... on earlier drafts of this paper. Prasanta. K. Pattanaik also acknowledges his intellectual debt to Salvador Barbera, Tapas Majumdar, and Amartya Sen.
Journal of Economic Theory 84, 95110 (1999) Article ID jeth.1998.2499, available online at http:www.idealibrary.com on
Stochastic Revealed Preference and the Theory of Demand* Taradas Bandyopadhyay Department of Economics, University of California, Riverside, Riverside, California 92521 tbandyucrac1.ucr.edu
Indraneel Dasgupta School of Economics, Deakin University, Geelong, Victoria 3217, Australia iguptamail-g.deakin.edu.au
and Prasanta K. Pattanaik Department of Economics, University of California, Riverside, Riverside, California 92521 ppatucrac1.ucr.edu Received November 26, 1997; revised August 7, 1998
This paper extends the deterministic revealed preference theory of consumers' behavior to permit stochastic choices. In this extended framework, we introduce a restriction on stochastic choices that constitutes a stochastic counterpart of the weak axiom of revealed preference. This restriction allows us to derive stochastic versions of nonpositivity of the own substitution effect and the demand theorem. 1999 Journal of Economic Literature, Classification Number Number: D11. Academic Press
I. INTRODUCTION The purpose of this paper is to explore certain aspects of the choice behavior of a consumer who is assumed to choose in a stochastic fashion * We thank Robert Breunig, Indranil Dutta, Gary Dymsky, Keith Griffin, Subodh Kumar, Robert Russell, Kunal Sengupta, an associate editor of Journal of Economic Theory, and an anonymous referee for their many helpful comments on earlier drafts of this paper. Prasanta K. Pattanaik also acknowledges his intellectual debt to Salvador Barbera, Tapas Majumdar, and Amartya Sen.
95 0022-053199 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
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from alternative budget sets. While, in spirit, our analysis is akin to the standard theory of revealed preference (see Samuelson [19], Houthakker [13], Arrow [2], and Richter [18]), our main point of departure is that we allow the consumer's choice to be stochastic. The classical theory of consumers' behavior either assumes a deterministic demand functioncorrespondence (as in the revealed preference approach) or derives it from the prior assumption of a deterministic preference ordering. However, experimental evidence suggests that the observed choices of individuals are often stochastic in nature. In response to this, and in contrast to the classical theory, a large literature has developed, with contributions by economists as well as psychologists, which deals with agents with stochastic preference andor stochastic choice behavior (see, among others, Barbera and Pattanaik [3], Block and Marschak [4], Falmagne [5], Fishbum [68], GeorgescuRoegen [911], Halldin [12], Luce [1416] and Quandt [17]). In this literature, given a set, say [x, y], of feasible options, the agent's choice behavior may be described by a probability distribution that assigns probability % to x and probability (1&%) to y where % is to be interpreted as the probability that the agent will choose x from [x, y] and where it is permissible to have 0q(A), that is, suppose the probability that the consumer will choose a bundle in A goes up when the budget set changes from goh to g$oh$. What can be the intuitive reason for this increase? In switching from goh to g$oh$ we are throwing out G from the original set and adding H. The addition of H brings in new competitors against the
FIGURE 1
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bundles in A. Intuitively, the increase in the probability of choosing something in A should not be due to the entry of these new competitors (if anything, this new competition should tend to reduce the probability of the chosen bundle lying in A). Then the increase in the probability of choosing a bundle in A must be due to the fact that some bundles, namely, the bundles in G, which were earlier competing against the bundles in A, are no longer available. Intuitively, therefore, the increase in the probability of choosing a bundle in A should not exceed the maximum amount of probability that can possibly be ``diverted'' from G to A (as the budget set changes from goh to g$oh$). This maximum is, clearly, q(G), the probability of choosing a bundle in G given the original budget set goh. This intuition suggests that [q$(A)&q(A) q(G)], the restriction imposed by WASRP. Remark 3.4. Formally, WARP for a deterministic demand function is different from WASRP for a degenerate SDF, since, formally, a deterministic demand function is different from a degenerate SDF. However, it can be checked that, if one identifies a degenerate SDF with a deterministic demand function, then the restrictions imposed by WASRP on a degenerate SDF turn out to be identical to those imposed by WARP (see Remark 2.3). The following proposition shows the relation between RSO and WASRP. Proposition 3.5. Rationalizability in terms of stochastic orderings implies, but is not implied by, the weak axiom of stochastic revealed preference. Proof.
See the Appendix.
Remark 3.6. It is easy to see that, if the SDF satisfies WASRP, then the probability measure, yielded by the SDF, over the set of all subsets of the budget set remains unchanged, when there is a proportionate change in all prices and the wealth of the consumer. This is the counterpart of the deterministic result that, if the (deterministic) demand function satisfies Samuelson's WARP, then it must be homogeneous of degree zero in all prices and wealth. In view of Remarks 3.4 and 3.6, it is worth exploring the analogy between WARP for a deterministic demand function and WASRP somewhat further. Suppose, given a deterministic demand function, x and x$ are the chosen consumption bundles corresponding to the pricewealth situations ( p, W) and ( p$, W$), respectively. Then, assuming that the deterministic demand function satisfies WARP, we know that, if x is available under ( p$, W$), then either x=x$ or x$ is unavailable under ( p, W) (i.e.
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p } x$>W). It is natural to ask whether some stochastic version of this holds under WASRP. Consider first the deterministic notion of the continued availability of the original chosen bundle x in the new budget situation. Clearly, when the demand function is stochastic, the counterpart of this notion is q(E)=1, where E=B( p, W) & B( p$, W$). Suppose we have q(E)=1. What, then, is the stochastic notion of the consumer's choice behavior remaining the same in the two situations? Given q(E)=1, it is intuitive to argue that the consumer's choice behavior remains the same in the two budget situations if and only if, for every A E, q(A)=q$(A). Then, according to the spirit of WARP outlined above, if the consumer's choice behavior does change for some A E when we go from the pricewealth situation ( p, W) to the pricewealth situation ( p$, W$), then the change in the consumer's choice behavior must take the form of a ``transfer'' of a probability mass from A to some set A$ of consumption bundles which are available under ( p$, W$) but not under ( p, W). It can be checked that this is indeed implied by WASRP. We state this implication of WASRP more formally in Remark 3.7. Remark 3.7. Suppose a stochastic demand function satisfies WASRP. Let ( p, W) and ( p$, W$) be two pricewealth situations such that q(B( p, W) & B( p$, W$))=1. Then, for all A B( p, W) & B( p$, W$), [q$(A) q(A) q$(A)+q$(B( p$, W$)&B( p, W))].
IV. NON-POSITIVITY OF THE OWN SUBSTITUTION EFFECT AND THE STOCHASTIC DEMAND THEOREM In order to investigate the implications of WASRP for the effects of changes in the price of a commodity on its demand, we first need to introduce stochastic counterparts of the following (deterministic) notions: nonpositivity of the own substitution effect, a normal good and a regular good. Notation 4.1. For every pricewealth situation ( p, W), every : # R + and every i # N, B i :( p, W), B i
