Revealed Preference and Stochastic Demand Correspondence: A Unified Theory

Revealed Preference and Stochastic Deand Correspondence: A Unified Theory Indraneel Dasgupta School of Econoics, University of Nottingha, Nottingha NG7 2RD, UK. E-ail: indraneel.dasgupta@nottingha.ac.uk Prasanta K. Pattanaik Departent.of Econoics, University of California, Riverside, CA 92521, USA. E-ail: prasanta.pattanaik@ucr.edu This draft:18 April 2007 Abstract: We expand the theory of consuer s behavior, based on the Weak Axio of Revealed Preference, to perit siultaneously both rando choice and choice sets, which are not singletons. We provide a consistency postulate for deand behavior when such behavior is represented in ters of a stochastic deand correspondence. We show that, when the consuer spends her entire wealth, our rationality postulate is equivalent to a condition we ter stochastic substitutability. This equivalence generates the following results as special cases: (i Sauelson s Substitution Theore, (ii the central result in Bandyopadhyay, Dasgupta and Pattanaik (2004 and (iii a version pertinent to deterinistic deand correspondences (which independently yields Sauelson s Substitution Theore as alternative special cases. Relevant versions of the non-positivity of the own substitution effect and the deand theore also follow as corollaries in each case. Keywords: Stochastic deand correspondence, weak axio of revealed preference, weak axio of stochastic revealed preference, general substitution theore. JEL Classification Nuber: D11 Corresponding author. 4/19/2007 1:22 PM 1

1. Introduction The purpose of this paper is to extend the classical revealed preference theory of consuers deand behavior in several directions. A central result in the classical theory is the inter-relationship between Sauelson s Weak Axio of Revealed Preference (WARP and the various epirically testable restrictions on the deand behavior of a copetitive consuer. We extend this result by peritting siultaneously both stochastic choices and non-singleton choice sets for consuers. In a seinal paper, Sauelson (1938 introduced his Weak Axio of Revealed Preference (WARP and deduced fro it the conclusion that, if the consuption bundle chosen by a copetitive consuer in soe initial price-wealth situation costs exactly the consuer s wealth in soe altered price-wealth situation, then the products of price change and quantity change for each coodity will su to a non-negative nuber. This conclusion, which we shall call Sauelson s Inequality, suarizes uch of the epirical content of the theory of consuers behavior. It iplies the nonnegativity of the own-price substitution effect as a special case; the non-negativity of the own-price substitution effect, in its turn, yields the law of deand which tells us that, other things reaining the sae, a fall in the price of a noral good increases the quantity of it purchased by the consuer. Not only does WARP iply Sauelson s Inequality, but, for a consuer who always spends her entire wealth, WARP also turns out to be equivalent to Sauelson s Inequality (see Sauelson (1947 and Mas-Colell et al. (1995, pp. 28-32. The equivalence of WARP and Sauelson s Inequality for a consuer who always exhausts her budget constitutes a basic result in the standard choice-based theory of consuers behavior; for convenience, we shall ter this equivalence Sauelson s Substitution Theore. The choice-based theory of consuer s behavior takes, as its priitive, soe coplete description of deand behavior for every possible price-wealth configuration. Its particular ethod of representation, however, iposes two ajor constraints at the very outset. First, faced with a given price-wealth situation, the consuer is assued to choose a single consuption bundle. Thus, the consuer s deand behavior is constrained to representation by eans of a deand function, rather than a deand correspondence. As a consequence of this, the analytical fraework cannot, handle the choice counterpart of flat indifference surfaces in the preference-based theory, even though it is not intuitively clear why, a priori, one should rule out the possibility that there ay be several consuption bundles in the consuer s budget set that she considers to be equally worthy of being chosen fro that budget set. 1 Second, a consuer s deand behavior is represented by eans of a unique deand function. This assues away the possibility of any probabilistic eleent in the consuer s choice. Modeling consuer s behavior by eans of stochastic deand correspondences, 1 Of course, observed arket choices, the focus of Sauelson s theory, would not reveal such ultiplicity of optial consuption bundles for a given budget set since, by definition, the consuer can buy only one consuption bundle belonging to her budget set. One ay, however, think of the proble of a consuer s choice in ters of consuer survey experients, where consuers are asked to choose a (possibly ultieleent subset out of soe feasible set of alternatives. 4/19/2007 1:22 PM 2

instead of the traditional deterinistic deand functions, would evidently allow one to accoodate these neglected aspects, thereby peritting a fraework with greater flexibility and epirical scope. The need to perit ulti-eleent choice was noticed early. Both Arrow (1959 and Sen (1971 specified versions of WARP applicable to general choice correspondences, though they did not address the specific issue of consuers deand behavior. Richter (1966 explicitly developed the revealed preference theory of consuers' behavior in ters of deand correspondences, rather than deand functions. Afriat (1967 and Varian (1982 developed the generalized axio of revealed preference (GARP to accoodate flat indifference surfaces. None of these contributions however addressed the issue of expanding the central result in the traditional theory, i.e., Sauelson s Substitution Theore, to encopass deand correspondences. Their focus instead was on identifying restrictions that would allow deand behavior to be rationalized in ters of axiization on the basis of soe coplete and transitive binary preference relation or even a utility function, requireents stronger than the satisfaction of WARP. 2 Thus, the priary objective of this line of enquiry was to construct a preference ordering or, possibly, a utility function fro soe given specification of deand behavior. It is not even clear fro these contributions exactly how one should interpret either the non-positivity of the own-price substitution effect or the deand theore, when the consuer s choice set contains ultiple consuption bundles. Furtherore, this literature restricted itself to deterinistic choice behavior. Pursuing a parallel, probabilistic, line of enquiry, Bandyopadhyay, Dasgupta and Pattanaik (2004, 1999 have recently presented a rationality postulate for stochastic deand behavior, the weak axio of stochastic revealed preference, and developed a stochastic version of Sauelson s Substitution Theore. 3 Their result encopasses the traditional version as a special case. Their analysis, however, was carried out in ters of stochastic deand functions. Thus, while Bandyopadhyay, Dasgupta and Pattanaik (2004, 1999 departed fro the classical fraework by allowing the consuer to choose in a probabilistic fashion, they nevertheless constrained her to randoize only aong alternative singleton sets of consuption bundles. Integration of these different strands of analysis would thus appear to be of considerable interest. Suppose one took, as one s theoretical priitive, a representation of consuer s behavior in ters of a stochastic deand correspondence (see Section 4 below for an exaple of the type of observed behavior of a consuer that can be accoodated in the conceptual fraework of a 2 Houthakker (1950 developed the strong axio of revealed preference (SARP precisely to bridge this gap. GARP is a generalization of SARP. Thus, uch of the core epirical content of the standard theory of consuer s behavior, as suarized by Sauelson s Inequality, does not require rationalizability, just as uch of the core epirical content of the traditional theory of copetitive fir behavior does not entail rationalizability in ters of profit axiization (Dasgupta (2005. 3 Bandyopadhyay, Dasgupta and Pattanaik (2002 also apply this rationality postulate to the proble of deand aggregation, while Dasgupta and Pattanaik (2006 explore its connection with a weaker rationality postulate, viz. regularity, in a general stochastic choice context. For a recent survey, see McFadden (2005. 4/19/2007 1:22 PM 3

stochastic deand correspondence but not in the available alternative conceptual fraeworks. Can one then develop a predictive theory of consuer s behavior, which expands Sauelson s WARPbased fraework, to cover siultaneously both rando choice and the choice of sets of ultiple consuption bundles? Such a theory would: (i subsue the central result of Bandyopadhyay, Dasgupta and Pattanaik (2004 as a special case, and (ii generate a version pertinent to deterinistic deand correspondences as a second special case (one which, in turn, would subsue Sauelson s Substitution Theore as its own special case. We seek to provide such an integrated theoretical fraework. We develop a rationality, or consistency, postulate for deand behavior when such behavior is represented in ters of a stochastic deand correspondence. We show that, given that the consuer spends her entire wealth with probability one, this rationality postulate turns out to be equivalent to a condition we ter stochastic substitutability. This equivalence generates the following as alternative special cases: (i Sauelson s Substitution Theore, (ii the central result in Bandyopadhyay, Dasgupta and Pattanaik (2004, and (iii a version pertinent to deterinistic deand correspondences, which independently yields Sauelson s Substitution Theore. The relevant versions of the non-positivity of the own-price substitution effect and the deand theore also fall out as corollaries in each of these special cases. Thus, our central result expands the traditional WARP-based fraework to cover siultaneously both rando choice of the consuer and nonsingleton choice sets, thereby enhancing both its epirical scope and analytical coverage. Section 2 presents the basic notation. Section 3 develops the idea of representing deand behavior via stochastic deand correspondences. Section 4 defines soe possible properties of stochastic deand correspondences. In particular, we define our notions of a noral good and the non-positivity property of the own substitution effect in this expanded context. We also present and discuss our rationality postulate in Section 4. Our results are presented in Section 5. Section 6 concludes. Proofs are presented in the appendix. 2. Soe Notation Let 2 be the nuber of coodities, and let M { 1,2,...,} = denote the set of coodities. R +, R ++ will denote, respectively, the set of all non-negative real nubers and the set of all positive real nubers. R + is the consuption set. The eleents of the consuption set will be denoted by x, x etc. Given any consuption bundle of coodity i contained in the bundle x. x R+ The set of all possible price vectors is vectors. For any given coodity [ i i i M p p and, for every coodity j i,, and given any i M ++, x i will denote the aount R, with p, p etc. denoting individual price, we say that two price vectors p and p are i-variant iff p = p ]. The set of all possible wealth levels of the j j 4/19/2007 1:22 PM 4

consuer is R +, with W, W etc. denoting specific wealth levels. A price-wealth situation is a pair ( W R + + R+ price-wealth situation ( p, W, the consuer s budget set, ( p W p,. Thus, the set of all possible price-wealth situations is Z R+ + R+. Given a For brevity, we shall typically write, etc., instead of B ( p W, B( p, W B B B,, is the set { x R + p. x W },, etc.. Given any non-epty set T, r ( T will denote the set of all possible non-epty subsets of T and R ( T will denote the power set of T ( thus, RT ( rt ( { φ }, where φ denotes the epty set. Given two sets, T and T, [ T \ T ] will denote the set of all eleents of T that do not belong to T. 3. Stochastic Deand Correspondence The first step in our analysis is to foralize the idea of odeling a consuer s deand behavior by eans of a stochastic deand correspondence, and to locate this idea in relation to other possible, ore traditional, representations. Given a price-wealth situation, the flexibility we seek intuitively involves allowing the consuer: (i to have a choice set containing ultiple consuption bundles (rather than a single bundle as in the classical fraework, and (ii to do so in a (non-trivially rando fashion, so that, in contrast to the classical fraework, no single choice set is assigned probability 1. Definition 3.1.,, (i A stochastic deand correspondence (SDC is a rule C which, for every ( p W Z specifies exactly one finitely additive probability easure Q on ( rb (, RrB ( ( (thus, rb ( is the set of outcoes and R(( rb is the relevant algebra in rb. (,, specifies (ii A stochastic deand function (SDF is a rule D which, for every ( p W Z exactly one finitely additive probability easure q on ( B, RB ( (thus, B is the set of outcoes here and R ( B is the relevant algebra B.,, (iii A deterinistic deand correspondence (DDC is a rule c which, for every ( p W Z specifies exactly one non-epty subset of B. (iv A deterinistic deand function (DDF is a rule d which, for every ( p W Z exactly one eleent of B.,, specifies Consider any price-wealth situation ( p, W. Let B denote the budget set corresponding to ( p, W. Let Q = C( p, W, where C is an SDC. Given the price-wealth situation ( W A R( r( B, ( A p,, for any Q is the probability that the (possibly ulti-eleent set of chosen bundles will lie 4/19/2007 1:22 PM 5

in the class A. C ( p W, C( p, W, etc. will be denoted, respectively, by Q, Q etc. Thus, an SDC captures the idea that, given a budget set: (i the consuer ay choose a subset with ultiple eleents, and (ii she ay choose aong the alternative subsets available in a probabilistic fashion. An SDF, introduced by Bandyopadhyay, Dasgupta and Pattanaik (1999, restricts the consuer to choosing a single consuption bundle, albeit allowing her to do so in a probabilistic fashion. Let q D p, W R B q A is the probability that the chosen = (, where D is an SDF. For any A (, ( bundle will lie in the set A. Thus, the intuitive idea captured by an SDF is that the consuer always chooses only one consuption bundle; however, exactly which bundle is going to be chosen is deterined by soe probabilistic rule. A DDC, in contrast, assues away the probabilistic eleent in choice, but allows the possibility that the consuer will choose ultiple bundles. Thus, given a DDC, c, c( p, W r( B is the (possibly ulti-eleent set of consuption bundles the consuer will choose fro the budget set corresponding to ( p, W. A DDF is the ost restrictive, yet also the ost coon, for of representation of deand behavior in consuer theory. This constrains the consuer s choice fro a budget set to a single consuption bundle, chosen according to soe deterinistic (i.e. non-probabilistic decision rule. Given a DDF, d, d( p W B consuer will choose fro the budget set corresponding to the price-wealth situation ( p, W., is the bundle the It is evident fro the preceding discussion that an SDC is the ost flexible tool available for odeling a consuer s deand behavior. Intuitively, SDFs, DDCs and DDFs are all special classes of SDCs. We now proceed to provide a foral stateent of this idea. Definition 3.2. (i An SDC, C, is singular, iff, for all ( p, W Z, and for every R( r( B Q ( A Q( { A A A = 1}. = i i (ii An SDC, C, is degenerate, iff, for all ( p, W Z, there exists r( B ({ A} = 1 Q. A, A such that A singular SDC is one where the consuer s probability of choosing a set with ultiple consuption bundles is zero. A degenerate SDC is one where the consuer chooses, in effect, in a deterinistic fashion. A singular SDC corresponds to an SDF, a degenerate SDC corresponds to a DDC, and a singular and degenerate SDC corresponds to a DDF. More forally, we define the following. 4/19/2007 1:22 PM 6

Definition 3.3. (i An SDC, C, induces an SDF, D, iff, for every ( p, W Z, and for every R( B q ( A = Q( { x} x A}. An SDF, D, induces an SDC, C, iff, for every ( p W Z A R( r( B, Q( A q( { x B { x} A} =. (ii An SDC, C, induces a DDC, c, iff, for every ( p, W Z, ( p W bundles A B such that { A} ({ c( p, W } = 1 Q. A,,, and for every c, is the set of consuption Q ( = 1. A DDC, c, induces an SDC, C, iff, for every ( p W Z,, Reark 3.4. Let C be a non-singular SDC and let D be an SDF. Given that C is non-singular, we ust have Q ({{} x x B} < 1; yet it ust always be the case that q ( B = 1. Therefore, C cannot possibly induce D. Thus, only singular SDCs can induce SDFs. Now notice that, an SDC that is induced by an SDF ust satisfy Q ({ x} x B} = 1 ; thus, every SDC induced by an SDF ust be singular. Notice further that every singular SDC induces soe SDF, and every SDF induces soe singular SDC. If a singular SDC, C, induces soe SDF, D, then the singular SDC induced by D ust be C itself. Analogous relations can easily be seen to hold between degenerate SDCs and DDCs, and also between DDFs and SDCs that are both singular and degenerate. Reark 3.5. There is a ethodological difference between the notions of DDF and SDF on the one hand and the notions of SDF and SDC on the other. The concept of choice involved in a DDF and an SDF can be interpreted in ters of the consuer s actual purchase of a consuption bundle in the arket place, though, in the case of a DDF, the consuer always purchases the sae consuption bundle fro the sae budget set, while, in the case of the SDF, the consuer s purchase fro a given a budget set can be stochastic in nature. In contrast, SDFs, as well as SDCs, perit ulti-eleent choice sets, which cannot be directly observed by watching the consuer s actual purchases in the arket place: the consuer cannot siultaneously choose ore than one consuption bundle in the arket even when his choice set corresponding to a given budget set contains ore than one consuption bundle. This iplies that the axios regarding the consuer s behavior, which are forulated in ters of possibly ulti-eleent choice sets, are not directly testable when we confine ourselves to observations of the actual arket behavior of the consuer. If, however, one does not confine oneself to the consuer s arket choices, it ay be possible to observe a ulti-eleent choice set of the consuer. For exaple, in response to a questionnaire, the consuer can indicate that, given the price-wealth situation, (p, W, he does not ind choosing either bundle x or bundle y but he will not choose any other consuption bundle in the budget set. 4/19/2007 1:22 PM 7

Of course, a theoretical structure, constructed in ters of either a DDC or an SDC can be used to derive conclusions about the consuer s arket behavior (in that case, though the observation of the consuer s arket choices cannot be used to test the assuptions of the theory directly, such observations can be used to test the assuptions indirectly by directly testing the conclusions. This will require soe assuption, however weak, to link the consuer s observed arket behavior to his possibly ulti-eleent choice set. As we shall argue later on, though we use the notion of an SDC as our priitive concept, our results have obvious intuitive interpretation in ters of the observed arket behavior of the consuer even under the weakest conceivable assuption regarding the link between ulti-eleent choice sets and the consuer s observed arket behavior. 4. Soe properties of SDCs We now forulate soe properties that an SDC ay conceivably have. Our subsequent analysis in Section 5 will clarify the interconnections aong these properties. Definition 4.1. An SDC, C, is tight iff, for every ( p W Z,, Q( r( { x R + p. x = W } = 1. Tightness is the intuitively appealing requireent that the probability of the consuer s having a choice set, such that each consuption bundle in the choice set exhausts her entire wealth, is always one. In essence, this is the counterpart of the standard global non-satiation presuption. We shall derive our substantive results in Section 5 under the assuption that the SDC satisfies this restriction. Consider now the following scenario. Suppose, starting fro soe initial price-wealth situation ( p, W, the price of coodity i falls fro p i to p i, all other prices reaining invariant. Consider any arbitrary aount of the i-th coodity, α, that the consuer could possibly have bought in the initial situation. The price fall reduces the cost of every consuption bundle containing this aount of the i-th coodity by ( p α. Suppose now that the consuer s wealth is i p i exactly copensated for this cost reduction. Thus, suppose that the consuer s wealth is reduced by exactly ( p α. Then all consuption bundles containing at least α aount of the i-th i p i coodity that were initially available continue to be available in the new price-wealth situation. Furtherore, the consuer can now afford soe consuption bundles containing ore than α aount of the i-th coodity, that she initially couldn t. However, soe consuption bundles containing less than α aount of the i-th coodity, that were initially available, now becoe unaffordable. How would the consuer respond to the new price-wealth situation? More specifically, what is the probability that the consuer will choose, in the new situation, a set of consuption bundles where every bundle contains at least α of the i-th coodity? 4/19/2007 1:22 PM 8

Suppose that this probability is not less than the initial probability of choosing a set of consuption bundles where at least one consuption bundle contains at least α of the i-th coodity. Suppose further that an analogous relationship holds for sets of consuption bundles containing ore than α of the i-th coodity. In such a case, we shall say that the SDC satisfies non-positivity of the own substitution effect. Thus, in expanding this failiar notion far beyond its original classical context, we essentially: (i ipose a set-doinance criterion, and (ii require this setdoinance criterion to be satisfied in a probabilistic fashion. Definition 4.2. A tight SDC, C, satisfies non-positivity of the own substitution effect (NPS iff, for every i M, every ordered pair ( p, W, ( p, W Z Z, and every α, such that [ p, p are i- variant with and. W p i < pi ; α [0, ]; and W = W ( p i p i α ], p i ({ s x > α for all x s } Q( { s B x > for soe x s} Q α, (4.1 i ({ s x α for all x s } Q( { s B x for soe x s} Q α. (4.2 i i i Our next step is to expand the failiar notion of a noral good fro its classical context. As before, we shall utilize a probabilistic set-doinance criterion in order to do so. Suppose a consuer s wealth increases, while all prices reain constant. Consider any arbitrary aount of the i-th coodity, α. What is the probability that the consuer will choose, in the new situation, a set of consuption bundles where every bundle contains at least α of the i-th coodity? Suppose this probability is not less than the initial probability of choosing a set of consuption bundles where at least one consuption bundle contains at least α of the i-th coodity. Suppose also that an analogous restriction holds for sets of consuption bundles containing ore than α of the i-th coodity. We shall then call the i-th coodity a noral good. Definition 4.3. Given an SDC, C, a coodity, i, is noral, iff, for all α R+, all all W, W R+ such that W > W ( p W B = B,. p R + +, (4.1 and (4.2 both hold when B B( p, W, and = and We shall call a good regular if, intuitively, its deand does not fall with a fall in its own price, wealth and all other prices reaining invariant. As before, we shall use a probabilistic set- 4/19/2007 1:22 PM 9

doinance criterion to foralize the notion of deand not falling. Thus, in our subsequent analysis, the failiar deand theore will siply constitute the clai that every noral good is also regular. Definition 4.4. Given an SDC, C, a coodity, i, is regular iff, for every ordered pair ( p, such that [[ p, p are i-variant with p i < pi ], for every W R+, and for all p R + + α, (4.1 and (4.2 both hold when B = ( p, W and B = B( p, W R +. We now define a restriction for SDCs when ore than one, possibly all, prices are allowed to change siultaneously. This condition is essentially an expansion of a condition introduced by Bandyopadhyay, Dasgupta and Pattanaik (2004 in the context of SDFs. Following their terinology, we call our condition stochastic substitutability. At this stage, it sees easier to otivate stochastic substitutability in ters of its instruental value, than through any transparent intuitive interpretation. As we shall show in Section 5 below, this condition allows us to deduce a nuber of existing results in the theory of consuer s behavior, as special cases within a unified fraework, that allows the consuer to both choose ultiple consuption bundles and do so in a rando fashion. Notation 4.5. Given two price-wealth situations ( p W, ( p, W { x R p. x = W and p x = W }, { x R p. x = W and p x > W }, { x R p. x = W and p x < W }, { x R p. x > W and p x = W }, { x R p. x < W and p x = W }, { s { x R p. x = W } [ s ] φ} I = +. G = +. H = +. G = +. H = +. G = + G, and H = + H. { s { x R p. x = W} [ s ] φ},, let: Definition 4.6. A tight SDC, C, satisfies stochastic substitutability (SS iff, for every ordered pair ( p W, ( p, W Z Z,, ( r( G Q( H Q ; (4.3 and, for every non-epty r( I A, Q ( r( G + Q ({ s ( G I ( s I A} Q H ({ A} ( + Q s ( G I ( s I. (4.4 4/19/2007 1:22 PM 10

The last ajor building block for our substantive analysis in Section 5 is also perhaps the central eleent. This is a rationality, or consistency, postulate for deand behavior. We now introduce this rationality postulate. Definition 4.7. An SDC, C, satisfies the weak axio of stochastic revealed preference (WASRP iff, for all ( p, W, ( p, W Z, and for every non-epty A r( B B Q, ( r( B B + Q( { s B ( s B A} Q ({ s ( s B A} \, (4.5 where B = ( p, W and B = B( p W,. WASRP iposes a consistency requireent. To see the intuitive logic underpinning this condition, consider two budget sets B, B. Let A denote any collection of sets of consuption bundles that are available under both budget sets. Consider the collection of all subsets of B, whose overlap with B consists of soe eber of A. What should be the axiu probability that the consuer s chosen subset lies in this collection under B? Every subset of B, whose overlap with B consists of a eber of the class [ ( B B \ A] r, continues to have that part of it available under B. By choosing any such subset under B, the consuer, in effect, rejects all ebers of A. Thus, consistency appears to require that the probability ass ascribed, under B, to the collection of all subsets of B, whose overlap with B consists of soe eber of A, can have only two sources. These are: (i subsets of B that have no overlap with B, and (ii subsets of B whose overlap with B consists of soe eber of A. Hence, if deand behavior is to exhibit consistency, the probability that the consuer will choose, under B, soe subset whose overlap with B consists of a eber of A, should not exceed the probability that the consuer s choice under B lies in one of these two classes ((i and (ii above. This is the nature of the restriction iposed by WASRP. Rationality postulates analogous to our WASRP for SDCs have been developed for SDFs, DDCs and DDFs (recall Definition 3.1. We now note these specifications and clarify their connections with our rationality postulate for SDCs. Definition 4.8. (i An SDF, D, satisfies the weak axio of stochastic revealed preference (WASRP iff, for all ( p, W, ( p, W Z, and for every A ( B B ( B B + q( A q ( A, q \. (4.6 (ii A DDC, c, satisfies the weak axio of revealed preference (WARP iff, for all ( p, W, ( p, W Z, if [ c( p, W B ] φ, then c ( p W [ r( B \ B { s ( s B = ( c( p, W B }],. (4.7 4/19/2007 1:22 PM 11

(iii A DDF, d, satisfies the weak axio of revealed preference (WARP iff, for all ( p, W, ( p, W Z, if ( p W B d,, then: ( p, W [ ( B \ B { d( p W }] d,. WASRP for SDFs (Definition 4.8(i was introduced by Bandyopadhyay, Dasgupta and Pattanaik (1999. This requires that the probability, under B, of the chosen consuption bundle lying in soe subset of ( B B, cannot exceed the probability, under B, of the chosen bundle either lying in that subset or being unavailable under B. WARP for DDCs, as specified in Definition 4.8(ii, is equivalent to Richter s (1966 weak congruence axio. It is also equivalent to Sen s (1971 specification of WARP (except that Sen considers general choice probles, not the specific proble of choice by a copetitive consuer. This condition requires the following. Suppose the (possibly ulti-eleent set of consuption bundles chosen under B has soe overlap with B. Then the set of bundles chosen under B ust either have the sae overlap with B, or have no overlap with B at all. This condition is a straightforward extension of the classical WARP for DDFs. Our stateent of WARP for DDFs (Definition 4.8(iii is equivalent to the original weak axio of revealed preference due to Sauelson (1938. This siply requires, when the (unique consuption bundle chosen under B is also available under B, the (unique consuption bundle chosen under B ust either be identical to that chosen under B, or else be unavailable under B. Reark 4.9. If a degenerate SDC, C, satisfies WASRP, then the DDC induced by C ust satisfy WARP. If a singular SDC, C, satisfies WASRP, then the SDF induced by C ust satisfy WASRP. If a singular and degenerate SDC, C, satisfies WASRP, then the DDF induced by C ust satisfy WARP. 4 Reark 4.10. In Definition 4.6, we introduced the concept of tightness for an SDC. Corresponding notions of tightness, intuitively iplying that the consuer always exhausts her wealth in every pricewealth situation, can be easily forulated for SDFs, DDCs, and DDFs, but we do not introduce the foral definitions here. The iplications of WASRP and tightness for the SDC of a consuer are our central concern and we turn to this in the next section. Before concluding this section, however, it ay be worth noting that an SDC satisfying WASRP and tightness can accoodate a wider range of arket behavior than can be accoodated by any of the following: (i a DDF satisfying WARP and tightness; (ii a DDC satisfying WASRP and tightness; and (iii an SDF satisfying WASRP and tightness. 4 The last clai is easy to check. Lea N.1(i and Lea N.2(i in the Appendix provide foral proofs of the first two clais. 4/19/2007 1:22 PM 12

Consider a world with ore than two coodities. Suppose that, under the price-wealth situation ( p *, W * pick two bundles, situation ( p ', W ', the consuer s actual arket purchase is observed repeatedly. She is observed to x and y, with identical frequency. Siilarly, under another price-wealth, the consuer is observed as picking two bundles, x and y, with identical frequency. All four bundles are distinct. Furtherore, (i p x = p y = W, (iii p x = W, p y < W, and p x = p y = W, (ii (iv = > W p x W, py. Evidently, the consuer s arket purchases could not have been generated by a DDF satisfying WARP. What happens if one assues that the consuer has an SDF satisfying WASRP? If the consuer s observed arket purchases are generated by an SDF, then, given ( p *, W * SDF ust assign probability ½ to { x } and probability ½ to { * y }. Siilarly, given ( p ', W ' SDF ust assign probability ½ to { x }. But then the SDF violates WASRP. Hence, no SDF satisfying WASRP can generate the deand behavior specified in our exaple. What if, in the spirit of Afriat (1967 and Varian (1982?????, one tries to rationalize the consuer s observed arket purchases deterinistically, in ters of an SDC? Intuitively, the idea then is that, given a budget set, the consuer s choice set ay have several eleents and that, when this happens to be the case, the consuer picks a bundle out of this ulti-eleent set for arket purchase according to soe unspecified behavioral rule (essentially, soe rule of thub since he cannot possibly purchase in the arket place ore than one consuption bundle belonging to the budget set. The behavioral rules for deterining the actual arket purchase when the choice set has ultiple consuption bundles can take various fors such as: (i given that y choice set corresponding to ( p, W is { x, y }, half of the ties when the price-wealth situation is (,, the, the p W, I shall purchase x, and the other half of the ties I shall purchase y ; (ii given that y choice set corresponding to ( p, W is { x, y }, 99 % of the ties when the price-wealth situation is (, p W, I shall purchase x, and the rest of the ties I shall purchase y ; and so on (in principle, there is an infinite nuber of such behavioral rules. Thus, what appears to be stochastic arket behavior of the consuer ay really be the result of a deterinistic deand correspondence together with soe rules of thub for deterining the arket purchases when the choice set happens to have ultiple eleents. The question then arises whether the observed arket deand behavior described in our exaple above could have been generated by soe tight deterinistic deand correspondence satisfying WARP, together with soe behavioral rules linking actual arket choices to choice sets with ultiple eleents? If the observed arket behavior in the exaple is generated by an SDC in 4/19/2007 1:22 PM 13

this sense, then it is clear that the SDC will satisfy WARP only if the choice set defined by the SDC p W contains x, y, x and y. Given p y < W, this would ean that the SDC is not for ( ', ' tight. Thus, no SDC satisfying WARP and tightness could have generated the observed arket behavior in the exaple, no atter what rules the consuer ight have followed to decide her arket purchases when his choice set has several eleents As we have seen, the deand behavior in our exaple cannot be generated by either a DDF satisfying WARP and tightness or an SDF satisfying WASRP and tightness or an SDC satisfying WARP and tightness. Yet, on closer scrutiny, the deand behavior specified in our exaple does not appear outrageously inconsistent. Consider two deand correspondences, a and b. Under a, the consuer s choice set is choice set is * * * { x, x '} for each of (, * * y for ( p, W and { y '} for ( p ', W ' * { } p W, and ( p ', W '. Under b, the consuer s. Notice that a as well as b individually satisfies both WARP and tightness. Furtherore, the consuer s observed arket purchases are copatible with randoization between a and b with equal probability if the behavioral rules that the consuer follows in the case of ulti-eleent choice sets are such that, whenever the price-wealth * * situation is (, p W and the corresponding choice set is whenever the price-wealth situation is ( ', ' * { x, x '}, she always buys x *, and, p W and her corresponding choice set is * { x, x '}, she always buys x '. Thus, in soe core intuitive sense, the consuer confors to the underlying intuition of WARP and always exhausts her wealth. Can her behavior be captured by an SDC satisfying WASRP and tightness? It can be checked that, if the consuer follows the behavioral rules entioned above, then an SDC, which, given ( p *, W * sets, * * { x, x '} and { } y, and which, given ( p ', W ', assigns probability ½ to each of the choice, assigns probability ½ to each of the choice sets, * { x, x '} and { y '}, will satisfy WASRP and tightness and will generate the observed arket purchases of the consuer. (INDRANEEL, AFTER THINKING FURTHER ABOUT THIS EXAMPLE, IT SEEMS TO ME THAT IT HAS SOME FEATURES, WHICH MAY CAUSE SOME DIFFICULTIES WITH THE REFEREE. FIRST, THE RULES OF THUMB ARE RATHER ARBITRARY. WHILE THINGS WORK OUT WORKS FOR THESE RULES OF THUMB, THEY WILL NOT WORK FOR OTHER MORE PLAUSIBLE RULES OF THUMB. MORE WORRYING IS SOMETHING ELSE. EVEN WITH THESE SPECIFIC BEHAVIORAL RULES, WE DO NOT REALLY KNOW WHETHER THERE EXISTS AN SDC THAT SATISFIES WASRP 4/19/2007 1:22 PM 14

AND TIGHTNESS AND GENERATES THE MARKET PURCHASES. WE HAVE NOT SPECIFIED SUCH AN SDC COMPLETELY. I AM WORRIED THAT THE REFEREE WILL PICK UP THIS GAP AND MAKE A FUSS. 5. Results We are now ready to present our substantive results. Proposition 5.1. (General Substitution Theore A tight SDC, C, satisfies WASRP if and only if it also satisfies SS. Proof: See the Appendix. The General Substitution Theore (Proposition 5.1 is our central result. Under the assuption of tightness, it copletely specifies the restrictions on deand behavior iposed by WASRP when applied to SDCs. It provides a central unifying result, in that a nuber of key results in the theory of consuer s behavior can be shown to ste fro this result. First notice that Proposition 5.1 yields two basic results in deand theory, non-positivity of the own substitution effect and the deand theore, for SDCs. Corollary 5.2. Suppose a tight SDC, C, satisfies WASRP. Then: (i C ust satisfy NPS; (ii every noral good ust also be regular. (iii C is hoogeneous of degree 0 in prices and wealth. Proof: See the Appendix. Counterparts of these results for DDCs can also be derived fro the General Substitution Theore, as we now forally note. Since the theory of deand behavior with degenerate SDCs, i.e. DDCs, appears to have escaped attention in the literature, this case is of independent interest as well. Corollary 5.3. Let c be a DDC such that, for every ( p, W Z, c ( p W ({ x B p. x = W} (i,. (Deterinistic Substitution Theore c satisfies WARP iff, for every ordered pair ( p W, ( p, W Z Z for all,, such that p x W x I, [( p (. x x 0]. for soe x c( p, W, for all x c( p W,, and p ; the inequality holding strictly when either (a p. x < W for soe x c( p, W, or (b [ c( p, W I ] [ c( p, W I ]. (5.1 4/19/2007 1:22 PM 15

(ii Let c satisfy WARP. Then, for every i M, for every ordered pair ( p, W, ( p, W Z Z such that [[ p p soe x c( p, W ]], and for all x c( p W [ x ] i x i, are i-variant with p i < pi ], and [ p. x = W for,, ; the inequality holding strictly when x < sup{ x x c( p W } (iii Let c satisfies WARP, and suppose i M induced by c. Then, for every ordered pair ( p i < p i ], for every R+ W, and for all x c( p, W [ { x x c( p, W } ] sup. i x i Proof: See the Appendix. i i, is noral according to the degenerate SDC p, p R + + such that [ p, p are i-variant with,. Notice the fors the failiar non-positivity property of the own substitution effect and the deand theore acquire in the context of deterinistic deand correspondences (Corollary 5.3(ii and Corollary 5.3(iii. Suppose the price of coodity i falls, all other prices reaining constant. Suppose one took any arbitrary consuption bundle that was initially chosen as the reference bundle, and reduced the consuer s wealth, so that this reference bundle cost exactly the consuer s wealth in the new situation. Then, given WARP, no consuption bundle chosen in the new situation can contain less of the i-th coodity than the aount contained in the reference bundle. If at least one bundle chosen in the initial situation contains strictly ore, then every bundle chosen in the new situation ust do so. An analogous set-doinance condition characterizes the deand theore. The central result of Bandyopadhyay, Dasgupta and Pattanaik (2004 also follows as a special case fro Proposition 5.1, as we now specify. Recall Notation 4.5. Corollary 5.4. (Stochastic Substitution Theore Let D be an SDF such that, for every ( p, W Z, q ({ x B p. x = W} = 1. D satisfies WASRP iff it also satisfies the following: for every ordered pair ( p W, ( p W Z Z [ q ( G + q ( A q( H + q( A ],,, and for every A I,. (5.2 Proof: See the Appendix. Bandyopadhyay, Dasgupta and Pattanaik (2004 show that the Stochastic Substitution Theore (Corollary 5.4 generates: (i non-positivity of the own substitution effect and the deand theore for SDFs, (ii equivalence of WARP with Sauelson s Inequality for DDFs (which we have tered the Sauelsonian Substitution Theore and (iii non-positivity of the own substitution effect and the deand theore for DDFs. It follows that these key results in the theory of consuer s behavior all follow as special cases of our General Substitution Theore (Proposition 5.1. Notice that the results 4/19/2007 1:22 PM 16

for DDFs, i.e. (ii and (iii above, can also be alternatively generated fro our Corollary 5.3 as the special case where the DDC is additionally constrained to be singular. Furtherore, (i and (iii above, and Corollary 5.3 ((ii and (iii can all be generated as special cases of Corollary 5.2. The interconnections between these various results are suarized in Figure 1. As is evident fro Figure 1, our General Substitution Theore (Proposition 5.1 provides the core unifying result, which yields all the other results as special cases. Insert Figure 1 Before concluding this section, we would like to coent on one aspect of our results. Our results establish properties of stochastic deand correspondences. The question arises whether, to interpret these results in ters of the observed arket behavior of the consuer, we need any specific behavioral rule for linking a realized ulti-eleent choice set and the consuption bundle that the consuer buys in the arket place given that realized choice set? The answer is that we do not need to assue any specific behavioral rule beyond the obvious rule that the purchased bundle ust be a eber of the realized choice set. For an exaple, consider our Corollary 5.2 (i, which tells us that, if a tight SDC, C, satisfies WASRP, then C ust satisfy NPS. It is straightforward to interpret the property of NPS for the stochastic deand correspondence, C, in ters of observed arket behavior of the consuer. Recall that C satisfies NPS if and only if, starting with any price-wealth situation ( p, W, if the price of a coodity j falls to ' p j (other prices reaining the sae and, siultaneously, the consuer s wealth is reduced by ' ( pj j p α, where α is any quantity of coodity j that the consuer could possibly buy in the initial situation, then the probability attached, after the fall in the price of coodity j, to the class of all possible choice sets, such that every bundle in the choice set has at least α aount of coodity j, is at least as great as the probability attached, in the initial situation, to the class of all choice sets, such that at least one bundle in the choice set contains at least α aount of coodity j. Given this definition, an obvious interpretation of NPS in ters of the observed arket purchases is as follows: when the price of coodity j falls and, siultaneously, the consuer s incoe is reduced by ' ( pj j p α, where α is soe quantity of coodity j that the consuer could possibly buy in the initial situation, the frequency with which she will buy at least α aount of coodity j in the new price-wealth situation will be greater than or equal to the frequency with which she buys at least α aount of coodity j in the initial situation, and this is true no atter what behavioral rules the consuer adopts to deterine which bundle she will choose to buy fro any given realized ulti-eleent choice set. Corollary 5.2 (i, therefore, has a straightforward 4/19/2007 1:22 PM 17

intuitive interpretation in ters of the consuer s actually observed arket behavior. Our other results have siilar interpretations. 6. Conclusion In this paper, we have expanded the standard WARP-based theory of consuer s behavior to siultaneously cover both rando consuer choice and choice of ultiple consuption bundles. We have offered a consistency postulate for deand behavior when such behavior is represented in ters of a stochastic deand correspondence. We have shown show that, when the consuer spends her entire wealth with probability one, our rationality postulate is equivalent to a condition we have tered stochastic substitutability. This equivalence generates: (i the Sauelsonian Substitution Theore, (ii the central result in Bandyopadhyay, Dasgupta and Pattanaik (2004 and (iii a version pertinent to deterinistic deand correspondences, which independently yields the Sauelsonian Substitution Theore, as alternative special cases. Relevant versions of the non-positivity property of the own substitution effect and the deand theore also fall out as corollaries in every case. Thus, we have provided a core unifying result, which subsues and expands available results. This result ay perhaps be seen as providing a logical closure to the WARP-based analysis of deand behavior initiated by Sauelson (1938. Extension of our analysis to the issue of rationalizability in ters of stochastic orderings would appear to be the natural next step. Application of our rationality postulate to the proble of aggregating deand correspondences, along the lines of Bandyopadhyay, Dasgupta and Pattanaik (2002, ay constitute another useful line of investigation. We leave these issues for the future. Appendix Throughout the proofs, we shall use the notation presented in Notation 4.5. Proof of Proposition 5.1. Let the SDC, C, be tight. First suppose C satisfies WASRP. Let A r( I. Then [( r ( H I \ A] r( B B ; [( r ( H I \ A] φ. Hence, by WASRP, Q ({ s ( s B [ ( r( H I \ A] } By tightness, when ({ s B ( s B (( r( H I \ A } + Q( r( B B Q \. (N1 A = φ, ({ s ( s B [ ( r( H I A] } Q ( r( G Q \ ({ s B ( s B (( r( H I \ A } + Q( r( B \ B Q Cobining (N1-(N3, we get (4.3. = 1, (N2 Q( H = 1. (N3 4/19/2007 1:22 PM 18

Again, by tightness, when A φ, ({ s ( s B [ ( r( H I A] } Q \ ({ s ( G I ( s I A} Q ( r( G = 1 Q, (N4 ({ s B ( s B (( r( H I \ A } + Q( r( B \ B = Q { s ( G I ( s I A} Q H Q ( ( 1. (N5 Cobining (N1, (N4 and (N5, we get (4.4. Hence, given tightness, WASRP iplies SS. A Now suppose C satisfies SS. Let A be any non-epty subset of r ( B B 0 = A r( I. First suppose A = r( I Since Q 0. Then, tightness and (4.3 together yield, ( r( B B + Q( { s B ( s B r( I } Q ({ s [ s B] φ}. We define: \. (N6 Q ({ s [ s B] } Q ({ s } [ s B] A and, noting that, since r( I A, φ, Q( { s B ( s B A} Q( { s B ( s B r( I }, it follows fro (N6 that (4.5 ust hold for any A r( B B such that ( I A Now suppose A 0 r( I. Then ( r ( I \ A 0 φ. Hence, by SS (4.4, Q ({ s ( G I ( s I [ ( r( I \ A0 ]} + Q ( r( G Q H + Q( { s G I s I [ r I \ A ]} By tightness, (N7 yields: ( ( ( ( 0 r.. (N7 ({ s ( G I ( s I A0 } + Q ( H Q( { s ( G I ( s I A } Q( r( G where H = { s ( G I H [ s H ] φ} 0 Q +. (N8 Again, by tightness, Q ({ s ( s B A} Q s ( G I ( s I A 0 ({ } Q H It follows fro (N8-(N9 that (4.5 ust hold for any non-epty A r( B B ( I ( +. (N9 such that A 0 r. Hence, SS iplies WASRP, given tightness of the SDC., Proof of Corollary 5.2. (i Consider any ordered pair ( p W, ( p W Z Z i-variant with,, such that, for soe i M, [ p, p are W p i < pi ] and [ W = W ( p i p i α for soe α [0, ] ]. Let the tight SDC, C, p satisfy WASRP. Then, by Proposition 5.1, C ust also satisfy SS. It is easy to check that, (i for all x G, x > α, (ii for all x H, x > α, (iii for all x I, x = α, (iv for all x H, i i i i 4/19/2007 1:22 PM 19

x < α, and (v for all x G, x < α. Then, noting that C is tight, (4.3 yields (4.1, and, putting i ( I A = r, (4.4 yields (4.2. i (ii Part (ii follows fro Corollary 5.2(i, Definition 4.3 and Definition 4.4. (iii Noting B( pw, = B( λp, λw for all ( p, W that, if B ( pw, B( λp, λw for soe ( p, W Z and all λ > 0, it can be easily checked Z and soe λ > 0, then we would have a violation of WASRP. We shall prove Corollary 5.3 via the following Lea. Lea N.1. (i A degenerate SDC, C, satisfies WASRP iff the DDC induced by C satisfies WARP. (ii A tight and degenerate SDC, C, satisfies SS iff the DDC induced by C satisfies the following: for every ordered pair ( p W, ( p W Z Z c( p W, for all x c( p W x,,,, such that p. x W for soe,, and for all x I, (5.1 holds. Proof of Lea N.1. Let C be a degenerate SDC, and let c be the DDC induced by C. Consider any,,. Then, for soe non-epty B ordered pair ( p W, ( p W Z Z soe non-epty a a, Q ({} a = 1, Q ({ a } = 1. Furtherore, c ( p, W = a and ( p W = a c,. (i First suppose C satisfies WASRP, and suppose [ ] φ [ r( B B ( a B ] t \. By WASRP, ({ s B ( s B = t} + Q( r( B \ B Q ({ s ( s B t} Q =., and for a B. Consider any Since the LHS is 0 by construction, it follows that c ust satisfy WARP. Now suppose c satisfies WARP. Notice that the degenerate SDC induced by c ust be C. If [ B ] = φ [ B ] φ a, then evidently the requireent for WASRP is trivially satisfied. Suppose a. Then, satisfaction of WARP by c iplies: (i for every non-epty [ r( B B \ ( a B ], the RHS of (4.5 is 0, and (ii for A { a B } A Thus, in all cases, (4.5 ust hold. (ii Let C be tight. First suppose C satisfies SS. Suppose p x W tightness of C, [ p. x = W ] for every x c( p, W. If p x < W It follows fro SS (4.3 that Q ( r( G = 1. Hence, ( p W G required strict inequality in (5.1 follows. Now suppose =, the LHS of (4.5 is 1.. for soe x c( p, W. for soe x c( p, W. By, ( H = 1 Q. c,. Noting Notation 4.5, the. for every x c( p, W p x W. Since we 4/19/2007 1:22 PM 20

also have p x W. for soe x c( p, W, in this case ( H = 0 ( c( p, W B = ( c( p, W I φ. Putting A {( c( p, W I } [ Q ({ s ( G I ( s I A} + Q ( r( G = 1] Lastly, suppose. for every x c( p, W p x W Q, whereas =, it follows fro SS (4.4 that. The required weak inequality in (5.1 follows., p x W. for soe x c( p, W [ c( p, W I ] [ c( p, W I ]. Putting A {( c( p, W I } [ Q( { s ( G I ( s I A} = 1] by construction, SS (4.4 then yields Q ( r( G = 1 ( p W G c,. The required strict inequality in (5.1 follows. Now suppose c satisfies the following: if ( p W p x W, and =, and noting. for soe x c( p, W. Hence, then, for all x c,, and for all x I, (5.1 holds. Recall that the tight and degenerate SDC induced by c ust be C. Notice also that, since C satisfies tightness, [ p x = W [ p x = W. for all x c( p W, ]. First suppose p x < W (5.1, c ( p, W G. Hence, Q ( r( G = 1. for all x c( p, W. for soe x c( p, W ], and. Then, by. Both (4.3 and (4.4 ust therefore be satisfied for all, and p. x W for every non-epty A r( I. Now suppose p. x W for soe x c( p, W x c( p, W. Then, by (5.1, c( p, W [ G I ]. First suppose [ c( p, W I ] [ c( p, W I ] Then, by (5.1, c ( p, W G. Hence, Q ( r( G = 1.. Both (4.3 and (4.4 ust therefore be satisfied for all non-epty A r( I. Now suppose [ c( p, W I ] = [ c( p, W I ] non-epty A r( I. If [ c( p W I ] A [ Q ({ s ( G I ( s I A} + Q ( r( G = 1]. If [ c( p W I ] A Q ({ s ( G I ( s I A} = 0. Consider any,, by (5.1,,,. Hence, in either case, (4.4 ust hold. Since ( H = 0 ust hold as well. Lastly, if. for every x c( p, W p x > W Q, (4.3, then the RHS of (4.4 ust be 0. Hence, (4.3-(4.4 ust trivially hold. Proof of Corollary 5.3. (i Suppose c satisfies WARP. Then, by Lea N.1(i, the tight and degenerate SDC, C, induced by c ust satisfy WASRP. By Proposition 5.1, hence, C ust satisfy SS. Thus, by Lea N.1(ii the DDC induced by C, c, ust satisfy (5.1 for every ordered pair ( p, W (, p W Z Z such that. for soe x c( p, W, for all x c( p W p x W c satisfies (5.1 for every ordered pair ( p W, ( p W Z Z c( p W, for all x c( p W x,,, and for all x I. Now suppose,, such that p. x W for soe,, and for all x I. Then C ust satisfy SS (Lea N.1(ii. Hence, by Proposition 5.1, C ust satisfy WASRP. By Lea N.1(i, c ust then satisfy WARP., 4/19/2007 1:22 PM 21