Optimisation of Production Under Uncertainty: The State-Contingent Approach

SpringerBriefs in Economics

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Svend Rasmussen

Optimisation of Production Under Uncertainty The State-Contingent Approach

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Dr. Svend Rasmussen Institute of Food and Resource Economics University of Copenhagen Rolighedsvej 25 1958 Frederiksberg C, Copenhagen Denmark e-mail: [email protected]

ISSN 2191-5504 ISBN 978-3-642-21685-5 DOI 10.1007/978-3-642-21686-2

e-ISSN 2191-5512 e-ISBN 978-3-642-21686-2

Springer Heidelberg Dordrecht London New York Ó Svend Rasmussen 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book presents a new approach to the optimisation of production under uncertainty. It has its roots in a leave which I spent at the School of Economics, University of New England, Armidale, Australia, in 2002. At that time, Robert G. Chambers and John Quiggin had just presented their book Uncertainty, Production, Choice, and Agency—The State-Contingent Approach, which included a new theoretical approach to the description of production under uncertainty. As a result of discussions about the book with my colleagues, I realised that the concepts developed and presented by Chambers and Quiggin could be used as the basis for deriving criteria for optimal production under uncertainty. The state-contingent approach differs from the traditional approach to planning under uncertainty, which has its foundation in the von Neumann-Morgenstern utility function and the theory of expected utility, including the EV-model. To throw light on the differences and similarities between the two approaches, the introductory chapters give a very general introduction to decision making under uncertainty, including the theory of expected utility and the classic EV model. The last part of the book uses the concepts of the state-contingent approach to derive criteria for optimal application and allocation of inputs when producing under uncertainty. This work is a natural extension of the author’s book Production Economics. The Basic Theory of Production Optimisation, which deals with the optimisation of production under certainty. Its primary audience is university students at the graduate level, but the book is also suitable as a hand-book for scientists seeking insight into the problem formulation and solution methods of production decisions under uncertainty. Copenhagen, June 2011

Dr. Svend Rasmussen

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Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2

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Decision Theory Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Utility Function . . . . . . . . . . . . . . 3.1 Linear Utility Function. . . . . . 3.2 Leontief Utility Function . . . . 3.3 EU Utility Function . . . . . . . . 3.4 EV-Utility Function . . . . . . . . 3.5 Cobb-Douglas Utility Function 3.6 Example . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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5 6 7 7 12 15 16 17

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State-Contingent Outcome and Preferences. . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preferences and Subjective Probabilities . . . . . . . . . . . . . . . . . . . .

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Formulation of the Decision Problem . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Connections Between Input and 7.1 State-General Inputs. . . . . . 7.2 State-Specific Input . . . . . . 7.3 State-Flexible Input . . . . . . 7.4 Summary . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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Contents

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Optimising Production Under Uncertainty . . . . . . . . . . . . 8.1 ‘‘Good’’ and ‘‘Bad’’ States of Nature . . . . . . . . . . . . . 8.2 Criteria for the Optimal Use of Inputs . . . . . . . . . . . . . 8.2.1 General Criteria . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Generic Example . . . . . . . . . . . . . . . . . . . . . . 8.2.3 State-General, Variable Inputs . . . . . . . . . . . . . 8.2.4 State-Specific Input . . . . . . . . . . . . . . . . . . . . . 8.2.5 State-Flexible Input . . . . . . . . . . . . . . . . . . . . . 8.3 Optimisation from the Output Side . . . . . . . . . . . . . . . 8.3.1 General Criteria . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Specific Criteria . . . . . . . . . . . . . . . . . . . . . . . 8.4 Optimisation Under Constraints. . . . . . . . . . . . . . . . . . 8.4.1 Allocation of a Fixed Input to Multiple Outputs . 8.4.2 Minimising Costs . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Minimising Costs for Given Utility Level . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Summary and Conclusion. 9.1 Definitions . . . . . . . . 9.2 Main Results. . . . . . . Reference . . . . . . . . . . . . .

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Chapter 1

Introduction

Abstract This chapter is an introduction to the book. It explains why decision making under uncertainty involves special challenges which cannot be dealt with using the traditional production economic theory. Keywords Positive deviations


Negative deviations
Asymmetry

In the basic theory of production economic as described in Rasmussen (2011), it is understood that the producer acts as if output and future prices are known with certainty: The relationship between the supply of inputs and the production of output is uniquely described in terms of a production function and output prices are given or can at least be inferred / calculated from a known functional relationships between supply and demand. However, such ideal conditions usually do not exist in practice. There is often a certain degree of uncertainty associated with the production, especially in agriculture, where wind and weather and other not entirely controllable factors, such as disease among livestock and pests in arable crops, come into play and affect yield positively or negatively. Also, product prices can differ significantly from the expectations manufacturers might have at the moment when production is initiated and production-economic decisions are made. This applies not least in agriculture and forestry, where there is often a long period of time between the commitment of inputs and resources and the harvesting and sale of the crop. This publication analyses and describes how the producer can build a basis for making decisions under such uncertain conditions in a systematic way. One way is to plan and just ignore uncertainty, which can be done by using average (statistically expected) prices and yields in the models used for the optimisation of production under certainty (Rasmussen 2011). Many practitioners probably use this kind of planning procedure and in many cases there may not be much to gain from using more sophisticated planning procedures. However, as a general procedure, this approach is not recommended as it does not allow one to exploit the

S. Rasmussen, Optimisation of Production Under Uncertainty, SpringerBriefs in Economics, DOI: 10.1007/978-3-642-21686-2_1, Ó Svend Rasmussen 2011

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possibility of actively responding to uncertainty and to possibly exploit the opportunities that uncertainty itself offers. One should not forget that uncertainty not only has a negative side (it can go worse than expected), but also a positive side (it can go better than expected). The development of production economic models that actively address uncertainty is also necessary if the objective is to explain the behaviour of producers. Observations from practice show that uncertainty often plays an important role when making production economic decisions. To base planning solely on averages often provides false results, since there is asymmetry in the decision maker’s assessment of positive and negative deviations. A project with a significant possibility of ‘‘going wrong’’ is often evaluated as being worse than a project that provides a riskless1 outcome–even if the two projects provide the same average yield. The explanation is that there can be disastrous consequences associated with things ‘‘going wrong,’’ such as the forced auction of the firm, while there are no similar fortunate consequences associated with things ‘‘going well.’’ Hence, there is no symmetry when the decision maker assesses deviations from the average and thus planning based solely on average data does not provide a sufficient basis for decisions. The book takes its point of departure in the traditional theory of expected utility, described by for instance Anderson, Dillon, Hardaker (1977). In so doing, Chap. 3 provides a brief description of alternative utility functions. Chaps. 4, 5 provide an introduction to the recent theory based on state-contingent events (state-contingent approach), which is relatively well explained in Hirshleifer and Riley (1992) and later extended and generalised by Chambers and Quiggin (2000). After the formalisation of decision problems under uncertainty in Chaps. 6, 7 provides a review of alternative production technologies (input–output relationships) by describing various types of input. In Chap. 8, which is the real key section, criteria for optimisation of production using alternative types of inputs are derived. Chapter 9 provides a summary of the results and a conclusion.

References Anderson, J., Dillon, J., & Hardaker, J. (1977). Agricultural decision analysis. Ames: Lowa State University Press. Chambers, R., & Quiggin, J. (2000). Uncertainty, production, choices, and agency. The statecontingent approach.. Cambridge: Cambridge University Press. Hirshleifer, J., & Riley, J. (1992). The analytics of uncertainty and information. New York: Cambridge University Press. Rasmussen, S. (2011). Production economics. the basic theory of production optimisation. Berlin: Springer.

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In the following, the term ‘riskless’ is used to describe projects where the outcome is received with certainty.

Chapter 2

Decision Theory Elements

Abstract This chapter provides a short introduction to the basic concepts of decision theory. Keywords States of nature
Probabilities
Utility function
Decision making
Projects The theory that describes how producers make decisions under uncertainty (decision theory), has historically been based on the following elements (Anderson, Dillon and Hardaker 1977): 1. A number of alternative decisions (call them production projects or just projects for convenience), a1, ..., aj, …, aJ, among which the decision maker may choose. 2. A number of events or states of nature (or simply states that are mutually exclusive) described by a set X = {1, 2… s… S} which ‘‘nature’’ may choose among. 3. A set of probabilities ps (1, 2… s… S) which associates a probability of occurrence for each incident/state of nature. 4. A set of consequences which associates a financial gain or outcome ysj for each project j and state of nature s. 5. A utility function U(y1j, yj2, …, ySj) which associates a utility value for a given project j (j = 1,…, J), based on which the decision maker chooses the project. The decision maker chooses the project with the highest utility. Decision making is now the process by which the decision maker selects the project j which maximises the utility in terms of utility function U(yj) = U(y1j, …, ySj). This process is based on the state-probabilities, ps, and the potential (expected) economic consequences, yjs. When the decision maker has decided on a project j = q, nature chooses a state of nature s = t, and the subsequent economic gain or outcome is therefore ytq.

S. Rasmussen, Optimisation of Production Under Uncertainty, SpringerBriefs in Economics, DOI: 10.1007/978-3-642-21686-2_2,  Svend Rasmussen 2011

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Table 2.1 Economic consequences of different production projects and states of nature State of nature Probability Project 1 Project 2 Project 3 Project 4 1 2 3 Utility (U(yj))

p1 p2 p3

y11 y21 y31 U(y1)

y12 y22 y32 U(y2)

y13 y23 y33 U(y3)

y14 y24 y34 U(y4)

The stated concepts and relationships can be illustrated as shown in Table 2.1, where there are three possible states of nature (S = 3) and four possible projects (J = 4). In the following, the index j is dropped for convenience. The vector y = (y1, y2… yS) hereafter describes the vector of state-contingent economic consequences, which associates financial gain (income) ys (s = 1… S) (while having understood that this applies to a given project j) for every state of nature s.

Reference Anderson, J., Dillon, J., & Hardaker, J. (1977). Agricultural decision analysis. Ames: Lowa State University Press.

Chapter 3

Utility Function

Abstract The utility function is an important concept when modelling decision making under uncertainty. This chapter provides a description of different forms of the utility function including the expected utility model (EU-model) and its derived forms. Keywords Expected utility  Utility function  Linear utility function  Leontief utility function  Cobb-Douglas utility function  Cardinal utility function  Neumann-Morgenstern utility function  CARA  DARA  Rate of substitution in utility (RSU)  Risk aversion  Risk-neutral  EV-utility function

The utility function U ðyÞ ¼ Uðy1 ; y2 ; . . .; yS Þ represents the producer’s (the decision maker’s) utility or welfare associated with the expectation of obtaining an income of y1 if state 1 occurs, y2 if state 2 occurs, etc. It is assumed that U is a non-decreasing function, which provides a complete description of the producer’s preferences in the sense that if Uðy0 Þ [ U ðyÞ; then the producer prefers y0 to y, and conversely, if the producer prefers y0 to y, then U ðy0 Þ [ U ðyÞ: The functional form of the utility function U depends on the decision maker’s preferences. There is no generally accepted functional form which can be said to be the one that best describes all decision makers’ preferences, and historically, prolific research has been conducted to empirically identify relevant functional forms and to estimate their parameters. Within agricultural economic research, theory development and empirical application has been based on the theory of expected utility (EU model) to a large extent (see e.g. Anderson et al. (1977); Hardaker et al. (2004)). The EU utility function and a few of the other most well-known functional forms will be mentioned briefly in the following. One would expect that the utility function U includes at least the probabilities p1 ; p2 ; . . .; pS as exogenous variables: If one of two alternative projects provides a S. Rasmussen, Optimisation of Production Under Uncertainty, SpringerBriefs in Economics, DOI: 10.1007/978-3-642-21686-2_3,  Svend Rasmussen 2011

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high income if state 1 occurs, and a low income if state 2 occurs, while the second project offers a low income if state 1 occurs, and a high income if state 2 occurs, then the preference of the two projects clearly depends on the probabilities associated with these two states. If there is a high probability of state 1, then the producer will probably choose the first project, while she will probably choose the second project if there is a high probability of state 2.

3.1 Linear Utility Function The simplest functional form is a linear utility function: X U ðyÞ ¼ p1 y1 þ p2 y2 þ    þ pS yS ¼ ps ys

ð3:1Þ

s2X

As shown, this utility function expresses the statistically expected value of the income from the project in question. This utility function reflects risk-neutral behaviour, i.e. the producer is indifferent between a project which provides a riskless outcome, and a project with uncertain consequences, but which has the same statistical expected outcome as the riskless project. Producers who are risk-neutral thus conceive negative and positive deviations as being symmetric in the sense that the pleasure (utility) of positive deviations exactly counterbalances the displeasure (negative utility) of the corresponding negative deviations. It is generally recognised that producers are risk averse, i.e. prefer safe projects rather than uncertain projects provided that the two have the same expected value. A decision maker is thus risk averse if the following applies: Uðy; . . .; yÞ  U ðy1 ; . . .; yS Þ

ð3:2Þ

where y is the expected income, i.e. y  p1 y1 þ p2 y2 þ    þ pS yS and ðy; . . .; yÞ is an S-dimensional vector of expected incomes. The condition (3.2) expresses that a decision maker is risk averse if he/she always experiences greater utility from a safe project that yields the expected income y in all states of nature than the project itself, which gives the state-contingent outcomes ðy1 . . .yS Þ: The linear utility function (3.1) represents one of the borderline cases of risk aversion, namely risk-neutrality. Risk-neutrality means that the condition (3.2) applies with equal sign, i.e. that a safe project that gives the expected income (average income) in all states of nature provides the same utility as an uncertain project, which offers the same expected income. When the decision maker is riskneutral, the utility of a project can be expressed by the expected income, as shown in (3.1). In the following, I will refer to decision makers as being risk-neutral where (3.2) applies with equal sign, while decision makers are called risk averse where (3.2) applies with unequal sign.

3.2 Leontief Utility Function

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3.2 Leontief Utility Function The other extreme, or borderline cases, of risk aversion can be expressed by a Leontief utility function: UðyÞ ¼ minfy1 ; y2 . . .yS g

ð3:3Þ

This utility function is special, because it does not have probabilities as exogenous variables. The probabilities are subordinate, as the decision maker only focuses on the worst possible outcome (the state with the lowest income) regardless of whether this worst outcome has a high or low probability of occurrence attached to it. This extreme form of risk aversion is also called maximin preferences because, under such preferences, the decision maker chooses the project that maximises the minimum income (i.e. chooses the project that maximises income under the worst imaginable circumstances).

3.3 EU Utility Function The most well-known utility function used to model decision makers’ behaviour under uncertainty is based on so-called ‘von Neumann–Morgenstern utility functions’ developed and described by John von Neumann and Oscar Morgenstern in their famous book from 1944: Theory of Games and Economic Behaviour. The EU utility function has the following form: X ps vðys Þ ð3:4Þ U ðyÞ ¼ p1 vðy1 Þ þ p2 vðy2 Þ þ    þ pS vðyS Þ ¼ s2X

where v() is called a von Neumann–Morgenstern (NM) utility function.1 An NM utility function v is a non-decreasing function which attaches a utility measure v(yi) to an income yi. An example of such an NM utility function is provided in Fig. 3.1 which shows an NM utility function for a risk averse decision maker. An NM utility function is a special form of utility function which is cardinal, unlike the ordinal utility function used in e.g. consumer theory. This implies that it must meet a number of restrictions, as it must be able to do more than just rank alternatives. The utility values themselves are significant because they are used when carrying out the calculation (the weighting with the probabilities) in (3.4).

1 Historically, there has been some confusion about names. Some authors refer to both U(y) and v(•) in (3.4) as a ‘‘utility function’’ and some even use the same name (Hardaker et al. 2004). Others explicitly denote the expression on the left side in (3.4) as ‘‘expected utility’’ (EU refers to Expected Utility). This is consistent with the expression on the right side of (3.4) which is the statistically expected value of the utility units obtained by the utility function v(•).

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3 Utility Function v(y) (utility)

1

NM-utility function 0,7

0 -600

450

2.300

y (income)

Fig. 3.1 A von Neumann-Morgenstern utility function

An NM-utility function is constructed as follows: First assign the utility value 0 (zero) to the least of the potential incomes y1, y2... yS. Subsequently, assign the utility value of 1 (one) to the largest of the potential incomes y1, y2... yS. Suppose, for instance, that the minimum income is MU -600 and the maximum income is MU2 2,300. This establishes the end points of preference function (NM utility function) shown in Fig. 3.1. With this as the basis, two alternatives are presented to the decision maker: 1. A lottery that involves a gain equal to the larger y (here 2,300) with a probability ph [ 0, and a gain equal to the smallest y (here -600) with a probability (1 – ph). 2. A riskless income equivalent to the lowest income (here equal to -600). It is clear that the decision maker will choose alternative 1 from these two alternatives because it always yields at least as much as the second alternative. But now the decision maker is presented with the following crucial question: To which level must the riskless income in alternative 2 increase for the two alternatives to be equally attractive? Suppose the decision maker’s answer is A. In this case, the utility of A (v(A)) is equal to ph! The approach can be illustrated by taking the example from Fig. 3.1. Suppose, for instance, that in alternative 1 the probability of the largest income (MU 2,300)

2

MU means Monetary Units

3.3 EU Utility Function

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is equal to 0.7 (ph = 0.7) and that the probability of the lowest income (MU -600) therefore equals 0.3 (1-ph = 0.3). Furthermore, the decision maker states that for a riskless income of A = MU 450, the two alternatives are equally attractive (the decision maker is indifferent between the two alternatives). In this case, the utility of MU 450 equals 0.7, as shown by the following calculation: vð450Þ ¼ 0:7vð2300Þ þ 0:3vð600Þ ¼ 0:7: This means that we now have three points on the NM utility function. More points can be determined by repeated application of the aforementioned procedure. Eventually, one is able to draw a curve through the points so that the total NM utility function appears as illustrated in Fig. 3.1. It is crucial to understand that an NM utility function is defined exactly as indicated here. It should be noted that an NM utility function has nothing to do with the utility function, which is a familiar tool in the theory used for the description of consumer behaviour! An NM utility function is called a cardinal utility function, while the classical utility function used to derive a consumer demand curve is a so-called ordinal utility function (see e.g. Hicks 1946; Gravelle and Rees (2004), p. 462). If you look at the formula for the EU-utility function in (3.4), it appears that, in this case, the utility is calculated as the statistically expected value of the utility values obtained by applying the NM utility function on each of the S possible outcomes of income y. Hence, the designation EU (Expected Utility). The theory of expected utility is based on 3–4 relatively simple assumptions (axioms).3 These assumptions, each of which seem quite reasonable, imply the central result that the utility of a given income in a (future) possible state s only depends on how large an amount (ys) will be available in this state. Similarly, the income (yt) in another state t is uninteresting (the utility of MU 100 in case of the state of nature ‘‘sunshine’’ is independent of whether we would have received MU 50 or MU 1,000 in the case of ‘‘rain’’). These results seem rather logical, since two states can never occur together (they exclude each other) and thus per definition must be independent. The fact that the alternative states appear with a different frequency is taken into account by weighting the utility in the individual states by the relative frequency (probability). The reason why the theory of expected utility has historically been used so heavily in empirical and theoretical work is that the theory is based on relatively convincing assumptions (independence between states) and a consequent relatively simple (additive) form of the utility function.4 The application of the EU utility function (3.4) for empirical work is more complicated than, say, using the linear utility function or a Leontief utility

3

The theory will not be dealt with here (refer to e.g. Gravelle and Rees (2004) Chap. 17). Over the years, there has been some criticism of the theory because it has been shown that the apparently convincing assumptions do not always hold in practice. For a good review of the criticism of the EU model, see e.g. Schoemaker (1982). 4

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function. The empirical application of the EU utility function presupposes that the NM utility function is known and has been estimated. In both theoretical and empirical work, several alternative functional forms for the NM-utility function v have been suggested. Daniel Bernouilli, who is the true founder of the NM-utility function, proposed the logarithmic cardinal utility function v(y) = ln(y) in his work in 1738. Later, the discussion about the choice of functional form has mainly been based on the Arrow (1965) and Pratt (1964) definition of the absolute risk aversion coefficient: ð3:5Þ RA ð yÞ ¼ v00 ð yÞ=v0 ð yÞ where v0 and v00 are the first and the second derivatives of the NM utility function v, and the relative risk aversion coefficient: RR ð yÞ ¼ yRA ð yÞ

ð3:6Þ

If the selected functional form of v implies that (3.5) is constant for all values of y, then the decision maker has constant absolute risk aversion (CARA), which implies that the decision makers’ preferences among risky projects are unchanged if a constant is added to all the incomes ðy1 ; . . .; yS Þ: Similarly, if (3.6) is constant for all values of y, then the decision maker has constant relative risk aversion (CRRA), which implies that the decision maker’s preferences among risky projects are unchanged if all the incomes ðy1 ; . . .; yS Þ are multiplied by a constant k (k [ 0). The much-used negative exponential function: vðyÞ ¼ 1  eky

ð3:7Þ

has constant absolute risk aversion (CARA), with RA(y) = k. By contrast, the exponent utility function: vðyÞ ¼

1 ð1rÞ y 1r

ð0\r\1Þ

ð3:8Þ

has decreasing absolute risk aversion (DARA), but constant relative risk aversion (CRRA). For r = 1, the exponent utility function reduces to the logarithmic, v(y) = ln(y). The properties of the utility functions obtained using different types of von Neumann–Morgenstern utility functions (NM utility functions) in the EU model, can be illustrated by the so-called rate of utility substitution (Rate of Substitution in Utility (RSUst)) defined as the absolute value of the slope –dys/dyt of an iso-utility curve in state-space. Thus, an EU utility function U(y) based on an NM-utility function in (3.7) has the following property: RSUst 

dys oU=oys ps kðys yt Þ ¼ ¼ e dyt oU=oyt pt

ð3:9Þ

An EU utility function based on the logarithmic NM utility function v(y) = ln(y) has the following property:

3.3 EU Utility Function

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RSUst 

dys oU=oys ps yt ¼ ¼ dyt oU=oyt pt ys

ð3:10Þ

Finally, an EU utility function based on the NM utility function in (3.8) has the following property:  r dys oU=oys ps yt RSUst  ¼ ¼ ð3:11Þ dyt oU=oyt pt ys It can be seen in (3.9)–(3.11) that the marginal rate of substitution (how much more income you want in state s if you have to give up some income in state t, whilst maintaining constant utility) increases the greater the difference is in income between the two states of nature. Further, it follows directly from (3.10) and (3.11) that the marginal rate of substitution in EU models based on a logarithmic NM-utility function, or an exponent utility function, does not change when income in all states of nature are multiplied by a constant (CRRA). A consequence of (3.9) is that the marginal rate of substitution for EU models based on an exponential NM utility function does not change when incomes in all states is increased by a constant (CARA). It can be shown relatively easily (the reader is invited to try!) that if v0 ð yÞ [ 0 and v00 ð yÞ\0; then the condition (3.2) applies with an unequal sign, i.e. that the decision maker is risk averse (i.e. prefers safe projects rather uncertain projects, even if the expected income is the same). Similarly, if v0 ð yÞ [ 0 and v00 ð yÞ [ 0; then the decision maker is attracted to risk, i.e. is risk seeking (prefers uncertain projects rather than riskless projects despite the expected income being the same). If v0 ð yÞ [ 0 and v00 ð yÞ ¼ 0 (linear NM utility function as in (3.1)) then the decision maker is risk-neutral (indifferent between projects which offer the same expected income).5 NM utility functions for risk averse and risk seeking decision makers are shown in Fig. 3.2a and b, respectively. It is assumed that income y1 is achieved with a probability p and the income y2 obtained with a probability (1-p). The expected income is y: It is easy to see that the utility of a riskless income equivalent to the expected income y is U ðyÞ; which for the risk averse decision maker (a) is higher than the expected utility of the uncertain project U(y). The opposite is the case for the risk seeking decision maker for whom the expected utility of the uncertain project U(y) is higher than the utility of a riskless income, which corresponds to the expected income. A relevant question is how large a riskless income the decision maker has to be offered to make him become just indifferent between the uncertain project and the riskless income. The solution is given in Fig. 3.2 as CE, which of course is the

5 Comparing with (3.5) it is easy to see that decision makers who are risk averse have positive absolute risk aversion coefficients, while decision makers who are risk seeking have negative absolute risk aversion coefficients.

12

3 Utility Function v(y)

v(y)

(a) Risk averse

(b) Risk seeking v(y)

v(y2) v(y)

v(y2)

U( y ) U(y) U(y) v(y1)

U( y ) v(y1)

y1

CE

y2 y

y

y1

y

CE

y2 y

Fig. 3.2 NM utility functions for a risk averse and b risk seeking decision maker

riskless amount which just gives the same utility U(y) as the uncertain project. This amount is designated the Certainty Equivalent (CE). As shown, the risk averse decision maker would be willing to accept a riskless amount that is lower than the expected income ðCE \yÞ from the project. This illustrates that it may be rational to take out insurance even if the cost of insurance reduces the expected returns. The opposite is the case for the risk seeking decision maker, i.e. CE [ y; which implies that the decision maker must receive compensation for not opting for the safe project. Such behaviour corresponds to the participation in games (e.g. Lotto), where the expected gain is negative. For a more detailed discussion of the EU utility function and the associated problem of the choice of functional form for the NM-utility function, see the extensive literature, e.g. Hardaker et al. (2004).

3.4 EV-Utility Function The EV-utility function has the following general form: U ðyÞ ¼ U ½Eð yÞ; V ð yÞ

ð3:12Þ

where E(y) stands for the (statistically) Expected value of y and V(y) stands for the statistical Variance of y. The utility function U is increasing in E(y) and decreasing in V(y). According to (3.12), the decision makers’ utility is solely a function of expected value and variance of the uncertain income. Although the EV-utility function seems to be based on fairly restrictive assumptions, it is nevertheless equivalent to the EU utility function under the assumption that the NM utility function is a quadratic function. Let us see why.

3.4 EV-Utility Function

13

A quadratic NM utility function has the following general form: vð yÞ ¼ a þ by þ cy2

ð3:13Þ

where a, b and c are parameters (constants) (b [ 0). According to von Neumann and Morgensterns theory of expected utility, a positive linear transformation of an NM utility function does not change its basic characteristics as a preference function. This corresponds to the fact that one can measure temperature in Celsius or in Fahrenheit, without the basic properties of the temperature scale being changed. If the baseline measure is the Celsius scale it can be converted into the Fahrenheit scale by a positive linear transformation (multiplication by a positive constant (here 1.6) and adding a constant (here 32)). Whether you measure temperature in Celsius or in Fahrenheit is a matter of taste. In a similar way, if we alter the scale of the preference function (3.13), first by multiplying by a positive constant 1/b and then by adding another constant -a/b, we get the following linearly transformed NM utility function: vð yÞ ¼ yi þ hy2i

ð3:14Þ

where h is a parameter (constant) given by c/b. If we insert the right hand side of (3.14) as the NM utility function v(y) in the formula for expected utility in (3.4), then the following expression of utility results:       U ðyÞ ¼ p1 y1 þ hy21 þ p2 y2 þ hy22 þ    þ pS yS þ hy2S   ð3:15Þ ¼ p1 y1 þ    þ pS yS þ h p1 y21 þ    þ pS y2S which can be written as: U ðyÞ ¼ Eð yÞ þ hEðy2 Þ

ð3:16Þ

where E is the expected value operator as before. The variance V(y) of a random variable y can be calculated as: V ð yÞ ¼ Eðy2 Þ  ½Eð yÞ2

ð3:17Þ

so (3.16) can be expressed as: U ðyÞ ¼ Eð yÞ þ hVðyÞ þ h½EðyÞ2

ð3:18Þ

which precisely expresses the utility U as a function of expected value (E) and variance (V). If an NM utility function is a quadratic function, then the EU utility function in (3.4) is equivalent to a EV-utility function as in (3.12). The relationship between utility (U), expected income (E) and variances in income (V) in an EV-utility function based on a quadratic NM utility function can be appropriately described in terms of indifference curves or iso-utility curves. For this purpose, we calculate the total differential (d) of U in (3.18) with respect to E(y) and V(y). The result is shown in (3.19).

14

(a) E(y)

3 Utility Function

(b) E(y)

h=0

V(y)

(c) E(y)

h>0

V(y)

h<0 and 1+2hE(y)<0

V(y)

Fig. 3.3 EV-utility function. Indifference curves for a risk-neutral, b risk seeking, and c risk averse decision maker. Increasing utility in the direction of the arrow

dU ðyÞ ¼ dEð yÞ þ hdV ð yÞ þ 2hEð yÞdEð yÞ

ð3:19Þ

If we set the right hand side of (3.19) equal to zero (0) and solve for dE(y)/dV(y) then we get the following expression of the slope of the indifference curves (curves with the same utility level) in an E-V diagram: dEðyÞ h ¼ dVðyÞ 1 þ 2hEðyÞ

ð3:20Þ

If h is zero, then the expression in (3.20) is also zero, and the slopes of the indifference curves thus are equal to zero. An h-value of zero implies that the utility in (3.16) is equal to the expected income E(y), and that the indifference curves are horizontal lines, as shown in Fig. 3.3a. For positive values of h, the decision maker has preferences for uncertainty. Positive values of h imply that the expression in (3.20) is negative and the indifference curves are like the ones shown in Fig. 3.3b. For negative values of h, the decision maker is risk averse. However, the sign of the term in (3.20) is not clear for negative values of h. If 1 ? 2hE (y) \ 0, then the expression in (3.20) is negative and the slope of the indifference curve is therefore also negative. If 1 ? 2hE(y) = 0 then the expression in (3.20) is infinity (?), and hence the indifference curve is vertical. If 1 ? 2hE(y) [ 0 then the expression in (3.20) is positive, and the slopes of the indifference curves are hence positive. The indifference curves are therefore like the ones outlined in Fig. 3.3c. As previously mentioned, the negative exponential function in (3.7) has been used extensively as a functional form in the EU utility functions in applied research. The reason for this is that if y is normally distributed, then the expected utility can be described as a simple function of expected value (E) and variance (V) in the following way: k UðyÞ ¼ EðyÞ  VðyÞ 2

ð3:21Þ

3.4 EV-Utility Function

15

where E(y) is the (statistically) expected value of y, V(y) is the statistical variance of y, and k is the absolute risk aversion coefficient (see Sect. 3.3). In very general terms, this result shows that an EV-utility function is sufficient for the ranking of projects where probability distributions of income only differ by scale and location (Gravelle and Rees 2004, p. 475). EV-utility functions have been used extensively in empirical research. The reason for this is that they are attractive to work with, because expected value (E) and variance (V) are well-known statistical quantities. The model has been used considerably in areas such as finance theory.

3.5 Cobb-Douglas Utility Function The last type of utility function which will be mentioned here is the Cobb-Douglas utility function, first mentioned in Rasmussen (2006). The Cobb-Douglas utility function has the following form: UðyÞ ¼ a0 ya11 . . .yaSS

ð3:22Þ

where the condition 0\at \1ðt ¼ 1; . . .; SÞ ensures that the function is quasiconcave in yt (convex iso-utility curves), which implies that decision makers are risk averse. As will be shown later,6 the relative probabilities associated with the S states are given by the slope of iso-utility curve measured at the values of yt, where there is complete certainty, i.e. where y1 ¼    ¼ yS : Therefore, the relative probabilities may be calculated as:  ps oU=oys  ¼ ðs; t 2 XÞ ð3:23Þ pt oU=oyt y1 ¼¼yS ¼y where X is the set of possible states of nature. If we carry out the differentiation of U in (3.22) and insert the result into (3.23), we achieve the following expression for the relative probabilities: ps a s ¼ ðs; t 2 XÞ pt a t

ð3:24Þ

This means that the choice of parameters in a Cobb-Douglas utility function (at, t = 1, …, S) is also an (implicit) choice of the relative (subjective) probabilities associated with the different states of nature. Conversely, if the probabilities ps (s = 1, …, S) have already been given/provided, then the relative parameter values as (s = 1, …, S) are determined by (3.24).

6

See Fig. 5.1.

16

3 Utility Function

Table 3.1 Economic consequences and utility of different projects and states of nature (MU) State of nature Probability Project 1 Project 2 Project 3 Project 4 1 2 3

0.20 0.50 0.30

100 150 180

180 200 80

10 150 250

80 100 120

Linear utility function Leontief utility function EU utility function

Utility Utility Utility

149 100 103

160 80 103

152 10 92

102 80 81

A Cobb-Douglas utility function has the following derivatives: oU as ¼ UðyÞ oys ys

ðs 2 XÞ

ð3:25Þ

Therefore, the utility rate of substitution (Rate of Substitution in Utility (RSUst)) is given by: RSUst 

oU=oys ps yt ¼ oU=oyt pt ys

ð3:26Þ

A comparison of (3.26) and (3.10) shows that the Cobb-Douglas utility function has the same marginal utility rate of substitution (slope of iso-utility curve) as an EU utility function based on a logarithmic form of the NM utility function v(y) = ln(y). The result in equation (3.26) also shows that the Cobb-Douglas utility function depicts constant relative risk aversion (CRRA—the expansion path is a straight line through the origin), and therefore decreasing absolute risk aversion (DARA).

3.6 Example Table 3.1 provides an example that uses the concepts described in this chapter. The utility of each of the four projects is calculated by using the linear utility function (3.1), the Leontief utility function (3.3) and the EU utility function (3.4), where the latter is based on an exponential NM-utility function v(y) = y - 0.002y2. Project 2 is selected by a decision maker who has a linear utility function, while a decision maker who has a Leontief utility function chooses project 1, which always provides an economic benefit of at least MU 100, regardless of which state nature chooses. The decision maker with the EU utility function is indifferent between projects 1 and 2.

References

17

References Anderson, J., Dillon, J., & Hardaker, J. (1977). Agricultural decision analysis. Ames: Iowa State University Press. Arrow, K. (1965). Aspects of the theory of risk bearing academic bookstore. Finland: Helsinki. Gravelle, H., & Rees, R. (2004). Microeconomics (3rd ed.). Harlow, UK: Pearson Education. Hardaker, J., Huirne, R., Anderson, J., & Lien, G. (2004). Coping with risk in agriculture (2nd ed.). Wallingford: CABI Publishing. Hicks, J. (1946). Value and capital (2nd ed.). Oxford, UK: Oxford University Press. Pratt, J. (1964). Risk aversion in the small and in the large. Econometrica 122–136. Rasmussen, S. (2006). Optimizing production under uncertainty. Generalization of the StateContingent approach and comparison with the EV model. FOI Working Papers no. 5/2006, Institute of Food and Resource Economics, The Royal Veterinary and Agricultural University, Copenhagen. Schoemaker, P. (1982). The expected utility model: Its variants, purposes, evidence and limitations. Journal of Economic Literature 529–563.

Chapter 4

State-Contingent Outcome and Preferences

Abstract This chapter explains what is meant by the state-contingent approach and state-contingent goods, and it introduces the indifference curves as important tool for graphical analysis. Keywords State-contingent goods Attributes  Expectation

 Indifference curves  Convex preferences 

As mentioned at the beginning of Chap. 2, the decision maker’s utility U is assumed to be a function of the vector of state-contingent incomes y1 . . .yS : This chapter provides a more detailed description of what is more accurately understood by the term state-contingent incomes. The vector y ¼ ðy1 . . .yS Þ is a vector of state-contingent incomes that expresses the income conditional or contingent on each of the S potential states of nature (hence the designation state-contingent). First, let us consider at a brief example. A farmer who grows grain has a certain amount of labour L available that he can use, either to improve the irrigation system, or to improve the drainage system. These improvement works must be undertaken before cereals are sown in the spring, and thus before the farmer knows the weather in the coming growing season. If he uses the workforce to improve the irrigation system he will obtain a grain yield of y1 = 100 if it is a dry growing season, and a cereal yield of y 2 = 40 if it is a wet (rainy) growing season. If he uses the workforce to improve the drainage system he will obtain a grain yield of y1 = 70 if it is a dry growing season, and a cereal yield of y2 = 60, if it is a wet (rainy) growing season. The question is: Should the farmer use his labour to improve the irrigation system or the drainage system? He estimates the probability of dry weather as p1 of 0.40, and the probability of wet weather as p2 of 0.60. The relationship is illustrated in the following Table 4.1: The answer depends on the utility function U(y) = U(y1, y2). Normally, one does not know the utility function. Here, and in what follows, we will make the

S. Rasmussen, Optimisation of Production Under Uncertainty, SpringerBriefs in Economics, DOI: 10.1007/978-3-642-21686-2_4,  Svend Rasmussen 2011

19

20

4 State-Contingent Outcome and Preferences

Table 4.1 Yield of grain for different activities and states of nature State of nature Probability Repair drainage

Repair irrigation

Dry Rainy

100 40

0.40 0.60

70 60

y2

Fig. 4.1 Indifference curves and state-contingent production

U

60

1

U

2

U

3

• •

40

70

100

y1

relatively minor assumption that farmers are risk averse, i.e. they prefer projects that provide a riskless income, rather than projects that provide an uncertain income, provided that the two plans give the same expected income (see (3.2) for a definition of risk aversion). The assumption of risk aversion implies that the decision maker has convex preferences. Convex preferences can be graphically illustrated by indifference curves that are convex, as shown in Fig. 4.1. Convex indifference curves also mean that the utility function is quasi-concave, which is precisely expressed by the condition (3.2). The shape of the indifference curves under the assumption of convex preferences, and hence risk aversion, is shown in Fig. 4.1. The figure shows three indifference curves U1, U2 and U3 which satisfy the condition that the decision maker is risk averse.1 The same figure illustrates, in the form of black dots, the two alternatives, ‘‘improvement of irrigation system’’ with the state-contingent yields for cereals (y1, y2) = (100, 40) and ‘‘improvement of drainage system’’ with the state-contingent yields for cereals of (y1, y2) = (70, 60). As shown in the figure, 1 The borderline case of convex indifference curves is linear indifference curves, which depicts risk neutral decision makers. Risk neutral decision makers have linear utility functions as shown in (Eq. 3.1).

4 State-Contingent Outcome and Preferences

21

the project ‘‘improvement of drainage system’’ is at a higher indifference curve than ‘‘improvement of irrigation system’’ and the farmer with the preference structure as illustrated in Fig. 4.1, will therefore prefer to use the workforce to improve the drainage system. It is crucial that the student understands the importance of state-contingent yields and the corresponding utility concept. Note especially that it is not the good itself (cereals), which provides utility. It is the expectation, at a later date, of receiving certain quantities of the good, conditional on the state chosen by ‘‘nature,’’ that provides utility. When the project to use the workforce to improve drainage facilities yields a utility U2 (see Fig. 4.1), then this utility U2 is attached to the expectation of receiving 70 units of grain if the growing season is dry, and 60 units of grain if the growing season is wet. So, it is the combination of the two states and the corresponding yields, which gives the stated utility. When the season is over and the grain is harvested with an actual yield of, say, 70 units because the growing season proved to be dry, then of course the 70 units of grain provide utility. But, it is a different type of utility. It is utility understood in the good oldfashioned sense of the word as described in terms of the classical utility functions used in economic theory to derive the demand for goods, or to describe the utility associated with the possession/consumption of a good (see any economic textbook, e.g. Nicholsen (1989) Chap. 3, or Varian (1984), Chap. 7). But is the utility function U(y) a completely new and different utility function compared with the ‘‘good old-fashioned’’ or ‘‘classic’’ utility function as just mentioned? And if so, what is the relation between the two? The classic utility function (here designated W(•)) depicts the consumer’s utility associated with the consumption of goods. If there are m goods, the consumption of the quantities x1, x2…xm gives a utility of W(x1, x2,…xm). The usual assumptions associated with W are that it is an increasing function (more of a good provides higher utility), with diminishing marginal utility and hence a diminishing marginal rate of substitution between goods. These assumptions imply that the utility function W with two goods x1 and x2 has indifference curves (curves, showing combinations of two goods x1, x2, which gives the same utility) as illustrated in Fig. 4.2, i.e. with convex indifference curves and where W2 [ W1. The existence of utility function W is based on the assumption that consumer preferences are complete, transitive and continuous (see Nicholsen 1989, p. 76). The utility/utility function U(y) mentioned in this chapter does not concern the utility related to the consumption of the good. Rather, it is linked to more ‘‘fluffy’’ types of goods, namely the quantities of the goods which one could consume at a future date, conditional on one of a number of possible states of nature occurring. If we assume that decision makers are risk averse, then the classical utility function W(x) illustrated in Fig. 4.2 could actually include the description of the utility attached to the state-contingent income/consumption opportunities (y1,…yS) described in this chapter. The only thing needed is to extend the interpretation of a good compared to the traditional definition of goods. The traditional definition of a good is related to the three properties, type, location and time: One kg of cereals (type) in Odense (place) today at 12 noon

22 Fig. 4.2 Indifference curves for classical utility function

4 State-Contingent Outcome and Preferences x2 W W

2

1

x1

(time) is a different type of good to one kg of grain in Hamburg today at 12 noon. Similarly, one kg of grain delivered to Odense in a month’s time, is a different good to one kg of grain delivered to Odense in six month’s time. In economic theory, distinguishing between objects that differ with respect to one or more of the three properties does not normally cause problems. One can thus easily assign different prices/values to these various goods. Even prices of future goods, i.e. goods that will be available at a future date, can be handled in economic theory by using the rate of interest for the price of time. If one broadens the definition of a good to include the property, incident or state of nature, so that a good is defined by the four attributes: type, location, time and event/state of nature, then it will actually be possible to describe and handle decision problems under uncertainty within the familiar theoretical framework based on the classical utility function W. The ‘‘only’’ thing required is that one is aware that: One kg of cereals delivered to Odense in a month’s time in the case of rain is a different good to one kg of cereals delivered to Odense in a month’s time in the case of sunshine. To put it another way: An umbrella in rainy weather is a different good to an umbrella in sunshine! Or more precisely: The utility one receives in the morning from bringing an umbrella to work, even though the sun is shining, is related to the benefit one would receive from having an umbrella to protect one from the rain when going home later that day. This utility depends on the preference for having an umbrella when it is raining and having an umbrella when it is sunny, combined with the probability of rain or sunshine at the time when one plans to leave work. It is exactly in this way—i.e. that the expectation of having an umbrella in the rain is a different good to the expectation of having an umbrella in the sunshine—that you should interpret the statement that an umbrella in the rain is a different good to an umbrella in the sunshine!

4 State-Contingent Outcome and Preferences

23

With this extended definition of a good, the vector of state-contingent (y1…yS) incomes (quantities of grain associated with the S different states of nature) may be regarded as the S different goods, which in principle can be used as arguments in the classical utility function W(y1,…,yS). This subsequently paves the way for talking about prices of state-contingent income (goods) and to speak about substitution between state-contingent incomes (goods). The two state-contingent yields of grain y1 and y2 are two different goods which are analogous to alternative arbitrary goods, such as milk and meat. The conclusion as to whether there is a difference between the classical utility function W(x) and the utility function U(y) referred to in this chapter is that there is no difference if the definition of a good xi is extended beyond the type, time and place to include a description of incident/state of nature. If the incident/state of nature is included, the entire model concept established on the basis of the classical utility function may be used unchanged. This means that the criteria for the optimal production of goods, which are derived assuming no uncertainty in Rasmussen (2011), may—in principle—be applied unchanged to state-contingent goods. This is further illustrated in Chap. 6.

References Nicholsen, W. (1989). Microeconomic theory–basic principles and extensions (4th ed.). Chicago, USA: The Dryden Press. Rasmussen, S. (2011). Production economics the basic theory of production optimisation. Berlin: Springer. Varian, H. (1984). Microeconomic analysis (2nd ed.). New York, USA: W.W. Norton & Company.

Chapter 5

Preferences and Subjective Probabilities

Abstract This chapter explains the relationship between preferences and subjective probabilities, and it shows how to model the relationship between the utility function, state-contingent goods, preferences and probabilities. Keywords Indifference curves  Certainty line  Probability line  Safety line  Subjective probability  Leontief utility function  Certainty equivalent  Risk premium Having discussed alternative utility functions in Chap. 3 and graphically illustrated the utility function for a risk averse decision maker as convex indifference curves in a coordinate system with state-contingent goods on the two axes in Chap. 4, this chapter will show how the probabilities associated with the possible states can be integrated into the model. The relationship between decision makers’ preferences, state-contingent income and probabilities associated with each of the S states of nature, can be illustrated graphically when there are two possible states of nature (S = 2). Figure 5.1 outlines an indifference curve and a production project y, which gives an income of b1 if state 1 occurs and an income b2 if state 2 occurs. As the chart shows, the project y gives a utility U1. Based on the model in Fig. 5.1, it is possible to illustrate a number of useful concepts associated with decisions under uncertainty. Corresponding to Figs. 4.1, 5.1 illustrates a project y and an indifference curve U1. But a number of other concepts are also outlined graphically. The certainty line is a line through the origin with an inclination of 45° to the x-axis. The certainty line indicates all the points where y1 = y2, and which therefore illustrates certainty, since income is the same regardless of which of the two states occur. Projects on the certainty line are riskless projects, i.e. projects where the outcome is received with certainty.

S. Rasmussen, Optimisation of Production Under Uncertainty, SpringerBriefs in Economics, DOI: 10.1007/978-3-642-21686-2_5, Ó Svend Rasmussen 2011

25

26

5 Preferences and Subjective Probabilities y2 Indifference curve

Certainty line

y CE •

y=(b1, b2) U1

Probability line: a =-p1/p2 45° CE

y1 y

Fig. 5.1 Different concepts related to decision making under uncertainty

The probability line is a straight line tangential to the indifference curve at the point where the indifference curve crosses the certainty line. The special characteristic about the probability line is that it has a slope that is equal to the negative of the probability that the state of nature 1 occurs (p1) divided by the probability that the state of nature 2 occurs (p2). The slope a of the probability line is thus given by a = -p1/p2. This requires an explanation. Firstly, the slope of the indifference curve where it cuts the certainty line is defined by: oUðc;cÞ

dy2 oy1 ¼  oUðc;cÞ dy1

ð5:1Þ

oy2

where c is an arbitrary constant. Equation (5.1) can be interpreted as follows: If the decision maker already has a riskless income of c, how much of good y2 (income in state 2) is she willing to give up if she, in return, is offered a (marginal) unit of good y1 (income in state 1)? The analysis can have two alternative starting points. 1. Suppose the decision maker knows the relative frequency of state 1 and state 2. Assume, for example, that state 1 occurs in 8 out of 10 cases, while state 2 occurs in 2 out of 10 cases. Thus, state 1 occurs 4 times as frequently as state 2. In this case, one would expect that the decision maker would require

5 Preferences and Subjective Probabilities

27

four units of y2 to give up one unit of y1. Why? Denote the relative frequency of state 1 as p1 and the relative frequency of state 2 as p2. In this case, p1 = 0.80 and p2 = 0.20. By demanding a compensation of 4 units of y2 in exchange for 1 unit of y1, the decision maker obtains a long-term gain of 0.80 9 1 0.20 9 4 = 0, which implies that the decision maker just remains on the indifference curve. 2. Suppose, as an alternative starting point, that the decision maker announces that she demands 4 units of y2 to give up 1 unit of y1. In this case, we can immediately infer that the decision maker associates a probability of state 1 occurring which is four times higher than that of state 2, i.e. p1 = 0.80 and p2 = 0.20. Why? Well, if the income in state 1 is estimated to be 4 times more important than the income in state 2, then it must be because state 1 is expected to occur with a frequency, which is 4 times that of state 2! The central prerequisite behind the results set out here is that the decision maker is indifferent concerning the state of nature where the income occurs. But, this condition is exactly satisfied here because we only assume marginal changes around the certainty line. Whether an increase or decrease in income is achieved in one state or the other is therefore secondary.1 It is interesting to note that the problem of the slope of the indifference curve where it crosses the certainty line can be viewed from two perspectives: (1) If the probabilities are known then it is possible to infer the slope of the indifference curve (preferences). (2) If preferences are known (slope of the indifference curve) then one can derive the probabilities. The results above imply that the probabilities associated with various states can be defined as follows: oUðc;cÞ

p1 oy1 ¼  oUðc;cÞ p2

and p1 þ p2 ¼ 1

ð5:2Þ

oy2

This key relationship is further illustrated by an example: Assume two possible states of nature ‘‘rain’’ (s = 1) and ‘‘sunshine’’ (s = 2), and assume as the starting point that we are considering a riskless project, i.e. y1 = y2 which refers to a project on the certainty line. The question now is: If the decision maker gets a little less income (Dy2) under the state ‘‘sunshine,’’ how much additional income (Dy1) should the decision maker receive in the case of ‘‘rain’’ in order to maintain the original utility, i.e. stay on the indifference curve? One can also ask the question as follows: If the decision maker, who at the starting point has an endowment of MU 100,000 whether it is rainy or sunny (y1 = y2 = 100,000) was to give up MU 1 (Dy2) in the case of sunshine, how much more (Dy1) would he need to receive in the case of rain to achieve the same

1

We further assume that the state itself is secondary, i.e. that it is the income in the case of dry weather that is important and not the dry weather as such.

28

5 Preferences and Subjective Probabilities

utility? If the decision maker responds that he must have MU 2 in extra income in the case of rain to give up MU 1 in the case of sunshine, then the marginal rate of substitution, i.e. the slope of the indifference curve (Dy2/(Dy1) is precisely equal to -‘ at the point where the indifference curve crosses the safety line. But, with this preference (MU 2 more in the case of rain to give up MU 1 in the case of sunshine), the decision maker has also just revealed that he considers the probability of sunshine is twice as large (2/1) as the probability of rain i.e. p2 = 2p1. Since probabilities must sum to 1, it is easy to calculate p1 = 1/3 and p2 = 2/3. The slope of the indifference curve where it crosses the certainty line is therefore equal to the negative of the ratio between the probability of state 1 (p1) and the probability of state 2 (p2). As it appears, the probabilities associated with each state of nature can be deduced from the decision maker’s utility function U(y1, y2). When the utility function U(y1, y2) is determined, you can calculate the slope of the indifference curve at points where y1 = y2, and this slope a is then just -p1/p2. Since the decision maker’s utility function is associated with the decision maker in question and is therefore by definition, is subjective by nature, the derived probabilities must, per definition, also be considered as subjective probabilities: When, for instance, the decision maker reveals that with a present wealth of MU 100,000 he is willing to give up MU 1 in the case of sunshine to get MU 2 extra in the case of rain, then this means that he (subjectively) evaluates the possibility of sunshine as being twice as large as the probability of rain! But does it not work just the opposite way? Is it not because there is an (objective) probability of 1/3 that it will rain and an (objective) probability of 2/3 that it will be sunny that the decision maker’s preference is to demand MU 2 more in the case of rain to give up MU 1 in the case of sunshine? Well, maybe. There is no doubt that in certain situations, things are actually related as just described. If, for instance, the states of nature are generated by the roll of a dice, and the uncertain project will have an income of y1 if the dice is a six, and an income of y2 if the dice is less than six, then one might well imagine that the decision maker’s preferences would be derived from the probabilities p1 = 1/6 and p2 = 5/6. This means that the decision maker, who at the outset has MU 100,000 whether it rains or is sunny, is willing to give up MU 5 if the state of nature is ‘‘a six’’ if the compensation awarded is MU 1 when the state is ‘‘less than a six.’’ In this context, the probabilities are known (objective) numbers, from which the marginal rate of substitution (slope of indifference curve) along the safety line is derived. For practical purposes, this distinction between what ‘‘comes first,’’ the probabilities or the preferences, is uninteresting in reality. The crucial point in a decision making situation is the decision maker’s (subjective) assessment of the probabilities associated with individual states of nature. Whether this (subjective) assessment is based on objective facts, such as the fact that there is a probability of 1/6 associated with the outcome ‘‘six’’ and a probability of 5/6 attached to the outcome ‘‘less than six,’’ or whether the assessment is based solely on the decision maker’s subjective belief in the various outcomes, is secondary!

5 Preferences and Subjective Probabilities

29

The above shows that preferences and (subjective) probabilities are two sides of same coin! As a general rule, for the starting point, it is appropriate to presume that the probabilities associated with individual states of nature ðp1 ; p2 . . .pS Þ are derived from the slope of the indifference curve where it crosses the certainty line. This approach makes it possible for the derived probabilities to be purely subjective probabilities, or to be purely ‘‘objective’’ probabilities. The latter will be true in situations where objective probabilities really exist combined with the assumption that the decision maker knows these objective probabilities and has rational preferences, i.e. has preferences which correspond to the marginal rate of substitution along the certainty line. For given functional forms of the utility function, the relationship (5.2) can be derived mathematically. For a linear utility function: UðyÞ ¼ p1 y1 þ p2 y2

ð5:3Þ

the relationship is shown immediately by differentiation of (5.3) with respect to y1 and y2 and subsequent insertion of the derivatives in the right side of (5.2). For the EU-utility function: UðyÞ ¼ p1 vðy1 Þ þ p2 vðy2 Þ

ð5:4Þ

the ratio between the derivative with respect to y1 and y2 is given by: ov

p1 oy1 ov p2 oy

ð5:5Þ

2

Along the certainty line, y1 = y2 = k (k is an arbitrary constant). Therefore qv/qy1 = qv/qy2 and (5.5) is reduced to: p1 =p2

ð5:6Þ

which confirms (5.2). To conclude, consider a Leontief utility function: UðyÞ ¼ min fy1; y2 g

ð5:7Þ

Leontief utility functions cannot be differentiated. However, it appears that a Leontief utility function has L-shaped indifference curves and that the corners of these coincide with the certainty line. When the figure in Fig. 5.1 has a Leontief utility function it becomes as shown in Fig. 5.2. The slope at the corner point where the indifference curves and the certainty line intersect is not unambiguous, and with this utility function, the relative probabilities p1 and p2 thus cannot be determined based on the decision maker’s preferences. This, however, is not a problem because the optimal decision in this borderline case does not depend on probabilities. The other concepts which can be deduced from Fig. 5.1, are the expected income and the certainty equivalent of the project. In addition, the concept risk

30

5 Preferences and Subjective Probabilities

Fig. 5.2 Leontief utility function

y2

Indifference curve Certainty line

U1

y

Probability line: a=-p1/p2 45 ° y

y1

premium is defined as the difference between the certainty equivalent and the expected income. Expected income: The expected income E(y) of a project that provides an income y1 with a probability p1 and an income of y2 with probability p2 is equal to: EðyÞ ¼ p1 y1 þ p2 y2

ð5:8Þ

In Fig. 5.1, the expected income of the project y equals y: The expected income is then obtained by drawing a line with the slope -p1/p2 through the project depicted by the state-contingent incomes, and then reading one of the coordinates of the intersection between this line and the certainty line. Note that the graphs in Fig. 5.1 show that the utility of the expected income for a risk-averse decision maker is always higher than, or equal to, the utility of the project itself (the point ðy; yÞ is at a higher indifference curve than the point y) (see definition of risk aversion in (3.2)). The certainty equivalent CE(y) is described as CE in Fig. 5.1, and it expresses the riskless income which provides the decision maker with the same utility as the project y itself. The certainty equivalent can be determined graphically as one of the coordinates of the intersection of the indifference curve on which the project y is placed, and the certainty line. The certainty equivalent is formally defined as CE(y) = max{k: U(k, k) B U(y)}. The Risk Premium r(y) is defined as the difference between the expected value E(y) and the certainty equivalent CE(y). In Fig. 5.1, the risk premium corresponds to the difference between y and CE. The risk premium expresses how much the decision maker is willing to pay to avoid uncertainty. It is clear from Fig. 5.1 that a risk-averse decision maker is always willing to pay an amount that is greater than or equal to zero to avoid uncertainty.

Chapter 6

Formulation of the Decision Problem

Abstract This chapter includes a formal definition of the decision problem under uncertainty and the concepts involved. Keywords Production  Controllable inputs  Non-controllable inputs  Net-income  Economic consequences

The description in the preceding has primarily focused on state-contingent income, preferences and probabilities and related concepts. In this chapter, we will demonstrate how these concepts and models can be used as a basis for the optimisation of production under uncertainty. But first a little systems analysis and notation. The relationship between state-contingent income y = (y1,…, yS) and the decision maker’s preferences [U(y)] on the one hand and the production-related decisions on the other is outlined in Fig. 6.1. The arrows leading to and from the box PRODUCTION in the figure show that production under uncertainty is based on both controllable and uncontrollable inputs. The amount of controllable inputs [the vector x = (x1,…, xn)] is determined by the decision maker, while the amounts of uncontrollable inputs such as weather, wind, disease, etc. are determined by the nature described in terms of a set X = {1, 2,…, s,…, S} of states which ‘‘nature’’ may choose. The output of production is described in terms of a vector of statecontingent output z = (z1… zS), which reflects the physical production under each of the S possible states of nature. The quantity zs of output in state s can, in quite general form, be written in implicit form by the function H(z, x) = 0, where H is an implicit description of the relationship between input and output. As shown in Rasmussen (2006), this general description of the production technology can be translated into an explicit form as a set of ‘‘normal’’ state-contingent production functions zs ¼ fs ðxÞðs 2 XÞ. Figure 6.1 also includes the possibility that product

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6 Formulation of the Decision Problem

Preferences U(y) Input prices (w) Net income (y) Costs (c)

Controllable input (x)

Output PRODUCTION (z) Product value (r)

Non-controllable (stochastic) inputs (States of nature)

Stochastic product prices (p)

Fig. 6.1 Production under uncertainty

prices may also be stochastic, and the product price for the product z is thus described in terms of a state-contingent price vector p = (p1,…, pS)1. The vector of state-contingent product values r is obtained by multiplying the volume of output and associated product prices. The vector of economic consequences y = (y1,…, yS) (state-contingent net incomes, or just incomes) is calculated by deducting the cost c, which is a scalar. These economic consequences are the arguments in the utility function U, the value of which ultimately forms the basis for the decision maker’s choice of project. It is noted that the cost c is a scalar, which only depends on how much controllable input the decision maker chooses to apply, as well as input prices described by the vector w of input prices w = (w1,…, wn). Costs are therefore not stochastic, and the amount of input is assumed to be decided before nature chooses its state. The costs are thus the same regardless of the state of nature. The vector of

1

Figure 6.1 includes only one output. With more (m) outputs the production is described in the form of an S9m matrix Z where the elements zsj are production of output j in state s. Output prices are, correspondingly, in the form of an S9m matrix of output prices psj .

6 Formulation of the Decision Problem

33

net incomes is obtained by subtracting the constant c from each of the elements of the production value vector r. The production economic decision when producing one output is to decide how much input to apply. The decision variable is thus the vector x = (x1… xn) of input. The optimal decision is the vector of controllable inputs that maximises the utility function U(y1,…, yS). In summary, the individual parts in Fig. 6.1 can be described as follows: Cost c ¼ w0 x Output zs ¼ fs ðxÞ s 2 X Product Value rs ¼ zs ps s 2 X Net income ys ¼ rs  c s 2 X Preferences U ¼ Uðy1 ; . . .; yS Þ; where U is quasi-concave In the following, we will analyse how to determine optimal decisions under alternative assumptions regarding the utility function. But first, the possible connections between input and state dependent output will be analysed.

Reference Rasmussen, S. (2006). Optimizing production under uncertainty. Generalization of the statecontingent approach and comparison with the EV model. FOI Working Papers no. 5/2006, Institute of food and resource economics. The Royal Veterinary and Agricultural University, Copenhagen.

Chapter 7

Connections Between Input and State-Contingent Output

Abstract This chapter focuses on the production process, and especially the relationship between state of nature, choice of input and production of output, and it introduces the concepts of state-general, state-specific and state-flexible inputs. Keywords State-contingent output  Joint production  State-specific input State-flexible input  State-general input  Transformation curve



The connection between input and output can generally be described graphically in the form of a production function (one input-one output), an isoquant (two inputsone output) or a transformation curve (production possibility curve) (one input-two outputs). As discussed in Chap. 4, state-contingent outputs can be considered as different outputs. Just as ‘‘barley’’ and ‘‘wheat’’ can be considered as two different outputs, so can ‘‘barley conditional upon a rainy year’’ and ‘‘barley conditional upon a dry year’’ be considered as two different outputs. And just as one can apply a transformation curve to describe the relationship between the production of barley and wheat using a given input quantity x, one can also apply a transformation curve to describe the relationship between the production of barley conditional upon the state of nature ‘‘rainy year’’ (s = 1) and the production of barley conditional upon the state of nature ‘‘dry year’’ (s = 2). As described in Rasmussen (2011, Fig. 17.2), transformation curves may have different shapes. Two of these shapes are reproduced in Fig. 7.1, where the amounts on the two axes (z1 and z2) now refer to the state-contingent output of a given product z. A transformation curve (production possibility curve) is a graphic illustration of the potential (maximum) output that can be achieved with a given amount of input x ¼ x: However, when production consists of state-contingent outputs, certain conditions apply to the shape of the transformation curves. In fact, it turns out that the transformation curves which describe state-contingent output have an archetype

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7 Connections Between Input and State-Contingent Output

z2

z2

Competitive production

z1

Joint production

z1

Fig. 7.1 Competitive and joint production

equivalent to the joint production in Fig. 7.1, i.e. the transformation curve has a right-angled shape as shown on the right side of the figure. The reason for this is that the pre-determined input quantities x ¼ ðx1 ; . . .; xn Þ are applied before nature reveals its choice of state.When the input volume x is applied, then output (production) will be z1 ðxÞ if nature chooses state 1 and z2 ðxÞ if nature chooses state 2 (corresponding to the dot in the corner of the curve in the figure ‘‘joint production’’ in Fig. 7.1). And, although the producer could choose to refrain from some of the output in state 1, once nature has selected its state, it would not be possible to obtain a higher yield in state 2. Similarly, if the producer could choose to refrain from some of the output in state 2, it would not be possible to obtain a higher yield in state 1. The production possibility curve is thus the right-angled curve with the dot drawn as the efficient corner point. However, based on this general form there are particular cases where other forms of transformation curve can occur. But, as will be shown later, such alternative forms are in reality special cases of the model called ‘‘joint production’’ in Fig. 7.1. The following first describes the general shape similar to joint production in Fig. 7.1. In this general form, there are no specific restrictions related to the inputs used, and correspondingly, the input is designated state-general input. Based on this general form, two special cases can be derived, namely state-specific input and state-flexible input. These are subsequently discussed. The different production methods are most easily described by considering some examples. The following examples assume that the product z is grain (e.g. barley), z1 is the yield of grain, subject to the growing season being rainy (state 1: Wet year), and z2 is the yield of grain, subject to the growing season being dry (state 2: Dry year).

7.1 State-General Inputs z1

37 z2

“wet” year

z2

“dry” year

Transformation curves z2 z1 z0

• x0

x1

x2

x0

x1

0

x2

z1

Fig. 7.2 Derivation of the transformation curve for state-general inputs

7.1 State-General Inputs In the first example, assume that the input x is fertiliser and yield z is grain. The production of grain if the growing season is wet is z1 = f1(x), and the production of grain if the growing season is dry is z2 = f2(x) (the supply of fertiliser is of course the same no matter what state of nature occurs afterwards, whilst the amount of applied fertiliser has to be decided before the state of nature transpires). Depending on the state of nature and the applied quantity of fertiliser, the production can be read in the two production functions shown to the left and in the centre of Fig. 7.2. As the amount of input is the same regardless of the state of nature, it is possible to derive the corresponding transformation curves for the two products, ‘‘grain, subject to wet weather’’ (z1) and ‘‘grain, subject to dry weather’’ (z2) from the two production functions. Transformation curves for the different levels of fertiliser illustrated in the two production functions are shown in the curve to the right in Fig. 7.2. If no fertiliser is applied, the production is equal to point 0 in the diagram to the right of the figure. By application of input quantity x0, the state-contingent outputs correspond to point z0 in the diagram on the right-hand side of Fig. 7.2. Similarly, production corresponding to the points z1 and z2 are achieved by the application of x1 and x2 units of fertilisers, respectively. The transformation curve for each of the input quantities 0, x0, x1 and x2 can be derived by calculating how much more of the ‘‘product’’ z2 could be produced if one produced slightly less of the ‘‘product’’ z1. This so-called rate of substitution describes the slope of the transformation curve. As the situation is described in Fig. 7.2, there is no possibility for substitution: If, for instance, we apply input quantity x0, thereby obtaining the conditional output z01 and z02 , then it is not possible to achieve higher yields in wet weather by simply producing inefficiently or even by throwing some of the output away in the case of good weather. There is simply no substitution, because with the statecontingent outputs we have an either-or situation: either the weather will be dry, or it will be wet. And no matter what one does when it turns out to be a wet year, one

38

7 Connections Between Input and State-Contingent Output “wet” year

z1

z2

z2 “dry” year Transformation curves

0

x0

x1

x2

x0

x1

x2

z0

z1 z2

z1

Fig. 7.3 Derivation of the transformation curve for state specific input

cannot affect the production one would obtain if it had turned out to be a dry year, because the supply of fertilisers has, of course, been chosen in advance! Therefore, the transformation curves—as shown by the illustration on the righthand side of Fig. 7.2—are lines at right angles with the corner point which corresponds to the state-contingent outputs associated with the chosen input volume in question. And, in fact, it is only such corner points that are interesting to the producer. Inputs, which have the properties described here, are called state-general inputs. A state-general input is thus one which is applied with a view to an overall increase in output, no matter what state of nature occurs. When farmers add fertiliser, the aim is to obtain a higher yield, both when the weather is good and when it is bad. As will be seen later, there are other types of input. For instance, input which is only effective in one of the possible states e.g. a specific spray, which is active only when it starts to rain. Such inputs may be treated as special cases.

7.2 State-Specific Input In the second example, assume that the input is a spray that protects against fungal attack in the grain crop when it’s raining. On the contrary, when the weather is dry, the spray (for instance a fungicide) has no effect. The spray is applied before one knows what the weather will be like. The effect of spray is illustrated in Fig. 7.3, where again the points 0, z0, z1 and z2 correspond to the input quantities 0, x0, x1 and x2. As shown on the left of the figure, the increased supply of spray increase production if it is a wet year. The middle part of the figure illustrates that, regardless of the applied quantity, the spray will have no effect on production if it turns out to be a dry growing season. The transformation curve is shown on the right of the figure. As shown, the transformation curves again are right-angled line segments (no ‘‘substitution’’) which, unlike Fig. 7.2, now have the property that the horizontal parts of four transformation

7.2 State-Specific Input

39

curves coincide (same value of output in the case of dry weather whatever input of fungicide). An input, which has the property described here, is called a state-specific input. A state-specific input is thus one which is applied with a view to increasing output in a specific state of nature, or which only works in a specific state. When farmers in the example here apply fungicide, it is to protect against fungal attack in order to achieve higher yields in the specific situation of wet weather.

7.3 State-Flexible Input The third example was used before at the beginning of Chap. 4. The example assumes that the input is labour, which farmers can apply either to improve the irrigation system or to improve the drainage system. If the farmer uses manpower to improve the irrigation system, in the case of a subsequent dry season, he will have a higher yield than normal because he will be able to irrigate more efficiently. This is shown by the production function in Fig. 7.4 (IV). If the farmer uses manpower to improve the drainage system, he will achieve a higher yield than normal in the case of a subsequent wet growing season because he will be able to drain more effectively. This is shown by the production function in Fig. 7.4 (I). These improvements must be undertaken before cereal is sown in the spring, and thus before the farmer knows what the weather will be like in the coming growing season. It should be noted that improving the irrigation system in a wet year [flat curve Fig. 7.4 (II)], or improving the drainage system in a dry year [flat curve in Fig. 7.4 (III)] will both have no effect. If the farmer decides to use 100% of his workforce (amount x2) to improve the drainage system, he will get state-contingent outputs as shown at point A in Fig. 7.4 (V). If he uses 100% of the workforce (x2) to improve the irrigation system, he will get state-contingent outputs as shown by point D. If he allocates the workforce so that 1/3 (x0) improves the irrigation system and 2/3 (x1) improves the drainage system, he will get state-contingent outputs as shown at point B. If he allocates the workforce using 2/3 (x1) to improve the irrigation system and 1/3 (x 0) to improve the drainage system, he will get state-contingent outputs as shown at point C (z1 and z2 are as before, the yield of grain in wet years and dry years, respectively). Assuming the possibility of other combinations, it is legitimate to connect the individual points. The transformation curve is, therefore, equal to the curve ABCD in Fig. 7.4 (V). Inputs, which can be targeted at more states of nature, thereby allowing for substitution between the state-contingent outputs as shown here, are called stateflexible inputs. A state-flexible input is thus an input the application of which can be targeted at the individual states of nature.1 When farmers choose to use the

1

Chambers and Quiggin (2000) describe such a technology as ‘‘State-allocable input technology’’

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7 Connections Between Input and State-Contingent Output (I)

z1

z2 (II)

z2 x0

x1

x2

x0

x1

x2

(V) D C B

z2

z1

(IV)

(III)

A z1

x0

x1

x2

x0

x1

x2

Fig. 7.4 Derivation of the transformation curve by allocation of input on independent, statespecific activities

input labour to establish/improve the drainage system, then the associated use of input is targeted at the state ‘‘wet year’’. If farmers instead choose to use labour to establish/improve the irrigation system, then the associated use of input is targeted at the state ‘‘dry year’’. A state-flexible input, as described here, may be considered as the sum of two (or more) state-specific inputs. State-flexible inputs should therefore be considered as special cases of state-general inputs as discussed earlier. We will return to this in Chap. 8.

7.4 Summary The previous examples show that input can be described by one of the following three types of inputs: 1. A State-general input is an input xi to which the following applies: •

ofs ðxÞ oxi

6¼ 0 for one or more states s [ X for relevant levels of xi

7.4 Summary

41

2. A State-specific input is an input xi to which the following applies: •

oft ðxÞ oxi

6¼ 0 and

ofs ðxÞ oxi

¼ 0 for s = t for relevant levels of xi (t, s [ X)

3. A State-flexible input is an input xi to which the following applies: • it can be targeted at two or more states (the volume targeted state s is called xis) 6¼ 0 for two or more states (s [ X) for relevant levels of xis • ofoxs ðxÞ is • if restrictions ðxi  xi Þ apply, then the restriction: xi1 þ    þ xiS  xi must be added • a state-flexible input xi where xis is a state-specific input V s [ X is called a strict state-flexible input. We have also discussed that state-specific inputs (inputs, which are only active in one state) can be regarded as a special case of state-general inputs. Furthermore, state-flexible inputs can be interpreted as a combination of two or more statespecific/state-general inputs. In reality, this means that all types of input can be analysed within the framework of the input type called state-general inputs.

References Chambers, R., & Quiggin, J. (2000). Uncertainty production, Choices, and agency. The statecontingent approach. Cambridge: Cambridge University Press. Rasmussen, S. (2011). Production economics the basic theory of production optimisation. Berlin: Springer.

Chapter 8

Optimising Production Under Uncertainty

Abstract This is the main chapter in which the concepts and tools developed in the previous chapters are used to derive criteria for optimal use and combination of inputs when producing under uncertainty. The chapter includes a formal definition of ‘good’ and ‘bad’ states of nature, and it derives criteria for optimal application of various types of input for risk-neutral and risk-averse decision makers. Although it is not possible to derive operational criteria for risk averse decision makers unless one knows the utility function, the analysis using the state-contingent approach provides interesting and operational results for risk neutral decision makers. It also reveals that even if one knows the form of the utility function, then it is not possible to determine uniquely whether risk-averse decision makers will use more or less input than risk-neutral decision makers. Keywords Good state  bad state  risk neutral  state-contingent income  stategeneral input  production technology  variable input  fixed input  outputcubical technology  state-contingent output set  income curve  cost function  constraints  risk averse After this review of possible relationships between input and state-contingent output, the subject of the following will be specific optimisation problems under alternative assumptions. Since any input can be considered a state-general input, or a special case of this, the derivations in the following will focus on optimising the application of state-general inputs. As will be seen, it is difficult to derive useful criteria for optimal production without knowing the utility function. For a risk neutral decision maker with a linear utility function as in (3.1), useful criteria are relatively easy to derive. But, for a risk-averse decision maker, it is not possible to derive operational criteria. However, it is possible to derive conditions that describe whether a risk-averse decision maker will use more or less of a variable input than a risk-neutral decision maker. The basis for this is a distinction between what may be termed ‘‘good’’ and

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8 Optimisation of Production Under Uncertainty

‘‘bad’’ states of nature. These two concepts must first be defined and this is done in Sect. 8.1. For each of the three types of input presented in the previous chapter, criteria for the optimal production under conditions where there are no restrictions attached to the optimisation problem are derived in Sect. 8.2. In Sect. 8.3, criteria for the optimisation of production seen from the output side are derived. Finally, in Sect. 8.4, criteria under alternative types of restrictions are derived.

8.1 ‘‘Good’’ and ‘‘Bad’’ States of Nature The immediate expectation is that a risk-averse decision maker will use more input than a risk-neutral decision maker, if the input in question enhances the income in those states of nature which are particularly unpleasant or relatively ‘‘bad’’ to riskaverse decision makers, and conversely if the states are relatively ‘‘good’’ states. But, how does one define a relatively ‘‘poor’’ state and a relatively ‘‘good’’ state? The intuitive explanation is that a relatively ‘‘poor’’ state is one in which the marginal utility of income is relatively high. Which states are considered ‘‘good’’ and which are considered ‘‘bad’’ thus depends on the decision maker’s preferences. Therefore, the starting point must be carefully defined. Since the aim is to compare a risk-neutral and a risk-averse decision maker, it is appropriate to take a vector of state-contingent incomes for a risk-neutral decision maker as the benchmark. Consider a risk-neutral decision maker who optimises production by applying the input vector xn.. The outcome is therefore the following vector of state-contingent incomes ðy1 ðxn Þ; . . .; yS ðxn ÞÞ; which yields a (maximum) utility of Uðy1 ðxn Þ; . . .; yS ðxn ÞÞ ¼ p1 y1 ðxn Þ þ p2 y2 ðxn Þ þ    þ pS yS ðxn Þ: Consider also a risk-averse decision maker with a general utility function Uðy1 ; . . .; yS Þ: Since the scale of the utility function is arbitrary, it can be re-scaled so that: S X oUðy1 ðxn Þ; . . .; yS ðxn ÞÞ 1 ð8:1Þ oys s¼1 The re-scaling of (8.1) implies that the sum of the derivatives with respect to ys ðs ¼ 1; . . .; SÞ (sum of marginal utilities) at the point y(xn) is equal to 1. Based on this scaling of the utility function, the following is defined: Definition For a risk-averse decision maker a ‘‘relatively good’’ (or simply ‘‘good’’) state s is a state in which: oUðy1 ðxn Þ; . . .; ys ðxn ÞÞ \ps oys

ð8:2Þ

and a ‘‘relatively bad’’ (or simply ‘‘bad’’) state s is a state where: oUðy1 ðxn Þ; . . .; ys ðxn ÞÞ [ ps oys

ð8:3Þ

8.1 ‘‘Good’’ and ‘‘Bad’’ States of Nature

45

where ys ðxn Þ ðs ¼ 1; . . .; SÞ is the state-contingent income in state s for a riskneutral decision maker who applies the optimal input quantity xn. A ‘‘good’’ state is thus defined as one in which a state-contingent marginal income of MU 1 gives a lower marginal utility than the probability of this state. Similarly, a ‘‘bad’’ state is defined as one in which a state-contingent marginal income of MU 1 gives a higher marginal utility than the probability of this state. The marginal incomes are measured at a level of income achieved by a risk-neutral decision maker who optimises the application of inputs. The marginal utility is measured on a utility function which is locally (at the point y(xn)) scaled so that the sum of marginal utilities over the S states of nature is equal to 1. Obviously, if U is linear (as in (3.1)), then the scaling in (8.1) has already been performed since the sum of probabilities is 1. Also, note that in this situation neither (8.2) nor (8.3) is fulfilled. States where the expression on the left side of (8.2) or (8.3) is equal to ps are defined as ‘‘neutral’’ states of nature. The definition used here corresponds to the definition which is used by Chambers and Quiggin in an informal way (2000, p. 64). It seems appropriate to use the term ‘‘bad’’ about a state where an additional monetary unit (MU) is worth more than the probability of the state in question. This definition is consistent with the view that a ‘‘bad’’ state is something you would like to avoid, and that therefore one more MU would make you happier in a ‘‘bad’’ state than in a ‘‘good’’ state. Defining ‘‘good’’ and ‘‘bad’’ within the context of income for a risk neutral decision maker is also appropriate, as can be seen in the following.

8.2 Criteria for the Optimal Use of Inputs 8.2.1 General Criteria As indicated at the end of Chap. 7, both state-specific and state-flexible inputs can be considered as special cases of state-general inputs. Thus, there is no reason to derive optimisation criteria for each of these inputs. The general optimisation problem can be formulated as follows: Maximise the utility function U given by: U ¼ U ðyÞ ¼ Uðy1 ; . . .; yS Þ

ð8:4Þ

where U is a continuous, differentiable, non-decreasing, quasi-concave function and y ¼ ðy1 ; . . .; yS Þ is a vector of (net) incomes in the S possible states of nature calculated as: n X ys ¼ zs ps  wi xi  cF þ ks ðs ¼ 1; . . .; SÞ ð8:5Þ i¼1

where zs is the production of output in state s, ps is the output price in state s, wi is the price of input i, xi is the quantity of input i, cF is fixed cost and ks is a given (base) income in state s.

46

8 Optimisation of Production Under Uncertainty

Fig. 8.1 Income without production

y2

Indifference curve

k 2 -c F

U(y)=w0

y1

k1-c F

Let us first consider the situation where there is no production. In this case, the decision maker’s consumption possibilities are given by the net income ys ¼ ks  cF ðs ¼ 1; ::; SÞ: This situation is illustrated for S = 2 in Fig. 8.1. In order to increase the ‘‘basic income’’ ks  cF ðs ¼ 1; . . .; SÞ; the decision maker may choose to start producing a product z (For simplicity we only consider the production of one output, but the model can be generalised to multiple outputs). Production is carried out by applying a number of inputs x ¼ ðx1 ; . . .; xN Þ of which the first n inputs are variable inputs and the last N - n are fixed inputs. For simplicity, we consider only one fixed input (xN) for which the following limitations apply: xN  xF  0

ð8:6Þ

where xN is the consumption of the fixed input and xF is the given (fixed) amount of input xN. The production technology is given by the following S implicit functions1): Hs ðzs ; xÞ ¼ zs  fs ðx1 ; . . .; xN Þ  0ðs ¼ 1; . . .; SÞ

ð8:7Þ

Moreover, we will allow for the possibility of expenditure restrictions, i.e.: w1 x1 þ    þ wn xn  C 0

1

ð8:8Þ

Notice that we assume that production zs in state s is independent of production zt in state t. This assumption is not tedious and implies the demand that one is meticulous when defining the inputs x1 ; . . .; xN (further mention in the following). This is especially the case with stateflexible inputs.

8.2 Criteria for the Optimal Use of Inputs

47

where C0 is the given budget. The Lagrange function for this optimisation problem is: L ¼ Uðy1 ; . . .; yS Þ 

S X

ls Hs ðzs ; xÞ  cðxN xF Þ  dð

s¼1

n X

w i xi  C 0 Þ

ð8:9Þ

i¼1

where ls ðs ¼ 1; . . .; SÞ; c and d are Lagrange multipliers associated with the three restrictions (8.6)–(8.8). The decision variables associated with the optimisation problem (8.4)–(8.8) is the input vector x, i.e. the amount of variable and fixed inputs x1 ; . . .; xN : To illustrate the problem of the distribution of the fixed inputs to activities, it is assumed that the fixed amount of input xF can be used for a number of different activities. Assume that the fixed input may be used for M different activities. Thus, the efficient number of inputs is n + M, and the interpretation of N therefore changes to N = n + M. This implies that the Lagrange function takes the following form: L ¼ Wðq1 ; . . .; qS Þ 

S X

ls Hs ðzs ; xÞ  c

s¼1

N X

! xj  x

F

j¼nþ1

d

n X

! wi xi  C

0

i¼1

ð8:10Þ Differentiation of (8.10) with respect to xi ði ¼ 1; . . .; NÞ and zz ðs ¼ 1; . . .; SÞ gives the following first order conditions (internal solutions are assumed):

8.2.1.1 Optimal Combination of Variable Inputs: S P oU

wi t¼1 oyt ¼ S wj P oU t¼1

oyt

VMPti ði; jÞ ¼ 1; . . .; nÞ

ð8:11Þ

VMPtj

where VMPti is pt ðoft =oxi Þ i.e. the value of the marginal product of input xi in state t. Assuming risk neutrality (i.e. a linear utility function (see Eq. (3.1)) oU oyt is equal S P oU to pt, which implies that oyt is equal to 1 (the sum of probabilities equals 1) t¼1

whereby (8.11) reduces to: wi EðVMPi Þ ¼ wj EðVMPj Þ

ði; j ¼ 1; . . .; nÞ

ð8:12Þ

The condition (8.12) expresses that for optimal production the risk-neutral producer must combine variable inputs so that the ratio between the expected marginal products is equal to the ratio of input prices for all combinations of inputs.

48

8 Optimisation of Production Under Uncertainty

8.2.1.2 The Optimal Supply of Variable Inputs The general condition for the optimal supply of variable inputs is: ! S S X X oU oft oU pt ¼ wi þd ði ¼ 1; . . .; nÞ oyt oxi oyt t¼1 t¼1

ð8:13Þ

which under risk neutrality (linear utility function) reduces to: EðVMPi Þ ¼ wi ð1 þ dÞ

ði ¼ 1; . . .; nÞ

ð8:14Þ

Equation (8.14) shows that risk-neutral producers should increase the supply of variable inputs as long as the expected value of the marginal product is higher than the input price, adjusted for any budget restrictions.

8.2.1.3 The Optimal Allocation of Fixed Inputs The general condition for the optimal allocation of fixed inputs is: S X

pt

t¼1

oU oft ¼c oyt oxj

ðj ¼ n þ 1; . . .; NÞ

ð8:15Þ

Under risk-neutrality (linear utility function) (8.15) reduces to: EðVMPj Þ ¼ c

ðj ¼ n þ 1; . . .; NÞ

ð8:16Þ

Condition (8.16) means that fixed inputs, which can be used in N - n different ways (different activities) should be divided between these activities in a manner so that the expected value of the marginal product is equal for all activities. In the earlier example of a division of labour between irrigation and drainage works, (8.16) means that the last hour used should provide the same expected profits under both activities. The conditions for optimum production, as shown in (8.11)–(8.16) are illustrated in Fig. 8.2 for S = 2. The figure assumes that ks  cF ðs ¼ 1; . . .; SÞ is equal to zero or, alternatively, that the axes of the coordinate system measure the change in income compared with the income without production. This then means that net income ys on the axes in Fig. 8.2 is calculated as: ys ¼

S X s¼1

zs ps 

n X

wi xi

ðS ¼ 1; 2Þ

ð8:17Þ

i¼1

and that the optimal project is the one which gives state-contingent incomes ðy1 ; y2 Þ (assuming that the set of net incomes Y is convex). The set of net incomes Y(xF, C0) (the y’s under the curve bb in Fig. 8.2) depends on the amount of fixed inputs (xF) and the budget (C0). The interpretation of this

8.2 Criteria for the Optimal Use of Inputs Fig. 8.2 Optimal statecontingent incomes

49

y2 b

y 2* Indifference curve Y (x F, C 0 )

U(y)=w*

45° y1*

b

y1

set is that it represents the combinations of potential (ex ante) state-contingent incomes that can be achieved with the fixed input quantity xF and the given budget C0, assuming an efficient use of variable inputs (see also details in Rasmussen (2006) and Rasmussen (2003)). It is important here to state that it is not sufficient just to determine the quantities of the N inputs to be used. It is also necessary to consider how the inputs are to be used, including for what activities and with what technology. The following three examples illustrate this. In the first example, assume that the effect of an input depends on the timing. Assume, for example, that the input xi is some kind of fertiliser (e.g. a given type of nitrogen fertiliser, such as calcium nitrate), and suppose further that if the weather during the growing season is ‘‘dry’’ (state u), then the farmer will obtain the highest yield by adding fertiliser on 1st April, while if the weather is ‘‘wet’’ (state v), then the farmer will obtain the highest yield by adding fertiliser on the 15th May. In the second example, assume that the effect of an input depends on the technique used. Assume, for example, that a pesticide applied with a spray nozzle that allows ‘‘fine’’ spray is the best if the weather subsequently is ‘‘dry’’ (state of nature s) and that a pesticide applied with a spray nozzle that allows ‘‘coarse’’ spray is best if the weather subsequently is ‘‘wet’’ (state of nature t). In the third and last example, assume that the effect of an input depends on what activity the input is used for. Assume, for example, that during the winter time you can use idle labour to either improve the property’s irrigation system, which would result in higher profits if it is a ‘‘dry’’ year, or to improve the drainage system, which would result in higher returns if it is a ‘‘wet’’ year.

50

8 Optimisation of Production Under Uncertainty

These examples show that it is not enough to simply decide the quantity of inputs (fertiliser, pesticide, labour) to be used. One must also decide how the input should be used, including the issues of timing, technique and activities. This problem is new in relation to decision making under certainty. The reason is that when it comes to decision making under certainty, and there are several alternative ways to use the input, then it is always (implicitly) assumed that the producer chooses the most efficient way. Take, for instance, the application of fertilisers under certainty. When optimising the amount of fertilisers, it is implicitly assumed that the fertiliser is applied at the most efficient time, and similarly that pesticides are applied by using the best spray nozzle. Under uncertainty, it is not possible to identify a priori which technique or timing would be the best (the most efficient). Surely it depends on the state of nature which only becomes apparent after the input is applied. Therefore, timing, technique and activity also become relevant decision variables under uncertainty. Decision making under uncertainty thus adds new dimensions to the decision making problem: It is necessary to determine not only the optimal amount of input xi ði ¼ 1; . . .; NÞ; but also timing and technique, and maybe even the distribution of these inputs according to activities. One way to integrate these decisions is to consider each input xi ði ¼ 1; . . .; NÞ as a specific combination of input type (such as nitrogen fertiliser, pesticide, labour, etc.), timing (date of application/use), technique (e.g. type of spray nozzle) and activity (used for drainage works or irrigation). Using the examples from before, it means that a pesticide applied with a spray nozzle A is regarded as a different input (xj) to the same pesticide applied with a spray nozzle B (xi). Nitrogen fertiliser applied on 1st April is considered a different input (xk) to nitrogen fertiliser added on 15th May (xl). And labour used to improve the irrigation system is considered a different input (xp) to labour used to improve the drainage system (xq). This extended interpretation of input vector x ¼ ðx1 ; . . .; xN Þ is crucial. It entails, among other things, that if the fixed input, which previously has been described xN, can be applied to M different activities, then these last M elements of the input vector x now express the amount of fixed inputs used at each of the M activities. This more explicit interpretation of the input vector ensures that the production technology can be expressed as already shown in (8.7). Such a technology is called an output-cubical technology2 corresponding to the graphical illustration in Fig. 8.3. The example shows a technology with an input x and an output z, which in state 1 yields z1, and in state 2 yields z2. The output set with the given amount of input x is labelled Z(x) in the figure with an efficient corner point A characterised by a yield of a1 if state 1 occurs and a2 if state 2 occurs (efficient production is assumed). It is not in any way possible to substitute the output in one state by output in another state (right-angled state transformation curves) and production,

2

After Chambers and Quiggin (2000).

8.2 Criteria for the Optimal Use of Inputs Fig. 8.3 Output-cubical technology

51

z2

A

a2

Z( x )

z1

a1

Fig. 8.4 Substitution between states

z2

A

a2 Z(xA)

B

b2 Z(xB)

C

c2

Z(xC) a1

b1

c1

z1

therefore, will always take place at the corner point. This kind of production technology also implies that it is not possible to target the inputs used towards any of the individual states of nature. Note that the statement ‘‘it is not possible to substitute the output in one state for output in another state’’ is an analytical formality. The reality is that it is possible (ex ante) to obtain substitution by choosing between alternative input vectors. This is illustrated for S = 2 in Fig. 8.4, where the rectangles with the corners A, B and C are the state-contingent output sets for the three input vectors, xA, xB and xC, respectively, where the curve through A, B and C is the state-

52

8 Optimisation of Production Under Uncertainty

contingent transformation curve (see Chambers and Quiggin (2000), pp. 40–41, 67). The state-contingent output set for an input vector x, is defined by: Z ðxÞ ¼ fz : zs  fs ðxÞ; s ¼ 1; . . .; Sg; and the associated state-contingent production function is defined by fs ðxÞ ¼ maxfzs g (see Chambers and Quiggin (2000), p. 54).

8.2.2 Generic Example An application of the general optimisation criteria derived in Eqs. (8.11), (8.13) and (8.15) will be illustrated in the following by a number of examples, which are all based on the following generic example illustrating, for instance, the cultivation of a cash crop (z) (e.g. cereals) using two inputs, nitrogen fertiliser (N) and pesticides (P). The following parameter values are used: • Selling price of z (pz): MU 1.00 per unit • Purchase price of nitrogen fertiliser N (wN): MU 4 per unit • Purchase price of pesticides P (wP): MU 3 per unit Besides the two variable inputs, fixed inputs are also used, which involves fixed costs (cF) of MU 4,000. There is no other income so that ks ¼ 0 ðs ¼ 1; . . .; SÞ: Traditional optimisation under the condition of complete certainty assumes ‘‘average weather’’ as the precondition. We assume that the ‘‘average weather’’ production function is: z ¼ 1700 þ 13P  0:031P2 þ 46N  0:12N 2

ð8:18Þ

As the reader may verify by applying the theory in Rasmussen (2011), the optimal application of nitrogen (N*) under complete certainty is 175 units whilst the optimal application of pesticides (P*) is 161 units. The overall optimum production (z*) is 7,356 units and the gross margin is MU 6,173. Now uncertainty is introduced. As the simple starting point, we assume three different states of nature (weather types), namely ‘‘fine weather’’ (state s = 1), ‘‘ordinary weather’’ (state s = 2) and ‘‘bad weather’’ (state s = 3). We also introduce the possibility to differentiate the application of pesticides, assuming two types of pesticides: Pesticide type 1 (P1), which only works if it is ‘‘fine weather’’ and pesticide type 2 (P2) which works especially well if it is ‘‘bad weather’’, but which also has a minor effect if it is ‘‘ordinary weather’’. P2 has no effect if it is ‘‘fine weather’’. The state contingent production functions are assumed to be the following: z1 ¼ 1300 þ 16P1  0:038ðP1 Þ2 þ 47N  0:10 N 2

ð8:19Þ

z2 ¼ 2100 þ 6P2  0:022ðP2 Þ2 þ 46N  0:12N 2

ð8:20Þ

z3 ¼ 1100 þ 18P2  0:035ðP2 Þ2 þ 45N  0:15N 2

ð8:21Þ

8.2 Criteria for the Optimal Use of Inputs

53

All types of input are represented in this example. Nitrogen (N) is a state-general input. Pesticides (P) can be considered a state-flexible input. Pesticide 1 (P1) is a state-specific input, and pesticide 2 (P2) is a state-general input. (As shown, the stateflexible input (P) may be represented by state-general and state-specific inputs). As a starting point, the (subjective) probabilities associated with the occurrence of the three states of nature are assumed to be the following: • Probability of ‘‘fine weather’’: • Probability of ‘‘normal weather’’: • Probability of ‘‘bad weather’’:

0.20 (p1) 0.60 (p2) 0.20 (p3)

If we knew in advance what the state of nature would be (perfect information— no uncertainty (but still variation!)) so that one could choose the correct amount of input (first-best-solution), then it would be optimal to apply the following amounts of input (please check yourself): State 1: N: 215 units P1: 171 units P2: 0 units State 2: N: 175 units P1: 0 units P2: 68 units State 3: N: 137 units P1: 0 units P2: 214 units The example above and the associated preconditions will be used as the benchmark for the following examples, which illustrate the use of the criteria derived in Sect. 8.2.1. As shown by the general criteria in Eqs. (8.11) (8.13) and (8.15), the use of optimisation criteria requires the knowledge of the utility function U (or at least the derivatives of the utility function). Since utility functions are of individual character, no one function is better than another. In the following examples, we assume linear utility functions, i.e. risk-neutral decision makers. This means that it is the criteria in Eqs. (8.12), (8.14) and (8.16) that are used. The reader is invited to try out alternative utility functions. Note also that the examples are based on the absence of budgetary constraints (d = 0).

8.2.3 State-General, Variable Inputs The general criteria are set out in formulas (8.13) and (8.14). As shown in (8.14), a risk-neutral decision maker increases the supply of variable inputs as long as the

54

8 Optimisation of Production Under Uncertainty

expected value of the marginal product is greater than the input price (no budget constraint (d = 0)). Example 8.1 Using the example in the beginning of Chapter 8 gives the following calculation of (8.14): For i = N For i = P1 For i = P2 For i = P

0:20ð47  0:20 NÞ þ 0:60ð46  0:24 NÞ þ 0:20ð45  0:30 NÞ ¼ 4 which gives the solution: N* = 172 units 0:20ð16  0:076 P1 Þ þ 0:60ð0Þ þ 0:20ð0Þ ¼ 3 which gives the solution: P1 ¼ 13 units 0:20ð0Þ þ 0:60ð6  0:044P2 Þ þ 0:20ð18  0:070P2 Þ ¼ 3 which gives the solution: P2 ¼ 104 units P ¼ P1 þ P2 ¼ 117 units

Comparing with earlier results (in Sect. 8.2.2), the conclusion is that one should apply slightly less nitrogen and much less pesticide than the amounts calculated under the assumption of complete certainty. Since the production functions fs(x) or the product price ps varies over states of nature, the following will typically apply to any input vector x:     ofs ðxÞ oft ðxÞ ps ð8:22Þ  wi 6¼ pt  wi oxi oxi i.e. that marginal income in state s is different from the marginal income in state t. Since the expected value of the marginal income at the optimal solution is equal to zero according to (8.14), the consequence is that the optimal solution is characofs terised by the fact that the marginal income ðps ox  wi Þ is positive in some states i and negative in others. In other words, the optimal application of a state-general input implies that the application in some states will be too high and in other states too low compared to the optimal input if the state had been known in advance. This applies to both risk-neutral and risk-averse decision makers. Example 8.2 Comparing the optimal quantities under perfect information in Sect. 8.2.2 with the calculated volumes of Example 8.1 shows that the applied 172 units N is 35 units more than would have been optimal to apply, if we had known in advance that state 3 would occur. The 13 units P1 are also too much if we had known in advance that state 2 or 3 would occur. Finally, the application of 102 units P2 is too much if we had known in advance that state 1 or 2 would occur. The interesting question is whether the optimal supply of input xi to a riskaverse decision maker (who optimises production according to Eq. (8.13)) differs from the optimal supply of xi to a risk-neutral decision maker (who optimises according to (8.14)). If all inputs are considered variable inputs, and therefore substitution between inputs is allowable, then it is not possible to give a general answer. However, if all other inputs than xi are considered fixed inputs, then a risk-averse decision maker will use more input than a risk-neutral decision maker if:

8.2 Criteria for the Optimal Use of Inputs

55

 oUðy1 ðxÞ; . . .; yS ðxÞÞ  n [0 oxi x¼x

ð8:23Þ

where - as before - xn is the optimal supply of input for a risk-neutral decision maker, and ys ðxÞ ¼ ps fs ðxÞ  wxðs ¼ 1; . . .; SÞ: If we carry out the differentiation in (8.23), the following condition results:    S X oU  oU ofs ðxÞ  ¼ ps  wi  [0 ð8:24Þ  n oxi x¼xn s¼1 oys oxi x¼x

It is hard to immediately see what condition (8.24) implies. But, based on the definitions of ‘‘good’’ and ‘‘bad’’ states in (8.1)–(8.3), the condition can be given a useful interpretation as follows. First assume that the S states of nature are ranked so that the first t states are ‘‘bad’’ states, while the remaining S - t states are ‘‘good’’ states. If, under these ofs circumstances the marginal income ðps ox  wi Þ is positive in all these first t ‘‘bad’’ i states and zero or negative in the remaining S - t ‘‘good’’ states, then condition (8.24) is fulfilled and a risk-averse decision maker thus uses more input than a riskneutral decision maker. The criterion can be expressed in words as follows: If the marginal profit from using a state-general input beyond xni is positive in those states which are ‘‘bad’’ for a risk-averse decision maker, then it will be optimal for a risk-averse decision maker to use more of this input than a risk-neutral decision maker. Conversely, if the marginal profit from using a state-general input beyond xni is negative in those states which are ‘‘bad’’ for a risk-averse decision maker, then it will be optimal for a risk-averse decision maker to use less of this input than a risk-neutral decision maker. The general validity of this result assumes that other inputs are considered fixed inputs. Example 8.3 In order to illustrate this criterion, it is necessary to first identify what is meant by ‘‘good’’ or ‘‘bad’’ states. Assume that the utility function has the following form: U ¼ Aya1 yb2 yc3

ðE1Þ

According to (5.2), the relationship between marginal utility of two (arbitrary) states calculated on the certainty line is equal to the ratio between the probabilities. Since the probabilities p1, p2 and p3 are respectively 0.20, 0.60 and 0.20, then the following applies: a p1 0:20 b p2 0:60 ¼ and ¼ ¼ ¼ b p2 0:60 c p3 0:20 This applies if the utility function U has the form: 0:54 0:18 U ¼ Ay0:18 1 y2 y3

ðE2Þ

56

8 Optimisation of Production Under Uncertainty

According to (8.1) the utility function must be scaled so that the sum of the partial derivatives with respect to the y’s is just 1 for the input supply which is optimal for a risk-neutral decision maker. This input supply is calculated in Example 8.1. On the basis of this and the other assumptions in the example listed in Sect. 8.2.2, each y for a risk-neutral decision maker can be calculated. For example, y1 is calculated as: y1 ¼ ð1; 300 þ 16  13  0:038  132 þ 47  172  0:10  1722  172  4  117  3  4; 000 ¼ 1; 589 The other values are calculated in a similar way (y2 = 1.809 and y3 = 856). If we insert these values in (E2) and use the resulting expression in (8.1), then (8.1) may be solved for A which has the value A = 2.20. The scaled utility function thus has the form: 0:54 0:18 ðE3Þ U ¼ 2:20y0:18 1 y2 y3 Then marginal utilities are calculated. For y1 the marginal utility equals: oU ¼ 2:20  0:18  1:5890:82  1:8090:54  8560:18 ¼ 0:18\0:20 oy1 Since the marginal utility (0.18) is less than the probability (p1) for the state in question (0.20), the state is a ‘‘good’’ state according to (8.2). Marginal utilities are calculated similarly for the other two states. This results shows that state 2 is a ‘‘good’’ state (0.48 \ 0.60 = p2) whilst state 3 is a ‘‘bad’’ state (0.34 [ 0.20 = p3). Then calculate marginal profit in the ‘‘bad’’ state 3 by application of additional input. First N: oy3 ¼ 45  0:30 N  4 oN which for N = 172 (optimal application for a risk-neutral decision maker) is equal to -10.6 (\ 0). This means (according to the above stated criteria) that a riskaverse decision maker will use less N than a risk-neutral decision maker, provided the other inputs are kept constant (fixed input). Similar calculations for P2: oy3 ¼ 18  0:07P2  3 oP2 which for P2 = 104 (optimal input for a risk-neutral decision maker) is equal to 7.7 ([0). This means (according to the above stated criteria) that a risk-averse decision maker will use more P2 than a risk-neutral decision maker. Due to the stated conditions, the criterion seems to be very weak. When will the specific situation when ‘‘bad’’ states are exclusively associated with positive marginal profits, or when ‘‘bad’’ states are exclusively associated negative marginal profits, occur? This criterion, therefore, can only give an indication of situations

8.2 Criteria for the Optimal Use of Inputs MU

57 MU f2(x)p2

f1(x)p1

wxi

wxi

xi y1

xi y2

Income in state 1

Income in state 2

a

xi

b

xi

y2 Income curve

b a y1

Fig. 8.5 Derivation of the income curve

where it is likely that one can say something relevant about the relationship between risk-neutral and risk-averse decision makers’ consumption of inputs. If there are only two states of nature (S = 2), the criterion can be supplemented with a graphical analysis as shown in Figs. 8.5 and 8.6. Figure 8.5 derives the so-called income curve. The income curve shows the relationship between income in state 1 and income in state 2 by increasing the application of input x. The income curve is derived on the basis of product value in the two states as shown in the upper part of Fig. 8.5. The product value fs(x)ps (s = 1, 2) minus the

58

8 Optimisation of Production Under Uncertainty y2 Indifference curves Certainty line

xa

U1

xn α=-π1/π2 K

45°

y1

Fig. 8.6 Derivation of xa and xn

cost of input purchases (wixi) gives the income in the two states of nature, as shown graphically in the middle part of Fig. 8.5. The income curve at the bottom of Fig. 8.5 is derived by increasing the application of input xi and then plotting the corresponding income in state 1 and 2 (y1 and y2), respectively, as points in an y1–y2 diagram. By connecting the individual points, we get the direction oriented graph as shown. The arrow on the graph at the bottom of Fig. 8.5 shows that the application of xi increases when moving in the direction of the arrow. The quantities of inputs a and b in the middle part of the figure are illustrated as points in the lower part of Fig. 8.5. When indifference curves are convex, the optimal supply of inputs will be in the range a \ xi \ b. This is outlined in more detail in Fig. 8.6. In 8.6, income curve K (derived as shown in Fig. 8.5) is plotted. The optimal supply of xi is where the income curve is tangent to the highest indifference curve. For a risk-neutral decision maker, this is the case when applying the amount xni , and for a risk-averse decision maker, when applying the amount xai . As shown in the example in Fig. 8.6, a risk-averse decision maker uses more input than a risk-neutral decision maker. This result is also obtained by using the above-mentioned criterion. First note that the following is valid at the point xn: oU oy1 oU oy2

\

p1 p2

8.2 Criteria for the Optimal Use of Inputs

59

Due to the scaling in (8.1): oUðy1 ðxn Þ; y2 ðxn ÞÞ=oy1 \p1 and oUðy1 ðxn Þ; y2 ðxn ÞÞ=oy2 [ p2 which according to (8.2) and (8.3) implies that state 1 is a ‘‘good’’ state and state 2 is a ‘‘bad’’ state. At the same time, Fig. 8.6 shows that the marginal income after having applied xn units of input is positive in state 2 (the ‘‘bad’’ state) and negative in state 1 (the ‘‘good’’ state). According to the above derived criterion, this implies that a risk-averse decision maker will use more input than a risk-neutral decision maker, which is precisely what is shown in Fig. 8.6 where xai represents the optimal input for a risk-averse decision maker. Figure 8.6 shows how difficult it is to give a general answer to the question whether a risk-averse decision maker will use more or less input than risk-neutral decision maker. The answer depends firstly on which state is the ‘‘bad’’ state and which state is the ‘‘good’’ state (whether the income curve is above or under the certainty line). It also depends on whether the marginal income after the application of xn input units is positive or negative in the ‘‘good’’/’’bad’’ state (which direction the arrow on the income curve points). If, for instance, the income curve in Fig. 8.6 had been pointing in the other direction (the arrow facing the opposite way), then xai \xni and a risk-averse decision maker would use less input than a risk-neutral decision maker. Finally, no concern is given to the possibility that when adjusting the production, input substitution may take place (the quantity of other inputs is assumed fixed).

8.2.4 State-Specific Input State-specific input is effective only in one state of nature. Assume that the input in question is only active in state t. In this case, the general condition (8.13) reduces to: S X oU oft oU pt ¼ wi ð8:25Þ oyt oxi oys s¼1 If the decision maker is risk-neutral, (8.25) is equivalent to: pt VMPit ¼ wi

ð8:26Þ

where VMPit is the value of the marginal product of input xi in state t. Consequently, a risk-neutral decision maker should increase the application of a state-specific input as long as the value of the marginal product weighted by the probability of being in state t (in which the input is active), is greater than, or equal to, input price wi.

60

8 Optimisation of Production Under Uncertainty

Example 8.4 In the example in Sect. 8.2.2, P1 is a state-specific input. If we insert the relevant parameter values in (8.26), the following condition appears: 0; 20  ð16  0:076P1 Þ ¼ 3 which has the solution P1 = 13. A risk-averse decision maker uses more input than a risk-neutral decision maker if: oUðy1 ðxÞ; . . .; yS ðxÞÞ jx¼xn [ 0 oxi

ð8:27Þ

where, as before, xni is the optimal application of input for a risk-neutral decision maker. Perform the differentiation of (8.27) and insert the value of VMPit calculated in (8.26). This gives the following condition: S X oU oU [ pt jx¼xn oyt oys s¼1

ð8:28Þ

Since the sum on the right side of (8.28), according to the scaling in (8.1), is equal to 1, (8.28) expresses that state t is a ‘‘bad’’ state (see (8.3)). Therefore it can be concluded that if the state-specific input xi is targeted towards a ‘‘bad’’ state, then a risk-averse decision maker will use more input than a risk-neutral decision maker. Conversely, if the state-specific input is directed towards a ‘‘good’’ state, then a riskaverse decision maker will use less input than a risk-neutral decision maker. The general validity of this result assumes that all other inputs are fixed inputs. Example 8.5 As shown in Example 8.3, state 3 (which the input P1 is targeted at) is a ‘‘good’’ state. Therefore, a risk-averse decision maker will use less P1 than the 13 units, which a risk-neutral decision maker would use.

8.2.5 State-Flexible Input A state-flexible input is characterised by the opportunity to direct the application of inputs towards the states which are evaluated as being the best. In principle, the same could be achieved if the decision maker had a series of state-specific inputs that were combined in varying proportions. It is, therefore, to be expected that the criteria for state-flexible inputs resemble the criteria for state-specific inputs. Assume that the input xi is a state-flexible input in the sense that there are alternative methods for the application of xi. The assumption is that these alternative methods are targeted at each individual state of nature, and for simplicity it is assumed here that there is a method specific to each state, and that if the method which is targeted at state t is used, then the input xi is only active in state t:

8.2 Criteria for the Optimal Use of Inputs

61

According to the discussion in Sect. 8.2, this condition implies that instead of one input xi, we may define S inputs, one for each of the state-specific application methods. Instead of input xi, we consider the vector ðxi1 ; . . .; xiS Þ of inputs, where xit is the amount of input applied using a method targeted at state t. The general condition (8.13) hereby reduces to: pt

X oU oU oft ¼ wi oyt oxit oys s2X

ðt ¼ 1; . . .; SÞ

ð8:29Þ

which, for a risk-neutral decision maker, implies the condition that: pt VMPit ¼ wi

ðt ¼ 1; . . .; SÞ

ð8:30Þ

The condition (8.30) expresses that, for a risk-neutral decision maker, for the optimal application of input targeted at state t (input active only in state t), the last input unit should provide an added value (marginal product value) which, when multiplied by the probability of this state, is equal to the input price. The condition is—as expected—completely identical to condition (8.26) which applies to statespecific inputs. The same applies to the question of whether a risk-averse decision maker uses more or less input targeted at a specific state of nature than a risk-neutral decision maker. This means that if the state specific input xit is targeted at a ‘‘bad’’ state (t = ‘‘bad’’), then a risk-averse decision maker will use more input than a riskneutral decision maker. Conversely, if the state-specific inputs is targeted at a ‘‘good’’ state, then a risk-averse decision maker will use less input than a riskneutral decision maker. As previously mentioned, these reported results concern strictly state-flexible inputs. The results are, as shown, completely analogous to the results concerning state-specific inputs. Similar conditions apply to generally state-flexible inputs (xis is a state-general input if it can affect production, not only in state s, but also in other states). The optimisation of generally state-flexible inputs is carried out for every ‘‘new’’ input xis according to the same criteria as applied to state-general inputs.

8.3 Optimisation from the Output Side 8.3.1 General Criteria As is the case of riskless production, the optimisation of production under uncertainty can also be considered from the output side. The criterion for optimal production under riskless conditions is: MC = p, where MC is marginal cost and p is product price. Under uncertainty, it is not sufficient to consider just one product. Indeed, under uncertainty, a product is not just one product. With S different states of nature, a

62

8 Optimisation of Production Under Uncertainty

given product type (e.g. barley) is equal to S different products since the product type which is expected to be produced in a state s, is different from the same product type which is expected to be produced in state t (This type of statement corresponds to the statement that ten kg of grain in one year is different from ten kg of grain in two years). The problem of production optimisation under uncertainty therefore includes the optimisation of production over S states of nature as follows: As the starting point, assume (as earlier) that the decision maker has the following general utility function: U ¼ U ðyÞ ¼ Uðy1 ; . . .; yS Þ

ð8:31Þ

where y ¼ ðy1 ; . . .; yS Þ is a vector of (net) incomes in the S possible states of nature. Assume that the income ys in state s is generated by cost minimisation, i.e.:   ðs ¼ 1; . . .; SÞ ð8:32Þ ys ¼ rs  cT r; P; w; xF where rs is the revenue (value of output) value in state s, calculated as: rs ¼

M X

zms pms

ðs ¼ 1. . .SÞ

ð8:33Þ

m¼1

and where zms is the yield of product m in state s and pms is the product price of product m in state s. Moreover, the function cT represents the (lowest) total cost of producing the revenue vector r ¼ ðr1 ; . . .; rS Þ; when product prices are given by an M 9 S matrix of product prices P, the input prices of variable input by input price vector w ¼ ðw1 ; . . .; wn Þ and the quantity of fixed input by the vector xF ¼ ðxnþ1 ; . . .; xN Þ; giving rise to fixed costs cF. Total costs (cT) are thus given by:     ð8:34Þ cT r; P; w; xF ¼ c r; P; w; xF þ cF where c(r, P, w, x,F) is variable costs given by: M X   c r; P; w; xF ¼ minfwx : x 2 XðZ; xF Þ; pms zms  rs ; s ¼ 1; . . .; Sg X

ð8:35Þ

m¼1

if there exists an M 9 S output matrix Z (with elements zms), which when using the fixed input xF can generate the revenue r, and ? otherwise. X ¼ ðx1 ; . . .; xn Þ is the vector of variable inputs, where the amount of input i allocated to product m is M P xim ; X is an N 9 M matrix of elements xim, called xim, and where therefore xi ¼ m¼1

and X(Z, xF) is the input set that can produce Z when the fixed input is given by the input vector xF. Since the formulation of (8.32) assumes that any revenue vector is produced as cheaply as possible within the given technology X(Z, xF) in (8.35), then one can be

8.3 Optimisation from the Output Side

63

sure that for every possible income vector, the application of the variable inputs has been optimised and that fixed inputs have been optimally allocated. The ‘only’ thing remaining is the choice of the output matrix Z (with elements zms) that maximises U. The derivative of the utility function (8.31) with respect to the production of product m in state s can be derived using (8.32) and (8.33) as: ! S oU oU ocT X oU ðm ¼ 1; . . .; M; s ¼ 1; . . .; SÞ ð8:36Þ ¼ pms  ozms oys ors t¼1 oyt However, the problem is that one cannot generally assume that the cost function cT is differentiable (see Chambers and Quiggin, 2000, Fig. 4.3, p. 135). If cT is (locally) differentiable in rs, then the right side of (8.36) can be interpreted as follows: The change in utility by producing one more product unit is equal to the price of this unit (pms) (for instance MU 5 per kg) multiplied by the extra utility achieved per MU of marginal income (the first part in the parenthesis, for instance an additional utility of 4 utils per MU of additional income in which case there would be an extra utility of 20 utils). But, we must also remember that there have been T additional costs of producing this extra income. These additional costs are oc ors (for instance MU 0.50) of producing an extra revenue of one MU in state s. With a total of 5 extra MUs, this implies an additional cost of MU 2.50. Given the marginal utility of 4 utils per MU in state s, this corresponds to a deduction of 2.50 times 4 = 10 utils. But, this is not enough because these additional costs will of course also have an effect in the other states. Therefore, we have to deduce the extra costs T of MU 2.50 times the marginal utility for each and every state. This is why oc ors is multiplied by the all marginal utilities. The optimal values of the elements zms C 0 are determined by the use of (8.36), which results in the following first order conditions, which only apply if cT is (locally) differentiable in rs: ! S oU ocT X oU  0 ðm ¼ 1; . . .; M; s ¼ 1; . . .; SÞ ð8:37Þ  oys ors t¼1 oyt zms

S oU ocT X oU  oys ors t¼1 oyt

! ¼0

ðm ¼ 1; . . .; M; s ¼ 1; . . .; SÞ

ð8:38Þ

oU ¼ ps ) and assuming an internal Assuming risk neutrality (which implies that oy s solution, the conditions (8.37) and (8.38) reduce to:

ps ¼

ocT ors

ðm ¼ 1; . . .; M; s ¼ 1; . . .; SÞ

ð8:39Þ

64

8 Optimisation of Production Under Uncertainty

which means that one should expand production in state s as long as the cost per MU of product value is less than the probability of the state in question. Optimal production is characterised by the condition that the extra costs of producing output with a value of one MU in state s are equal to the probability of this state.

8.3.2 Specific Criteria As shown in the previous section, it is difficult to derive operational criteria based on the most general model formulation in (8.31)–(8.35). In the following, we will simplify the problems with a view to deriving operational criteria for optimal production. Assuming only one output, the optimisation problem under uncertainty can be formulated as follows: MaxUðy1 ; . . .; yS Þ with respect to z1 ; . . .; zS

ð8:40Þ

where: ys ¼ ps zs  cs ðw; zs Þ and where cs(w, zs) is the cost function: cs ðw; zs Þ  minfwx : fs ðxÞ  zs ; s ¼ 1; . . .; Sg

ð8:41Þ

and x is a vector of state-general inputs. As shown by Chambers and Quiggin (2000), the lowest costs of producing the entire state contingent output vector z satisfy the following condition: cðw; zÞ  maxfcs ðw; zs Þg ðs 2 XÞ

ð8:42Þ

Since state-specific input and state-flexible inputs are just special cases of stategeneral inputs, the condition (8.42) can be considered a general condition concerning the cost of producing the output vector z. Even under conditions where (8.42) holds with an equal sign, (8.42) is not necessarily differentiable. One, therefore, cannot apply the usual optimisation tools (differentiation) for solving (8.40). However, if one considers the special case where all inputs ðx1 ; . . .; xn Þ are state-specific inputs, i.e. each input affects output only in one state of nature, in this case the cost function cs(w, zs) is: cs ðw; zs Þ ¼ minfws xs : fs ðxs Þ  zs ; s ¼ 1; . . .; Sg ¼ cs ðws ; zs Þ

ð8:43Þ

where xs is the (sub)vector of inputs specifically targeted at state s and ws is the corresponding (sub)vector of input prices. As production in state s is completely independent of the input applied with a view of producing output in state t (and vice versa), the cost function for the entire vector of state-contingent output has the following simple additive form:

8.3 Optimisation from the Output Side

65

cðw; zÞ ¼

S X

cs ðws ; zs Þ

ð8:44Þ

s¼1

and since (8.43) is differentiable, so is (8.44). Under the condition that all inputs are state-specific, it is therefore possible to differentiate (8.40) with respect to each element ðz1 ; . . .; zS Þ: If this is done the partial derivatives are set equal to zero, the following S conditions for optimal production appear: S X oU oU ps  MCs ¼ 0 oys oyt t¼1

ðs 2 XÞ

ð8:45Þ

MCs is the marginal cost with respect to zs i.e. the derivative of cost function cs in (8.43) with respect to zs. For a risk-neutral decision maker (8.45) reduces to: ps ps ¼ MCs

ð8:46Þ

Thus, if the input is state-specific, then a risk-neutral decision maker should increase the production of output in state s as long as the marginal cost is lower than the product price in state s multiplied by the probability of state s. The criterion is similar to the criterion for optimal production under certainty, where the criterion for optimal production is that the product price p should be equal to the marginal cost MC. Under uncertainty, the output price is ‘‘adjusted’’ by the probability that the state in question—and hence the price—is obtained. Summation over s on both sides of the equal sign in (8.46) gives: X X ps p s ¼ MCs ðws ; zÞ ð8:47Þ s2X

s2X

which, if the left side is calculated, gives: X EðpÞ ¼ MCs ðws ; zÞ

ð8:48Þ

s2X

The sum on the right side of Eq. (8.48) is the cost of producing a (marginal) unit more in all states of nature. It will be optimal to increase production for a riskneutral producer as long as this cost is lower than the expected product price (left side). It is possible to derive conditions which describe the circumstances under which it would be advantageous for a risk-averse decision maker to produce more output in state s (s = 1,.., S). The general condition that it is optimal for a riskaverse producer to produce more output in state s is: oUðy1 ðzn Þ; . . .; yS ðzn ÞÞ [0 ozs

ð8:49Þ

66

8 Optimisation of Production Under Uncertainty

where: zn ¼ ðzn1 ; zn2 ; . . .; znS Þ If the differentiation of (8.49) is carried out and the expression of MCs in (8.46) is inserted, then the following condition is obtained:  ps

X S oU oU  ps [0 oys oyt t¼1

ð8:50Þ

which is fulfilled when qU/qys [ ps. But, according to (8.3) this is precisely the definition of a ‘‘bad’’ state. One can therefore conclude that if there is a ‘‘bad’’ state of nature, then a risk-averse decision maker will produce more in the state in question than a risk-neutral decision maker. Similarly, if s is a ‘‘good’’ state, then a risk-averse decision maker will produce less in state s than a risk-neutral decision maker. The general validity of this result assumes that all inputs are statespecific. As for the relationship between outputs, the following relationship between output in state s and t can be derived from (8.45): oU oys ps oU oyt pt

¼

MCs MCt

ð8:51Þ

Assuming that ps = pt = p (i.e. no price uncertainty, but only production uncertainty), (8.51) can be reduced to the following: oU oys oU oyt

¼

MCs MCt

ð8:52Þ

which means that the condition for optimal production is that the absolute value of the slope of the indifference curve is equal to the ratio between the marginal cost when producing in state s and state t, respectively. Condition (8.52) implies that a risk-neutral decision maker, who only uses state-specific inputs and where there is only production uncertainty (no price uncertainty), should allocate input to the various states of nature in such a way that the relationship between the marginal costs is equal to the ratio of the probabilities of the states. If a risk-neutral decision maker, who produces the optimal amount of output, is producing more in state s than in state t (state s is the ‘‘good’’ and state t is the ‘‘bad’’ state), then the following will apply for a risk-averse decision maker who produces the same: The left side of (8.52) is smaller than the right side of (8.52). Thus, to fulfil the condition (8.52), production must be adjusted so that the right side is reduced, i.e. marginal costs in state s decrease or marginal costs in state t increase (or both). This implies that a risk-averse decision maker will reduce production in state s relative to production in state t. If the risk-neutral decision maker already produces more in state t than in state s, then it will be optimal for a

8.3 Optimisation from the Output Side

67

risk-averse decision maker to increase production in state s relative to production in state t. The relationship between productions in different states of nature will thus be different when comparing the risk-neutral and the risk-averse decision maker. The risk-averse decision maker will tend to produce more in the ‘‘bad’’ states compared to production in the ‘‘good’’ states. Finally, assume that the state-specific inputs considered are strictly state-flexible inputs to be allocated on S states of nature. In this case, (8.52) may be used to derive the optimal allocation of a limited amount of input on the different states of nature for a risk-neutral decision maker. In this context, the interpretation of (8.52) is that if a decision maker is risk-neutral and there is no price uncertainty (only production uncertainty), then a limited amount of state-flexible input should be allocated between the states of nature in such a way that the ratio of the marginal costs in each state of nature is equal to the ratio between the probabilities of being in these states of nature.

8.4 Optimisation Under Constraints The general criterion for the allocation of fixed inputs to alternative uses is given by (8.15) and (8.16) in Sect. 8.2.1. In the present section, the problem of optimal allocation of a given amount of input will be discussed in more detail, whilst a few additional optimisation problems under constraints will be addressed.

8.4.1 Allocation of a Fixed Input to Multiple Outputs Under certainty, the classical problem consists of the allocation of a given amount of input to a number of production opportunities (output) in such a way that the value of the total output is maximised. Under uncertainty, a given type of output produced under S different states of nature is considered as being S different outputs. Therefore, the optimal use of inputs when producing under uncertainty is a matter of the optimal allocation of inputs both to products and to states of nature. With m different outputs, S different states of nature and state-flexible inputs, the problem is to optimally allocate the given amount of input to a total of m times the S different ‘‘products’’. The following assumes that the amount of a single input x is given as a fixed quantity x. 8.4.1.1 State-General Inputs For state-general inputs, it is not possible to target the application of input to states of nature. The optimisation problem is, therefore, ‘only’ to allocate the given amount of input to the m different products.

68

8 Optimisation of Production Under Uncertainty

The optimisation problem is formulated as follows: Max U ðy1 ; . . .; yS Þ with respect to x1 ; . . .; xm

ð8:53Þ

subject to: m X

xj  x

j¼1

where xj is the application of input x to the production of product j. The state-contingent income ys is: ys ¼

m X

fsj ðx j Þpsj

ðs 2 XÞ

ð8:54Þ

j¼1

where fsj is the production function for product j in state s and psj is the price of product j in state s. The Lagrange function for this problem is: L ¼ Uðy1 ; . . .; yS Þ  kðx1 þ    þ xm  xÞ

ð8:55Þ

The conditions for the optimal application of input xj to each of the m products are derived from the Kuhn-Tucker conditions as: S oL X oU j ofsj ðx j Þ ¼ p  k0 j ox oys s ox j s¼1

x j ¼ 0 for

ðj ¼ 1; . . .; mÞ

oL \0 ðj ¼ 1; . . .; mÞ ox j

oL ¼ ðx  ðx1 þ    þ xm ÞÞ  0 ok k ¼ 0 for

oL [0 ok

k0 

j

ð8:56Þ

ð8:57Þ ð8:58Þ ð8:59Þ ð8:60Þ

If the marginal product ofsj ðx j Þ ox is positive in all states ðs 2 X and x j  xÞ for just one of the m products, then (8.56) will be fulfilled only if k [ 0 since both qU/qys and the product prices psj are positive. In this case, (8.58) applies with equal sign, and it is optimal to use the entire amount of input x: As long as the marginal product is positive in all states for just one of the m products, it is advantageous to increase the application of input x, and if so, it would be optimal to use the entire input amount x —regardless of the shape of the utility function (as long as it is

8.4 Optimisation Under Constraints

69

non-decreasing). In this context, condition (8.56) also shows that the relationship between products i and j at the optimal allocation of input is characterised by: S X oU s¼1

oys

psj

S ofsj ðx j Þ X oU i ofsi ðxi Þ ¼ p k [ 0 ðj; i ¼ 1; . . .; mÞ j ox oys s oxi s¼1

ð8:61Þ

If - on the contrary  - input x has the potential for a ‘‘toxic effect’’ in certain j states of nature, i.e. ofs oxj \0 for x j  x in at least one of the S states of nature for all products, then there is no unique solution. In this case, one cannot be sure that it is optimal to use the entire amount x of input. It may be optimal, but it depends on the severity of this ‘‘toxic effect’’ (diminishing returns, and shape of the utility function). Regarding the relationship between the individual products, the following applies to those products for which the optimal solution of x j is greater than zero: S X oU s¼1

oys

psj

S   ofsj ðx j Þ X oU i ofsi ðxi Þ ¼ ps  0 for x j ; xi [ 0 j i ox oys ox s¼1

ð8:62Þ

If there is a possibility of a toxic effect for xj  x for all products in one or more states of nature, then it is not immediately possible to derive the optimal input application. For those products to which input is applied, the marginal utility has to be equal. Assuming a linear utility function (risk neutrality) the condition (8.56) can be written as: S j j oL X j ofs ðx Þ ¼ p p  k0 s s ox j ox j s¼1

ðj ¼ 1; . . .; mÞ

ð8:63Þ

The expression under the summation sign is the expected value of the marginal product for product j (E(VMPj)). The condition therefore can also be written as:  oL ¼ E VMP j  k  0 j ox

ðj ¼ 1; . . .; mÞ

ð8:64Þ

If E[VMPj] is strictly positive for at least one j (one product type), then fulfilment of (8.64) implies that k is also positive. Therefore, it will be optimal to apply the entire amount of input x: The criterion for the optimal allocation of inputs to the m products in this case also appears from (8.64) because for positive values of k the following must be true:   ð8:65Þ E VMP j ¼ E VMPi  0 ði; j ¼ 1; . . .; mÞ The good old optimisation criterion, i.e. that one should continue the supply of inputs as long as the value of the marginal product (VMPj) is positive, is now—under uncertainty, and with the assumption that the decision maker is risk neutral— replaced by the optimisation criterion that you should continue the application of

70

8 Optimisation of Production Under Uncertainty

input as long as the expected value of the value of the marginal product is positive. Moreover, the last unit of input applied should provide the same expected marginal value in the production of product i as in the production of product j. For a riskneutral decision maker, the following is therefore true: application of input should continue as long as the expected value of the marginal product is positive and the expected value of the marginal product must be the same for all products. If product prices are not uncertain (the price of output j (pj) is the same in all states of nature), then the condition (8.65) implies: pi EðMPP j Þ ði; j ¼ 1; . . .; mÞ ¼ p j EðMPPi Þ

ð8:66Þ

where E(MPPi) is the expected marginal product of product i. A risk-neutral producer who only faces production uncertainty should, therefore, allocate a given amount of input in such a way that the relationship between product prices is equal to the reciprocal relationship between the expected marginal products. This criterion corresponds closely to the optimisation criterion under riskless production, the only difference being that the term ‘‘marginal product’’ is replaced by ‘‘expected marginal product.’’

8.4.1.2 State-Specific Inputs A state-specific input is only effective in one of the S states of nature. For instance, a crop protection spray which is only active in the state of nature ‘‘wet weather’’. How one should interpret the concept state-specific inputs when there are several outputs has not previously been discussed. Does the ‘state-specific’ apply to all outputs, or to only one of the outputs? The following assumes that the state-specific applies to all the considered outputs, and thus the fixed input x is effective only in one state of nature, and that this is the same state of nature for all outputs. The optimisation problem is the following: Max U ðy1 ; . . .; yS Þ with respect to x1 ; . . .; xm under the constraint:

m X

xj  x

j¼1

where: yt ¼

m X j¼1

ft j ðx j Þptj

ðj ¼ 1; . . .; mÞ

ð8:67Þ

8.4 Optimisation Under Constraints

71

ys ¼

m X

ksj for s 6¼ t

j¼1

We assume that input x is only effective in state t for all products. For other m P ksj Þ: states, the income ys is constant ðys ¼ j¼1

The Lagrange function for this problem is: m X

L ¼ Uðy1 ; . . .; yS Þ  k

! j

x x

ð8:68Þ

j¼1

The conditions for the optimal application of x are derived from the KuhnTucker conditions as: oL=ox j ¼

oU j oft j ðx j Þ p  k0 oyt t ox j

x j ¼ 0 for

oL \0 ox j

ðj ¼ 1; . . .; mÞ

ðj ¼ 1; . . .; mÞ

oL ¼ ðx  ðx1 þ . . . þ xm ÞÞ  0 ok k ¼ 0 for

oL [0 ok

k0

ð8:69Þ ð8:70Þ ð8:71Þ ð8:72Þ ð8:73Þ

j

oft If the marginal product ox j in (8.69) is strictly positive for at least one j, then k is also strictly positive, and (8.71) thus applies with equal sign. For those products where the optimal solution involves a strictly positive application of the input (xj [ 0), the optimal allocation of inputs to the two products i and j is according to (8.69) is characterised by the following condition:

ptj

i i oft j ðx j Þ i oft ðx Þ ¼ p 0 t ox j oxi

ðj ¼ 1; . . .; mÞ

ð8:74Þ

This result shows that, regardless of the form of the utility function (increasing), it would be optimal to increase the supply of the inputs-specific input x as long as the marginal product is positive, and if so to use the entire input volume x: The allocation of inputs to products follows the criterion (8.74), i.e. that the last input unit should give the same value of product i as of product j, or alternatively, that the relationship between the price of product i and product j must be equal to the ratio of the marginal product for product j and product i.

72

8 Optimisation of Production Under Uncertainty

8.4.1.3 State-Flexible Inputs For state-flexible inputs, the decision maker has the opportunity to target the available amount of input x both to the m products and the S states of nature. If we use the term xsj to describe the use of input x to output j and targeted state s, then the optimisation problem can be stated as follows: Max U ðy1 ; . . .; yS Þ with respect to x11 ; x21 ; . . .; xsj ; . . .; xm S

ð8:75Þ

subject to the constraint: m X S X

xsj  x

j¼1 s¼1

where: ys ¼

m X

fsj ðxsj Þpsj ðs 2 XÞ

j¼1

The Lagrange function for this problem is: L ¼ Uðy1 ; . . .; yS Þ  kð

m X S X

xsj  xÞ

ð8:76Þ

j¼1 s¼1

The conditions for the optimal application of x are derived from the KuhnTucker conditions: oL oxsj

¼

oU j ofsj ðxsj Þ p  k0 oys s oxsj

xsj ¼ 0 for

oL oxsj

\oL=oxsj \0

ðs ¼ 1; . . .; S; j ¼ 1; . . .; mÞ

ð8:77Þ

ðs ¼ 1; . . .; S; j ¼ 1; . . .; mÞ

ð8:78Þ

m X S X oL xsj Þ  0 ¼ ðx  ok j¼1 s¼1

k ¼ 0 for

oL [0 ok

k0

ð8:79Þ

ð8:80Þ ð8:81Þ

If the quantity x is really  limiting (internal solution), then the optimal solution is characterised by ofsj ðxsj Þ oxsj [ 0 and therefore k [ 0. In this case, (8.77) applies with an equal sign, and the optimal solution is therefore characterised by the following:

8.4 Optimisation Under Constraints

psj

ofsj ðxsj Þ oxsj

¼ pis

73

ofsi ðxis Þ 0 oxis

ðs ¼ 1; . . .; SÞ

oU j ofsj ðxsj Þ oU j oft j ðxtj Þ p ¼ p 0 oys s oxsj oyt t oxtj

ðj ¼ 1; . . .; mÞ

ð8:82Þ

ð8:83Þ

As it appears in (8.82), the allocation of the inputs to the two products i and j within a given state does not depend of the utility function. According to (8.82), input in each state of nature s must be allocated to outputs in such a way that the value of the last input unit is the same for the two products Condition (8.83) shows that the allocation of inputs to states depends on the utility function. Assuming a linear utility function (risk-neutrality), the condition (8.83) may, however, be written as: ps psj

ofsj ðxsj Þ oxsj

¼ pt ptj

oft j ðxtj Þ oxtj

0

ðs ¼ 1; . . .; SÞ

ð8:84Þ

which shows that, for a given product, input should be allocated to states in such a way that the value of the marginal products weighted by the probability of the state in question are equal. The following, therefore, applies to a risk-neutral decision maker: A given amount of input used to produce a given product should be allocated to the S states of nature in such a way that the marginal product in any state of nature multiplied by the probability of the state of nature in question is the same for all states. Example 8.6 In the example presented in Sect. 8.2.2, the pesticides (P) may be regarded as a state-flexible input. If, for instance, the firm is subject to a restriction  units of pesticides, then the which implies that it is not allowed to use more than P problem is how many units of pesticide 1 (P1) and how many units of pesticide 2 (P2) should be used. The example in Sect. 8.2.2 cannot be used directly, since P2 is a state-general (and not state specific) input. However, suppose that P2 has no effect in state 2, but does have an effect in state 3. In this case, both P1 and P2 are state-specific inputs. According to the recently derived criterion (8.84), the following must apply for optimal use of pesticides: p1 ð16  0:076P1 Þ ¼ p3 ð18  0:070P2 Þ

ðE4Þ

 P1 þ P2  P

ðE5Þ

and:

where the parentheses in (E4) are the marginal product in state 1 and state 3, respectively.  ¼ 50; i.e. a limit of 50 units of pesticides. The Assume, for instance, that P simultaneous solution of (E4) and (E5) gives the result P1 = 10 and P2 = 40.

74

8 Optimisation of Production Under Uncertainty

8.4.2 Minimising Costs A classic problem in production economics is to determine the combination of inputs which produces a given amount of output in the cheapest possible way. The corresponding problem under uncertainty is the production of a given expected amount of output, or alternatively, a given vector of state-contingent outputs in the cheapest possible way. In the former case, (the production of a given expected amount of output) the problem is formulated as follows: Minfw1 x1 þ    þ wn xn Þ subject to:

X

ð8:85Þ

ps zs ¼ z

s2X

where zs is a function of x1 ; . . .; xn i.e. zs ¼ zs ðx1 ; . . .; xn Þ assuming that each of the n inputs is a state-general input. The Lagrange function is given by: ! X ps z s  z ð8:86Þ L ¼ w1 x1 þ    þ wn xn  k s2X

Differentiating (8.86) with respect to xi ði ¼ 1; . . .; nÞ and setting the derivative equal to zero produces the following condition for the optimal combination of xi og xj: wi EðMPPi Þ ð8:87Þ ¼ wj EðMPPj Þ where E(MPPi) is equal to: EðMPPi Þ ¼

X

ps

s2X

ozs ðx1 ; ::; xn Þ oxi

ð8:88Þ

The condition is analogous to the riskless condition, the only difference being that the ratio of marginal products is now replaced by the ratio of expected marginal products. If the task is instead to produce a given vector of state-contingent output in the cheapest possible way, then the problem is: Minfw1 x1 þ    þ wn xn Þ subject to: z1  z1  zs  zs

ð8:89Þ

8.4 Optimisation Under Constraints

75

where: zs ¼ zs ðx1 ; . . .; xn Þ ðs ¼ 1; . . .; SÞ and the assumption is that all n inputs are state-general inputs. The Lagrange function is: L ¼ w1 x1 þ . . . þ wn xn þ

S X

ks ðzs  zs Þ

ð8:90Þ

s¼1

which implies the following Kuhn-Tucker conditions: wi 

S X

ks

s¼1

xi ðwi 

S X s¼1

ozs 0 oxi

ks

ði ¼ 1; . . .; nÞ

ozs Þ¼0 oxi

z s  zs  0 ks ðzs  zs Þ ¼ 0

ði ¼ 1; . . .; nÞ

ðs ¼ 1; . . .; SÞ ðs ¼ 1; . . .; SÞ

ð8:91Þ

ð8:92Þ ð8:93Þ ð8:94Þ

Since all inputs are assumed to be necessary inputs, there will be an internal solution ðxi [ 0; i ¼ 1; . . .; nÞ: The conditions in (8.91) therefore apply with equal signs. The optimal production will therefore be characterised by: PS ozs wi s¼1 ks oxi ¼ PS ði; j ¼ 1; . . .; nÞ ð8:95Þ ozs wj s¼1 ks oxj Moreover, not more than n of the S conditions in (8.93) will apply with equal sign. This means that not more than n of the S values of ks will be strictly positive. This latter fact can be illustrated graphically when there are two inputs, as shown in Fig. 8.7. For simplicity, assume that there are only two states of nature. The relationship between input prices (-w1/w2) is given as the slope of the dotted line in the figure. The left side of the figure (A) shows that the lowest cost at which the statecontingent output quantities z1 og z2 can be produced, is achieved by using a units x1 and b units x2. The condition (8.93) is satisfied with an equal sign both for s = 1 and s = 2. The right side of the figure (B) shows that the lowest cost at which the statecontingent output quantities z1 and z2 can be produced using c units x1 and d units x2. The condition (8.93) is, however, satisfied with equality sign only for s = 1, because the stated amount of input production is higher than z2 in state s = 2.

76

8 Optimisation of Production Under Uncertainty x2

x2 ⎯z 2

A

B ⎯z 1

⎯z 1 ⎯z 2 b

d

a

c

x1

x1

Fig. 8.7 Derivation of minimum costs for given production

In situation (B) in Fig. 8.7, the condition (8.95) can be written as: oz

w1 ox11 ¼ oz 1 w2 ox

ð8:96Þ

2

since k2 is equal to zero. In the - perhaps not completely unrealistic - case that it is exactly one of the S states of nature, which effectively determines the necessary quantity of all inputs, the optimisation conditions reduce to the familiar expression that the relationship between input prices should equal the ratio of marginal products for production in the state in question.

8.4.3 Minimising Costs for Given Utility Level In this final part of the section, criteria for the optimisation of production when the objective is to achieve a given utility level as cheaply as possible are derived. The problem is formulated as follows, assuming only two inputs for simplicity: Min fw1 x1 þ w2 x2 g subject to:  U ðy1 ; . . .; yS Þ  U where again: ys ¼ zs ðx1 ; x2 Þps  w1 x1  w2 x2

ð8:97Þ

8.4 Optimisation Under Constraints

77

The Lagrange function is:  L ¼ w1 x1 þ w2 x2 þ k½U ðy1 ; . . .; yS Þ  U

ð8:98Þ

Differentiate the Lagrange function and set the derivatives equal to zero. This produces the following three optimisation conditions, assuming an internal solution: S X oU ozs ð ps  w 1 Þ k oys ox1 s¼1 S X oU ozs k ð ps  w 2 Þ oys ox2 s¼1

! ¼ w1

ð8:99Þ

¼ w2

ð8:100Þ

!

 U ðy1 ; . . .; yS Þ ¼ U

ð8:101Þ

Assuming that the decision maker is risk-neutral, then the conditions (8.99) and (8.100) can be combined into: EðVMP1 Þ  w1 w1 ¼ EðVMP2 Þ  w2 w2

ð8:102Þ

where E(VMPi) is the expected value of the marginal product of input i, calculated as: EðVMPi Þ ¼

S X s¼1

ps

ozs ps oxi

The condition (8.102) expresses that a risk-neutral decision maker applies an optimal amount of input when the relationship between the expected marginal return is equal to the ratio of input prices. Condition (8.102) can be generalised to more inputs. Example 8.7 We use again the generic example from Sect. 8.2.2. In this context, Eq. (8.102) implies the following condition for the optimal combination of N and P2: 0:20  ð47  0:20 NÞ þ 0:60  ð46  0:24 NÞ þ 0:20  ð45  0:30 NÞ  4 4 ¼ 0:20  ð0Þ þ 0:60  ð6  0:044P2 Þ þ 0:20  ð18  0:07P2 Þ  3 3 This relationship actually expresses the expansion path in the N-P2 dimension for a risk-neutral decision maker. The question is whether the optimal application of inputs is greater or less for a risk-averse decision maker. If we consider Fig. 8.8, the answer is clear. Regardless of how you increase income in state 1 and state 2, you will first cross the indifference curve for the risk-

78

8 Optimisation of Production Under Uncertainty y2

⎯U

A

B

⎯U

45° y1

Fig. 8.8 Utility for risk-averse and risk-neutral decision makers

neutral decision maker, as shown in the figure by following the two ‘‘expansion’’ paths A and B. Therefore, the cost of achieving a given level of utility is always lower for a risk-neutral decision maker than it is for a risk-averse decision maker.

References Chambers, R., & Quiggin, J. (2000). Uncertainty, Production, Choices, and Agency. The StateContingent Approach. Cambridge: Cambridge University Press. Rasmussen, S. (2003). Criteria for Optimal Production under Uncertainty. The State-Contingent Approach. Aust. J. Agric. Res. Econ. p. 447–476. Rasmussen, S. (2006). Optimizing Production under Uncertainty. Generalization of the StateContingent Approach and Comparison with the EV Model. FOI Working Papers no. 5/2006, Institute of Food and Resource Economics, The Royal Veterinary and Agricultural University, Copenhagen. Rasmussen, S. (2011). Production Economics. The Basic Theory of Production Optimisation. Berlin: Springer.

Chapter 9

Summary and Conclusion

Abstract This chapter provides a short summary in the form of a list of the main results, and it concludes that the state-contingent approach is a useful approach to planning under uncertainty because it explicitly considers the possibility that the outcome of uncertain events may be controlled by the decision maker herself Keywords Summary
Risk-averse decision maker
Risk-neutral decision maker
Conclusion The purpose of this book has been to give a brief introduction to the theory of decision making under uncertainty, including the most central concepts of production under uncertainty, as Chambers and Quiggin (2000) have presented in their book, and based on this to derive operational criteria for optimal production under uncertainty. The key concept in the book by Chambers and Quiggin is the use of the concept of state-contingent goods. With this concept, the definition of the concept ‘‘a good’’ is extended, so that in addition to type, place and time, the description of a good also includes a statement concerning the state of nature that occurs in the future when the good becomes available. With this expansion of the definition of what is meant by a good, it is in principle possible to treat decision making under uncertainty within the familiar framework of economic optimisation under certainty. Based on a general utility function which has state-contingent incomes as arguments, and which only requires that the decision-maker is risk averse, concepts such as subjective probabilities, expected income, certainty equivalents, and risk premiums have been described. Decision problems associated with production under uncertainty have been formalised and described, and we have seen that in the context of optimisation, there is a need to differentiate between different types of input, depending on the effect they have in the different states of nature. Definitions are provided for three different types of input, namely the state-general inputs, state-specific inputs and state-flexible inputs. Subsequently, we have seen

S. Rasmussen, Optimisation of Production Under Uncertainty, SpringerBriefs in Economics, DOI: 10.1007/978-3-642-21686-2_9,  Svend Rasmussen 2011

79

80

9 Summary and Conclusion

that from an analytic point of view, it is not necessary to distinguish between these three types of inputs, since state-specific and state-flexible inputs are just special cases of state-general inputs. For each of these three types of inputs, criteria for the optimal application under uncertainty for risk-averse and risk-neutral decision makers have been derived, and we have analysed whether there are differences in the optimal use of inputs for, risk-neutral and risk-averse decision makers, respectively. The main definitions and results can be summarised as follows:

9.1 Definitions • State-general inputs: Inputs, which affect the production of one or more states of nature. • State-specific inputs: Inputs which affect production in just one state. • State-flexible inputs: Inputs which can be targeted at alternative states. • Good state: A state in which a state-contingent additional income of one MU gives a lower marginal utility than the probability of that state occurring. • Bad state: A state in which a state-contingent additional income of one MU gives a higher marginal utility than the probability of that state occurring.

9.2 Main Results 1. Risk-averse decision makers: • Derivation of operational criteria for the optimal use of inputs requires that one knows the decision maker’s utility function. • Even under highly simplified assumptions, it is difficult to give general answers to the question of whether risk-averse decision makers will use more or less input than risk-neutral decision makers. Even when one knows the form of the utility function, it is not possible to derive operational criteria which describe the circumstances under which a risk-averse and a risk-neutral decision maker would use different quantities of a given input. • If state-specific input is targeted at a ‘‘bad’’ state, then a risk-averse decision maker will use more input than a risk-neutral decision maker. Conversely, if the state-specific inputs are targeted at a ‘‘good’’ state, then a risk-averse decision maker will use less input than a risk-neutral decision maker. The general validity of this result assumes that all other inputs are fixed inputs. • If all inputs are state-specific, then a risk-averse decision maker will produce more in a ‘‘bad’’ state of nature than a risk-neutral decision maker. Similarly, a risk-averse decision maker will produce less in a ‘‘good’’ state of nature than a risk-neutral decision maker.

9.2 Main Results

81

2. Risk-neutral decision makers • A risk-neutral decision maker optimises the application of a state-general input by increasing the application as long as the expected value of the marginal product is greater than the input price. • A risk-neutral decision maker should increase the application of a statespecific input as long as the value of the marginal product of using this input, weighted by the probability of being in the state in which the input is active, is greater than or equal to the input price. • A risk-neutral decision maker should increase the production of output in state s as long as the additional cost is lower than the product price in state s multiplied by the probability of state s. • If the input is state-specific, a risk-neutral decision maker should increase the production of output in state s as long as the additional cost of this is lower than the product price in state s multiplied by the probability of state s. • Risk-neutral decision makers should continue the supply of inputs as long as the expected value of the marginal product is positive and the expected value of the marginal product should be the same for all products. • If the decision maker is risk-neutral, then a given amount of input applied to a production should be allocated to S states of nature in such a way that the marginal product in the state in question multiplied by the probability of the state in question is the same for all states. • The cost of achieving a given level of utility will always be lower for a riskneutral than for a risk-averse decision maker. 3. Other results • Concerning state-flexible inputs, if one considers an input i targeted at given state s as a separate input (xis), then such a separate input may be treated as a state-general or state-specific input depending on whether the input in question affects production in only one, or in more than one state. • The cost function for state-contingent outputs is not in general differentiable. • The cost function is differentiable if all inputs are state-specific inputs. • A fixed quantity of state-general inputs should be allocated to alternative productions so that the condition (8.15) (or (8.16)) is satisfied. • Fixed amounts of state-flexible inputs can be allocated both to alternative productions and alternative states of nature. The allocation to productions (within states) follow the familiar criterion for the distribution of allocation of inputs under certainty, i.e. VMPi = VMPj, where VMPi is the value of the marginal product of input i. The allocation to states of nature depends on the shape of the utility function.

Reference Chambers, R., & Quiggin, J. (2000). Uncertainty production, Choices, and agency. The statecontingent approach. Cambridge: Cambridge University Press.