Revealed Preference with Stochastic Deand Correspondence 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 Draft: 13 October 2006 Abstract: We expand the WARP based theory of consuer s behavior to siultaneously cover both rando choice and choice of ultiple consuption bundles. 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: (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. Keywords: Stochastic deand correspondence, weak axio of revealed preference, weak axio of stochastic revealed preference, general substitution theore. JEL Classification Nuber: D11 Corresponding author. 1
1. Introduction Sauelson s Inequality largely suarizes the epirical content of the standard theory of copetitive consuer behavior. It predicts that, if the consuption bundle chosen in soe initial price-wealth situation (i costs exactly the consuer s wealth in soe altered price-wealth situation, and (ii differs fro the bundle chosen in the new situation, then the products of price change and quantity change for each coodity will su to a negative nuber. Sauelson s Inequality yields the non-positivity property of the own-price substitution effect as a special case; this, in turn, yields the law of deand. Sauelson (1938 introduced the behavioral postulate, the so-called weak axio of revealed preference (WARP, which leads to this inequality. The central result in the standard choice-based theory of consuer s behavior is the equivalence of Sauelson s Inequality and WARP, given that the consuer spends her entire wealth (Sauelson (1947, Mas-Colell et al. (1995, pp. 28-32. For convenience, one ay ter this equivalence the Sauelsonian Substitution Theore. The standard 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 restraints 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. This akes the fraework conceptually ill equipped to address flat indifference surfaces, even though one would intuitively expect consuers to often exhibit such preferences. 1 Second, deand behavior is represented by eans of a unique deand function. This assues away the possibility of a probabilistic eleent in the consuer s choice behavior. Modeling consuer s behavior by eans of stochastic deand correspondences, 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. However, 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. the Sauelsonian 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 1 Of course, observed arket choices, the focus of Sauelson s theory, would not reveal indifference aong ultiple best eleents. However, one ay conceptualize the revelation of indifference in ters of consuer survey experients, where consuers are asked to choose a (possibly ulti-eleent subset out of soe feasible set of alternatives. This is how, for exaple, Arrow (1959 conceptualizes choice in a general context. 2
on the basis of soe coplete and transitive binary preference relation, even a utility function, requireents stronger than the satisfaction of WARP. 2 Thus, the priary objective of this line of enquiry was to consistently construct utility functions fro soe given specification of deand behavior. It is not even clear fro these contributions exactly how one should interpret the nonpositivity property of the own substitution effect, nor, indeed, the deand theore, when the consuer chooses (ulti-eleent sets of consuption bundles, rather than a single consuption bundle. 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 expansion of the Sauelsonian Substitution Theore. 3 Their result encopasses the traditional version as a special case. However, their analysis 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 parallel 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. Can one then develop a predictive theory of consuer s behavior, which expands the Sauelsonian WARP-based fraework, to siultaneously cover both rando consuer choice and choice 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 the Sauelsonian Substitution Theore as its own special case. Providing such an integrated theoretical fraework is the purpose of this paper. 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. Our central result 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 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. 3
Sauelsonian Substitution Theore, as alternative special cases. Relevant versions of the nonpositivity property of the own substitution effect and the deand theore also fall out as corollaries in every case. Thus, our central result expands the traditional WARP-based fraework to siultaneously cover both rando consuer choice and choice of ultiple consuption bundles, 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. Basic 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 +, and given any i M, will denote the aount x i The set of all possible price vectors is R ++, with p, p etc. denoting individual price vectors. For any given coodity i M, we say that two price vectors p and p are i-variant iff [ p and, for every coodity i p i j i, p = p ]. The set of all possible wealth levels of the j consuer is R, with W, W etc. denoting specific wealth levels. A price-wealth situation is a pair ( + + p, W R+ + R. Thus, the set of all possible price-wealth situations is Z R+ + R+. Given a price-wealth situation ( p, W, the consuer s budget set, ( p W For brevity, we shall typically write B B j, etc., instead of B ( p W, B( p, W B,, is the set { x R + p. x W },, etc.. Given R, T will denote the set of all possible non-epty subsets of T. R T will T + denote the power set of T; thus, r ( ( ( T [ r( T U { φ} ] R, where φ denotes the epty set. Given two sets T, 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 4
involves allowing the consuer to: (i choose a set of ultiple consuption bundles (rather than a single bundle as in the classical fraework, and (ii do so in a rando fashion, i.e. to choose no set of consuption bundles with probability 1 (rather than the certain choice in the classical fraework. 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 over ( r( B R.,, (ii A stochastic deand function (SDF is a rule D which, for every (, specifies exactly one finitely additive probability easure q over R ( B. p, W (iii A deterinistic deand correspondence (DDC is a rule c which, for every ( p W Z 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, specifies exactly one eleent of B. ( W Consider any price-wealth situation ( p, W. Let B denote the budget set corresponding to. Let Q = C( p, W, where C is an SDC. Given the price-wealth situation ( p, W, for any p, A R( r( B (, Q A is the probability that the (possibly ulti-eleent set of chosen bundles will lie ( ( W in the class A. C p, W, C p, 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, where D is an SDF. For any A R( B, q ( A is the probability that the chosen 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, is the bundle the consuer will choose fro the budget set corresponding to the price-wealth situation ( p,w. 5
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 ( A Q( { A A A = 1} Q. = i i (ii An SDC, C, is degenerate, iff, for all ( p W Z Q ({ A} = 1. A, (,, there exists A r B 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. Definition 3.3. (i An SDC, C, induces an SDF, D, iff, for every ( p, W Z, and for every R( B q ( A { } A, = Q( { x x A}. An SDF, D, induces an SDC, C, iff, for every ( p, W Z, and for every 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 c, is the set of consuption A Q ({ } 1 ( p W Z bundles B such that A =. A DDC, c, induces an SDC, C, iff, for every Q ({ c( p, W } = 1.,, Reark 3.4. If an SDC is non-singular, then Q ({ x} x B} < 1 q ( B = 1 ; yet it ust always be the case that. Hence, 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: (i degenerate SDCs and DDCs, and (ii DDFs and SDCs that are both singular and degenerate. 6
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 choosing soe subset of consuption bundles that exhaust her entire wealth is always one. In essence, this siply reflects 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 pi 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 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? 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. 7
Definition 4.2. A tight SDC, C, satisfies non-positivity of the own substitution effect (NPS iff, for every i M, and for every ordered pair ( p, W, ( p, W Z Z such that [[ p, p are i-variant W with p i < p i ], and [ W = W ( p i p i α for soe α 0, ]], p i and x > α for all x s } Q( { s B x > for soe x s} Q α, (4.1 i x α for all x s } Q( { s B x for soe x s} Q α, (4.2 W W where α =. ( p i p i 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 4 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 p R + +, and all W, W R + such that W > W, (4.1 and (4.2 both hold when B = B( p, W and ( p W B = B,. 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 setdoinance 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 R + such that [[ p p are i-variant with p + R +, i i α, (4.1 and (4.2 both hold when B = ( p, W and B = B( p, W p < p ], for every W R +, and for all. 4 Notice that we adopt a weak definition of norality. We could adopt a stronger definition, by insisting on a greater than relationship, without altering our results in any way. 8
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 I ] φ} I = +. G = +. H = +. G = +. H = +. G = + G, and H = + H. { s { x R p. x = W } [ s I ] φ},, let: Insert Figure 1 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 A r I, Q ( r( G + Q ( G I ( s I I A} Q H ({ A} U ( + Q s ( G I ( s I I U. (4.4 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. 9
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 I, ( r( B B + Q( { s B ( s I B A} Q ( s I 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 I 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 I, q ( B \ B + q( A q (A. (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 I B ] φ, then c ( p W [ r( B \ B U { s ( s I B = ( c( p, W I B }],. (4.7 (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 U { d( p W }] d,. 10
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 I 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. 5 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. 5 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. 11
Corollary 5.2. Suppose a tight SDC, C, satisfies WASRP. Then: (i C ust satisfy NPS; (ii every noral good ust also be regular. 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 (Substitution Theore for DDCs c satisfies WARP iff, for every ordered pair p, W, p, W Z, such that p. x W for soe x c( p, W, for all x c( p, W, and ( ( Z for all x I, [( p (. x x 0] p ; the inequality holding strictly when either (a p. x < W for [ ] [ ( ] soe x c( p, W, or (b c( p, W I I c p, W I I. (5.1 (ii Let c satisfy WARP. Then, for every i M, for every ordered pair ( p W, ( p, W Z Z soe, such that [[ p, p are i-variant with p i < pi ], and [ p. x = W for x c( p, W ]], and for all x c ( p, W, [ x ] i x i ; the inequality holding strictly when x < sup{ x x c( p W } (iii Let c satisfies WARP, and suppose i M p p R + + by c. Then, for every ordered pair ( such that [ every R +, and for all x c p, W, W ( [ { x x c( p, W } ] sup. i x i Proof: See the Appendix. i i, is noral according to the degenerate SDC induced, p, p are i-variant with p < ], for. i p i 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 12
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. (Substitution Theore for SDFs 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 ]. (5.2 Proof: See the Appendix.,,, and for every A I, Bandyopadhyay, Dasgupta and Pattanaik (2004 show that the Substitution Theore for SDFs (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 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 2, our General Substitution Theore (Proposition 5.1 provides the core unifying result, which yields all the other results as special cases. Insert Figure 2 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 13
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 U I \ A] r( B I B ; [( r ( H U I \ A] φ. Hence, by WASRP, Q ( s I B [ ( r( H U I \ A] } By tightness, when ({ s B ( s B (( r( H U I \ A } + Q( r( B B Q I \. (N1 A = φ, ( s I B [ ( r( H U I A] } Q ( r( G ({ s B ( s I B (( r( H U I \ A } + Q( r( B \ B Q( H Q \ Q Cobining (N1-(N3, we get (4.3. Again, by tightness, when A φ, ( s I B [ ( r( H U I A] } Q \ ( G U I ( s I A} Q ( r( G = 1, (N2 = 1. (N3 = 1 Q I, (N4 ({ s B ( s I B (( r( H U I \ A } + Q( r( B \ B = Q( { s ( G U I ( s I I A} Q( H Q 1. (N5 Cobining (N1, (N4 and (N5, we get (4.4. Hence, given tightness, WASRP iplies SS. Now suppose C satisfies SS. Let A be any non-epty subset of r ( B I B. We define: A 0 = A I r( I 0 ( Q( r( B B + Q( { s B ( s I B r( I } Q [ s I B] φ} Since. First suppose A = r I. Then, tightness and (4.3 together yield, \. (N6 14
[ s I B] } Q } [ s B] A Q φ I, and, noting that, since ( I Q r A, ({ s B ( s I B A} Q( { s B ( s I B r( I }, it follows fro (N6 that (4.5 ust hold for any A r( B I B such that ( I A Now suppose A 0 r( I. Then ( r ( I \ A 0 φ. Hence, by SS (4.4, ( G U I ( s I I [ ( r( I \ A ]} + Q ( r( G Q 0 Q By tightness, (N7 yields: ( H + Q { s ( G I ( s I I [ r( I \ A ] ( 0 r. U }. (N7 ( G U I ( s I I A } + Q ( H Q( { s ( G U I ( s I I A } Q( r( G 0 0 Q + { U φ}. (N8 where H = s ( G I U H [ s I H ] Again, by tightness, Q ( s I B A} Q ( G U I ( s I I A 0 } + Q (H. (N9 It follows fro (N8-(N9 that (4.5 ust hold for any non-epty A r( B I B such that ( A 0 r I. 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,, such that, for soe i M, [[ p, p are W i-variant with p i < p i ], and [ W = W ( p i p i α for soe α 0, ]]. Let the tight SDC, C, p i 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, i x < α, and (v for all x G, i A = r( I, (4.4 yields (4.2. x i x > α, (iii for all x I, x = α, (iv for all x H, i < α. Then, noting that C is tight, (4.3 yields (4.1, and, putting (ii Part (ii follows fro Corollary 5.2(i, Definition 4.3 and Definition 4.4. i 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: 15
for every ordered pair ( p W, ( p W Z Z,,, such that p. x W for soe x c( p, W, for all x c( p, W, 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 ordered pair ( p, W, ( p, W Z Z. Then, for soe non-epty a B, Q( {} a = 1, and for soe non-epty Q ({ } 1 (, a and ( p W = a a, a =. Furtherore, c p W = (i First suppose C satisfies WASRP, and suppose [ ] φ [ r( B B ( a I B ] t I \. By WASRP, ({ s B ( s I B = t} + Q( r( B \ B Q ( s I B t} Q =. c,. a I 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 [ r( B I B \ ( a I B ], the RHS of (4.5 is 0, and (ii for A { a B } a I, then evidently the requireent for WASRP is trivially satisfied. Suppose a I. Then, satisfaction of WARP by c iplies: (i for every non-epty = I, the LHS of (4.5 is 1. Thus, in all cases, (4.5 ust hold. (ii Let C be tight. First suppose C satisfies SS. Suppose p. x W for soe x c( p, W. By tightness of C, [ W ] p. x = for every x c( p, W. If p. x < W for soe x c( p, W, Q ( H = 1. It follows fro SS (4.3 that Q ( r( G = 1. Hence, ( p W G c,. Noting Notation 4.5, the required strict inequality in (5.1 follows. Now suppose p. x W for every x c( p, W. Since we also have. for soe x c( p, W p x W, in this case ( H = 0 ( c( p, W I B = ( c( p, W I I φ. Putting A {( c( p, W I I } [ Q ( G U I ( s I I A} + Q ( r( G = 1] Q, whereas =, it follows fro SS (4.4 that. The required weak inequality in (5.1 follows. Lastly, suppose p. x W for every x c( p, W, p. x W for soe x c( p, W, and [ c( p, W I I ] [ c( p, W I I ]. Putting A = {( c( p, W I I }, and noting [ Q({ s ( G U I ( s I I A} = 1] by construction, SS (4.4 then yields Q ( r( G = 1. Hence ( G c p, W. The required strict inequality in (5.1 follows. Now suppose c satisfies the following: if p. x W for soe x c( p, W, then, for all x c( p, W, 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 for all x c( p, W ], and [ p. x = W for all x c( p, W ]. First suppose p. x < W for soe x c( p, W. Then, by 16
(, G ( ( = 1 (5.1, c p W. Hence, Q r G non-epty A r( I. Now suppose x c( p, W. Then, by (5.1, c( p W [ G U I ]. Both (4.3 and (4.4 ust therefore be satisfied for all p. x W for soe x c( p, W,. First suppose [ c( p, W I I ] [ c( p, W I I ], and p. x W for every. Then, by (5.1, c ( p, W G. Hence, Q ( r( G = 1 satisfied for all non-epty A r( I. Now suppose [ c( p, W I I] = [ c( p, W I I] non-epty A r( I. If [ c( p W I I ] A [ Q ( G U I ( s I I A} + Q ( r( G = 1]. If [ c( p W I I ] A Q ({ s ( G I ( s I I A} = 0. Both (4.3 and (4.4 ust therefore be. Consider any,, by (5.1,,, U. 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 Hence, (4.3-(4.4 ust trivially hold. Q, (4.3, then the RHS of (4.4 ust be 0. 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 p, Z N.1(ii the DDC induced by C, c, ust satisfy (5.1 for every ordered pair (, W (, p W Z c p x c( p W such that p. x W for soe x (, W, for all c satisfies (5.1 for every ordered pair,, and for all x I. Now suppose ( p W, ( p, W Z Z, such that p. x W for soe x c( p, W, for all x c( p, W, 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. Part (i yields part (ii, while part (iii follows fro part (ii and Definition 4.3. We shall prove Corollary 5.4 via the following Lea. Lea N.2. (i A singular SDC, C, satisfies WASRP iff the SDF induced by C satisfies WASRP. (ii A tight and singular SDC, C, satisfies SS iff the SDF induced by C satisfies (5.2. Proof of Lea N.2. Let C be a singular SDC, and let D be the SDF induced by C. Then the,,. singular SDC induced by D ust be C. Consider any ordered pair ( p W, ( p W Z Z (i First suppose C satisfies WASRP. For any non-epty A [ B B ] singular, and since D is induced by C, we get: ({ s B ( s B r( A } + Q( r( B \ B = q( A + q( B \ B ( I, consider r A. Since C is Q I, (N10 17
({ s ( s B r( A } = q ( A Q I. (N11 Noting that q ( φ ( = 0, it follows fro (N10-(N11 that D ust satisfy WASRP. ~ A r B B A = x B I B x A. Now suppose D satisfies WASRP. Let ( I. Let { ( { } } ~ ( = q ( A ~ [ Q( { s B ( s B A} + Q( r( B \ B = q( B \ B + q( A ] Since C is singular, Q ( s B A} I, and I. Since D satisfies WASRP, noting ~ that A [ B I B ], it follows that C ust satisfy WASRP. (ii Let C be tight. Suppose C satisfies SS. Consider ( A r( I Since C is singular and tight, and since D is the SDF induced by C, ( G I ( s I r( A } + Q ( r( G = q ( G + q ( A r for any non-epty A I. Q U I, (N12 ( H + Q s ( G I ( s I I r( A ({ } = q( H q( A Q U +, (N13 ( r( G = q ( G Q, (N14 ( H q( H Q =. (N15 Since C satisfies SS, it follows fro (N12-(N15 that D ust satisfy (5.2. Now suppose D satisfies (5.2. Consider any non-epty A r( I ~ { x I {} x A} a =. Then, since C is singular, ~ ({ A} = q ( G + q ( a ~ ({ s G I s I I A} = q( H q( a ( r( G + Q s ( G I ( s I I ~. Let Q U, (N16 ( H + Q ( ( Q U +. (N17 Since D satisfies (5.2, it follows fro (N16-(N17 that C ust satisfy (4.4. Since (5.2 iplies [ q ( G q( H ], and C is singular, C ust also satisfy (4.3. Proof of Corollary 5.4. Suppose D satisfies: for every ( p, W Z, q( { x B p. x = W} = 1. The singular SDC, C, induced by D ust then satisfy tightness. Suppose D satisfies WASRP. Then, by Lea N.2(i, C satisfies WASRP. Hence, by Proposition 5.1, C satisfies SS. By Lea N.2(ii, D then satisfies (5.2. Now suppose D satisfies (5.2. Then, by Lea N.2(ii, C satisfies SS, and thus, by Proposition 5.1, WASRP. Hence, by Lea N.2(i, D satisfies WASRP. References Afriat, S.N. (1967: The construction of a utility function fro expenditure data, International Econoic Review 8: 67-77. Arrow, K. (1959: Rational choice functions and orderings, Econoica (N.S. 26: 121-127. 18
Bandyopadhyay, T., I. Dasgupta and P.K. Pattanaik (2004: A general revealed preference theore for stochastic deand behavior, Econoic Theory 23: 589-599. Bandyopadhyay, T., I. Dasgupta and P.K. Pattanaik (2002: Deand aggregation and the weak axio of stochastic revealed preference, Journal of Econoic Theory 107: 483-489. Bandyopadhyay, T., I. Dasgupta and P.K. Pattanaik (1999: Stochastic revealed preference and the theory of deand, Journal of Econoic Theory 84: 95-110. Dasgupta, I. (2005: Consistent fir choice and the theory of supply, Econoic Theory 26: 167-175. Dasgupta, I. and P.K. Pattanaik (2006: 'Rational choice and the weak axio of stochastic revealed preference, Econoic Theory (forthcoing. Houthakker, H.S. (1950: Revealed preference and the utility function, Econoica (N.S. 17: 159-174. Mas-Colell, A., M.D. Whinston and J. Green (1995: Microeconoic theory, New York: Oxford University Press. McFadden, D. (2005: Revealed stochastic preference: a synthesis, Econoic Theory 26: 245-264. Richter, M.K. (1966: Revealed preference theory, Econoetrica 34: 635-645. Sauelson, P.A. (1947: Foundations of econoic analysis, Cabridge MA: Harvard University Press. Sauelson, P.A. (1938: A note on the pure theory of consuer s behavior, Econoica 5: 61-71. Sen, A.K. (1971: Choice functions and revealed preference, Review of Econoic Studies 38: 307-317. Varian, H.R. (1982: The nonparaetric approach to deand analysis, Econoetrica 50: 945-972. 19
Figure 1 x 2 a a' g o b b' x1 oab is the budget set, B, corresponding to the price-wealth situation ( p, W.,. oa'b' is the budget set, B', corresponding to the price-wealth situation ( p W I is the singleton set containing g. G is ag excluding the point g. H is gb excluding the point g. G' is gb' excluding the point g. H' is a'g excluding the point g. 20
Figure 2 SAMUELSONIAN SUBSTITUTION THEOREM SUBSTITUTION THEOREM FOR DDCs SUBSTITUTION THEOREM FOR SDFs GENERAL SUBSTITUTION THEOREM [NPS+DT+HGN(0] [NPS+DT+HGN(0] [NPS+DT+HGN(0] for DDCs for SDCs for SDFs [NPS+DT+HGN(0] for DDFs 21
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