Conditional Logit Analysis of Qualitative Choice Behavior,” in Frontiers in Econometrics, ed. by P. Zarembka. Academic Press. [216] NEAL, D. (1999): “The Complexity of Job Mobility of Young Men,” Journal of Labor Economics, 17, 237–261. [212] PESSINO, C. (1991): “Sequential Migration Theory and Evidence From Peru,” Journal of Development Economics, 36, 55–87. [218] REAGAN, P. B., AND R. OLSEN (2000): “You Can Go Home Again: Evidence From Longitudinal Data,” Demography, 37, 339–350. [221]
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RUST, J. P. (1987): “Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher,” Econometrica, 55, 999–1033. [216] (1994): “Structural Estimation of Markov Decision Processes,” in Handbook of Econometrics, Vol. 4, ed. by R. Engle and D. McFadden. Elsevier. [212,215] SCHULTZ, T. P. (1982): “Lifetime Migration Within Educational Strata in Venezulea: Estimates of a Logistic Model,” Economic Development and Cultural Change, 30, 559–593. [219] SJAASTAD, L. A. (1962): “The Costs and Returns of Human Migration,” Journal of Political Economy, 70, 80–89. [239] SWEETING, A. (2007): “Dynamic Product Repositioning in Differentiated Product Industries: The Case of Format Switching in the Commerical Radio Industry,” Working Paper 13522, NBER. [234] TOPEL, R. H. (1986): “Local Labor Markets,” Journal of Political Economy, 94, S111–S143. [212, 214] TUNALI, I. (2000): “Rationality of Migration,” International Economic Review, 41, 893–920. [212] UNITED STATES NATIONAL CLIMATIC DATA CENTER (2002): “Historical Climatography Series 5-1 and 5-2,” Report, Asheville, NC. Available at http://lwf.ncdc.noaa.gov/oa/climate/normals/ usnormals.html. [230]
Dept. of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706, U.S.A. and NBER; [email protected] and Dept. of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706, U.S.A. and NBER; [email protected]. Manuscript received May, 2003; final revision received December, 2009.
Econometrica, Vol. 79, No. 1 (January, 2011), 253–302
THE DIFFUSION OF WAL-MART AND ECONOMIES OF DENSITY BY THOMAS J. HOLMES1 The rollout of Wal-Mart store openings followed a pattern that radiated from the center outward, with Wal-Mart maintaining high store density and a contiguous store network all along the way. This paper estimates the benefits of such a strategy to WalMart, focusing on the savings in distribution costs afforded by a dense network of stores. The paper takes a revealed preference approach, inferring the magnitude of density economies from how much sales cannibalization of closely packed stores Wal-Mart is willing to suffer to achieve density economies. The model is dynamic with rich geographic detail on the locations of stores and distribution centers. Given the enormous number of possible combinations of store-opening sequences, it is difficult to directly solve Wal-Mart’s problem, making conventional approaches infeasible. The moment inequality approach is used instead and works well. The estimates show the benefits to Wal-Mart of high store density are substantial and likely extend significantly beyond savings in trucking costs. KEYWORDS: Economies of density, moment inequalities, dynamics.
1. INTRODUCTION WAL -MART OPENED ITS FIRST STORE in 1962, and today there are over 3,000 Wal-Mart stores in the United States. The rollout of stores illustrated in Figure 1 displays a striking pattern. (See also a video of the rollout posted as Supplemental Material for this article (Holmes (2011)).2 ) Wal-Mart started in a relatively central spot in the country (near Bentonville, Arkansas) and store openings radiated from the inside out. Wal-Mart never jumped to some far-off location to later fill in the area in between. With the exception of store number 1 at the very beginning, Wal-Mart always placed new stores close to where it already had store density. This process was repeated in 1988 when Wal-Mart introduced the supercenter format (see Figure 2). With this format, Wal-Mart added a full-line grocery store alongside the general merchandise of a traditional Wal-Mart. Again, the diffusion of the supercenter format began at the center and radiated from the inside out. 1 I have benefited from the comments of many seminar participants. In particular, I thank Glenn Ellison, Gautam Gowrisankaran, and Avi Goldfarb for their comments as discussants, Pat Bajari and Kyoo-il Kim for suggestions, and Ariel Pakes for advice on how to think about this problem. I thank the referees and the editor for comments that substantially improved the paper. I thank Junichi Suzuki, Julia Thornton Snider, David Molitor, and Ernest Berkas for research assistance. I thank Emek Basker for sharing data. I am grateful to the National Science Foundation under Grant 0551062 for support of this research. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. 2 The video of Wal-Mart’s store openings can also be seen at www.econ.umn.edu/~holmes/ research.html.
© 2011 The Econometric Society
DOI: 10.3982/ECTA7699
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FIGURE 1.—Diffusion of Wal-Mart stores and general distribution centers.
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FIGURE 2.—Diffusion of supercenters and food distribution centers.
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This paper estimates the benefits of such a strategy to Wal-Mart, focusing on the logistic benefits afforded by a dense network of stores. Wal-Mart is vertically integrated into distribution: general merchandise is supplied by Wal-Mart’s own regional distribution centers; groceries for supercenters are supplied through its own food distribution centers.3 When stores are packed closely together, it is easier to set up a distribution network that keeps stores close to a distribution center, and when stores are close to a distribution center, Wal-Mart can save on trucking costs. Moreover, such proximity allows WalMart to respond quickly to demand shocks. Quick response is widely considered to be a key aspect of the Wal-Mart model. (See Holmes (2001) and Ghemawat, Mark, and Bradley (2004).) Wal-Mart famously was able to restock its shelves with American flags on the very day of 9/11. A challenge in estimating the benefits of density is that Wal-Mart is notorious for being secretive. I cannot access confidential data on its logistics costs, so it is not possible to conduct a direct analysis relating costs to density. Even if Wal-Mart were to cooperate and make its data available, the benefits of being able to quickly respond to demand shocks might be difficult to quantify with standard accounting data. Instead, I pursue an indirect approach that exploits revealed preference. Although density has a benefit, it also has a cost, and I am able to pin down that cost. By examining Wal-Mart’s choice behavior of how it trades off the benefit (not observed) versus the cost (observed with some work), it is possible to draw inferences about how Wal-Mart values the benefits. The cost of high store density is that when stores are close together, their market areas overlap and new stores cannibalize sales from existing stores. The extent of such cannibalization is something I can estimate. For this purpose, I bring together store-level sales estimates from ACNielsen and demographic data from the U.S. Census at a very fine level of geographic detail. I use this information to estimate a model of demand in which consumers choose among all the Wal-Mart stores in the general area where they live. The demand model fits the data well, and I am able to corroborate its implications for the extent of cannibalization with certain facts Wal-Mart discloses in its annual reports. Using my sales model, I determine that Wal-Mart has encountered significant diminishing returns in sales as it has packed stores closely together in the same area. I write down a dynamic structural model of how Wal-Mart rolled out its stores over the period 1962–2005. The model is quite detailed and distinguishes the exact location of each individual store, the location of each distribution center, the type of store (regular Wal-Mart or supercenter), and the kind of distribution center (general merchandise or food). The model takes into account wage and land price differences across locations. The model takes into account that although there might be benefits of high store density to Wal-Mart, there 3 According to Ghemawat, Mark, and Bradley (2004), over 80 percent of what Wal-Mart sells goes through its own distribution network.
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also might be disadvantages of high population density—beyond high wages and land prices—because the Wal-Mart model might not work so well in very urban locations. Given the enormous number of different possible combinations of storeopening sequences, it is difficult to directly solve Wal-Mart’s optimization problem. This leads me to consider a perturbation approach that rules out deviations from the chosen policy as being nonoptimal. When the choice set is continuous, a perturbation approach typically implies equality constraints (i.e., first-order conditions). Here, with discrete choice, the approach yields inequality constraints. To average out measurement error, I aggregate the inequalities into moment inequalities; for inference, I follow Pakes, Porter, Ho, and Ishii (2006). Identification is partial, that is, there is a set of points satisfying the moment inequalities, rather than just a single point. There have been significant developments in the partial identification literature. (See Manski (2003) for a monograph treatment.) Much of the recent interest is driven by its application to game-theoretic models with multiple equilibria (e.g., Ciliberto and Tamer (2009)). The possibility of multiple equilibria is not an issue in the decisiontheoretic environment considered here. Nevertheless, the moment inequality approach is useful because of the discrete choice nature of the problem. A concern with any partial identification approach is that the identified set may potentially be so wide as to be relatively uninformative. In practice, this concern should be ameliorated by the imposition of a large number of constraints that narrow the identified set to a tight region. This is the case here. For my baseline specification with the full set of constraints imposed, I estimate that when a Wal-Mart store is closer by 1 mile to a distribution center, over the course of a year, Wal-Mart enjoys a benefit that lies in a tight interval around $3,500. This estimate extends significantly beyond likely savings in trucking costs alone. Given the many miles involved in Wal-Mart’s operations and its thousands of stores, the estimate implies that economies of density are a substantial component of Wal-Mart profitability. An economy of density is a kind of economy of scale. Over the years, various researchers have made distinctions among types of scale economies and noted the role of density. For the airline industry, Caves, Christensen, and Tretheway (1984) distinguished an economy of density from traditional economies of scale as arising when an airline increases the frequency of flights on a given route structure (as opposed to increasing the size of the route structure, holding fixed the flight frequency per route). (See also Caves, Christensen, and Swanson (1981).) The analogy here would be Wal-Mart expanding by adding more stores in the same markets it already serves (as opposed to expanding its geographic reach and keeping store density the same). Roberts (1986) made an analogous distinction in the electric power industry. This paper differs from the existing empirical literature in three ways. First, there is rich micromodeling with an explicit spatial structure. I do not have lumpy market units (e.g.,
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a metro area) within which I count stores; rather, I employ a continuous geography. Second, I explicitly model the channel of the density benefits through the distribution system, rather than leaving them as a “black box.” Third, rather than directly relate costs to density, I use a revealed preference approach as explained above. (See also Holmes and Lee (2009).) There is a large literature on entry and store location in retail.4 There is also a growing literature on Wal-Mart itself.5 This paper is most closely related to the recent parallel work of Jia (2008).6 Jia estimated density economies by examining the site selection problem of Wal-Mart as the outcome of a static game with K-Mart. Jia’s paper features interesting oligopolistic interactions that my paper abstracts away from. My paper highlights (i) dynamics and (ii) cannibalization of sales by nearby Wal-Marts that Jia’s paper abstracts away from. 2. MODEL A retailer (Wal-Mart) has a network of stores supported by a network of distribution centers. The model specifies how Wal-Mart’s revenues and costs in a period depend on the configuration of stores and distribution centers that are open in the period. It also specifies how the networks change over time. There are two categories of merchandise: general merchandise (abbreviated by g) and food (abbreviated by f ). There are two kinds of Wal-Mart stores: A regular store sells only general merchandise; a supercenter sells both general merchandise and food. There is a finite set of locations in the economy. Locations are indexed by = 1 L. Let d denote the distance in miles between any given pair of locations and . At any given period t, a subset BtWal of locations have a WalSuper Mart. Of these, a subset Bt ⊆ BtWal are supercenters and the rest are regular stores. In general, the number of locations with Wal-Marts will be small relative to the total set of locations, and a typical Wal-Mart will draw sales from many locations. Sales revenues at a particular store depend on the store’s location and its g proximity to other Wal-Marts. Let Rjt (BtWal ) be the general merchandise sales revenue of store j at time t given the set of Wal-Mart stores open at time t. If Super f store j is a supercenter, then its food sales Rjt (Bt ) analogously depend on the configuration of supercenters. The model of consumer choice from which this demand function will be derived will be specified below in Section 4. With this demand structure, Wal-Mart stores that are near each other will be regarded as substitutes by consumers. That is, increasing the number of nearby stores will decrease sales at a particular Wal-Mart. 4
See, for example, Bresnahan and Reiss (1991) and Toivanen and Waterson (2005). Recent papers on Wal-Mart include Basker (2005, 2007), Stone (1995), Hausman and Leibtag (2007), Ghemawat, Mark, and Bradley (2004), and Neumark, Zhang, and Ciccarella (2008). 6 See also Andrews, Berry, and Jia (2004). 5
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I abstract from price variation and assume Wal-Mart sets constant prices across all stores and over time. In reality, prices are not always constant across Wal-Marts, but the company’s every day low price (EDLP) policy makes this a better approximation for Wal-Mart than it would be for many retailers. Let μ g denote the gross margin. Thus, for store j at time t, μRjt (BtWal ) is sales receipts less the cost of goods sold for general merchandise. In the analysis, three components of cost will be relevant besides the cost of goods sold: (i) distribution costs, (ii) variable store costs, and (iii) fixed costs at the store level. I describe each in turn. Distribution Costs Each store requires distribution services. General merchandise is supplied by a general distribution center (GDC) and food is supplied by a food distribution center (FDC). For each store, these services are supplied by the closest g distribution center. Let djt be the distance in miles from store j to the closf est GDC at time t and analogously define djt . If store j is a supercenter, its distribution cost at time t is g
f
DistributionCostjt = τdjt + τdjt where the parameter τ is the cost per mile per period per merchandise segment (general or food) of servicing this store.7 If j carries only general merchandise, g the cost is τdjt . The distribution cost is a fixed cost that does not depend on the volume of store sales. This would be an appropriate assumption if Wal-Mart made a single delivery run from the distribution center to the store each day. The driver’s time is a fixed cost and the implicit rental on the tractor is a fixed cost that must be incurred regardless of the size of the load. To keep a tight rein on inventory and to allow for quick response, Wal-Mart aims to have daily deliveries even for its smaller stores. So there clearly is an important fixed cost component to distribution. Undoubtedly, there is a variable cost component as well, but for simplicity I abstract from it. Variable Costs The larger the sales volume at a store, the greater the number of workers needed to staff the checkout lines, the larger the parking lot, the larger the required shelf space, and the bigger the building. All of these costs are treated as variable in this analysis. It may seem odd to treat building size and shelving as a variable input. However, Wal-Mart very frequently updates and expands its 7 More generally, the distribution costs for the two segments might differ. I constrain them to be the same because it greatly facilitates my estimation procedure later in the paper.
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stores. So in practice, store size is not a permanent decision that is made once and for all; rather, it is a decision made at multiple points over time. Treating store size as a variable input simplifies the analysis significantly. Assume that the variable input requirements at store j are all proportionate to sales volume Rj : Laborj = ν Labor Rj Landj = ν Land Rj Otherj = ν Other Rj Wages and land prices vary across locations and across time. Let Wagejt and Rentjt denote the wage and land rental rate that store j faces at time t. Other consists of everything left out so far that varies with sales, including the rental on structure and equipment in the store (the shelving, the cash registers, etc.). The other cost component of variable costs is assumed to be the same across locations and the price is normalized to 1. Fixed Costs We expect there to be a fixed cost of operating a store. To the extent the fixed cost is the same across locations, it will play no role in the analysis of where Wal-Mart places a given number of stores. We are only interested in the component of fixed cost that varies across locations. From Wal-Mart’s perspective, urban locations have some disadvantages compared to nonurban locations. These disadvantages go beyond higher land rents and higher wages that have already been taken into account above. The Wal-Mart model of a big box store at a convenient highway exit is not applicable in a very urban location. Moreover, Sam Walton was very concerned about the labor force available in urban locations, as he explained in his autobiography (Walton (1992)). To capture potential disadvantages of urban locations, the fixed cost of operating store j is written as a function c(Popdenj ) of the population density Popdenj of the store’s location. The functional form is quadratic in logs, (1)
c(Popdenj ) = ω0 + ω1 ln(Popdenj ) + ω2 ln(Popdenj )2
A supercenter is actually two stores, a general merchandise store and a food store, so the fixed cost is paid twice. It will be with no loss of generality in our analysis to assume that the constant term ω0 = 0, since the only component of the fixed cost that will matter in the analysis is the part that varies across locations.
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Dynamics Everything that has been discussed so far considers quantities for a particular time period. I turn now to the dynamic aspects of the model. Wal-Mart operates in a deterministic environment in discrete time where it has perfect foresight. The general problem Wal-Mart faces is to answer the following questions for each period: Q1. How many new Wal-Marts and how many new supercenters should be opened? Q2. Where should the new Wal-Marts and supercenters be put? Q3. How many new distribution centers should be opened? Q4. Where should the new distribution centers be put? The main focus of the paper is on part Q2 of Wal-Mart’s problem. The analysis conditions on Q1, Q3, and Q4 being what Wal-Mart actually did, and solves Wal-Mart’s problem of getting Q2 right. Of course, if Wal-Mart’s actual behavior solves the true problem of choosing Q1–Q4, then it also solves the constrained problem of choosing Q2, subject to Q1, Q3, and Q4 fixed at what Wal-Mart did. Getting at part Q1 of Wal-Mart’s problem—how many new stores Wal-Mart opens in a given year—is far afield from the main issues of this paper. In its first few years, Wal-Mart added only one or two stores per year. The number of new store openings has grown substantially over time, and in recent years they sometimes number several stores in 1 week. Presumably, capital market considerations have played an important role here. This is an interesting issue, but not one I will have anything to say about in this paper. Problems Q3 and Q4 regarding distribution centers are closely related to the main issue of this paper. I will have something to say about this later in Section 7. Now for more notation. To begin with, the discount factor each period is β. The period length is a year, and the discount factor is set to β = 95. Super As defined earlier, BtWal is the set of Wal-Mart stores in period t, and Bt ⊆ Wal Bt is the set of supercenters. Assume that once a store is opened, it never shuts down. This assumption simplifies the analysis considerably and is not inconsistent with Wal-Mart’s behavior because it rarely closes stores.8 Then Wal Wal we can write BtWal = Bt−1 + AWal is the set of new stores opened t , where At Super Super in period t. Analogously, a supercenter is an absorbing state, Bt = Bt−1 + Super ASuper for At , the set of new supercenter openings in period t. A supercenter t can open two ways. It can be a new Wal-Mart store that opens as a supercenter as well or it can be a conversion of an existing Wal-Mart store. Super Let NtWal and Nt be the number of new Wal-Marts and supercenters Super opened at t, that is, the cardinality of the sets AWal and At . Choosing these t 8
Wal-Mart’s annual reports disclose store closings that are on the order of two per year.
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values is defined as part Q1 of Wal-Mart’s problem. These are taken as given here. Also taken as given is the location of distribution centers of each type and their opening dates (parts Q3 and Q4 of Wal-Mart’s problem). There is exogenous productivity growth of Wal-Mart according to a growth factor ρt in period t. If Wal-Mart were to hold fixed the set of stores, and demographics also stayed the same, then from period 1 to period t, revenue and all components of costs would grow by (an annualized) factor ρt . As will be discussed later, the growth of sales per store of Wal-Mart has been remarkable. Part of this growth is due to the gradual expansion of its product line, initially from hardware and variety items to food, drugs, eyeglasses, tires, and so on. The part of growth due to food through the expansion into supercenters is explicitly modeled here, but expansion into drugs, eyeglasses, tires, and so on is not modeled explicitly. Instead, this growth is implicitly picked up through the exogenous growth parameter ρt . The role ρt plays in Wal-Mart’s problem is like a discount factor. Super Super A policy choice of Wal-Mart is a vector a = (AWal AWal ) 1 A1 2 A2 that specifies the locations of the new stores opened in each period t. Define a choice vector a to be feasible if the number of store openings in period t under policy a equals what Wal-Mart actually did, that is, NtWal new stores in Super a period and Nt supercenter openings. Wal-Mart’s problem at time 0 is to pick a feasible a to maximize (2)
max a
∞ g g g f f f (ρt β)t−1 [πjt − cjt − τdjt ] + [πjt − cjt − τdjt ] t=1
j∈BtWal
Super
j∈Bt
where the operating profit for merchandise segment e ∈ {g f } at store j in time t is πjte = μRejt − Wagejt Laborejt − Rentjt Landejt − Otherejt and where djte is the distance to the closest distribution center at time t for merchandise segment e. No explicit mention has been made about the presence of sunk costs. Implicitly, sunk costs are large, and that is why no store is ever closed once opened. Sunk costs can easily be worked into the model by having some portion of the present value of the fixed cost in equation (1) paid at entry rather than in perpetuity each period. This leaves the objective in equation (2) unchanged. 3. THE DATA Five main data elements are used in the analysis. The first element is storelevel data on sales and other characteristics. The second is opening dates for stores and distribution centers. The third is demographic information from the
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DIFFUSION AND ECONOMIES OF DENSITY TABLE I SUMMARY STATISTICS OF STORE-LEVEL DATAa Store Type
Variable
N
Mean
All Regular Wal-Mart Supercenter
Sales ($millions/year) Sales ($millions/year) Sales ($millions/year)
3,176 1,196 1,980
70.5 47.0 84.7
All Regular Wal-Mart Supercenter
Employment Employment Employment
3,176 1,196 1,980
254.9 123.5 333.8
Std. Dev.
Min
Max
30.0 20.0 25.9
9.1 9.1 20.8
166.4 133.9 166.4
127.3 40.1 91.5
31.0 57.0 31.0
812.0 410.0 812.0
a End of 2005, excludes Alaska and Hawaii. Source: Trade Dimensions retail data base.
Census. The fourth is data on wages and rents across locations. The fifth is other information about Wal-Mart from annual reports. The first data element comprising store-level variables was obtained from Trade Dimensions, a unit of ACNielsen. These data provide estimates of storelevel sales for all Wal-Marts open at the end of 2005. These data are the best available and the primary source of market share data used in the retail industry. Ellickson (2007) recently used these data for the supermarket industry. Table I presents summary statistics of annual store-level sales and employment for the 3,176 Wal-Marts in existence in the contiguous United States as of the end of 2005. (Alaska and Hawaii are excluded in all of the analysis.) Almost two-thirds of all Wal-Marts (1,980 out of 3,176) are supercenters that carry both general merchandise and food. The remaining 1,196 are regular WalMarts that do not have a full selection of food. The average Wal-Mart has annual sales of $70 million. The breakdown is $47 million per regular Wal-Mart and $85 million per supercenter. The average employment is 255 employees. The second data element is opening dates of the four types of Wal-Mart facilities. The table treats a supercenter as two different stores: a general merchandise store and a food store. The two kinds of distribution centers are general (GDC) and food (FDC). Table II tabulates opening dates for the four types of facilities by decade. Appendix A explains how this information was collected. Note that if a regular store is later converted to a supercenter, it has an opening date for its general merchandise store and a later opening date for its food store. This is called a conversion. The third data element, demographic information, comes from three decennial censuses: 1980, 1990, and 2000. The data are at the level of the block group, a geographic unit finer than the Census tract. Summary statistics are provided in Table III. In 2000, there were 206,960 block groups with an average population of 1,350. I use the geographic coordinates of each block group to draw a circle of radius 5 miles around each block group. I take the population within this 5-mile radius and use this as my population density measure. Table III reports that the mean density in 2000 across block groups equals 219,000 people
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THOMAS J. HOLMES TABLE II DISTRIBUTION OF WAL -MART FACILITY OPENINGS BY DECADE AND OPENING TYPEa General Merchandise (Including Supercenters)
Decade Open
1962–1969 1970–1979 1980–1989 1990–1999 2000–2005
Opened This Decade
15 243 1,082 1,130 706
Food Store (Part of Supercenter)
Cumulative
Opened This Decade
15 258 1,340 2,470 3,176
0 0 4 679 1,297
General Distribution Centers
Cumulative
Opened This Decade
0 0 4 683 1,980
1 1 8 18 15
Food Distribution Centers
Cumulative
Opened This Decade
Cumulative
1 2 10 28 43
0 0 0 9 26
0 0 0 9 35
a Source: See Appendix A.
within a 5-mile radius. The table also reports mean levels of per capita income, share old (65 or older), share young (21 or younger), and share black. The per capita income figure is in 2000 dollars for all the Census years, using the consumer price index (CPI) as the deflator.9 The fourth data element is information about local wages and local rents. The wage measure is the average retail wage by county from County Business Patterns, a data set from the U.S. Census Bureau. This measure is payroll divided by employment. I use annual data over the period 1977–2004. It is difficult to obtain a consistent measure of land rents at a fine degree of geographic detail over a long period of time. To proxy land rents, I use information about residential property values from the 1980, 1990, and 2000 decennial censuses. For each Census year and each store location, I create an index of property TABLE III SUMMARY STATISTICS FOR CENSUS BLOCK GROUPSa
N Mean population (1,000) Mean population density (1,000 in 5-mile radius) Mean per capita income (thousands of 2000 dollars) Share old (65 and up) Share young (21 and below) Share black
1980
1990
2000
269,738 .83 165.3 14.73 .12 .35 .10
222,764 1.11 198.44 18.56 .14 .31 .13
206,960 1.35 219.48 21.27 .13 .31 .13
a Source: “Summary File 3, Census of Population and Housing” (U.S. Bureau of the Census (1980, 1990, 2000)).
9
Per capita income is truncated from below at $5,000 in year 2000 dollars.
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values by adding up the total value of residential property within 2 miles of the store’s location and scaling it so the units are in inflation-adjusted dollars per acre. See Appendix A for how the index is constructed. Interpolation is used to obtain values between Census years. The Census data are supplemented with property tax data on property valuations of actual Wal-Mart store locations in Iowa and Minnesota. As discussed in Appendix A, there is a high correlation between the tax assessment property valuations of a Wal-Mart site and the property value index. The fifth data element is information from Wal-Mart’s annual reports, including information about aggregate sales for earlier years. I also use information provided in the Management Discussion section of the reports on the degree to which new stores cannibalize sales of existing stores. The specifics of this information are explained below when the information is incorporated into the estimation. 4. ESTIMATES OF OPERATING PROFITS This section estimates the components of Wal-Mart’s operating profits. Section 4.1 specifies the demand model and Section 4.2 estimates it. Section 4.3 treats various cost parameters. Section 4.4 explains how estimates for 2005 are extrapolated to other years. 4.1. Demand Specification A discrete choice approach to demand is employed, following common practice in the literature. I separate the general merchandise and food purchase decisions, and begin with general merchandise. A consumer at a particular location chooses between shopping at the “outside option” and shopping at any Wal-Mart located within 25 miles. Formally, the consumer’s choice set for Wal-Marts is
B¯Wal = {j j ∈ B Wal and Distancej ≤ 25} where Distancej is the distance in miles between location and store location j. (The time subscript t is implicit throughout this subsection.) If the consumer chooses the outside alternative 0, utility is (3)
u0 = b(Popden ) + LocationChar α + ε0
The first term is a function b(·) that depends on the Popden at the consumer’s location . Assume b (Popden) ≥ 0; that is, the outside option is better in higher population density areas. This is a sensible assumption because we would expect there to be more substitutes for a Wal-Mart in larger markets for the usual reasons. A richer model of demand would explicitly specify the alternative shopping options available to the consumer. I do not have sufficient
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data to conduct such an analysis, so instead account for this mechanism in a reduced-form way.10 The functional form used in the estimation is b(Popden) = α0 + α1 ln(Popden) + α2 (ln(Popden))2 where (4)
Popden = max{1 Popden}
The units of the density measure are thousands of people within a 5-mile radius. By truncating Popden at 1, ln(Popden) is truncated at 0. All locations with less than 1,000 people within 5 miles are grouped together.11 The second term of equation (3) allows demand for the outside good to depend on a vector LocationChar of location characteristics that impact utility through the parameter vector α. In the empirical analysis, a location is a block group. The characteristics will include the demographic and income characteristics of the block group. The third term of the outside good utility in equation (3) is a logit error term. Assume this is drawn independently and identically distributed (i.i.d.) across all consumers living in block group . Next consider the utility from buying at a particular Wal-Mart j ∈ B¯Wal . It equals (5)
uj = −h(Popden )Distancej + StoreCharj γ + εj
for h(Popden) parameterized by h(Popden) = ξ0 + ξ1 ln(Popden) The first term of equation (5) is the disutility of commuting Distancej miles to the store from the consumer’s home. The second term of equation (5) allows utility to depend on an additional characteristic StoreCharj of store j. In the empirical analysis, this characteristic will be store age. In this way, the demand model will capture that brand-new stores have fewer sales, everything else being the same. The last term is the logit error εj . g The probability pj a consumer at location shops at store j can be derived using the standard logit formula. The model’s predicted general merchandise revenue for store j is then g g Rj = (6) λg × pj × n {|j∈B¯ Wal }
10 This is in the spirit of recent work such as Bajari, Benkard, and Levin (2007), who estimated policy functions and equilibrium relationships directly. 11 This same truncation is applied throughout the paper.
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where λg is spending per consumer. In words, there are n consumers at locag tion and a fraction pj of them are shopping at store j where they will each spend λg dollars. Spending on food is modeled the same way. The parameters are the same f except for the spending λf per consumer. The formula for food revenue Rj at store j is analogous to (6). Note that when calculating food revenue, it is Super necessary to take into account that the set of alternatives for food B¯ is, Wal ¯ in general, different from the set of alternatives B for general merchandise. 4.2. Demand Estimation Recent empirical papers on demand typically use data sets with quantities directly determined from sales records. In these analyses, quantities are treated as being measured without error. Following Berry, Levinsohn, and Pakes (1995), demand models are estimated to perfectly fit these sales data, with unobserved product characteristics soaking up discrepancies. In contrast, the store-level sales information used here is estimated by Trade Dimensions, using proprietary information it has acquired. There is certainly measurement error in these estimates that needs to be incorporated into the demand estimation. For simplicity, I attribute all of the discrepancies between the model and the data to classical measurement error. Given a vector ψ of parameters from the demand model, we can plug in the demographic data and obtain predicted values of general merchandise sales g Rj (ψ) for each store j from equation (6) and predicted values of food sales f
Rj (ψ). Let RData be the sales volume in the data. Let ηSales be the discrepancy j j between measured log sales and predicted log sales. For a regular store, this equals g
= ln(RData ) − ln(Rj (ψ)) ηSales j j For a supercenter, this equals g
f
ηSales = ln(RData ) − ln(Rj (ψ) + Rj (ψ)) j j are i.i.d. normally distributed. The model is estimated usAssume the ηSales j ing maximum likelihood, and the coefficients are reported in Table IV in the column labeled “Unconstrained.” Getting right the extent to which new stores cannibalize sales of existing stores is crucial for the subsequent analysis. Fortunately, Wal-Mart has provided information that is helpful in this regard. Wal-Mart’s annual report
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THOMAS J. HOLMES TABLE IV PARAMETER ESTIMATES FOR DEMAND MODEL
Parameter
Definition
Unconstrained
Constrained (Fits Reported Cannibalization)
λg
General merchandise spending per person (annual in $1,000)
1686 (056)
1938 (043)
λf
Food spending per person (annual in $1,000)
1649 (061)
1912 (050)
ξ0
Distance disutility (constant term)
642 (036)
703 (039)
ξ1
Distance disutility (coefficient on ln(Popden))
−046 (007)
−056 (008)
α
Outside alternative valuation parameters Constant
−8271 (508)
−7834 (530)
ln(Popden)
1968 (138)
1861 (144)
ln(Popden)2
−070 (012)
−059 (013)
Per capita income
015 (003)
013 (003)
Share of block group black
341 (082)
297 (076)
Share of block group young
1105 (464)
1132 (440)
Share of block group old
563 (380)
465 (359)
183 (024)
207 (023)
065 (002)
065 (002)
γ
Store-specific parameters Store age 2 + dummy
σ2
Measurement error
N Sum of squared error R2 (Likelihood)
3,176 205117
3,176 206845
755 −155749
753 −169072
for 2004 disclosed (Wal-Mart Stores, Inc. (1971–2006) (Annual Report 2004, p. 20)) the following information: As we continue to add new stores domestically, we do so with an understanding that additional stores may take sales away from existing units. We estimate that comparative store
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sales in fiscal year 2004, 2003, 2002 were negatively impacted by the opening of new stores by approximately 1%.
This same paragraph was repeated in the 2006 annual report with regard to fiscal year 2005 and 2006. This information is summarized in Table V.12 To define the model analog of cannibalization, first calculate what sales would be in a particular year for preexisting stores if no new stores were opened in the year and if there were no new supercenter conversions. Next calculate predicted sales to preexisting stores when the new store openings and supercenter conversions for the particular year take place. Define the percentage difference to be the cannibalization rate for that year. This is the model analog of what Wal-Mart is disclosing. Table V reports the cannibalization rates for various years using the estimated demand model. The parameter vector is the same across years. What varies over time are the new stores, the set of preexisting stores, and the demographic variables. The demand model—estimated entirely from the 2005 cross-sectional store-level sales data—does a very good job fitting the cannibalization rates reported by Wal-Mart. For the years that Wal-Mart disclosed that the rate was “approximately 1%,” the estimated rates range from .67 to 1.43. It is interesting to note the sharp increase in the estimated cannibalization rate beginning in 2002. Evidently, Wal-Mart reached some kind of saturation point in 2001. Given the pattern in Table V, it is understandable that Wal-Mart has felt the need to disclose the extent of cannibalization in recent years. TABLE V CANNIBALIZATION RATES, FROM ANNUAL REPORTS AND IN MODELa
Year
From Annual Reports
Demand Model (Unconstrained)
Demand Model (Constrained)
1998 1999 2000 2001 2002 2003 2004 2005
n.a. n.a. n.a. 1 1 1 1 1
.62 .87 .55 .67 1.28 1.38 1.43 1.27
.48 .67 .40 .53 1.02 1.10 1.14 1.00b
a Source: Estimates from the model and Wal-Mart Stores, Inc. (1971–2006) (Annual Reports 2004, 2006). b Cannibalization rate is imposed to equal 1.00 in 2005.
12 Wal-Mart’s fiscal year ends January 31, so the fiscal year corresponds (approximately) to the previous calendar year. For example, the 2006 fiscal year began February 1, 2005. In this paper, I aggregate years like Wal-Mart (February through January), but I use 2005 to refer to the year beginning February 2005.
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In what follows, the estimated upper bound on the degree of density economies will be closely connected to the degree of cannibalization. The more cannibalization Wal-Mart is willing to tolerate, the higher the inferred density economies. The estimated cannibalization rates of 1.38, 1.43, and 1.27 for 2003, 2004, and 2005 qualify as approximately 1, but one may worry that these rates are on the high end of what would be consistent with Wal-Mart’s reports. To explore this issue further, I estimate a second demand model where the cannibalization rate for 2005 is constrained to be exactly 1. The estimates are reported in the last column of Table IV. The goodness of fit under the constraint is close to the unconstrained model, although a likelihood ratio test leads to a rejection of the constraint. In the interests of being conservative in my estimate of a lower bound on density economies, I will use the constrained model throughout as the baseline model. The parameter estimates reveal that, as hypothesized, the outside good is better in more dense areas and that utility decreases in distance traveled to a Wal-Mart. To get a sense of the magnitudes, Table VI examines how predicted demand in a block group varies with population density and distance to the closest Wal-Mart, with the demographic variables in Table III set to their mean values, and with only one Wal-Mart (2 or more years old) in the consumer choice set. Consider the first row, where the distance is zero and the population density is varied. The negative effect of population density on demand is substantial. A rural consumer right next to a Wal-Mart shops there with a probability that is essentially 1. At a population density of 50,000, this falls to .72 and falls to only .24 at 250,000. The model captures in a reduced-form way that in a large market, there tend to be many substitutes compared to a small market. A rural consumer who happens to live 1 mile from a Wal-Mart is unlikely to have many other choices. In contrast, an urban consumer who lives 1 mile from a Wal-Mart is likely to have many nearby discount-format stores to TABLE VI COMPARATIVE STATICS WITH DEMAND MODELa
Distance (Miles)
0 1 2 3 4 5 10
Population Density (Thousands of People Within a 5-Mile Radius) 1
5
10
20
50
100
250
.999 .999 .997 .995 .989 .978 .570
.989 .979 .962 .933 .883 .803 .160
.966 .941 .899 .834 .739 .615 .083
.906 .849 .767 .659 .531 .398 .044
.717 .610 .490 .372 .268 .184 .020
.496 .387 .288 .206 .142 .096 .011
.236 .172 .123 .086 .060 .041 .006
a Uses constrained model.
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choose among and, in addition, have nearby substitute formats like a Best Buy, Home Depot, or shopping mall. Next consider the effect of distance, holding population density fixed. In a very rural area, increasing distance from 0 to 5 miles has only a small effect on demand. This is exactly what we would expect. Raising the distance further from 5 to 10 miles has an appreciable effect, .98 to .57, but still much demand remains. Contrast this with higher density areas. At a population density of 250,000, an increase in distance from 0 to 5 miles reduces demand on the order of 80 percent.13 A higher distance responsiveness in urban areas is just what we would expect. A few remarks about the remaining parameter estimates are in order. Recall that λg and λf are spending per consumer in the general merchandise and food categories. The estimates can be compared to aggregate statistics. For 2005, per capita spending in the United States was 1.8 in general merchandise stores (North American Industry Classification System [NAICS] 452) and 1.8 in food and beverage stores (NAICS 445) (in thousands of dollars). The aggregate statistics match the model estimates well (λg = 19 and λf = 19 in the constrained model; λg = 17 and λf = 16 in the unconstrained model), although for various reasons we would not expect them to match exactly. The only store characteristic used in the demand model (besides location) is store age. This is captured with a dummy variable for stores that have been open 2 or more years. This variable enters positively in demand, so everything else being the same, older stores attract more sales. 4.3. Variable Costs at the Store Level In the model, required labor input at the store level is assumed to be proportionate to sales. In the data, on average there are 3.61 store employees per million dollars of annual sales. I use this as the estimate of the fixed labor coefficient, ν Labor = 361. In the empirical work below, I examine the sensitivity of the results to this calibrated parameter and to the other calibrated parameters. To determine the cost of labor at a particular store, the coefficient νLabor is multiplied by average retail wage (annual payroll per worker) in the county where the store is located. Table VII reports information about the distribution of labor costs across the 2005 set of Wal-Mart stores. The median store faces a labor cost of $21,000 per worker. Given ν Labor = 361, this translates into a labor cost of 361 × 21000 per million in sales or, equivalently, 7.5 percent of sales. The highest labor costs can be found at stores in San Jose, California, where wages are almost twice as high as they are for the median store. 13 The reader may note that the distance coefficient ξ1 on ln(m) in Table IV is actually slightly negative. This effect is overwhelmed by the nonlinearity in the logit model combined with the impact of density on the outside utility. It is possible to rescale units so distance cost is constant or increasing in density and get the same implied demand structure.
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THOMAS J. HOLMES TABLE VII DISTRIBUTION OF VARIABLE INPUT COSTSa Estimated 2005 Labor Costs
Quartile
Minimum 25 50 75 Maximum
Store Location
Pineville, MO Litchfield, IL Belleville, IL Miami, FL San Jose, CA
Annual Payroll per Worker ($)
12,400 19,300 21,000 23,000 37,900
Wages as Percentage of Sales
4.5 7.0 7.6 8.3 13.7
Estimated 2005 Land Value–Sales Ratios
Quartile
Minimum 25 50 75 Maximum
Store Location
Index of Residential Property Value per Acre ($)
Land Value as Percentage of Sales
Lincoln, ME Campbellsville, KY Cleburne, TX Albany, NY Mountain View, CA
1,100 32,100 67,100 137,300 1,800,000
.0 1.2 2.4 5.0 65.0
a Percentiles of distribution are weighted by sales revenue.
An issue that needs to be raised about the County Business Patterns wage data is measurement error. Dividing annual payroll by employment is a crude way to measure labor costs because it does not take into account potential variations in hours per worker (e.g., part time versus full time) or potential variations in labor quality. The empirical procedure used below explicitly takes into account measurement error. Turning now to land costs, Appendix A describes the construction of a property value index for each store through the use of Census data. As discussed in the Appendix, this index, along with property tax data for 46 Wal-Mart locations in Minnesota and Iowa, is used to estimate a land value to sales ratio for each store. The distributions of this index and ratio are reported in Table VII. Perhaps not surprisingly, the most expensive location is estimated to be the Wal-Mart store in Silicon Valley (in Mountain View, California), where the ratio of the land value for the store relative to annual store sales is estimated to be 65 percent. The rental cost of the land, including any taxes that vary with land value, is assumed to be 20 percent of the land value. For the median store from Table VII (the Wal-Mart in Cleburne, Texas), this implies annual land costs of about half a percent of sales (5 ≈ 2 × 24). It is important to emphasize that this rental cost is for the land, not structures. (Half of a percent of sales would be a very low number for the combined rent on land and struc-
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tures and equipment.) The rents on structures and equipment are separated out because they should be approximately the same across locations, as least as compared to variations across stores in land rents. The cost of cinderblocks for walls, steel beams for roofing, shelving, cash registers, asphalt for parking lots, and so on are all assumed to be the same across locations. I now turn to those aspects of variable costs that are the same across locations. I begin with cost of goods sold. Wal-Mart’s gross margin over the years has ranged from .22 to .26. (See Wal-Mart’s annual reports.) To be consistent with this, the gross margin is set equal to μ = 24. Over the years, Wal-Mart has reported operating, selling, general, and administrative expenses that are in the range of 16–18 percent of sales. Included in this range is the store-level labor cost discussed above, which is on the order of 7 percent of sales and has already been taken into account. Also included is the cost of running the distribution system, the fixed cost of running central administration, and other costs that I do not want to include as variable costs. I set the residual variable cost parameter ν Other = 07. Netting this out of the gross margin μ yields a net margin μ − ν Other = 17. In the analysis, the breakdown between μ and ν Other is irrelevant; only the difference matters. The analysis so far has explained how to calculate the operating profit of store j in 2005 as (7)
g
f
πj2005 = (μ − ν Other )(Rj2005 + Rj2005 ) − LaborCostj2005 − LandRentj2005
where the sales revenue comes from the 2005 demand model, and labor cost and land rent are explained above. The next step is to extrapolate this model to earlier years. 4.4. Extrapolation to Other Years We have a demand model for 2005 in hand, but need models for earlier years. To get them, assume demand in earlier years is the same as in 2005 except for the multiplicative scaling factor ρt introduced above in the definition of Wal-Mart’s problem in equation (2). For example, the 2005 demand model with no rescaling predicts that, at the 1971 store set and 1971 demographic variables, average sales per store (in 2005 dollars) is $31.5 million. Actual sales per store (in 2005 dollars) for 1971 is $7.4 million. The scale factor for 1971 adjusts demand proportionately so that the model exactly matches aggregate 1971 sales. Over the 1971–2005 period, this corresponds to a compound annual real growth rate of 4.4 percent. Wal-Mart significantly widened the range of products it sold over this period (to include tires, eyeglasses, etc.). The growth factor is meant to capture this. The growth factor calculated in this manner has leveled off in recent years to around 1 percent a year. Wal-Mart has also been expanding by converting regular stores to supercenters. This expansion
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is captured explicitly in the model rather than indirectly through exogenous growth. Demographics change over time, and this is taken into account. For 1980, 1990, and 2000, I use the decennial census for that year.14 For years in between, I use a convex combination of the censuses.15 5. PRELIMINARY EVIDENCE OF A TRADE-OFF This section provides some preliminary evidence of an economically significant trade-off to Wal-Mart. Namely, the benefits of increased economies of density come at the cost of cannibalization of existing stores. This section puts to work the demand model and other components of operating profits compiled above. Consider some Wal-Mart store j that first opens in time t. Define the increof store j to be what the store adds to total Wal-Mart sales in mental sales Reinc jt segment e ∈ {g f } in its opening year t, relative to what sales would otherwise be across all other stores open that year. The incremental sales of store number 1 opening in 1962 equals Rej1962 that year. For a later store j, however, the ≤ Rejt , because incremental sales are, in general, less than store j’s sales, Reinc jt some part of the sales may be diverted from other stores. Using the demand model, we can calculate Reinc for each store. jt Table VIII reports that the average annual incremental sales at opening in general merchandise across all stores equals $36.3 million (in 2005 dollars throughout). Analogously, average incremental sales in food from new supercenters is $40.2 million. (Note that conversions of existing Wal-Marts to supercenters count as store openings here.) To make things comparable across years, the 2005 demand model is applied to the store configurations and demographics of the earlier years with no multiplicative scale adjustment ρt . In an analogous manner, we can use equation (7) to determine the incremental operating profit of each store at the time it opens. The average annual incremental profit in general merchandise from a new Wal-Mart is $3.1 million and in food from new supercenters is $3.6 million. Finally, we can ask how far a store is from the closest distribution center in the year it is opened. On average, a new Wal-Mart is 168.9 miles from the closest regular distribution center when it opens, and a new supercenter is 137.0 miles from the closest food distribution center. Incremental sales and operating profit can be compared to what sales and operating profit would be if a new store were a stand-alone operation. That is, what would sales and operating profits be at a store if it were isolated so 14
I use 1980 for years before 1980 and 2000 for years after 2000. For example, for 1984 there is .6 weight on 1980 and .4 weight on 1990, meaning 60 percent of the people in each 1980 block group are assumed to be still around as potential Wal-Mart customers, and 40 percent of the 1990 block group consumers have already arrived. This procedure keeps the geography clean, since the issue of how to link block groups over time is avoided. 15
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DIFFUSION AND ECONOMIES OF DENSITY TABLE VIII INCREMENTAL AND STAND -ALONE VALUES OF NEW STORE OPENINGSa
N
All
Incremental Sales
Operating Profit
Incremental Distribution Center Distance (Miles)
Stand-Alone Sales
Part A: General Merchandise (New Wal-Marts Including Supercenters) 3,176 36.3 3.1 168.9 41.4
By state’s Wal-Mart age at opening 1–2 3–5 6–10 11–15 16–20 21 and above All By state’s Wal-Mart age at opening 1–2 3–5 6–10 11–15 16–20
288 614 939 642 383 310 1,980
202 484 775 452 67
38.0 39.5 37.6 36.1 32.9 29.5
3.5 3.5 3.3 2.9 2.8 2.4
3.6
38.7 41.5 40.9 42.2 41.2 44.4
3.6 3.7 3.6 3.4 3.5 3.6
Part B: Food (New Supercenters) 40.2 3.6 137.0
44.8
4.0
42.4 42.7 41.0 36.7 30.1
43.9 44.7 45.3 45.3 38.6
3.9 4.1 4.0 3.9 3.5
3.9 4.0 3.6 3.2 2.8
343.3 202.0 160.7 142.1 113.7 90.2
Stand-Alone Operating Profit
252.9 171.2 113.5 95.3 94.0
a All evaluated at 2005 demand equivalents in millions of 2005 dollars.
that none of its sales is diverted to or from other Wal-Mart stores in the vicinity? Table VIII shows for the average new Wal-Mart, there is a big difference between stand-alone and incremental values, implying a substantial degree of market overlap with existing stores. Average stand-alone sales is $41.4 million compared to an incremental value of $36.3 million, approximately a 10 percent difference. Two considerations account for why the big cannibalization numbers found here are not inconsistent with the 1 percent cannibalization rates reported earlier in Table V. First, the denominator of the cannibalization rate from Table V includes all preexisting stores, including those areas of the country where Wal-Mart is not adding any new stores. Taking an average over the country as a whole understates the degree of cannibalization taking place where Wal-Mart is adding new stores. Second, stand-alone sales include sales that a new store would never get because the sales would remain in some existing store (but would be diverted to the new store if existing stores shut down).
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Define the Wal-Mart Age of a state to be the number of years that Wal-Mart has been in the state.16 The remaining rows in Table VIII classify stores by the Wal-Mart age of their state at the store’s opening. Those stores in the row labeled “1–2” are the first stores in their respective states. Those stores in the row labeled “21 and above” are opened when Wal-Mart has been in their states for over 20 years. Table VIII shows that incremental operating profit in a state falls over time as Wal-Mart adds stores to a state and the store market areas increasingly begin to overlap. Things are actually flat the first 5 years at $3.5 million in incremental operating profit for general merchandise, but it falls to $3.3 million in the second 5 years and then to $2.9 million and lower beyond that. An analogous pattern holds for food. This pattern is a kind of diminishing returns. Wal-Mart is getting less incremental operating profit from the later stores it opens in a state. The table also reveals a benefit from opening stores in a state where WalMart has been for many years. The incremental distribution center distance is relatively low in such states. It decreases substantially as we move down the table to stores opening later in a state. The very first stores in a state average about 300 miles from the closest distribution center. This falls to less than 100 miles when the Wal-Mart age of the state is over 20 years. There is a tradeoff here: the later stores deliver lower operating profit but are closer to a distribution center. The magnitude of the trade-off is on the order of 200 miles for $1 million in operating profit. This trade-off is examined in a more formal fashion in the next three sections, and the results are roughly on this order of magnitude. 6. BOUNDING DENSITY ECONOMIES: METHOD It remains to pin down the parameters relating to density. There are three such parameters, θ = (τ ω1 ω2 ). The τ parameter is the coefficient on distance between a store and its distribution center. It captures the benefit of store density. The parameters ω1 and ω2 relate to how fixed cost varies with population density in equation (1). The estimation task is spread over three sections. This section lays out the set identification method. Section 7 presents the baseline estimates and interprets them. After that, Section 8 tackles the issue of inference, addressing the specific complications that arise here. Section 8 also discusses the robustness of the findings to alternative specifications and assumptions. 16 For the purposes of this analysis, the New England states are treated as a single state. Maryland, Delaware, and the District of Columbia are also aggregated into one state.
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6.1. The Linear Moment Inequality Framework My approach follows the partial identification literature initiated by Manski (2003). The contributions to this literature have been extensive. In my application, I follow Pakes, Porter, Ho, and Ishii (2006) (hereafter PPHI). In the first part of this section, I lay out the general linear moment inequality framework. In the second part, I map Wal-Mart’s choice problem into the framework. Let there be a set of M linear inequalities, with each inequality indexed by a, (8)
ya ≥ xa θ
a ∈ {1 2 M}
for scalar ya , 3 × 1 vector xa , and parameter vector θ ∈ Θ¯ ⊆ R3 . It is known that at the true parameter θ = θ◦ , equation (8) holds for all a. In what follows, a indexes deviations from the actual policy a◦ that Wal-Mart chose, ya is the incremental operating profit from doing a◦ rather than a, and xa θ is the incremental cost. The revealed preference that Wal-Mart chose a◦ implies equation (8) must hold at the true parameter θ◦ for all deviations a. Let {zak k = 1 2 K} be a set of K instruments for each a. Assume the instruments are nonnegative, zak ≥ 0. Hence, at the true parameter θ = θ◦ , zak ya ≥ zak xa θ
for all a and k
Suppose the M observations {(ya xa za1 za2 zaK ) a = 1 M} are drawn randomly from an underlying population, and that the population averages E[zak ya ] and E[zak xa ] are well defined. Assume xa and zak are directly observed, but there is measurement error on ya . In particular, we observe y˜a = ya + ηa where E[ηa |xa zak ] = 0. Taking expectations, we obtain a set of K moment inequalities that are satisfied at the true parameter θ◦ , that is, (9)
mk (θ) ≥ 0
for k ∈ {1 2 K}
for mk (θ) defined by mk (θ) ≡ E[zak y˜a ] − E[zak xa ]θ The identified set ΘI is the subset of points satisfying the K linear constraints in equation (9). Defining Q(θ) by (10)
Q(θ) =
K
2 min{0 mk (θ)}
k=1
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THOMAS J. HOLMES
the identified set can equivalently be written as the points θ ∈ ΘI solving 0 = min Q(θ) θ∈Θ¯
(As an aside, taking the square in equation (10) is common in the literature, but I also consider an alternative formulation that leaves out the square, summing the absolute value of any deviations. An attractive feature of this linear version is that it can be minimized through linear programming; see footnote 22 in Section 8.) ˜ ˜ k (θ) and Q(θ) be the sample analogs of mk (θ) and Q(θ): Let m (11)
˜ k (θ) ≡ m
M zak y˜a
M
a=1
˜ Q(θ) ≡
−
M zak x
a
a=1
M
θ
K
2 ˜ k (θ)} min{0 m
k=1
We can define an analog estimate of the identified set ΘI as the set of θ that solves (12)
˜ Θˆ I = arg min Q(θ) θ∈Θ¯
If the sample moments are consistent estimates of the population moments, then Θˆ I is a consistent estimate of the identified set ΘI . If there were no measurement error, ηa = 0, a preferable estimation strategy would be to use the information content in the M disaggregated inequalities in equation (8), rather than aggregate to the K moment inequalities and lose the individual information. Bajari, Benkard, and Levin (2007) considered just such an environment. There is an error term in the first stage of their procedure but not in the second stage where they exploit inequalities derived from choice behavior. Their method minimizes the analog of equation (10) applied to the M disaggregated inequalities in equation (8). Here, with measurement error in the second stage, a disaggregated approach like this runs into problems. Consider a simple example where the right-hand-side variable is just a constant and it is known that ya ≥ θ (where θ is a scalar, for the example). Suppose we observe y˜a = ya + ηa (i.e., there is measurement error on the left-hand-side variable). If we pick θ to minimize the disaggregated analog of equation (11), the solution is the set of θ that satisfies (13) θ | θ ≤ min{ya + ηa } a
That is, we require each individual inequality to hold. This works fine with no measurement error. If there is measurement error with full support, then
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asymptotically the estimated set goes to minus infinity. In large samples, significantly negative outlier draws of ηa pin down the estimate. In contrast, by aggregating to moment inequalities, the measurement error averages out in large samples. 6.2. Applying the Approach to Wal-Mart Looking again at Wal-Mart’s objective function in equation (2) and noting the log-linear form in equation (1) of how the fixed cost c varies with population density, we can readily see that Wal-Mart’s objective is linear in the parameters τ, ω1 , and ω2 , consistent with the structure above. Define ya by ya ≡ Π(a◦ ) − Π(a) for (14)
∞ g t−1 (ρt β) πjt (a) + Π(a) ≡ t=1
j∈BtWal (a)
f jt
π (a)
Super j∈Bt (a)
Thus, ya is the increment in the present value of operating profit from implementing the chosen policy a◦ instead of a deviation a. (Note that equation (14) includes periods into the distant future that I do not have information on. In calculating ya , these future terms are differenced out, because the deviations considered involve past behavior, not future.) Define the three-element vector xa by x1a ≡ Da x2a ≡ C1a x3a ≡ C2a The first element is the present value Da of the difference in distributiondistance miles between the two policies. This is calculated by substituting djte (a) for πjte (a) in equation (14), e ∈ {f g}. The second and third elements are the analogous summed present value differences of ln(Popden) and ln(Popden)2 . (These are the present values of the differences in the first and second terms of the fixed cost c, given in (1).) Since a◦ solves the problem in equation (2), at the true parameter θ◦ = (τ◦ ω◦1 ω◦2 ), ya ≥ xa θ must hold for each deviation a, just as in equation (8) above. There are two categories of error in the analysis. The first comes from the demand estimation in the first stage. The second arises from measurement
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error on the wages and land rents discussed in Section 4. Call this the secondstage error. To begin to account for these two sources of error, let Rejt (a) be the sales revenue under policy a at store j at time t for segment e ∈ {f g} at policy a, evaluated at the true demand parameter vector ψ◦ . Let Rˆ ejt (a) be the estimated value using ψˆ from the first stage. It is useful to initially isolate the second-stage error and then account for the first-stage error later. Evaluating at the true demand parameter ψ◦ , the observed operating profit at a particular segment e, store j, and time t equals (15)
Wage
π˜ jte (a) = (μ − ν Other )Rejt (a) − (Wagejt +εjt
)ν Labor Rejt (a)
− (Rentjt +εjtRent )ν Land Rejt (a) Wage
where εjt and εjtRent are the measurement errors on wages and the rents alluded to earlier. The present value of the measurement error, analogous to (14), equals ⎞ ⎛ Wage g (εjt ν Labor + εjtRent ν Land )Rjt (a) ∞ ⎟ ⎜ j∈BWal (a) t ⎟ ⎜ (16) (ρt β)t−1 ⎜ a ≡ ⎟ f Wage Labor Rent Land ⎠ ⎝ + (ε ν + ε ν )R (a) t=1 jt jt jt
Super
j∈Bt
(a)
Define the differenced measurement error associated with deviation a to be ηa ≡ a◦ − a Finally, let y˜a ≡ ya + ηa be the observed differenced operating profits at deviation a, evaluated at the true demand vector ψ◦ . Assume the underlying second-stage measurement erWage rors εjt and εjtRent are mean zero and independent of the other variables in the analysis. Then E[ηa |xa ] = 0, so that with the first-stage error ignored, the analysis maps directly into the linear moment inequality framework outlined above, for now putting aside the selection of instruments. So that the set estimate Θˆ I defined by equation (12) is a consistent estimate of ΘI , it is necessary that the differenced measurement error ηa entering in the moment equalities average out when the number of stores N is large. Assuming the underlying Wage store-level measurement errors εjt and εjtRent are independent across stores is sufficient but not necessary for this to be true.
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To take into account the first-stage error, let y˜ a be defined the same way as y˜a , but using the estimated demand parameter vector ψˆ instead of the true value ψ◦ . We can write it as (17)
y˜ a = y˜a + [ y˜ a − y˜a ]
The error term in brackets arises because of the measurement error in storelevel sales encountered in the demand estimation. It was assumed earlier that this store-level measurement error is drawn i.i.d. across stores. Hence, taking asymptotics with respect to the number of stores N, the estimate ψˆ is a consistent estimate of ψ◦ and y˜ a is a consistent estimate of y˜a . Then Θˆ I constructed by using y˜ a instead of y˜a is a consistent estimate of ΘI . It remains to describe the choice of deviations to consider and the selection of the instruments. Like Fox (2007) and Bajari and Fox (2009), I restrict attention to pairwise resequencing, that is, deviations a in which the opening dates of pairs of stores are reordered. For example, store number 1 actually opened in 1962 and number 2 opened in 1964. A pairwise resequencing would be to open store number 2 in 1962, store number 1 in 1964, and to leave everything else the same. I define groups of deviations that share characteristics. An indicator variable for membership in the group plays the role of an instrument, so taking means over inequalities within each group creates a moment inequality for each group. The groups are chosen to be informative about the parameters and thus narrow the size of the identified set ΘI . There are 12 different groups, which are formally defined in Table IX. Summary statistics for each group are reported in Table X. Each of the 12 deviation groups is in one of three broad classifications. The first broad classification is “Store density decreasing” deviations. To construct these deviations, I find instances where Wal-Mart at some relatively early time period (call it t) is adding another store (call it j) near where it already has a large concentration of stores, and there is some other store location j opened at a later period t > t that would have been far from Wal-Mart’s store network if it had opened at time t instead. In the deviation, Wal-Mart opens the farther-out store sooner (j at t) and the closer-in store later (j at t ), and this decreases store density in the earlier period. Analogously, the second broad classification consists of “Store density increasing” deviations that go the other way. The third broad classification, “Population density changing” deviations, holds store density roughly constant by flipping stores opened in the same state. For these, the stores involved in the flip come from different population density locations. Let χka be an indicator variable equal to 1 if deviation a is in group k defined in Table IX and equal to 0 otherwise. The group definitions depend only on store locations and opening dates, and these are all assumed to be measured without error, that is, there is no measurement error in χka . Moreover, the error
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THOMAS J. HOLMES TABLE IX DEFINITIONS OF DEVIATION GROUPSa
Deviation Category
Deviation Group
Description (Store j Flips with Store j )
1 2 3
Find the set of stores, S = {j tj ≥ tjstate + 10}. For each j ∈ S, find all j , where (i) tj ≥ tj + 3, (ii) j is in a different state than j, and (iii) tj ≤ tjstate + 4. Take all of these and further classify by group on the basis of Da as follows: −75 ≤ Da < 0 −150 ≤ Da < −75 Da < −150
4 5 6
Find the set of stores, S = {j tj ≤ tjstate + 5}. For each j ∈ S, find all j , where (i) tj ≥ tj + 3, (ii) j is in a different state than j, and + 10. Take all of these and further (iii) tj ≥ tjstate classify by group on the basis of Da as follows: 0 < Da ≤ 75 75 < Da ≤ 150 150 < Da
Store density decreasing
Store density increasing
Population density changing
7 8 9 10 11 12
Take pairs of stores (j j ) opening in the same state, where tj ≤ tj + 2. Classify based on Popdenj (in units of 1,000 people within 5-mile radius). Define density classes 1, 2, 3, and 4 by Popdenj < 15 15 ≤ Popdenj < 40, 40 ≤ Popdenj < 100, and 100 ≤ Popdenj . j in class 4, j in class 3 j in class 3, j in class 2 j in class 2, j in class 1 j in class 1, j in class 2 j in class 2, j in class 3 j in class 3, j in class 4
a Notes: The table uses the following notation: t is the opening date of store j , t state is the opening date of the j j
first store in the state where j is located, Da is the present value of the increment in distribution distance miles (in 1,000s) from doing the actual policy a◦ instead of deviating and doing a. In words, to construct group 1, take the set of all stores opening when there is at least one store in their state that is 10 years old or more. For each such store, find alternative stores that open 3 or more years later in different states, where Wal-Mart has been in the different state no more than 4 years when the alternative store opens. Openings for general merchandise stores and food stores are considered two different opening events. In cases where a supercenter opens from scratch rather than as a conversion of an existing Wal-Mart, there are two opening events. In all the pairs considered, a general merchandise opening is paired with another general merchandise opening, and a food opening with another food opening.
ηa is mean zero conditional on χka , given the independence assumption already Wage made about εjt and εjtRent . Hence, χka is a valid instrument. Let the set of basic instruments be defined by zak = Weighta ×χka
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DIFFUSION AND ECONOMIES OF DENSITY TABLE X SUMMARY STATISTICS OF DEVIATIONS BY DEVIATION GROUP Mean Values
Π˜ D Number of (Millions of (Thousands of C1 C2 Deviations 2005 Dollars) Miles) (log Popden) (log Popden2 )
Deviation Brief Description Group of Group
1 2 3
Store density decreasing −75 ≤ D < 0 −150 ≤ D < −75 D < −150
64,920 61,898 114,588
−2.7 −3.6 −4.7
−.4 −1.1 −3.0
−.6 −1.5 −3.4
−3.0 −9.0 −22.2
4 5 6
Store density increasing 0 < D ≤ 75 75 < D ≤ 150 150 < D
158,208 34,153 16,180
3.0 3.7 5.9
.3 1.0 2.1
−1.9 −3.6 −4.8
−17.2 −28.9 −37.7
7,048 10,435 14,399 12,053 14,208 14,877
1.2 3.7 5.3 −2.4 .6 2.5
.0 .0 −.1 .0 −.1 .0
3.2 3.4 3.5 −3.4 −3.9 −4.6
31.1 25.7 19.3 −19.3 −29.4 −44.9
522,967
−.2
−.6
−2.1
−15.6
7 8 9 10 11 12
Population density changing Class 4 to class 3 Class 3 to class 2 Class 2 to class 1 Class 1 to class 2 Class 2 to class 3 Class 3 to class 4
All
Weighted mean
for a weighting variable Weighta =
1 taFirst
(ρt β)t−1
t=1
where taFirst is the first period that deviation a is different from a◦ . This rescales things to the present value at the point when the deviation actually begins. Additional instruments are obtained by interacting the basic instruments with positive transformations of the xa . Define x+ia ≡ xia − xmin i , where xia is − the ith element of xa and xmin = min x . Analogously, x ≡ xmax − xia for a ia ia i i max xi = maxa xia . Level 1 interaction instruments are obtained by multiplying the x+ia and x−ia , of which there are six, by each of the 12 basic moments for a total of 72 = 6 × 12 level 1 interaction moments. Analogously, we can take the various second-order combinations, such as x+1a x+1a x+1a x+2a , and so on, and multiply them times the basic instruments to create level 2 interaction moments, of which there are 252 = 21 × 12. In the full set of all three types of moment inequalities, there are 336 = 12 + 72 + 252 restrictions. Before moving on to the baseline results, I make a comment about optimization error. The above discussion models the choice a◦ as the true solution
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to Wal-Mart’s problem (2). For any deviation a that has higher measured discounted profit, the discrepancy is attributed to a combination of first-stage estimation error and second-stage measurement error on wages and land rentals. In principle, one could incorporate optimization error on the part of Wal-Mart as a further source of discrepancy (and PPHI discussed how to do this), but I have not done that here. 7. BOUNDING DENSITY ECONOMIES: BASELINE RESULTS This section presents the baseline density economy estimates and provides a discussion of the results. The next section addresses issues of confidence intervals and robustness. The complete set of all deviations in the 12 groups defined above consists of 522,967 deviations. Table X presents summary statistics by deviation group. In particular, it reports means of Π˜ a (the y˜a variable) as well as the means of Da , C1a , and C2a (which make up the xa vector). The variables are all rescaled by Weighta before taking means, so present values are taken as of the point when the deviations begin. Consider the Store density decreasing deviations, groups 1–3. These deviations open farther-out stores sooner and closer-in stores later. Thus, distribution miles are fewer when the actual policy is chosen instead of deviations in these groups (i.e., Da < 0). We also see that by choosing the actual policy rather than these deviations, Wal-Mart is sacrificing operating profit. These losses average −$2.7, −$3.6, and −$4.7, in millions of 2005 dollars, for groups 1, 2, and 3. We can see evidence of a trade-off across groups 1, 2, and 3 as the absolute value of mean Π˜ a increases (the sacrifices in operating profit) and as the absolute value of mean Da increases (the savings in miles). For the sake of illustration, suppose we only consider the deviations in group 1. Also, temporarily zero out the ω1 and ω2 coefficients on population density. Then using the information in Table X, the moment inequality for group 1 reduces to E[Π˜ 1 ] − τE[D1 ] = −27 + τ4 ≥ 0 or τ ≥ $675, where the units are in thousands of 2005 dollars per mile year. Freeing up ω1 and ω2 loosens the constraint. For example, suppose we plug in ω1 = 428 and ω2 = −50 (this choice is explained shortly). Then the moment inequality from group 1 is instead E[m1 ] = E[Π˜ 1 ] − τE[D1 ] − ω1 E[C1 ] − ω2 E[C2 ] = −27 + τ4 − 428(−6) − (−50)(−30) = −14 + τ4 ≥ 0
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DIFFUSION AND ECONOMIES OF DENSITY TABLE XI BASELINE ESTIMATED BOUNDS ON DISTRIBUTION COST τ a Specification 1 Basic Moments (12 Inequalities)
Specification 2 Basic and Level 1 (84 Inequalities)
Specification 3 Basic and Levels 1, 2 (336 Inequalities)
Lower
Upper
Lower
Upper
Lower
Upper
Point estimate
3.33
4.92
3.41
4.35
3.50
3.67
Confidence thresholds With stage 1 error correction PPHI inner (95%) PPHI outer (95%)
2.69 2.69
6.37 6.41
2.89 2.86
5.40 5.45
3.01 2.97
4.72 5.04
No stage 1 correction PPHI inner (95%) PPHI outer (95%)
2.84 2.84
5.74 5.77
2.94 2.93
5.11 5.13
3.00 2.99
4.62 4.97
a Units are in thousands of 2005 dollars per mile year; number of deviations M = 522967; number of store locations N = 3176.
or τ ≥ $333.17 This is substantially looser than when ω1 = ω2 = 0 is imposed. Now turn to the general problem of bounding τ. Let τ and τ¯ be the lower and upper bounds of τ in the identified set ΘI . When a solution satisfying all of the sample moment inequalities exists, as is the case here, the estimates of these bounds are obtained through linear programs that impose the moment inequalities and the a priori restrictions ω1 ≥ 0 and ω2 ≤ 0. Table XI presents the results. The first set of estimates imposes only the 12 basic moment inequalities. Table X contains all the information needed to do this. The estimated lower bound is in fact τˆ = $333, and this is obtained when ω1 = 428 and ω2 = −50, the values used above. In the solution to the linear programming problem, moment 1 is binding, as are moments 9 and 12, and the remaining inequalities have slack. The estimated upper bound is $4.92. By adding in interaction moments, additional restrictions are imposed, narrowing the identified set. The additional moments created when the basic moments [Ey]k − [Ex θ]k ≥ 0 are multiplied by positive transformations of the x are analogous to the familiar moment conditions for ordinary least squares (OLS), (y − xθ) x = 0. With both level 1 and level 2 interactions included, the estimate of the identified set is narrowed to the extremely tight range of $3.50–$3.67. This case with the full set of interactions will serve as my baseline estimate. 17
Because of rounding, there is a slight discrepancy in these two inequalities.
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7.1. Discussion of Estimates The parameter τ represents the cost savings (in thousands of dollars) when a store is closer to its distribution center by 1 mile over the course of a year. At the baseline estimate of τ in a tight range at $350, if all 5,000 Wal-Mart stores (here, supercenters are counted as two stores) were each 100 miles farther from their distribution centers, Wal-Mart’s costs would increase by almost $2 billion per year. To get a sense of the direct cost of trucking, I have talked with industry executives and have been quoted marginal cost estimates of $1.20 per truck mile for “in-house” provision. If a store is 100 miles from the distribution center (200 miles round trip) and if there is a delivery every day throughout the year, then the trucking cost is $1.20 × 200 × 365 = $85,400, or $.85 in thousands of dollars per mile year. Thus, the baseline estimated cost savings in a tight range around $3.50 is approximately four times as large as the savings in trucking costs alone.18 The difference includes the valuations Wal-Mart places on the ability to quickly respond to demand shocks. My industry source on trucking costs emphasized the value of quick turnarounds as an important plus factor beyond savings in trucking costs. A second perspective on the τ parameter can be obtained by looking at Wal-Mart’s choice of when to open a distribution center (DC). An in-depth analysis of this issue is beyond the scope of this paper, but some exploratory calculations are useful. Recall that Wal-Mart’s problem specified in (2) held DC opening dates fixed and considered deviations in store opening dates. Now hold store opening dates fixed and consider deviations in DC openings. Deopen note tk to be the year DC k opens. Define Dinc kt to be DC k’s incremental contribution in year t to reduction in store distribution miles. This is how much higher total store distribution miles would be in year t if distribution center k were not open in that year. Assume there is a fixed cost φk of operating distribution center k in each year. Optimizing behavior implies that the following open inequalities must hold for the opening year t = tk : (18)
φk ≤ τDinc kt φk ≥ τDinc kt−1
The first inequality says that the fixed cost of operating the distribution center open in year tk must be less than the distribution cost savings from it being open.19 Otherwise, Wal-Mart can increase profit by delaying the opening by a year. 18
My distances are calculated “as the crow flies,” whereas the industry trucking estimate is based on the highway distance, which is longer. The discrepancy is small relative to the magnitudes discussed here. 19 There is also a marginal cost involved with distribution, but assume this is the same across distribution centers, so shifting volume across distribution centers does not affect marginal cost.
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DIFFUSION AND ECONOMIES OF DENSITY TABLE XII
MEAN INCREMENTAL MILES SAVED AND STORES SERVED FOR DISTRIBUTION CENTERS ACROSS ALTERNATIVE OPENING DATES INCLUDING ACTUAL 1 Year Prior to Actual
Actual Year Opened
1 Year After Actual
2 Years After Actual
All distribution centers (N = 78) Mean incremental miles saved Mean stores served
4.4 23.6
5.8 52.1
6.7 58.4
7.1 62.9
By type of DC Regional distribution centers (N = 43) Mean incremental miles saved Mean stores served
6.1 37.1
7.7 68.6
8.7 76.1
8.9 79.0
2.3 6.9
3.4 31.8
4.3 36.5
5.0 43.0
Food distribution centers (N = 35) Mean incremental miles saved Mean stores served
The second inequality states that the fixed cost exceeds the savings of opening it the year before (otherwise it would have been opened a year earlier). Now if inc Dinc kt changes gradually over time, then equation (18) implies φk ≈ τDkt holds approximately at the date of opening. (Think of this as a first-order condition.) Table XII reports the mean values of the Dinc kt statistic across distribution centers. The statistic is reported for the year the distribution center opens, as well as the year before opening and the 2 years after opening. For example, to calculate this statistic in the year before opening, the given DC is opened 1 year early, everything else the same, and the incremental reduction in store miles is determined. I also report how the mean number of stores served varies when the DC opening date is moved up or pushed back. At opening, the mean incremental reduction in store miles of a distribution center is 5.8 thousand miles and the DC serves 52.1 stores. The later the DC opens, the higher the incremental reduction in store miles and the more stores served. This happens because more stores are being built around it. If we knew something about the fixed cost, then the condition φk ≈ τDinc kt provides an alternative means of inferring τ. A very rough calculation suggests a ballpark fixed cost of $18 million per year.20 Since the mean value of Dinc kt when DCs open equals 5.8 thousand miles, we back out an estimate of τ equal to τˆ =
φ $18 ($million per year) = $310 = inc 58 (thousands of miles) mean Dkt
20 Distribution centers are on the order of 1 million square feet. Annual rental rates including maintenance and taxes are on the order of $6 per square foot, so $6 million a year is a rough approximation for the rent of such a facility. A typical Wal-Mart DC has a payroll of $36 million. If a third of labor is fixed cost, then we have a total fixed cost of $18 = $6 + $12 million.
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This estimate is close to the baseline estimate from above on the order of $3.50. It is encouraging that these two approaches—coming from two very different angles—provide similar results. 8. BOUNDING DENSITY ECONOMIES: CONFIDENCE INTERVALS AND ROBUSTNESS
To apply the PPHI method of inference to this application, two issues need to be confronted. First, there is correlation in the error terms across deviations when two deviations involve the same store. Second, first-stage demand estimation error needs to be taken into account. Section 8.1 explains how the two issues are addressed. Section 8.2 discusses the confidence intervals for the baseline estimates. Section 8.3 examines the robustness of the baseline results to alternative assumptions. There are a variety of different alternative approaches for inference with moment inequalities in addition to PPHI, including Imbens and Manski (2004), Chernozhukov, Hong, and Tamer (2007), Romano and Shaikh (2008), and Andrews and Soares (2010). The main focus of this paper is on the sample analog estimates of the identified set, as presented in Section 7. PPHI complemented this focus, as they simulated the construction of sample analogs. (See also Luttmer (1999).) In the recent literature, a distinction is made between constructing a confidence interval for the identified set versus a confidence interval for the true parameter. The work below is in the first category; the confidence intervals are for extreme points of the identified set. 8.1. Procedure To explain the PPHI method, it is useful to introduce additional notation. Let w˜ a be a vector that stacks the moment inequality variables for deviation a into a column vector that is (K + 3K) × 1. The first K elements contain the zak y˜a and the remaining 3K elements contain the zak xa in vectorized form, where again K is the number of moment inequalities. PPHI assumed the w˜ a are drawn independently and identically. (This will not be true here, but ignore this for now to explain what they do.) Let Σ be the variance–covariance matrix of the distribution of w˜ a . The sample mean of w˜ a over the M deviations equals (19)
w¯ =
M w˜ a M a=1
and it has a variance–covariance matrix equal to Σ/M. Take the sample variance–covariance matrix Σˆ as a consistent estimate of Σ. PPHI proposed a way to simulate inner and outer confidence intervals that asymptotically bracket the true confidence interval for τ. Begin with the inner
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approximation first. Consider a set of simulations of the moment inequality exercise indexed by s from s = 1 to S. For each simulation s, draw a random column vector s with K + 3K elements from the normal distribution with ˆ mean zero and variance–covariance Σ/M. Then define ws = w¯ + s Put ws into moment inequality form, letting the first K elements be w1s and reshaping the remaining elements into a K × 3 matrix w2s . Analogous to equation (9), we then have K moment inequalities for each simulation s, (20)
w1s − w2s θ ≥ 0
˜ Define Q˜ inners (θ) to be the analog of Q(θ) in equation (11). Define τinners by τˆ
inners
≡ inf{τ (τ ω1 ω2 ) ∈ arg min Q˜ inners (θ)}
which produces the analog estimate of τ in the simulation. The inner α percent confidence interval for τ is obtained by taking the α2 and 1 − α2 percentiles of inners the distribution of the simulated estimates, {τˆ s = 1 S}. The construction of the outer approximation is the same, with one difference. For each simulation s, reformulate the set of moment inequalities in equation (20) by adding a nonnegative vector ζ: w1s − w2s θ + ζ ≥ 0 for ζ defined by ˆ ζ ≡ max{0 w1 − w2 θ} The vector ζ is the “slack” in the nonbinding moment inequalities evaluated at the actual data (as opposed to the simulated data) and at the estimate in ˆ Adding the slack term the identified set θˆ ∈ Θˆ I containing the lower bound τ. makes it easier for any given point to satisfy the inequalities. PPHI showed outers is stochastically dominated by the that, asymptotically, the distribution of τˆ inners distribution of τˆ and that the distribution of τˆ lies in between. One thing to note is that Andrews and Guggenberger (2009) and Andrews and Soares (2010) forcefully advocated a criterion of uniform consistency for confidence intervals, and PPHI did not address that. Two issues arise in applying the method here. First, PPHI’s assumption of independence does not hold here. There are M = 523000 deviations used in the analysis, but these are derived from only N = 3176 different store locaˆ tions. If Σ/M is used to estimate the variance of w¯ as in PPHI, the amount of averaging that is taking place is exaggerated, since the measurement error is at
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the store level. The issue is addressed by the following subsampling procedure (see Politis, Romano, and Wolf (1999)). Draw a store location subsample of size b N from the N store locations (here b = N3 ). The deviation subsample consists of those deviations where both stores being flipped are in the store location subsample. Next calculate mean w¯ b in the deviation subsample. Repeat, subsampling with replacement, and use the different subsamples to estimate var cov(w¯ b ). Next use var cov(w¯ N ) =
b var cov(w¯ b ) N
to rescale the estimate of var cov(w¯ b ) into an estimate of var cov(w¯ N ) (see Apˆ pendix B1 for a discussion). Use this rather than Σ/M to draw normal random s variables in the PPHI procedure. The second issue is variance from the first-stage demand estimation. Using equation (17), the left-hand-side term of the kth moment inequality can be written as (21)
w¯ k1 =
M zak y˜a a=1
M
+
M zak [ y˜ a − y˜a ] M a=1
The second term is, in general, not zero because the first-stage estimate ψˆ of demand is different from the true parameter ψ◦ . Thus, the second term is an additional source of variance. To take this into account, I use a simulation procedure that is valid based on the following asymptotic argument. Appendix B2 shows, first, that for large N, the covariance between the two terms goes to zero relative to the variances of the two terms. Hence, the covariance can be ignored in large samples. It shows, second, that for large N, the variance of the second term is approximately what it would be with only first-stage variation in ψˆ and no second-stage variation. In the simulation procedure, for each subsample s, I take the subsample mean as above and plug this in for the first term. To get a draw for the second term, let Σˆ ψ be the estimated variance–covariance matrix from the first-stage demand estimation. For each subsample s (again of size b), take a normal draw ψsb centered at ψˆ with variance Nb Σˆ ψ and use this to construct a y˜ sa − y˜ a ] to plug in y˜ sa . Then take an estimate of the mean of zak [ s s ˆ Finally, for the second term of equation (21), where y˜ a uses ψ. y˜ a uses ψb and use the different subsamples s to estimate var cov(w¯ b ) as above. Splitting equation (21) into two pieces this way makes it possible to economize on deviations used to construct the second term.21 (These deviations are expensive because ψ varies.) 21 To estimate the second term, a fixed set of 100 deviations is used for each of the 12 basic groups, and all variation is driven by differences in ψs . It would have been infeasible to calculate
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The procedure can be summarized as follows. The basic PPHI approach simulates the distribution of extreme points of the identified set. It does this by calculating sample analogs of the identified set for simulated moments. It constructs an estimate of the variance–covariance matrix of the asymptotic normal distribution of the moments using the variance–covariance of the raw data. The procedure used here changes this last step in two respects. First, to take into account that two deviations involving the same store are not independent, a subsampling approach is used. In particular, subsamples of stores are drawn and the moment inequalities are constructed for each subsample. Second, to take into account first-stage estimation error of the demand model, draws are taken from the asymptotic distribution of the demand parameters. The simulated errors from each such draw are added to the means from a subsample draw just discussed, with adjustments made for subsample size. The distribution of this composite of two simulation draws is used to estimate the variance–covariance matrix needed for the PPHI procedure. 8.2. Estimates of Confidence Thresholds Table XI reports confidencethresholds for the baseline estimates. One set of estimates takes into account first-stage demand error and the second set does not.22 The left-side 95 percent confidence thresholds of the lower bounds are reported (the 2.5 percentile points) and the right-side thresholds of the upper bound are reported (i.e., the 97.5 percentile points). The first thing to note is that the inner and outer PPHI thresholds (which asymptotically bracket the true thresholds) are similar. Henceforth, I focus on the outer threshold, the conservative choice. The second thing to note is that, like the point estimates, the PPHI thresholds narrow as we move across the table and add more constraints from interactions. Third, the point estimates of the bounds are relatively precise. For example, take Specification 3 where the point estimates of the lower and upper bounds are $3.50 and $3.67. The corresponding confidence thresholds are $2.97 and $5.04.23 Fourth, taking into account the firststage demand error makes a difference, particularly in Specification 1. 8.3. Robustness Results I turn now to the robustness of the results to alternative assumptions and samples. The results are reported in Table XIII. All specifications in the table this term for all the different deviations associated with subsample s, so the asymptotic argument making it valid to vary only ψs is very useful. 22 For the second set, the simulated draws over the demand parameter ψˆ that add to the variance in the second term of equation (21) are shut down. 23 I have also experimented by seeing how the results change if I leave the square out of the criterion function in equation (10). The results are very similar and are not reported.
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THOMAS J. HOLMES TABLE XIII ROBUSTNESS ANALYSIS OF ESTIMATED BOUNDS ON DISTRIBUTION COST τ a Lower Bound
Upper Bound
Number of Sample Deviations
Point Estimate
PPHI Outer Thresh.
Point Estimate
PPHI Outer Thresh.
Baseline
522,967
3.50
(2.97)
3.67
(5.04)
Split sample by opening date 1962–1989 1990–2005
331,847 191,120
3.51 3.43
(2.92) (1.85)
4.81 3.43
(7.19) (3.90)
Vary operating profit parameters from first stage: net margin (baseline = .17) .15 .19
522,967 522,967
2.62 4.36
(2.18) (3.78)
3.05 4.36
(3.90) (6.22)
Labor cost factor 20% higher 20% lower
522,967 522,967
4.12 2.80
(3.64) (2.29)
4.12 3.38
(5.66) (4.38)
Rent cost factor 20% higher 20% lower
522,967 522,967
3.60 3.41
(3.04) (2.89)
3.79 3.55
(5.12) (5.03)
Exclude any deviations with change in supercenters
366,412
3.78
(2.66)
3.78
(5.61)
Include only deviations with change in supercenters
156,555
3.79
(1.52)
3.79
(4.72)
59,693 96,862
.70 4.38
(.00) (2.48)
.70 4.38
(3.48) (5.39)
Specification
By supercenter opening 1988–1997 1998–2005
a Units are in dollars per mile year; all specifications include basic, level 1, and level 2 interactions (336 moment inequalities).
impose the full set of moment conditions (basic plus level 1 plus level 2). The top row is the baseline from Table XI. The first exercise partitions the perturbations based on the opening date of the first store in the deviation pair. The first sample contains the deviations where the first store of the pair opens before 1990, the second sample where it opens 1990 and later. One thing to note is that for the later time period, τ is point identified. That is, there exists no θ vector that simultaneously solves the full set of 336 empirical moment inequalities, and in such a case, the point estimate of the lower and upper bounds minimizes the squared deviations in (11). The main take-away quantitative point here is that the estimate for the first time period τ ∈ [351 481] is very similar to the estimate for the second time period τ = 343 and to the baseline estimate τ ∈ [350 367].
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As already explained, the confidence intervals reported here incorporate error from the first-stage estimate of demand parameters, but there are other parameters estimated in the first stage that are not incorporated into the confidence intervals. A particularly important parameter is the net margin μ − ν other that equals variable profit per dollar of sale, excluding labor costs and land rents. It was argued that a plausible estimate for this is μˆ − νˆ other = 17, and this is used in the baseline. Table XIII shows what happens when instead the estimate is lowered to .15 or raised to .19. Significant changes in this key parameter do lead to significant changes in the estimates. For example, when the parameter is lowered from 17 to 15, the estimated lower bound on τ falls from $3.50 to $2.62. Nevertheless, even this low-end estimate for τ is quite large relative to the $.85 industry estimate discussed above that would be based purely on trucking considerations. Analogous to changes in μ − ν other , Table XIII also reports the results with alternative labor cost and rental cost assumptions (increasing or decreasing each by a factor of 20 percent). The results for alternative labor costs are similar to the results for alternative μ − ν other . The results for alternative rental costs are not much different from the baseline, because rents are a relatively small share of costs. An issue that has been ignored up to this point is how to treat the distribution of groceries during the early phase of Wal-Mart’s entry into the grocery business. Wal-Mart initially outsourced grocery wholesaling and then later built up its own grocery distribution system. In particular, it opened its first supercenter selling groceries in 1988, but did not open its first food distribution center until 1993, 5 years later, and likely did not become fully integrated until several years beyond that. For the baseline estimates, for lack of a better alternative, I assume that food distribution costs are constant before 1993 (i.e., invariant to store locations) and as of 1993 Wal-Mart is fully integrated with distances calculated according to Wal-Mart’s own internal network. I expect there is some measurement error for grocery distribution distances during the early phase of Wal-Mart’s supercenter business. This is a concern because my procedure yields inconsistent estimates of the identified set ΘI when there is measurement error in the x variables. To examine the robustness of the results to this issue, I first exclude all deviations that impact grocery distribution (i.e., any deviations involving supercenter opening dates). This eliminates 157,000 deviations. Estimating the model on the remaining 366,000 deviations, this “supercenter-excluded” estimate is τ = 378, consistent with the baseline. Next I consider those deviations that do include changes in supercenter opening dates. The result of τ = 379 is also similar to the baseline. Differences emerge when this subsample is broken up by opening date. In the later period when Wal-Mart is completely integrated into grocery wholesaling, the results are similar to the baseline, but for the early period when Wal-Mart was building its network, the point estimate is only τ = 70. The measurement error during this period noted above is one potential explanation for this discrepancy. Also, the sample is small and the
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PPHI confidence bounds are relatively far apart—0 at the bottom and 3.48 at the top. 9. CONCLUDING REMARKS This paper examines the dynamic store location problem of Wal-Mart. Using the moment inequality approach, the paper is able to bound a technology parameter relating to the benefits Wal-Mart obtains when stores are close to distribution centers. The paper illustrates the power of this type of approach in getting a sensible analysis out of what would otherwise be complex and likely intractable. Wal-Mart has attracted much attention, and various interest groups have attempted to slow its growth, for example, by trying to get local governments to use zoning restrictions to block entry of stores. These kinds of policies limit store density. The analysis here is not at the stage where it is possible to run a policy experiment to evaluate the welfare effects of limiting Wal-Mart’s growth. Among other things, that would require uncovering how such limits would impact Wal-Mart’s DC network, and (except briefly in Section 7) this has been held fixed in the analysis. Nevertheless, the estimates of this paper suggest any policy that would substantially constrain store density would result in significant cost increases. Although the analysis is rich in many dimensions—notably in its fine level of geographic detail and in the way it incorporates numerous data objects—it has limitations. One is that all economies of density are channeled through the benefits of stores being close to distribution centers. Benefits can potentially emerge through other channels, including management (it is easier for upperlevel management to oversee a given number of stores when the stores are closer together) and marketing (satisfied Wal-Mart customers might tell their friends and relatives on the other side of town about Wal-Mart—this benefits Wal-Mart only if it has a store on the other side of town).24 A caveat, then, is that my estimate of τ may be picking up some economies of density from management and marketing. I have chosen to focus on distribution both because (i) I can measure it (i.e., the locations of distribution centers), but cannot measure management and marketing activities, and because (ii) my priors tell me distribution is very important for Wal-Mart. My findings in Section 7 regarding DC opening dates that corroborate my baseline findings are particularly helpful here. DC openings should be unrelated to management and marketing sources of economies of density. Although the analysis allows for several varieties of measurement error, it simplifies by leaving out a structural error, that is, location-specific factors that 24 Neumark, Zhang, and Ciccarella (2008) quoted Sam Walton on the importance of the marketing benefit.
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influence Wal-Mart’s behavior but are not in the data. For example, an alternative explanation to density economies for the pattern of Wal-Mart’s rollout is that there is simply something good about the Wal-Mart model in Bentonville, Arkansas, that deteriorates away from this point for whatever reason. This is not plausible, but an alternative possibility of unobserved heterogeneity across store locations within markets is plausible. In such a case, the diminishing returns from adding additional stores to the same market exhibited in Table VIII may be understated. (Locations picked last in a market may be worse in unobservable ways.) In this way, my results potentially underestimate density economies. The analysis does not take explicit account of the location of competitors. If one region is empty of competition, while other regions are crowded with competitors, a chain might concentrate its stores in the empty region first and the crowded regions later. This is not a plausible explanation for the qualitative pattern of Wal-Mart’s rollout, of starting in the center and going to the outside later. As of 1977, when Wal-Mart was still a regional operation with only 181 stores, the national leader K-Mart had over 1,000 stores distributed across all of the contiguous states. Although K-Mart’s stores did not uniformly follow population, it is a good first approximation, and K-Mart continued its relatively uniform coverage as it doubled its store count over the next 5 years.25 There was no void in the discount industry in the middle of the United States. Even if the location of rivals is not the primary driver of the overall qualitative pattern, keeping track of competitor locations matters for the quantitative analysis, as it impacts the relative profitability of different sites. The analysis accounts for the presence of rivals only implicitly, allowing the outside alternative to get better in denser areas in a reduced-form way. Following the logic of Bresnahan and Reiss (1991), there tend to be more rivals in larger markets. To the extent the reduced form does not fully capture differences in rival presence across store locations, there is unobserved heterogeneity and the comments just made above apply. Finally, the analysis ignores the issue of preemption. This is a particularly interesting area for future research, as large density economies suggest possibilities for preemption might extend beyond the store level to the regional level. APPENDIX A: DATA The selected data and programs used in the paper are posted as Supplemental Material (Holmes (2011)) and also at my website. In particular, data on facility locations and opening dates are posted there. The store-level sales and employment data used in this paper can be obtained from Trade Dimensions. 25
A regression of K-Mart store counts and population at the state level (in logs and weighted by log population) has a slope of 9 and R2 = 86 for both 1977 and 1983. This is a period over which the total store count increased from 1,089 to 1,941.
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Facility Locations and Opening Dates The data for Wal-Mart stores were constructed as follows. In November 2005, Wal-Mart made public a file which listed for each Wal-Mart store the address, store number, store type (supercenter or regular store), and opening date. These data were combined with additional information posted at Wal-Mart’s website about openings through January 31, 2006 (the end of fiscal year 2006). The opening date mentioned above is the date of the original store opening, not the date of any later conversion to a supercenter. To get the date of supercenter conversions, I used two pieces of information. From announcements at Wal-Mart’s website, I determined the dates of conversions taking place in 2001 and after. To get the dates of conversions taking place before 2001, I used data collected by Basker (2005) based on published directories. The data on Wal-Mart’s food distribution centers (FDCs) are based on reports that Wal-Mart is required to file with the Environmental Protection Agency (EPA) as part of a risk management plan.26 (The freezers at FDCs use chemicals that are potentially hazardous.) Through these reports, all FDCs are identified. The opening dates of most of the FDCs were obtained from the reports. Remaining opening dates were obtained through a search of news sources. Data on Wal-Mart general distribution centers (GDCs) that handle general merchandise were cobbled together from various sources including Wal-Mart’s annual reports and other direct Wal-Mart sources, as well as Mattera and Purinton (2004) and various web and news sources. Great care was taken to distinguish GDCs from other kinds of Wal-Mart facilities such as import centers and specialty distribution centers such as facilities handling Internet purchases. The longitude and latitude of each facility were obtained from commercial sources and manual methods. In the analysis, I aggregate time to the year level, where the year begins February 1 and ends January 31 to follow the Wal-Mart fiscal year. January is a big month for new store openings (it is the modal month), and February and March are the main months for distribution center openings. New January-opened stores soon obtain distribution services from new February-opened DCs. To be conservative in not overstating the number of distribution store miles, I assume that the flow of services at a DC begins the year prior to the opening year. Put differently, in my analysis I shift down the opening year of each DC by 1 year. Wage Data The wages are the average retail wage in the county containing the store (payroll divided by March employment). The source is County Business Patterns, U.S. Bureau of the Census (1977, 1980, 1985, 1990, 1995, 1997, 1998, 26
The EPA data are distributed by the Right-to-Know Network at http://www.rtknet.org/.
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2000, 2002, 2004) with interpolation in intervening years and years with missing values. For 2000 and beyond, the NAICS definition of the retail sector does not include eating and drinking establishments. For 1997 and earlier, the Standard Industrial Classification (SIC) code definition of retail does include eating and drinking, and these are subtracted out for these years. Property Value Data For each store location and each census year, I created an index of residential property value as follows. I identified the block groups within a 2-mile radius of each store and called this the store’s neighborhood. Total property value in the neighborhood was calculated as the aggregate value of owner-occupied property plus 100 times monthly gross rents of renter-occupied property. This was divided by the number of acres in a circle with radius of 2 miles, and the consumer price index was used to convert it into 2005 dollars. County property tax records were obtained from the Internet for 46 WalMart locations in Minnesota and Iowa. (All stores in these states were searched, but only for these locations could the records be obtained.) Define the land value–sales ratio to be land value from the tax records as a percentage of the (fitted) value of 2005 sales for each store. For these 46 stores, the correlation of this value with the 2000 property value index is .71. I regressed the land value–sales ratio on the 2000 property value index without a constant term and obtained a slope of .036 (standard error of .003). The regression line was used to obtain fitted values of the land value–sales ratio for all Wal-Mart stores. APPENDIX B1: VARIANCE–COVARIANCE OF THE SUBSAMPLE I will show that var cov(w¯ N ) decreases at rate N1 (where N is the number of stores) rather than rate M1 (where M is the number of deviations). To make the point, for simplicity consider the following simple case. Suppose there are only two periods, t = 1 2 and that Wal-Mart has chosen to open N/2 stores in period 1, with store numbers A1 = {1 2 N2 }, and the remaining N/2 stores in period 2, with store numbers A2 = {N/2 + 1 N}. Consider all deviations where Wal-Mart instead opens store k ∈ A2 in period 1 and delays the opening of a store j ∈ A1 until period 2. There are M = ( N2 )2 deviations, and let a = (j k) index them. ˜ that It is sufficient to look at the component of w¯ N that is the mean of the y, is, w¯ 1 =
M M y˜a ya + ηa = M M a=1 a=1
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where ηa is the difference in store-level measurement errors, ηjk = εj − εk We can write the measurement error component of the above as N/2 N M ηa a=1
M
=
[εj − εk ]
j=1 k=N/2+1
N 2
2
It is straightforward to calculate that the variance is M ηa 1 = 4σε2 var M N a=1 APPENDIX B2: VARIANCE–COVARIANCE RELATED TO THE FIRST STAGE I show here that (i) the variance of each of the two terms of equation (21) goes to zero at rate 1/N, while the covariance goes to zero at a faster rate, and (ii) for large N, the variance of the second term is approximately equal to what the variance would be with just first-stage error. To simplify the exposition, assume for each deviation a we can write y˜a (ψ) as y˜a (ψ) = Ra1 (ψ)(1 + εa1 ) − Ra2 (ψ)(1 + εa2 ) where Ra1 (ψ) is the revenue of the store whose opening is being delayed due to deviation a and Ra2 (ψ) is the revenue of the store being opened early. This is a simplification of equation (15), but it is sufficient. For now, assume there are only M = N/2 deviations (where again N is the number of stores) and that each store only appears in one deviation (I address the general case below). The εak are all independent classical measurement errors, across a and k ∈ {1 2}, that combine the measurement error of the wages and rents of the given store into one place. For each (a k) there is a corresponding store j, and I make the substitution shortly. Define w by (22)
w=
N/2 y˜a (ψ◦ ) a=1
N/2
+
N/2 ˆ − y˜a (ψ◦ ) y˜a (ψ) a=1
N/2
I need to show that for large N, the covariance of these two terms goes to zero relative to their variances. Since y˜a (ψ◦ ) is nonrandom, the covariance of the
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two terms can be expanded to E
Ra1 (ψ◦ )εa1 − Ra2 (ψ◦ )εa2 N/2
×
Ra1 (ψ)ε ˆ a1 − Ra2 (ψ)ε ˆ a2 − Ra1 (ψ◦ )εa1 − Ra2 (ψ◦ )εa2
+E ×
N/2 Ra1 (ψ◦ )εa1 − Ra2 (ψ◦ )εa2 N/2
Ra1 (ψ) ˆ − Ra2 (ψ) ˆ − E[Ra1 (ψ) ˆ − Ra2 (ψ)] ˆ
N/2 ◦ Ra1 (ψ )εa1 − Ra2 (ψ◦ )εa2 −E N/2
×
E[Ra1 (ψ)ε ˆ a1 − Ra2 (ψ)ε ˆ a2 − Ra1 (ψ◦ )εa1 − Ra2 (ψ◦ )εa2 ] N/2
The independence of the εa1 and εa2 (from stage 2) from the measurement error in store sales (from stage 1) implies that the second and third terms of this expanded expression are zero. The second term is zero, because independence implies ˆ − Ra2 (ψ)] ˆ
Ra1 (ψ◦ )εa1 − Ra2 (ψ◦ )εa2 [Ra1 (ψ) = 0 cov N/2 N/2 the third term is zero, because εa1 and εa2 are mean zero. Hence, we can write the covariance between the two terms of w in (22) as E
Ra1 (ψ◦ )εa1 − Ra2 (ψ◦ )εa2 ×
N/2
ˆ a1 − Ra2 (ψ)ε ˆ a2 ] [Ra1 (ψ)ε ◦
◦
− [Ra1 (ψ )εa1 − Ra2 (ψ )εa2 ] /(N/2) =E
N/2
2 2 ˆ a1 (ψ◦ )εa1 ˆ a2 (ψ◦ )εa2 + Ra2 (ψ)R ] [Ra1 (ψ)R
a=1
◦ 2 2 a1
− [Ra1 (ψ ) ε
− Ra2 (ψ )ε ] /(N 2 /4) ◦
2 a2
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=
1 E N
N ˆ − Rj (ψ◦ )]Rj (ψ◦ )4εj2 [Rj (ψ) j=1
N
where I have substituted j for (a k) in the last step. The estimate ψˆ is a consistent estimate of ψ◦ , which implies the covariance above goes to zero at a faster rate than 1/N. It is immediate that the first term of (22) is of order 1/N, so now turn to the variance of the second term, N/2 y˜a (ψ) ˆ − y˜a (ψ◦ ) var N/2 a=1 N/2 Ra1 (ψ) ˆ − Ra2 (ψ) ˆ = var N/2 a=1 ˆ − Ra2 (ψ◦ )]εa1 − [Ra1 (ψ) ˆ − Ra2 (ψ◦ )]εa2 [Ra1 (ψ) + N/2 ˆ − The variable εj εk is mean zero and independent of the variable [Rj (ψ) ˆ − Rk (ψ◦ )], and the variable εj (which depends on measurement Rj (ψ◦ )][Rk (ψ) error in wages and rents) is independent of ψˆ (which depends on measurement error of store-level sales). Together these imply that the variance of the second term of (22) equals N N/2 ˆ − Rj (ψ◦ )]2 εj2 Ra1 (ψ) [Rj (ψ) ˆ − Ra2 (ψ) ˆ 1 + E var N/2 N N/4 a=1 j=1 The first term, which contains the variance just from the first stage, is of order 1/N. The second term goes to zero faster than 1/N. Now return to the issue that an individual store may appear in multiple deviations. The same argument holds, as all of the terms involving a given store can be grouped together, and this will show up in the final terms as a scaling coefficient for store j. REFERENCES ANDREWS, D. W. K., AND P. GUGGENBERGER (2009): “Validity of Subsampling and ‘Plug-In Asymptotic’ Inference for Parameters Defined by Moment Inequalities,” Econometric Theory, 25, 669–709. [289] ANDREWS, D. W. K., AND G. SOARES (2010): “Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection,” Econometrica, 78, 119–157. [288,289]
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NEUMARK, D., J. ZHANG, AND S. M. CICCARELLA, JR. (2008): “The Effects of Wal-Mart on Local Labor Markets,” Journal of Urban Economics, 63, 405–430. [258,294] PAKES, A. J. PORTER, K. HO, AND J. ISHII (2006): “Moment Inequalities and Their Application,” Working Paper, Harvard University. [257,277] POLITIS, D. N., J. P. ROMANO, AND M. WOLF (1999): Subsampling. New York: Springer-Verlag. [290] ROBERTS, M. J. (1986): “Economies of Density and Size in the Production and Delivery of Electric Power,” Land Economics, 62, 378–387. [257] ROMANO, J. P., AND A. M. SHAIKH (2008): “Inference for Identifiable Parameters in Partially Identified Econometric Models,” Journal of Statistical Planning and Inference, 138, 2786–2807. [288] STONE, K. E. (1995): “Impact of Wal-Mart Stores on Iowa Communities: 1983–93,” Economic Development Review, 13, 60–69. [258] TOIVANEN, O., AND M. WATERSON (2005): “Market Structure and Entry: Where’s the Beef?” RAND Journal of Economics, 36, 680–699. [258] U.S. BUREAU OF THE CENSUS (1977, 1980, 1985, 1990, 1995, 1997, 1998, 2000, 2002, 2004): “County Business Patterns,” available at http://www.census.gov/econ/cbp/index.html. [296] (1980, 1990, 2000): “Summary File 3, Census of Population and Housing,” available at http://factfinder.census.gov/servlet/DatasetMainPageServlet. [264] WAL -MART STORES, INC. (1971–2006): Annual Report, available at http://investors. walmartstores.com/phoenix.zhtml?c=112761&p=irol-reportsannual. [268,269] WALTON, S. (1992): Made in America: My Story. New York: Doubleday. [260]
Dept. of Economics, University of Minnesota, 4-101 Hansen Hall, 1925 Fourth Street South, Minneapolis, MN 55455, U.S.A., Federal Reserve Bank of Minneapolis, and NBER; [email protected]; www.econ.umn.edu/~holmes. Manuscript received January, 2008; final revision received March, 2010.
Econometrica, Vol. 79, No. 1 (January, 2011), 303–306
ANNOUNCEMENTS 2011 NORTH AMERICAN SUMMER MEETING
THE 2011 NORTH AMERICAN SUMMER MEETING of the Econometric Society will be held June 9–12, 2011, at Washington University in St. Louis, MO. The program will include submitted papers as well as the Presidential Address by Bengt Holmstrom (Massachusetts Institute of Technology), the Walras-Bowley Lecture by Manuel Arellano (CEMFI), the Cowles Lecture by Michael Keane (University of New South Wales and Arizona State University), and the following semi-plenary sessions: Behavioral economics Jim Andreoni, University of California, San Diego Ernst Fehr, University of Zurich Decision theory Bart Lipman, Boston University Wolfgang Pesendorfer, Princeton University Development economics Abhijit Bannerjee, Massachusetts Institute of Technology Edward Miguel, University of California, Berkeley Financial and informational frictions in macroeconomics George-Marios Angeletos, Massachusetts Institute of Technology Nobu Kiyotaki, Princeton University Game theory John Duggan, University of Rochester Ehud Kalai, Northwestern University Microeconometrics Guido Imbens, Harvard University Costas Meghir, University College London Networks Steven Durlauf, University of Wisconsin–Madison Brian Rogers, Northwestern University Time Series Econometrics Bruce E. Hansen, University of Wisconsin–Madison Ulrich Müller, Princeton University Urban Enrico Moretti, University of California, Berkeley Esteban Rossi-Hansberg, Princeton University © 2011 The Econometric Society
DOI: 10.3982/ECTA791ANN
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ANNOUNCEMENTS
The Society now welcomes submissions via Conference Maker at http:// editorialexpress.com/conference/NASM2011. The program committee invites submissions of papers (either complete or work in progress). Each person may submit only one paper and present only one paper. However, each person is allowed to be the co-author of several papers submitted to the conference. The deadline for submissions is January 31, 2011. Information on local arrangements will be available at http://artsci.wustl. edu/~econconf/EconometricSociety/. Meeting Organizers: Marcus Berliant, Washington University in St. Louis (Chair) David K. Levine, Washington University in St. Louis John Nachbar, Washington University in St. Louis Program Committee: Donald Andrews, Yale University Marcus Berliant, Washington University in St. Louis Steven Berry, Yale University Ken Chay, University of California, Berkeley Sid Chib, Washington University in St. Louis John Conley, Vanderbilt University Charles Engel, University of Wisconsin Amy Finkelstein, Massachusetts Institute of Technology Sebastian Galiani, Washington University in St. Louis Donna Ginther, University of Kansas Bart Hamilton, Washington University in St. Louis Paul J. Healy, Ohio State University Gary Hoover, University of Alabama Tasos Kalandrakis, University of Rochester David Levine, Washington University in St. Louis Rody Manuelli, Washington University in St. Louis John Nachbar, Washington University in St. Louis Ray Riezman, University of Iowa Aldo Rustichini, University of Minnesota Suzanne Scotchmer, University of California, Berkeley William Thomson, University of Rochester Chris Waller, Federal Reserve Bank of St. Louis Ping Wang, Washington University in St. Louis 2011 AUSTRALASIA MEETING
THE 2011 AUSTRALASIA MEETING of the Econometric Society in 2011 (ESAM11) will be held in Adelaide, Australia, from July 5 to July 8, 2011.
ANNOUNCEMENTS
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ESAM11 will be hosted by the School of Economics at the University of Adelaide. The program committee will be co-chaired by Christopher Findlay and Jiti Gao. The program will include plenary, invited and contributed sessions in all fields of economics. 2011 ASIAN MEETING
THE 2011 ASIAN MEETING of the Econometric Society will be held in Seoul, Korea, in the campus of Korea University in Seoul from August 11 to August 13, 2011. The program will consist of invited and contributed papers. Authors are encouraged to submit papers across the broad spectrum of theoretical and applied research in economics and in econometrics. The meeting is open to all economists including those who are not currently members of the Econometric Society. The program committee accepts electronic submission of manuscripts only. You can submit your paper through Conference Maker https://editorialexpress. com/cgi-bin/conference/conference.cgi?action=login&db_name=FEMES11 from 10:00AM December 15, 2010 PST to midnight of February 15, 2011 PST when the submission site will be closed. Submissions may be reassigned if some committee members are getting too many submissions and others are getting too few. The preliminary program is scheduled to be announced on March 31, 2011. Although the deadline for general registration is June 30, 2011, the authors of the papers in the preliminary program will be required to register early by April 30, 2011. Otherwise, their submissions will be understood to be withdrawn. We plan to announce the final program by the end of May. Please refer to the conference website http://www.ames2011.org for more information. IN-KOO CHO AND JINYONG HAHN Co-Chairs of Program Committee 2011 EUROPEAN MEETING
THE 2011 EUROPEAN MEETING of the Econometric Society (ESEM) will take place in Oslo, Norway, from 25 to 29 August, 2011. The Meeting is jointly organized by the University of Oslo and it will run in parallel with the Congress of the European Economic Association (EEA). Participants will be able to attend all sessions of both events. The Program Committee Chairs are John van Reenen (London School of Economics) for Econometrics and Empirical Economics, and Ernst-Ludwig von Thadden (University of Mannheim) for Theoretical and Applied Economics. The Local Arrangements Chair is Asbjørn Rødseth (University of Oslo).
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ANNOUNCEMENTS
Submissions will be open from 15 December, 2010. The deadline is 15 February, 2011. Papers can only be submitted electronically through the EEA–ESEM 2011 website, http://eea-esem2011oslo.org/. Each author may submit only one paper to the ESEM and only one paper to the EEA Congress. The same paper cannot be submitted to both ESEM and the EEA Congress. At least one co-author must be a member of the Econometric Society or join at the time of submission. Decisions will be notified by 15 April, 2011. Paper presenters must register by 1 May, 2011.
Econometrica, Vol. 79, No. 1 (January, 2011), 307
FORTHCOMING PAPERS THE FOLLOWING MANUSCRIPTS, in addition to those listed in previous issues, have been accepted for publication in forthcoming issues of Econometrica. ATTANASIO, ORAZIO, AND NICOLA PAVONI: “Risk Sharing in Private Information Models With Asset Accumulation: Explaining the Excess Smoothness of Consumption.” FAINGOLD, EDUARDO, AND YULIY SANNIKOV: “Reputation in ContinuousTime Games.” FIELER, ANA CECÍLIA: Non-Homotheticity and Bilateral Trade: “Evidence and a Quantitative Explanation.” FOGLI, ALESSANDRA, AND LAURA VELDKAMP: “Nature or Nurture? Learning and the Geography of Female Labor Force Participation.” GABAIX, XAVIER: “The Granular Origins of Aggregate Fluctuations.” GAGLIARDINI, PATRICK, CHRISTIAN GOURIEROUX, AND ERIC MICHEL RENAULT: “Efficient Derivative Pricing by the Extended Method of Moments.” MIDRIGAN, VIRGILIU: “Menu Costs, Multi-Product Firms, and Aggregate Fluctuations.” PARK, ANDREAS, AND HAMID SABOURIAN: “Herding and Contrarian Behaviour in Finanicial Markets.” SHAIKH, AZEEM M., AND EDWARD J. VYTLACIL: “Partial Identification in Triangular Systems of Equations With Binary Dependent Variables.”
© 2011 The Econometric Society
DOI: 10.3982/ECTA791FORTH
Econometrica, Vol. 79, No. 1 (January, 2011), 309–317
THE ECONOMETRIC SOCIETY ANNUAL REPORTS REPORT OF THE SECRETARY SHANGHAI, CHINA AUGUST 16, 2010
1. MEMBERSHIP AND CIRCULATION THIS REPORT STARTS by describing the evolution of the Society’s membership and of the number of institutional subscribers. Information is provided on both a midyear and an end-of-year basis. The latest information available, as of June 30 of the current year and of selected previous years, is provided in the top panel of Table I. The bottom panel of Table I reports the final number of members and subscribers as of the end of 2009 and selected previous years. For any given year the figures in the bottom half of Table I are larger than in the top half, reflecting those memberships and subscriptions that are initiated between the middle and the end of that calendar year. The membership of the Society has been very stable in the last five years, with a mean of 5,634 members and a standard deviation of 92 members. However, there has been a significant decrease in the number of student members, from 1,222 in 2005 to 867 in 2009, which has been compensated by the increase in ordinary members. The mid-year figure for 2010 suggests that this year there will be a jump in membership, possibly due to the very high number of submissions to the World Congress in Shanghai, China. In fact, it is very likely that there will be more than 6,000 members at the end of the year, a record figure in the history of the Society. At the same time, the number of institutional subscribers has continued its declining trend, reaching 1,761 subscribers in 2009. The mid-year figure for 2010 suggests that this year there will be a further reduction in the number of subscribers. This could be related to the significant increase in institutional subscription rates agreed by the Executive Committee in 2009. The Committee decided to move from a two-tier to a three-tier pricing scheme based on the World Bank classification of countries and to increase the rates for print subscribers, which implied a jump in institutional rates, especially for the new category of middle income countries. Table II displays the division between print and online and online only memberships and subscriptions. Since the choice between these two alternatives was offered in 2004, there has been a continued shift toward online only. This is especially significant for student members, 83.6 percent of whom chose this option as of June 2010, but the shift is also very significant for ordinary members, for whom the proportion of online only reached 59.7 percent. It is also noticeable in institutional subscriptions, for which the proportion of online only went up from 24.6 percent in June 2009 to 36.5 percent in June 2010, although this may be partly due to the increase in rates for print subscribers noted above. © 2011 The Econometric Society
DOI: 10.3982/ECTA791SEC
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ANNUAL REPORTS TABLE I INSTITUTIONAL SUBSCRIBERS AND MEMBERS Members
Year
Institutions
Ordinary
Student
Soft Currency
Freea
Life
Total Circulation
1. Institutional subscribers and members at the middle of the year 1980 2,829 1,978 411 53 45 1985 2,428 2,316 536 28 55 1990 2,482 2,571 388 57 73 1995 2,469 2,624 603 46 77 2000 2,277 2,563 437 — 112
74 71 69 66 62
5,390 5,434 5,643 5,885 5,471
2001 2002 2003 2004 2005
2,222 2,109 1,971 1,995 1,832
2,456 2,419 2,839 2,965 3,996
363 461 633 784 1,094
— — — — —
71 103 117 111 106
62 61 60 60 57
5,174 5,153 5,620 5,915 7,085
2006 2007 2008 2009 2010
1,776 1,786 1,691 1,686 1,477
4,020 4,393 4,257 4,268 4,684
1,020 916 759 744 949
— — — — —
110 97 89 81 86
58 58 56 56 56
6,984 7,250 6,852 6,835 7,252
47 61 74 96 77
74 70 68 66 62
6,018 6,123 6,608 6,651 6,316
2. Institutional subscribers and members at the end of the year 1980 3,063 2,294 491 49 1985 2,646 2,589 704 53 1990 2,636 3,240 530 60 1995 2,569 3,072 805 43 2000 2,438 3,091 648 — 2001 2002 2003 2004 2005
2,314 2,221 2,218 2,029 1,949
3,094 3,103 3,360 3,810 4,282
680 758 836 1,097 1,222
— — — — —
87 105 112 101 110
61 60 60 58 58
6,233 6,247 6,586 7,095 7,621
2006 2007 2008 2009
1,931 1,842 1,786 1,761
4,382 4,691 4,742 4,599
1,165 1,019 916 867
— — — —
93 86 89 81
58 56 56 56
7,629 7,694 7,589 7,364
a Includes free libraries.
Table III compares the Society’s membership and the number of institutional subscribers with those of the American Economic Association. (For the membership category these figures include ordinary, student, free, and life members for both the ES and the AEA.) The ES/AEA ratio for members has been very stable in the last five years, with an average of 32.9 percent. At the same time, the long-run proportional decline in the number of institutional subscribers has been similar for both organizations. Although the ES/AEA ratio has increased
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ANNUAL REPORTS TABLE II INSTITUTIONAL SUBSCRIBERS AND MEMBERS BY TYPE OF SUBSCRIPTION (MIDYEAR) 2009
2010
Total
Percent
Total
Percent
Institutions Print + Online Online only
1,686 1,271 415
1000 754 246
1,477 938 539
1000 635 365
Ordinary members Print + Online Online only
4,268 1,902 2,366
1000 446 554
4,684 1,888 2,796
1000 403 597
744 128 616
1000 172 828
949 156 793
1000 164 836
Student members Print + Online Online only
TABLE III INSTITUTIONAL SUBSCRIBERS AND MEMBERS ECONOMETRIC SOCIETY AND AMERICAN ECONOMIC ASSOCIATION (END OF YEAR) Institutions
Members
Year
ES
AEA
ES/AEA (%)
ES
AEA
ES/AEA (%)
1975 1980 1985 1990 1995 2000
3,207 3,063 2,646 2,636 2,569 2,438
7,223 7,094 5,852 5,785 5,384 4,780
44.4 43.2 45.2 45.6 47.7 50.8
2,627 2,955 3,416 3,972 4,082 3,878
19,564 19,401 20,606 21,578 21,565 19,668
13.4 15.2 16.0 18.4 18.9 19.7
2001 2002 2003 2004 2005
2,314 2,221 2,218 2,029 1,949
4,838 4,712 4,482 4,328 4,234
47.8 47.1 49.5 46.9 46.0
3,919 4,026 4,368 5,066 5,672
18,761 18,698 19,172 18,908 18,067
20.9 21.5 22.8 26.8 31.4
2006 2007 2008 2009
1,931 1,842 1,786 1,761
3,945 3,910 3,726 3,383
48.9 47.1 47.9 52.1
5,698 5,852 5,803 5,603
17,811 17,143 17,096 16,944
32.0 34.1 33.9 33.1
from 47.9 percent in 2008 to 52.1 percent in 2009, the data in Table I suggests that the ratio will probably go down in 2010. The geographic distribution of members (including students) by countries and regions as of June 30 of the current year and of selected previous years is shown in Table IV. The format of this table was slightly changed in 2008, and it now shows individual data on countries with more than 10 members
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ANNUAL REPORTS
TABLE IV GEOGRAPHIC DISTRIBUTION OF MEMBERSa (MIDYEAR) Region and Country
Australasia Australia New Zealand
1980
1985
1990
1995
2000
2005
2009
2010
57 52 5
60 57 3
95 84 11
98 88 10
90 78 12
162 137 25
209 182 27
245 218 27
Europe and Other Areas Austria Belgium Cyprus Denmark Finland Franceb Germany Greecec Hungary Ireland Israel Italyd Netherlands Norway Poland Portugal Russiae Spain Sweden Switzerland Turkey United Kingdom Other Europe Other Asia Other Africa Other Europe, Asia, and Africa
665 15 23 — 19 19 53 92 12 34 4 — 16 75 24 4 5 5 34 27 26 1 135 — — — 42
718 21 21 — 22 26 36 106 12 30 5 16 43 68 26 6 5 2 43 31 27 1 145 8 4 14 —
803 25 30 — 27 17 56 112 6 30 5 25 48 90 23 20 11 4 36 25 25 3 162 10 2 11 —
1,031 27 31 — 38 15 81 135 14 5 6 32 57 103 29 27 11 4 88 45 34 8 210 17 5 9 —
992 24 32 — 22 13 73 153 15 5 6 37 59 86 21 27 19 5 81 42 25 9 207 19 7 5 —
2,092 49 61 — 47 27 188 354 18 13 15 56 126 130 52 22 32 11 171 72 79 21 509 23 6 10 —
2,067 38 44 8 42 43 186 399 28 16 18 36 158 148 40 13 38 9 204 47 90 21 385 36 6 14 —
2,323 39 39 11 49 38 232 442 19 19 15 42 147 175 53 17 35 27 220 65 97 20 471 35 4 12 —
Far East China Hong Kongf Japan Korea Taiwan Other Far East
105 — — 83 — — 22
134 — — 114 — — 20
144 — — 101 — — 43
228 — — 143 — — 85
189 — — 130 — — 59
315 — — 203 — — 112
459 28 28 316 45 41 1
580 91 55 331 50 52 1
North America Canada United States
1,676 159 1,517
2,059 192 1,867
2,150 194 1,956
1,989 200 1,789
1,498 127 1,371
2,409 208 2,201
2,058 227 1,831
2,275 249 2,026
(Continues)
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ANNUAL REPORTS TABLE IV—Continued Region and Country
1980
Latin America Argentina Brazil Chile Colombia Mexico Other Latin America South and South East Asia India Philippines Singapore Other South and South East Asiaf Total
1985
1990
1995
2000
2005
2009
2010
42 — — — — — 42
53 — — — — 10 43
30 — — — — 1 29
87 — — — — 16 71
105 — — — — 15 90
180 — — — — 33 147
233 20 101 35 15 40 22
191 20 73 31 13 33 21
6 6
0
51 30 — — 21
42 18 — — 24
49 10 — — 39
31 14 — — 17
105 22 — — 83
76 21 8 41 6
115 34 10 56 15
2,551
3,075
3,264
3,482
2,905
5,263
5,102
5,729
— —
a Only countries with more than 10 members in 2010 are listed individually. Until 2005 some countries were grouped together, so their individual membership data is not available. b Until 2005 the data for France includes Luxembourg. c Until 2005 the data for Greece includes Cyprus. d Until 2005 the data for Italy includes Malta. e Until 2005 the data for Russia corresponds to the Commonwealth of Independent States or the USSR. f Until 2005 Hong Kong was included in South and South East Asia.
TABLE V PERCENTAGE DISTRIBUTION OF MEMBERS (MIDYEAR)
Australasia Europe and Other Areas Far East North America Latin America South and Southeast Asia Total
1980
1985
1990
1995
2000
2005
2009
2010
23 249 42 655 21 11
20 234 44 674 13 16
29 246 44 659 09 13
28 296 65 571 25 14
31 341 65 516 36 11
31 397 60 458 34 20
41 405 90 403 46 15
43 405 101 397 33 20
1000
1000
1000
1000
1000
1000
1000
1000
in 2010. Previously some countries were grouped together, so their individual membership data is not available. In comparison with the 2009 figures, the membership has significantly increased in South and South East Asia and in the Far East, which is probably explained by the organization of World Congress in Shanghai, while it has significantly decreased in Latin America. Table V shows the percentage distribution of members (including students) by regions as of June 30 of the current year and of selected previous years. The share of North America in total membership fell for the first time in 2009 below that of Europe and Other Areas, and it is now 39.7 percent.
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ANNUAL REPORTS TABLE VI PERCENTAGE DISTRIBUTION OF INSTITUTIONAL SUBSCRIBERS (MIDYEAR)
Australasia Europe and Other Areas Far East North America Latin America South and South East Asia Total
2008
2009
2010
12 351 290 260 42 45
17 360 297 253 42 31
20 350 310 253 31 36
1000
1000
1000
Finally, the new Table VI presents the percentage distribution of institutional subscribers by regions as of June 30 of the current and the previous two years. The largest share corresponds to Europe and Other Areas. Somewhat surprisingly, the share of the Far East is greater than that of North America, which is now only 25.3 percent. 2. FELLOWS Table VII displays the geographic distribution of Fellows as of June 30, 2010. As noted in previous reports, this distribution is very skewed, with 68.2 percent of the Fellows based in North America and 24.7 percent in Europe and Other Areas. The rules for the election of Fellows were reviewed by the Executive Committee at its 2009 meeting following a report prepared by Eric Maskin, Roger Myerson, and Torsten Persson. The report proposed some changes in the ballot and in the remit of the Nominating Committee with the aim to promote the election of Fellows from regions outside of North America. Specifically, it was proposed that (1) the ballot should list the candidates by region in reverse order of the number of existing Fellows, quoting the number of Fellows in each region, (2) the ballot should remind voters to carefully consider candidates outside of North America, and (3) the list of candidates nominated by the Nominating Committee should normally include at least one candidate from each region. The proposal was agreed by the Executive Committee, and the first two recommendations were implemented in the 2009 election. Table VIII provides information on the nomination and election of Fellows. Since 2006, the election has been conducted with an electronic ballot system. This has led to a very significant increase in the participation rate, which averaged 71.1 percent in the last four elections compared to an average of 55.8 percent in the previous five elections and a historical minimum of 45.5 percent in 2005. The number of votes needed to be elected in 2009 (30 percent of the number of ballots submitted) was 92, and the average number of votes per ballot was 14.8. The number of nominees was 56 and the number of new Fellows
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ANNUAL REPORTS TABLE VII GEOGRAPHIC DISTRIBUTION OF FELLOWS, 2010 Australasia Australia
7 7
Europe and Other Areas Austria Belgium Czech Republic Denmark Finland France Germany Hungary Israel Italy Netherlands Norway Poland Russia Spain Sweden Switzerland United Kingdom
157 2 8 1 1 3 29 9 4 22 4 6 1 2 4 6 4 2 49
Far East China Japan
19 2 17
North America Canada United States
444 11 433
Latin America Argentina Brazil Mexico
5 1 3 1
South and Southeast Asia India Singapore
3 2 1
Total (as of June 30, 2010)
635
TABLE VIII FELLOWS’ VOTING STATISTICS Late Ballots Returned but Not Counted
Inactive
Eligible to Vote
Returned Ballots
Percent Returning Ballots
Number of Nominees
Number Elected
Percent Ratio Elected to Nominee
197 299 354 422 499 546
26 49 57 47 119 147
171 251 301 375 380 399
100 150 164 209 225 217
58.5 59.8 54.4 55.7 59.2 54.4
63 73 60 44 52 59
21 18 13 23 15 14
33.3 24.7 21.7 52.3 28.8 23.7
n.a. n.a. 17 5 2 10
2001 2002 2003 2004 2005
564 577 590 582 604
170 189 200 145 140
394 388 390 437 464
245 236 217 239 211
62.2 60.8 55.6 54.7 45.5
55 45 53 51 50
10 17 20 15 14
18.2 37.8 37.7 29.4 28.0
0 2 10 8 16
2006 2007 2008 2009
601 599 610 617
154 166 163 184
447 433 447 433
325 305 310 311
72.7 70.4 69.4 71.8
55 50 61 56
5 16 15 21
9.1 32.0 24.6 37.5
— — — —
Year
Total Fellows
1975 1980 1985 1990 1995 2000
316
ANNUAL REPORTS
elected was 21, which included 3 Fellows from the Far East, 3 Fellows from Latin America, and 1 Fellow from South and South East Asia, regions which had been traditionally under-represented in the Fellowship. As it was the case in the previous two elections, the majority of the Fellows elected (12 out of the 21, including the 7 new Fellows from the under-represented regions) had been nominated by the Committee. 3. NEW ECONOMETRIC SOCIETY JOURNALS The Executive Committee decided in 2008 to propose to the Council, in its capacity as the highest decision-making body of the Society, to adopt the journal Theoretical Economics, which had started in 2006, and to publish a new journal called Quantitative Economics. It was the unanimous view of the Committee that the new journals would complement the Society’s efforts to broaden its impact and promote innovative research in theoretical and quantitative economics, in a manner consistent with Ragnar Frisch’s original vision of the Society. The proposal was approved by the Council in January 2009, and was ratified by the Fellows in February 2009. The two new journals are open access, so papers can be freely downloaded by both members and non-members of the Society. Each journal will initially publish three issues per year. The print version of Theoretical Economics will be mailed in January, May, and September, together with the corresponding issue of Econometrica. The print version of Quantitative Economics will be mailed in March, July, and November, together with the corresponding issue of Econometrica. The first issue Theoretical Economics as an Econometric Society journal was published in January 2010, while the first issue of Quantitative Economics was published in July 2010. 4. REGIONAL MEETINGS AND SHANGHAI WORLD CONGRESS On August 17–21, 2010, the Econometric Society will hold its 10th World Congress in Shanghai, China. The Congress will take place in the Shanghai International Convention Center and the Gran Melia Hotel Shanghai, and it will be organized by the Shanghai Jiao Tong University in cooperation with the Shanghai University of Finance and Economics, Fudan University, the China Europe International Business School, and the Chinese Association of Quantitative Economics. The Local Arrangements Chair is Lin Zhou, and the Program Chairs are Daron Acemoglu, Manuel Arellano, and Eddie Dekel. The Plenary Lectures will be given by John Moore (Presidential Address), Elhanan Helpman (Frisch Memorial), Orazio Attanasio (Walras-Bowley), Whitney Newey (Shanghai), and Drew Fudenberg (Fisher-Schultz). In addition, there will be 15 paired invited sessions and 3 plenary policy invited sessions, whose papers will later appear in the World Congress Volumes of the Econometric Society Research Monograph Series published by Cambridge
ANNUAL REPORTS
317
University Press. Almost 3,000 papers were submitted for the contributed sessions, of which 1,056 appear in the final program. This makes it the largest-ever event in the history of the Society. In World Congress years there are no regional meetings, except the North American Winter Meeting that took place in Atlanta, Georgia, January 3–5, 2010, and the European Winter Meeting that will take place in Rome, Italy, November 4–5, 2010. 5. A FINAL NOTE To conclude, I would like to thank the members of the Executive Committee, and in particular Roger Myerson, for their help and support during 2009. I am also very grateful to Claire Sashi, the Society’s General Manager in charge of the office at New York University, for her excellent work during this year. RAFAEL REPULLO
Econometrica, Vol. 79, No. 1 (January, 2011), 319–325
THE ECONOMETRIC SOCIETY ANNUAL REPORTS REPORT OF THE TREASURER
SHANGHAI, CHINA AUGUST 16, 2010
1. 2009 ACCOUNTS THE 2009 ACCOUNTS of the Econometric Society show a surplus of $245,284 (Table III, Line G). The surplus is slightly higher than the estimate of $235,000 at this time last year. Total revenues were $1,332,921 (Table II, Line D) and total expenses were $1,087,637 (Table III, Line F). Both figures are also slightly higher than last year’s estimates. There was a significant negative deviation in membership and subscription revenues (Table II, Line A), which was more than compensated by positive deviations in investment income and other revenues (Table II, Lines B and C). The net worth of the Society on 12/31/2009 went up to $1,354,865 (Table I, Line C). Consequently, the ratio of net worth to total expenses on 12/31/2009 was 125 percent, a figure which is at the lower end of the target range between 120 and 160 percent agreed by the Executive Committee in August 2007. Table I shows the balance sheets of the Society for the years 2005–2009, distinguishing between unrestricted assets and liabilities, whose difference gives the Society’s net worth, and five restricted accounts: The World Congress Fund, which is a purely bookkeeping entry that serves to smooth the expenses every five years on travel grants to the World Congress, the Jacob Marschak Fund, devoted to support the Marschak lectures at regional meetings outside Europe and North America, and the Far Eastern, Latin American, and European Funds, which are held in custody for the convenience of the corresponding Regional Standing Committees. Tables IV and V show the movements in the World Congress Fund and the other restricted accounts for the years 2005– 2009. Table II shows the actual revenues for 2008, the estimated and actual revenues for 2009, and the estimated revenues for 2010 and 2011. Total revenues for 2010 are expected to be 5.5 percent lower than those for 2009, mainly because of the reduction in investment income (Line C), which is not compensated by the increase in membership and subscription revenues (Line A). The budget for 2011 incorporates an increase of 7.1 percent in total revenues as a result of the recovery in investment income and a further increase in membership and subscription revenues, due to the proposed changes in individual membership rates. Table III shows the actual expenses for 2008, the estimated and actual expenses for 2009, and the estimated expenses for 2010 and 2011. Total expenses © 2011 The Econometric Society
DOI: 10.3982/ECTA791TREAS
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ANNUAL REPORTS TABLE I ECONOMETRIC SOCIETY BALANCE SHEETS, 2005–2009 12/31/05 $
12/31/06 $
12/31/07 $
12/31/08 $
12/31/09 $
1,707,036 123,106 1,114,981 450,198 7,067 5,791 5,893
1,801,710 63,854 1,548,878 167,360 1,884 2,459 17,275
2,203,312 117,574 1,753,807 245,755 7,913 3,161 75,102
1,896,510 96,245 1,428,604 266,921 26,587 1,545 76,608
2,565,714 357,540 1,882,888 260,669 0 1,157 63,460
B. Unrestricted Liabilities 1. Accounts Payable 2. Deferred Revenue 3. World Congress Fund
755,569 68,293 607,276 80,000
554,504 37,861 356,643 160,000
778,093 99,103 438,990 240,000
786,929 117,257 349,672 320,000
1,210,849 147,795 663,054 400,000
C. Unrestricted Fund Balance
951,467
1,247,206
1,425,219
1,109,581
1,354,865
80,000
160,000
240,000
320,000
400,000
A. Unrestricted Assets 1. Short Term Assets 2. Investments 3. Accounts Receivable 4. Back Issue Inventory 5. Furniture and Equipment 6. Other Assets
D. World Congress Fund Balance E. Jacob Marschak Fund Balance
29,011
26,560
24,926
21,649
17,806
F. Far Eastern Fund Balance
63,576
66,624
70,016
68,047
88,784
G. Latin American Fund Balance
22,046
23,103
21,941
22,577
22,717
H. European Fund Balance
—
—
64,903
38,011
48,838
TABLE II ECONOMETRIC SOCIETY REVENUES, 2008–2011 Actual 2008 $
A. Membership and Subscriptions 1,138,648 1. Total Revenue 1,049,330 2. Change in Deferred Revenue 89,318 B. Other Revenues 1. Back Issues 2. Reprints and Royalties 3. Advertising 4. List Rentals 5. Permissions 6. North American Meetings C. Investment Income 1. Interest and Dividends 2. Capital Gains (Losses) D. Total Revenues
Estimate 2009 $
Actual 2009 $
Estimate 2010 $
Budget 2011 $
1,050,000 900,563 1,050,000 1,090,000 1,150,000 1,213,945 1,100,000 1,120,000 (100,000) (313,382) (50,000) (30,000)
90,845 53,508 3,165 5,549 1,846 5,954 20,823
66,000 30,000 3,000 5,000 2,000 6,000 20,000
96,020 31,233 878 3,350 1,390 6,925 52,244
60,000 30,000 2,000 5,000 2,000 6,000 15,000
60,000 30,000 3,000 5,000 2,000 6,000 15,000
(560,286) 52,982 (613,268)
200,000 40,000 160,000
336,338 34,665 301,674
150,000 35,000 115,000
200,000 40,000 160,000
1,316,000
1,332,921
1,260,000
1,350,000
669,206
321
ANNUAL REPORTS TABLE III ECONOMETRIC SOCIETY EXPENSES, 2008–2011 Actual 2008 $
Estimate 2009 $
Actual 2009 $
Estimate 2010 $
Budget 2011 $
A. Publishing 1. Composition 2. Printing 3. Inventory 4. Circulation 5. Postage
270,567 49,323 61,347 7,913 98,625 53,359
290,000 55,000 70,000 0 100,000 65,000
279,318 85,415 57,912 26,587 61,525 47,879
365,000 140,000 90,000 0 80,000 55,000
380,000 145,000 90,000 0 90,000 55,000
B. Editorial 1. Editors 2. Editorial Assistants 3. Software 4. Meetings
333,745 231,540 89,940 6,000 6,265
408,000 299,000 95,000 6,000 8,000
404,326 294,375 100,362 6,924 2,665
485,000 357,500 100,000 21,500 6,000
477,000 357,500 105,000 8,500 6,000
C. Administrative 1. Salaries and Honoraria 2. Administrative Support 3. Accounting and Auditing 4. Office 5. Website 6. IRS
200,650 140,110 10,000 33,870 6,328 10,086 256
203,000 140,000 10,000 34,000 6,000 12,000 1,000
229,293 135,704 10,000 73,240 5,189 5,160 0
205,000 140,000 10,000 40,000 5,000 9,000 1,000
211,000 140,000 10,000 40,000 10,000 10,000 1,000
D. Executive Committee
51,770
52,000
42,241
30,000
40,000
E. Meetings 1. World Congress 2. Regional Meetings
128,111 80,500 47,611
128,000 80,000 48,000
132,459 80,000 52,459
100,000 115,000 20,000
142,000 80,000 62,000
F. Total Expenses
984,844
1,081,000
1,087,637
1,220,000
1,250,000
G. Surplus H. Unrestricted Fund Balance I. Ratio of Unrestricted Fund Balance
235,000
245,284
40,000
100,000
1,109,581
(315,638)
1,344,581
1,354,865
1,394,865
1,494,865
1.13
1.24
1.25
1.14
1.20
to Total Expenses
for 2010 are expected to be 12.2 percent higher than those for 2009, mainly because of the significant increase in publication and editorial expenses (Lines A and B) associated with the launching of Quantitative Economics and Theoretical Economics, the two new Econometric Society journals. The budget for 2011 incorporates a further increase of 2.5 percent in total expenses. These figures together with the estimates of total revenues imply an estimated surplus of $40,000 for 2010 and of $100,000 for 2010, after allocating $115,000 and $80,000, respectively, to the World Congress Fund. Consequently the ratio of net worth to total expenses is expected to go down to 114 percent in 2010
322
ANNUAL REPORTS TABLE IV WORLD CONGRESS FUND, 2005–2009 2005 $
2006 $
2007 $
2008 $
2009 $
A. Income 1. Transfer from General Fund
180,000 180,000
80,000 80,000
80,000 80,000
80,000 80,000
80,000 80,000
B. Expenses 1. Travel Grants 2. Transfer to General Fund
400,000 326,385 73,615
0 0 0
0 0 0
0 0 0
0 0 0
80,000
160,000
240,000
320,000
400,000
C. Fund Balance
TABLE V RESTRICTED ACCOUNTS, 2005–2009 2005 $
2006 $
2007 $
A. Jacob Marschak Fund 1. Investment Income 2. Expenses 3. Fund Balance
1,135 0 29,012
1,393 3,844 26,561
1,393 3,029 24,926
723 4,000 21,649
157 4,000 17,806
B. Far Eastern Fund 1. Transfer from Region 2. Investment Income 3. Expenses 4. Fund Balance
0 1,866 0 63,576
0 3,048 0 66,624
0 3,392 0 70,016
0 2,031 4,000 68,047
20,314 423 0 88,784
C. Latin American Fund 1. Investment Income 2. Expenses (net) 3. Fund Balance
558 (7,000) 22,046
1,057 0 23,103
1,162 2,324 21,941
636 0 22,577
140 0 22,717
— — —
— — —
62,612 2,291 64,903
0 −26,892 38,011
0 10,827 48,838
D. European Fund 1. Transfer from Region 2. Investment Income 3. Fund Balance
2008 $
2009 $
and to increase to 120 percent in 2011, just at the lower end of the target range agreed by the Executive Committee in August 2007. The 2009 financial statements have been compiled by David Ciciyasvili, 42 Vista Drive, Morganville, NJ 07751, and will be audited by Rothstein, Kass & Company, 1350 Avenue of the Americas, New York, NY 10019.
ANNUAL REPORTS
323
2. MEMBERSHIP AND INSTITUTIONAL SUBSCRIPTION RATES The Executive Committee agreed by e-mail in May 2009 to a proposal of a Committee chaired by Eddie Dekel on institutional subscription rates for 2010. The proposal was to move from the traditional two-tier to a three-tier pricing scheme based on the World Bank classification of countries, with a high income tier, a middle income tier, and a low income tier comprising those economies classified as low income by the World Bank plus the International Development Association (IDA) countries. Income classifications are set by the World Bank each year on July 1. In the latest classification, high (low) income economies are those with 2009 gross national income per capita (calculated using the World Bank Atlas method) higher (lower) than $12,195 ($996). IDA countries are those that had a per capita income in 2009 of less than $1,165 and lack the financial ability to borrow from the International Bank for Reconstruction and Development (IBRD). The Executive Committee agreed by e-mail in June 2010 to keep unchanged the institutional subscription rates for 2011, which will be the following: High income Print + Online Online only Middle income Print + Online Online only Low income Print + Online Online only
$650 $500 $175 $125 $60 $10
Print + Online subscribers receive hard copies of the three Econometric Society journals (Econometrica, Quantitative Economics, and Theoretical Economics) for the corresponding year, and have free online access to volumes of Econometrica back to 1999 (Quantitative Economics and Theoretical Economics are open access). Online only subscribers do not get the hard copies of the journals. Since 2006, institutional subscribers to Econometrica have perpetual online access to the volumes to which they subscribed. The Dekel Committee proposed an adjustment in individual membership rates for 2010, which was agreed by the Executive Committee at its meeting in August 2009. In the light of the expected increases in expenditures associated with the new journals, my proposal for 2011 (agreed by the Executive Commit-
324
ANNUAL REPORTS
tee) is to increase individual membership rates in the following manner: Ordinary member (High income) Print + Online Online only Ordinary member (Middle and low income) Print + Online Online only Student member Print + Online Online only
2010
2011
$90 $50
$100 $55
$50 $10
$60 $15
$50 $10
$60 $15
Members that choose the Print + Online option receive hard copies of the three Econometric Society journals (Econometrica, Quantitative Economics, and Theoretical Economics) for the corresponding year, and have free online access to volumes of Econometrica back to 1933 (Quantitative Economics and Theoretical Economics are open access). 3. INVESTMENTS The Society’s Investments Committee consists of the Executive VicePresident and two Fellows appointed by the Executive Committee for a term of three years that can be renewed once. The Executive Committee decided in August 2008 to reappoint John Campbell and Hyun Shin for a second term starting on January 1, 2009. In July 2009, John Campbell expressed his desire TABLE VI ECONOMETRIC SOCIETY INVESTMENT PORTFOLIO Market Value 7/31/2009 Name of Fund
Unrestricted Investment Portfolio Fidelity Money Market Spartan Interm. Treasury Bond Fidelity Total Bond Fidelity Inflation Protected Bond Spartan 500 Index Spartan International Index Fidelity Emerging Markets Restricted Investment Portfolio Fidelity Money Market Spartan International Index Total Investment Portfolio
Market Value 12/31/2009
Market Value 7/31/2010
$
%
$
%
$
%
1,827,410 344,279 167,700 — — 581,480 592,009 141,942
1000 188 92 — — 318 324 78
1,882,888 344,558 168,107 — — 622,315 598,228 149,680
1000 183 89 — — 331 318 79
2,029,927 434,637 — 107,301 104,038 611,936 622,070 149,945
1000 214 — 53 51 301 306 74
160,880 116,901 43,979
1000 727 273
157,831 108,993 48,838
1000 691 309
175,825 129,325 46,499
1000 736 264
1,988,290
2,040,719
2,205,752
ANNUAL REPORTS
325
to step down from the Committee, following his appointment as Chair of the Department of Economics at Harvard. The Executive Committee decided in August 2009 to appoint Darrell Duffie for a term of three years starting on January 1, 2010. In December 2009, Hyun Shin asked for a leave for 2010 in order to take up a senior advisory position in Korea. The Executive Committee decided to ask John Campbell to serve for another year, alongside Darrell Duffie, which he agreed. During 2009, the Committee maintained, with only marginal deviations, the reference asset allocation of 20 percent cash, 10 percent bonds, and 70 percent equities, of which 45 percent correspond to US equities, 45 percent to international equities, and 10 percent to emerging market equities. All investments are in no-load Fidelity mutual funds. In February 2010, the Committee decided to slightly rebalance the portfolio toward bonds, selling the position in the Fidelity Intermediate-term Bond Fund and transferring the balance to the Fidelity Total Bond Fund and the Fidelity Inflation-Protected Bond Fund. On 7/31/2010, the breakdown by type of asset was 21.4 percent cash, 10.4 percent bonds, 30.1 percent US equities, 30.6 percent international equities, and 7.4 percent emerging markets equities (Table VI, Column 3). A separate account fully invested in cash was opened in April 2010 to manage the liquidity of the Society without interfering with the allocation of its investments. The balance of this account on 7/31/2010 was $420,600, which will mostly be used to pay for the travel grants for the 2010 World Congress. The return of the unrestricted portfolio in the year ending July 31, 2010 was 9.3 percent, as compared to the return of the S&P 500 stock market index of 11.6 percent. RAFAEL REPULLO
Econometrica, Vol. 79, No. 1 (January, 2011), 327–330
THE ECONOMETRIC SOCIETY ANNUAL REPORTS REPORT OF THE EDITORS 2009–2010 THE THREE TABLES BELOW provide summary statistics on the editorial process in the form presented in previous editors’ reports. Table I indicates that we received 714 new submissions this year. This number is the second highest ever, and reflects a growing trend in the last few years, following a small drop last year. The number of accepted papers (61) is the highest in recent years. Table III gives data on the time to first decision for decisions made in this reporting year, with 49% of papers decided within three months and 88% decided within six months. Decisions on revisions were 47% within three months and 79% within six months. These times are comparable with times in recent years. Although not reported in the tables, we can report that papers published during 2009–2010 spent an average of one year in the hands of the journal (adding up all “rounds”) and one year in the hands of the authors (carrying out revisions). This year saw the announcement that the launch a new Econometrics Society journal, Quantitative Economics, and the incorporation of the journal Theoretical Economics. As we described in last year’s report, we decided to experiment with sharing editorial material with the other Econometric Society journals. Specifically, if both authors and the other journal request it, we will (1) forward (anonymous) referee reports and decision letters to the other journal’s editor; (2) ask referees if they would like their names and cover letters to be shared with the other journal’s editor. Since the policy has been implemented, we have received requests for referee report transfer for 31 papers from Theoretical Economics. Of 85 requests to referees, 82 have agreed and 3 have declined. TABLE I STATUS OF MANUSCRIPTS
In process at beginning of year New papers received Revisions received Papers accepted Papers conditionally accepted Papers returned for revision Papers rejected or active withdrawals [Of these rejected without full refereeing] Papers in process at end of year
© 2011 The Econometric Society
05/06
06/07
07/08
08/09
09/10
158 615 161 57
165 691 127 45 16 95 591 [163] 236
236 744 146 57 32 156 656 [154] 216
216 672 188 59 29 157 590 [123] 241
241 714 147 61 24 136 675 [163] 206
190 520 [146] 165
DOI: 10.3982/ECTA791EDS
328
ANNUAL REPORTS TABLE II DISTRIBUTION OF NEW PAPERS AMONG CO-EDITORS 05/06
Current Editors Acemoglu Morris Pesendorfer Robin Samuelson Stock Uhlig Guest
06/07
07/08
08/09
09/10
84 170
70 128 116
76 149 135 81 92 107 70
110
115
102
105
90 3
91 6
88
12
71 169 127
70 4 110
75
116
107
89
1
691
744
672
714
Previous Editors Berry Dekel Levine Meghir Newey
184 129 75 105
Total
615
1 2
1
Larry Samuelson served a term as Co-Editor from 2005 to 2009 and agreed to extend his term for an extra year through 2010. We admire his editorial dedication as he moves straight to a term as Co-Editor at the American Economic Review! He has played a vital role in handling a wide portfolio with efficiency and wisdom and Stephen has greatly valued the advice on many topics. We are delighted that Philippe Jehiel has agreed to join the board for a term from 2010–2014. Harald Uhlig was a Co-Editor from 2006–2010. He has TABLE III TIME TO DECISION Decisions on New Submissions
Decisions on Revisions
Decisions on All Papers
#
%
Cumulative %
#
%
Cumulative %
#
%
Cumulative %
In ≤ 1 months In 2 months In 3 months In 4 months In 5 months In 6 months In 7 months In 8 months In > 8 months
194 34 135 131 92 65 32 27 27
26% 5% 18% 18% 12% 9% 4% 4% 4%
26% 31% 49% 67% 80% 88% 93% 96% 100%
44 11 19 24 9 19 11 11 11
28% 7% 12% 15% 6% 12% 7% 7% 7%
28% 35% 47% 62% 67% 79% 86% 93% 100%
238 45 154 155 101 84 43 38 38
27% 5% 17% 17% 11% 9% 5% 4% 4%
27% 32% 49% 66% 77% 87% 92% 96% 100%
Total
737
159
896
ANNUAL REPORTS
329
greatly strengthened our coverage of macroeconomics and applied work more generally and thoughtfully and skillfully handled a diverse portfolio. The Associate Editors of Econometrica have always played a special role at Econometrica with their consistently high quality refereeing and advice. We try to balance a desire to have turnover and add new talents as AEs with the remarkable sustained input that we get from some long serving AEs. This year George Mailath (University of Pennsylvania) steps down after nine years and Uzi Segal (Boston College) steps down after fifteen years. Generations of Co-Editors have benefited from their wisdom. We are very grateful for all they have done for the journal. We are delighted that Yeon-Koo Che (Columbia University), Jianqing Fan (Princeton University), Johannes Hörner (Yale University), Sujoy Mukerji (University of Oxford), Joris Pinkse (Pennsylvania State University), and Andrew Schotter (New York University) will be joining us. We are very grateful also to those who have agreed to extend their service for another term: Yacine Aït-Sahalia (Princeton University), Joseph Altonji (Yale University), Jeffrey Ely (Northwestern University), Philip Haile (Yale University), Rosa Matzkin (UCLA), Wojciech Olszewski (Northwestern University), Marciano Siniscalchi (Northwestern University), and Asher Wolinsky (Northwestern University). Our referees also maintain a tradition of writing referee reports to a remarkably high standard. We offer them our sincere gratitude for their willingness to invest their time in offering us their insightful views on the submissions we receive. Following this report we list those who advised us this year; we apologize to anyone whom we have mistakenly omitted. Mary Beth Bellando continues to coordinate the editorial process of Econometrica from the editorial office at Princeton University. She also provides invaluable support to the Editor and Co-Editors in managing the review process. Princeton University provides us with facilities and backup services for the editorial office; we are grateful in particular to Matthew Parker (for technical support), Barbara Radvany and Laura Sciarotta and to the Economics Department Chair, Gene Grossman. We benefit from the help of the Co-Editors’ assistants: Emily Gallagher, Sharline Samuelson, and Lauren Fahey. John Rust and Sarbartha Bandyopadhyay of Editorial Express® continue to assist us by developing and maintaining the software we use for running the journal. The Managing Editor Geri Mattson and her staff at Mattson Publishing Services supervise an efficient publication process. Wiley-Blackwell manages the journal’s website and subscriptions. Michael Brown coordinated their work on Econometrica and we look forward to working his replacement, Hester Tilbury, in the coming years. We appreciate the assistance of Elisabetta O’Connell, Charu Jalota, and Sophie Gillanders at Wiley-Blackwell with the journal’s website. Vytas Statulevicius and his staff at VTeX continue their superb work typesetting the journal. The Econometric Society in the form of its General Manager, Claire Sashi, and its Executive Vice-President, Rafael Repullo, oversee the production process and the management of our editorial process. We thank
330
ANNUAL REPORTS
them for their efficiency in doing this as well as their input and advice on running the journal. STEPHEN MORRIS DARON ACEMOGLU WOLFGANG PESENDORFER JEAN-MARC ROBIN LARRY SAMUELSON JAMES H. STOCK HARALD UHLIG
Econometrica, Vol. 79, No. 1 (January, 2011), 331–339
THE ECONOMETRIC SOCIETY ANNUAL REPORTS ECONOMETRICA REFEREES 2009–2010
A Aaberge, R. Abbring, J. Abdulkadiroglu, A. Abe, K. Abrevaya, J. Ackerberg, D. Adda, J. Aghion, P. Aguirregabiria, V. Ahn, D. Ai, C. Ai, H. Akcigit, U.
Ali, S. N. Allen, F. Al-Najjar, N. Alos-Ferrer, C. Alvarez, F. Amador, M. Amir Ahmadi, P. Ambrus, A. Anderlini, L. Andersen, T. Andrews, D. Angrist, J. Antras, P.
Aradillas-Lopez, A. Arcidiacono, P. Armenter, R. Armstrong, M. Aruoba, S. Asheim, G. Ashraf, N. Atkeson, A. Aue, A. Avarucci, M. Avery, C.
B Bachmann, R. Back, K. Bagnoli, M. Backus, D. Bagwell, K. Bai, J. Bajari, P. Baker, M. Balkenborg, D. Bandi, F. Banerjee, S. Banks, J. Barbera, S. Barberis, N. Barelli, P. Bar-Isaac, H. Barrett, S. Barseghyan, L. Basak, S. Basov, S. Basso, A. Bayer, P. Beare, B. Belzil, C. © 2011 The Econometric Society
Benjamin, D. Benkard, L. Ben-Porath, E. Berentsen, A. Berk, J. Berliant, M. Bernhardt, D. Bernheim, B. Bester, H. Bhaskar, V. Biais, B. Bikhchandani, S. Billot, A. Bimpikis, K. Bisin, A. Bjork, T. Bloch, F. Blonski, M. Bloom, N. Blume, A. Board, S. Bolton, G. Bochet, O. Bohn, H.
Bolton, G. Bomze, I. Bond, E. Bond, P. Bonhomme, S. Bonnisseau, J. M. Bontemps, C. Boone, J. Bossaerts, P. Bouchaud, J. P. Bound, J. Bowlus, A. Boyarchenko, N. Boyd, J. Brandts, J. Breitung, J. Bresnahan, T. Bruche, M. Brunnermeier, M. Buchinsky, M. Buera, F. Bugni, F. Bursztyn, L. Bushnell, J. DOI: 10.3982/ECTA791REF
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ANNUAL REPORTS
C Cabrales, A. Cagetti, M. Camerer, C. Campbell, J. Campbell, S. Canay, I. Caner, M. Canova, F. Cantillon, E. Cao, D. Caplin, A. Carlsson, H. Carbonell-Nicolau, O. Carillo-Tudela, C. Carmona, G. Carneiro, P. Carranza, J. Carrasco, M. Cason, T. Cattaneo, M. Cerreia-Vioglio, S. Cesarini, D. Cespa, G. Chakravarty, S. Chalak, K. Chamberlain, G.
Chambers, C. Chamley, C. Chandrasekhar, A. Chaney, T. Chao, J. Charness, G. Chassang, S. Chatterjee, K. Che, Y. K. Chen, H. Chen, S. Chen, W. Chen, Y. Chen, Y. C. Chernov, M. Chesher, A. Chetty, R. Chiburis, R. Cho, I. K. Choi, S. Christoffersen, P. Chung, K. S. Ciccone, A. Ciliberto, F. Citanna, A. Cochrane, J.
Cogley, T. Coles, M. Collard, F. Collin-Dufresne, P. Comin, D. Conlon, J. Cook, P. Cooley, J. Cooper, D. Cooper, R. Cordoba, J. Cornes, R. Costinot, A. Cox, J. Cramton, P. Crawford, I. Crawford, V. Cremer, J. Cripps, M. Crockett, S. Croson, R. Cubitt, R. Cullen, J. Cunha, F. Cvitanic, J.
D Dal Bó, P. Daradanoni, V. Dasgupta, A. Davidson, C. Davidson, R. Davis, D. Dawid, H. de Castro, L. de Clippel, G. de Jong, R. Delbaen, F. Dell, M. Dellarocas, C.
Del Negro, M. De Loecker, J. de Marti Beltran, J. DeMarzo, P. Deneckere, R. de Palma, A. Dessein, W. Dette, H. D’Haultfoeuille, X. Diasakos, T. Dickens, W. Dierker, E. Diermeier, D.
Dillenberger, D. Doraszelski, U. Dubois, P. Duffy, J. Dufwenberg, M. Duggan, J. Duggan, M. Dupor, B. Dupuy, A. Duranton, G. Durlauf, S. Dutta, B.
333
ANNUAL REPORTS
E Easley, D. Eaton, J. Echenique, F. Eden, B. Edlund, L. Ehlers, L. Ekmekci, M. Elbers, C.
Elhorst, J. P. Ellickson, B. Ellickson, P. Ellingsen, T. Elliott, G. Engelmann, D. Epstein, L. Eraslan, H.
Erev, I. Escanciano, J. C. Eso, P. Esteban, J. Evans, G. Evans, R. Evdokimov, K. Eyster, E.
F Faingold, E. Falk, A. Fan, Y. Fang, H. Feddersen, T. Fehr, E. Feinberg, Y. Fernández, R. Fernandez-Val, I. Fernandez-Villaverde, J. Ferraz, C. Fibich, G.
Filipovic, D. Finan, F. Firpo, S. Fischbacher, U. Flinn, C. Fong, P. Fontaine, F. Forand, J. G. Fox, J. Francesconi, M. Frankel, D. Frechette, G.
Frederick, S. French, E. Friedenberg, A. Friedman, D. Friedman, E. Froot, K. Fruhwirth-Schnatter, S. Fuchs, W. Fudenberg, D. Fulop, A.
G Gabaix, X. Gaechter, S. Gailmard, S. Gale, I. Galeotii, A. Gali, J. Galichon, A. Gallant, A. R. Gancia, G. Gandhi, A. Gao, J. Garcia Pérez, J. I. Geanakoplos, J. Genicot, G. Gerardi, D. Gerhard, J. Gershkov, A. Geweke, J. Ghiglino, C. Ghiradato, P.
Ghysels, E. Giacomini, R. Giesecke, K. Gilboa, I. Gilchrist, S. Gine, X. Giorgi, G. Gjerstad, S. Glasserman, P. Gneezy, U. Goeree, J. Goette, L. Goldstein, I. Goldstein, R. Gollier, C. Gomes, F. Goncalves, S. Gonzalez, P. Gospodinov, N. Gossner, O.
Gottardi, P. Gottlieb, D. Gottschalk, P. Gowresandaran, G. Goyal, S. Grant, S. Green, B. Grether, D. Griffith, R. Grofman, B. Gromb, D. Grosskopf, B. Guerdjikova, A. Guerrieri, V. Guggenberger, P. Guha, B. Guiso, L. Guvenen, F.
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ANNUAL REPORTS
H Haas, M. Hagedorn, M. Halevy, Y. Haliassos, M. Hall, A. Hallin, M. Hanany, E. Hansen, B. Hansen, P. Harrison, G. Harvey, D. Hastings, J. Hatfield, J. Hawkins, W. Hayashi, T. Hazan, M. He, Z.
Healy, P. Heathcote, J. Heinermann, F. Henry, E. Hirano, K. Hobson, D. Hoderlein, S. Hofbauer, J. Hoffler, F. Holmstrom, B. Hong, H. Hong, H. Hong, Y. Honoré, B. Hopkins, E. Hörner, J. Hornstein, A.
Horowitz, J. Hortacsu, A. Hossain, T. Hosseini, R. Houser, D. Howitt, P. Hu, Y. Huck, S. Huffman, D. Huggett, M. Huillery, E. Hummel, P. Hurkens, S. Hurst, E. Hurvich, C.
I Ibragimov, R. Ichiishi, T. Ichino, A. Iglesias, E.
Illeditsch, P. Imbens, G. Imhof, L. Inoue, A.
Iriberri, N. Izmalkov, S.
J Jachno, G. Jacod, J. Jeitschko, T. Jermann, U.
Joslin, S. Jouini, E. Jovanovic, B.
Jia, P. Johari, R. Jolivet, G. Jordan, J. K
Kagel, J. Kajii, A. Kalai, A. Kalandrakis, T. Kapteyn, A. Kariv, S. Karlan, D. Karni, E. Kartik, N. Kastl, J. Keane, M.
Keister, T. Kelsey, D. Khan, A. Kiefer, N. King, I. Kircher, P. Kirchsteiger, G. Kirman, A. Klaus, B. Kleibergen, F. Klenow, P.
Kleven, H. Klibanoff, P. Kocherlakota, N. Koenker, R. Koessler, F. Kojima, F. Komunjer, I. Konishi, H. Koop, G. Kopylov, I. Koszegi, B.
335
ANNUAL REPORTS
Kramarz, F. Kranton, R. Kremer, I. Krishna, V.
Kristensen, D. Kuersteiner, G. Kurlat, P. Kuruscu, B.
Kuzmics, C. Kyle, A.
L Laeven, L. Lagos, R. Lagunoff, R. Lam, C. Lamba, R. Laroque, G. Lauermann, S. Laurence, P. Leahy, J. Lechner, M. Ledoit, O. Lee, D. Lee, J. Lee, S. Lee, S.
Leeper, E. Lentz, R. LeRoy, S. Lester, B. Lettau, M. Le Van, C. Levin, D. Levine, D. Lewbel, A. Li, C. Li, J. Li, N. Li, Q. LiCalzi, M. Lieberman, O.
Ligon, E. List, J. Liu, H. Liu, Q. Ljungqvist, L. Llavador, H. Lochner, L. Longin, F. Lopomo, G. Lorenzoni, G. Lovo, S. Lucas, R. Luchini, S.
M Maasoumi, E. Maccheroni, F. Machado, J. Machina, M. Macho-Stadler, I. MacLeod, W. B. Magnus, G. Mammen, E. Manea, M. Manovskii, I. Manski, C. Manso, G. Mansur, E. Manzini, P. Marcet, A. Marinacci, M. Mariotti, M. Mariotti, T. Martin, A. Martin, I. Martinelli, C. Martínez Gorricho, S.
Maskin, E. Massó, J. Mathevet, L. Matias, C. Matouschek, N. Maurin, E. Mavroeidis, S. McAdams, D. McCracken, M. McDonald, R. McLean, R. McNeil, A. Meilijson, M. Melly, B. Mercurio, F. Merlo, A. Merz, M. Meyer, M. Meyer-ter-Vehn, M. Miao, J. Michelangeli, V. Mikusheva, A.
Milgrom, P. Miller, R. Minelli, E. Mira, P. Moen, E. Molinari, F. Moll, B. Mongin, P. Monte, D. Monteiro, P. Montrucchio, L. Moon, H. Moore, D. Moreira, M. Morelli, E. Morton, R. Moscarini, G. Moulin, H. Mueller, C. Mueller, U. Mukerji, S. Müller, R.
336 Muthoo, A. Myerson, R.
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Mykland, P. Mylovanov, T. N
Nachbar, J. Nagel, S. Napp, C. Natenzon, P. Nau, R. Navarro, S.
Nehring, K. Nekipelov, D. Nesheim, L. Nevo, A. Neyman, A. Ng, S.
Nicodano, G. Noldeke, G. Noor, J. Norets, A. Norman, P.
O Obara, I. Ochs, J. O’Donoghue, T. Okhrin, O. Okui, R.
Olivertti, B. Onatski, A. Ortalo-Magne, F. Ossa, R. Österholm, P.
Ostrovsky, M. Otsu, T. Ottaviano, G. Oviedo, J.
P Paarsch, H. Pai, M. Pakes, A. Palfrey, T. Pan, J. Park, J. Pathak, P. Patton, A. Pavan, A. Pavoni, N. Pedersen, L. Pennacchi, G. Penta, A. Pepper, J.
Peracchi, F. Perea, A. Perron, P. Pesaran, H. Pesendorfer, M. Peski, M. Peters, H. Peters, M. Petrin, A. Pettersson-Lidbom, P. Piccione, M. Pinske, J. Pistaferri, L. Plug, E.
Podczeck, K. Polborn, M. Porter, J. Porter, R. Postel-Vinay, F. Pouget, S. Pouzo, D. Powell, J. Prat, A. Prelec, D. Primiceri, G. Prowse, V. Puppe, C.
Q Quadrini, V.
Quah, J.
Quinzii, M.
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R Raimondo, R. Rahman, D. Rampini, A. Ray, I. Rayo, L. Reichelstein, S. Reis, R. Renault, E. Ridder, G. Riedel, F. Riella, G. Rigobon, R.
Rigotti, L. Rios-Rull, J. V. Robinson, P. Robson, A. Rochet, J. C. Rocheteau, G. Rockoff, J. Rodriquez Mendizabal, H. Rogerson, R. Rosen, A. Rosenbaum, M.
Rossi-Hansberg, E. Rostek, M. Roth, A. Roughgarden, T. Roys, N. Rubio-Ramirez, J. Russell, J. Rust, J. Rustichini, A. Rutstrom, E. Ryan, S.
S Sabourian, H. Sacerdote, B. Sadowski, P. Safra, Z. Sagi, J. Sakovics, J. Sala, L. Salanie, B. Salant, Y. Sandholm, W. Sandroni, A. Sannikov, Y. Santos, A. Sargent, T. Sarver, T. Sauer, R. Scaillet, O. Scarsini, M. Schargrodsky, E. Scheinkman, J. Scheuer, F. Schied, A. Schlag, K. Schmidt, K. Schorfheide, F. Schott, P. Schotter, A. Schummer, J. Schwartzstein, J. Schwarz, M. Seater, J.
Segal, I. Seidmann, D. Sen, A. Sentana, E. Seo, K. Seo, M. H. Serrano, R. Sethuraman, J. Shaikh, A. Shaliastovich, I. Shalizi, C. Shapiro, J. Shaw, K. Shearer, B. Shephard, A. Shi, S. Shi, X. Shimizu, Y. Shin, H. Shmaya, E. Shneyerov, A. Siconolfi, P. Sieg, H. Simsek, A. Singleton, K. Skiadas, C. Skrzypacz, A. Slawski, A. Slonim, R. Smith, J. Smith, L.
Smith, R. Smith, V. Smorodinsky, R. Soares, R. Soderstrom, U. Song, K. Sonin, K. Sonsino, D. Sopher, B. Sørensen, P. Sorin, S. Sowell, F. Spear, S. Spiegler, R. Sprumont, Y. Stacchetti, E. Stakenas, P. Stavrakeva, V. Stein, J. Steiner, J. Stephen, R. Stewart, C. Stinchcombe, M. Stokey, N. Stole, L. Stoltenberg, C. Stoye, J. Strausz, R. Strulovici, B. Strzalecki, T.
338 Sturm, D. Su, C. L. Sugaya, T.
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Tebaldi, C. Teles, P. Tercieux, O. Tertilt, M. Teulings, C. N. Thesmar, D. Thomas, J. Thomas, J. Thomson, W.
Toikka, J. Tourky, R. Townsend, R. Trenkler, C. Tsiakas, I. Tsyvinski, A. Tuzel, S.
U Udry, C.
Ui, T.
V Valimaki, J. Van Biesebroeck, J. van den Akker, R. van den Berg, G. van der Klaauw, B. Vanhems, A. Vannetelbosch, V. van Ours, J. Van Zandt, T.
Vayanos, D. Velasco, C. Veldkamp, L. Venkateswaran, V. Verdelhan, A. Veronesi, P. Verrecchia, R. Vesterlund, L. Vickers, J.
Vincent, D. Visaria, S. Vives, X. Vogelsang, T. Vohra, R. Volij, O. von der Fehr, N. H. Vuong, Q.
W Wachter, J. Waelde, K. Wakker, P. Walker, M. Walker, T. Wallace, N. Wang, N. Wang, T. H. Washington, E. Watson, J. Weber, S. Weibull, J.
Weill, P. O. Weinstein, J. Weretka, M. Werker, B. Werner, J. Westerfield, M. Wettstein, D. Weyl, E. Weymark, J. Williams, N. Williams, S. Wilson, A.
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Xue, L.
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Z Zaffaroni, P. Zame, W. Zender, J. Zenou, Y. Zha, T. Zhang, J.
Zhang, L. Zhang, Q. Zhang, Y. Zheng, C. Zhu, S. Ziegler, K.
Ziliak, J. Zilibotti, F. Zinde-Walsh, V. Zweimüller, J.
Econometrica, Vol. 79, No. 1 (January, 2011), 341–343
THE ECONOMETRIC SOCIETY ANNUAL REPORTS REPORT OF THE EDITORS OF THE MONOGRAPH SERIES THE GOAL OF THE SERIES is to promote the publication of high-quality research works in the fields of Economic Theory, Econometrics, and Quantitative Economics more generally. Publications may range from more or less extensive accounts of the state of the art in a field to which the authors have made significant contributions to shorter monographs representing important advances on more specific issues. In addition to the usual promotion by the Publisher (Cambridge University Press) in their advertising and displays at conferences, it also arranges for members of the Econometric Society to receive monographs at a special discount. The publishing arrangement with Cambridge University Press specifies that the reviewing process and the decision to publish a monograph in the series rests solely in the hands of the Editors appointed by the Society, in the same way as for papers submitted to Econometrica. Our experience shows that this procedure generates quite valuable services to the authors. Referee reports are usually very professional, and contain detailed and specific suggestions on how to improve the manuscript. Such services, which are not normally offered by private publishing companies, are among the features that distinguish the Monograph Series of the Society from others. The complete list of publications in the series follows; the original publication dates of the hardcover (HC) and paperback (PB) versions are given. All 46 monographs are now available for electronic purchase, and available online to Econometric Society members free of charge. 1. W. Hildenbrand (ed.), Advances in Economic Theory, *HC:2/83, *PB:8/85. 2. W. Hildenbrand (ed.), Advances in Econometrics, *HC:2/83, *PB:8/85. 3. G. S. Maddala, Limited Dependent and Qualitative Variables in Econometrics, HC:3/83, PB:6/86. 4. G. Debreu, Mathematical Economics, HC:7/83, PB:10/86. 5. J.-M. Grandmont, Money and Value, *HC:11/83, *PB:9/85. 6. F. M. Fisher, Disequilibrium Foundations of Equilibrium Economics, HC:11/83, PB:3/89. 7. B. Peleg, Game Theoretic Analysis of Voting in Committees, *HC:7/84. 8. R. J. Bowden and D. A. Turkington, Instrumental Variables, *HC:1/85, *PB:1/90. 9. A. Mas-Colell, The Theory of General Economic Equilibrium: A Differentiable Approach, HC:8/85, PB:1/90. 10. J. Heckman and B. Singer, Longitudinal Analysis of Labor Market Data, *HC:10/85. 11. C. Hsiao, Analysis of Panel Data, *HC:7/86, *PB:11/89. 12. T. Bewley (ed.), Advances in Economic Theory: Fifth World Congress, *HC:8/87, *PB:7/89. 13. T. Bewley (ed.), Advances in Econometrics: Fifth World Congress, Vol. I, HC:11/ 87, PB:4/94. 14. T. Bewley (ed.), Advances in Econometrics: Fifth World Congress, Vol. II, HC:11/87, PB:4/94. 15. H. Moulin, Axioms of Cooperative Decision-Making, HC:11/88, PB:7/91. © 2011 The Econometric Society
DOI: 10.3982/ECTA791MONO
342 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
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L. G. Godfrey, Misspecification Tests in Econometrics, HC:2/89, PB:7/91. T. Lancaster, The Econometric Analysis of Transition Data, HC:9/90, PB:6/92. A. Roth and M. Sotomayor, Two-Sided Matching, HC:9/90, PB:6/92. W. Härdle, Applied Nonparametric Regression Analysis, HC:10/90, PB:1/92. J.-J. Laffont (ed.), Advances in Economic Theory: Sixth World Congress, Vol. I, HC:12/92, PB:2/95. J.-J. Laffont (ed.), Advances in Economic Theory: Sixth World Congress, Vol. II, HC:12/92, PB:2/95. H. White, Inference, Estimation and Specification Analysis, HC:9/94, PB:6/96. C. Sims (ed.), Advances in Econometrics: Sixth World Congress, Vol. I, HC:3/94, PB:3/96. C. Sims (ed.), Advances in Econometrics: Sixth World Congress, Vol. II, HC:10/94, PB:3/96. R. Guesnerie, A Contribution to the Pure Theory of Taxation, HC:9/95, PB:9/98. D. M. Kreps and K. F. Wallis (eds.), Advances in Economics and Econometrics: Theory and Applications (Seventh World Congress), Vol. I, HC:2/97, PB:2/97. D. M. Kreps and K. F. Wallis (eds.), Advances in Economics and Econometrics: Theory and Applications (Seventh World Congress), Vol. II, HC:2/97, PB:2/97. D. M. Kreps and K. F. Wallis (eds.), Advances in Economics and Econometrics: Theory and Applications (Seventh World Congress), Vol. III, HC:2/97, PB:2/97. D. Jacobs, E. Kalai, and M. Kamien (eds.), Frontiers of Research in Economic Theory—The Nancy L. Schwartz Memorial Lectures, HC & PB:11/98. A. C. Cameron and P. K. Trivedi, Regression Analysis of Count—Data, HC & PB:9/98. S. Strøm (ed.), Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium, HC & PB:2/99. E. Ghysels, N. Swanson, and M. Watson (eds.), Essays in Econometrics— Collected Papers of Clive W. J. Granger, Vol. I, HC & PB:7/01. E. Ghysels, N. Swanson, and M. Watson (eds.), Essays in Econometrics— Collected Papers of Clive W. J. Granger, Vol. II, HC & PB:7/01. M. Dewatripont, L. P. Hansen, and S. J. Turnovsky (eds.), Advances in Economics and Econometrics: Theory and Applications (Eighth World Congress), Vol. I, HC:2/03, PB:2/03. M. Dewatripont, L. P. Hansen, and S. J. Turnovsky (eds.), Advances in Economics and Econometrics: Theory and Applications (Eighth World Congress), Vol. II, HC:2/03, PB:2/03. M. Dewatripont, L. P. Hansen, and S. J. Turnovsky (eds.), Advances in Economics and Econometrics: Theory and Applications (Eighth World Congress), Vol. III, HC:2/03, PB:2/03. C. Hsiao, Analysis of Panel Data: Second Edition, HC & PB:2/03. R. Koenker, Quantile Regression, HC & PB:5/05. C. Blackorby, W. Bossert, and D. Donaldson, Population Issues in Social Choice Theory, Welfare Economics and Ethics, HC & PB:11/05. J. Roemer, Democracy, Education, and Equality, HC & PB:1/06. R. Blundell, W. Newey, and T. Persson, Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Vol. I, HC & PB:8/06. R. Blundell, W. Newey, and T. Persson, Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Vol. II, HC & PB:8/06.
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43. R. Blundell, W. Newey, and T. Persson, Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Vol. III, HC & PB:3/07. 44. F. Vega-Redondo, Complex Nextworks, HC & PB:3/07. 45. I. Gilboa, Theory of Decision Under Uncertainty, HC & PB:3/09. 46. K. Samphantharak and R. Townsend, Households as Corporate Firms: An Analysis of Household Finance Using Integrated Household Surveys and Corporate Financial Accounting, HC & PB:11/09. The editors welcome submissions of high-quality manuscripts, as well as inquiries from prospective authors at an early stage of planning their monographs. Note that the series now includes shorter more focussed manuscripts on the order of 100 to 150 pages, as well as the traditional longer series of 200 to 300 or more pages. Information on submissions can be found on the society webpage. George Mailath is the Co-Editor responsible for economic theory and Rosa Matzkin is the Co-Editor for theoretical and applied econometrics. GEORGE J. MAILATH ROSA L. MATZKIN
Econometrica, Vol. 79, No. 1 (January, 2011), 345
SUBMISSION OF MANUSCRIPTS TO THE ECONOMETRIC SOCIETY MONOGRAPH SERIES FOR MONOGRAPHS IN ECONOMIC THEORY, a PDF of the manuscript should be emailed to Professor George J. Mailath at [email protected]. Only electronic submissions will be accepted. In exceptional cases for those who are unable to submit electronic files in PDF format, three copies of the original manuscript can be sent to: Professor George J. Mailath Department of Economics 3718 Locust Walk University of Pennsylvania Philadelphia, PA 19104-6297, U.S.A. For monographs in theoretical and applied econometrics, a PDF of the manuscript should be emailed to Professor Rosa Matzkin at [email protected]. Only electronic submissions will be accepted. In exceptional cases for those who are unable to submit electronic files in PDF format, three copies of the original manuscript can be sent to: Professor Rosa Matzkin Department of Economics University of California, Los Angeles Bunche Hall 8349 Los Angeles, CA 90095 They must be accompanied by a letter of submission and be written in English. Authors submitting a manuscript are expected not to have submitted it elsewhere. It is the authors’ responsibility to inform the Editors about these matters. There is no submission charge. The Editors will also consider proposals consisting of a detailed table of contents and one or more sample chapters, and can offer a preliminary contingent decision subject to the receipt of a satisfactory complete manuscript. All submitted manuscripts should be double spaced on paper of standard size, 8.5 by 11 inches or European A, and should have margins of at least one inch on all sides. The figures should be publication quality. The manuscript should be prepared in the same manner as papers submitted to Econometrica. Manuscripts may be rejected, returned for specified revisions, or accepted. Once a monograph has been accepted, the author will sign a contract with the Econometric Society on the one hand, and with the Publisher, Cambridge University Press, on the other. Currently, monographs usually appear no more than twelve months from the date of final acceptance.
© 2011 The Econometric Society
DOI: 10.3982/ECTA791SUM
