Dispersed excess demands, the weak axiom and uniqueness of equilibrium

Journal of Mathematical Economics 31 Ž1999. 15–48

Dispersed excess demands, the weak axiom and uniqueness of equilibrium Michael Jerison

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Department of Economics, BA 108, SUNY, Albany, NY 12222, USA Received 15 October 1997; accepted 4 September 1998

Abstract This paper introduces an economically interpretable hypothesis that implies that mean excess demand satisfies the weak axiom and that competitive equilibrium is unique. The hypothesis requires, roughly, that the consumers’ excess demand vectors spread apart on average as their wealth increases. The hypothesis is potentially testable using cross-section data on consumer expenditures and endowments. It is satisfied in a robust class of economies, including those with suitable types of consumer heterogeneity. However, it implies stringent restrictions on the consumers’ Engel curves if it is required to hold for eÕery distribution of collinear consumer endowments. q 1999 Elsevier Science S.A. All rights reserved. JEL classification: D11; D12 Keywords: Weak axiom of revealed preference; Increasing dispersion; Average derivative estimation; Competitive equilibrium; Aggregation

1. Introduction In applied general equilibrium analysis, assumptions are often made to ensure that equilibrium prices and resource allocation are determined by ‘fundamentals’: policy variables, technology, preferences and the initial distribution of private ownership. Competitive equilibrium is assumed to be unique. But uniqueness of )

Tel.: q1-518-442-4287; fax: q1-518-442-4736; e-mail: [email protected]

0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 8 . 0 0 0 5 6 - 1

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M. Jerisonr Journal of Mathematical Economics 31 (1999) 15–48

equilibrium is not implied by competitive behavior on the part of individual consumers and firms, cf. Shafer and Sonnenschein Ž1982. and their references. A sufficient condition for uniqueness of equilibrium in regular production economies is that the consumers’ mean excess demand function satisfies the weak axiom of revealed preference. In fact, the weak axiom is the weakest demand-side restriction ensuring uniqueness of equilibrium in regular production economies. However, the weak axiom at the aggregate level is ad hoc. We would like to have reasonably general, economically interpretable conditions under which it is satisfied. The present paper presents a robust, interpretable sufficient condition under which mean consumer excess demand satisfies the weak axiom, and competitive equilibrium is unique. The condition is called increasing dispersion of excess demands ŽIDED.. It requires, roughly, that if the consumers were given additional money their excess demand vectors would move apart on average. IDED is potentially testable using methods of nonparametric statistics applied to cross-section data on household expenditures and endowments. It is satisfied in a broad class of economic models in which the consumers’ shares of aggregate wealth vary with prices. It is of interest in welfare theory since it is necessary in order for the wealth distribution to be optimal according to a single social welfare function no matter what prices prevail, cf. Jerison Ž1994.. IDED is a hypothesis concerning effects of changes in wealth. It is related to hypotheses about consumer heterogeneity that are shown by Grandmont Ž1992., Marhuenda Ž1995. and Quah Ž1997. to imply uniqueness of equilibrium. Mas-Colell Ž1991. offers an alternative sufficient condition for uniqueness based on large substitution effects, a condition that does not imply the weak axiom for aggregate excess demand. Relationships between IDED and all of these hypotheses are discussed in Section 7. IDED is also closely related to ‘spreading Engel curves,’ a property formulated by Jerison Ž1982.. The latter property Žreferred to below as increasing demand dispersion ŽIDD.. requires, roughly, that if the consumers become richer, their demand vectors move apart on average. IDD has empirical support, cf. Haerdle et al. Ž1991., Hildenbrand and Kneip Ž1993. and Hildenbrand Ž1994., but it only leads to the weak axiom in the aggregate and to uniqueness of equilibrium if the consumers’ shares of aggregate wealth do not vary with prices Žas when their endowment vectors are collinear.. IDD and increasing dispersion of excess demands are essentially equivalent if the consumers’ endowment vectors are collinear. IDED is a robust property leading to the weak axiom in the aggregate. It does not imply any restrictions on the functional forms of the consumers’ Engel curves or excess demand functions or on the forms of the mean demands or excess demands. The robustness of IDED might appear to conflict with results due to Freixas and Mas-Colell Ž1987. and Hildenbrand Ž1989.. Their results suggest that

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requiring the weak axiom in the aggregate imposes severe restrictions on the functional forms of the consumers’ Engel curves. One of the purposes of this paper is to clarify the sources of these restrictions. Hildenbrand Ž1989. shows that the weak axiom in the aggregate is generically violated when consumers’ wealth is derived from inelastically supplied primary factors of production. The problem is that changes in factor prices can cause a redistribution of wealth among consumers. But the weak axiom in the aggregate requires that pure wealth redistribution has no effect on aggregate demand. This can be assured only if the consumers’ Engel curves at any given price vector are linear and parallel, cf. Gorman Ž1953.. IDED for this reduced model is robust. IDED is typically violated in models with inelastically supplied primary factors. But the equilibria in such a model with constant returns, nonjoint production can be derived from the equilibria in a ‘reduced’ pure exchange model. The goods in the reduced model are the original primary factors, and the demands for these goods are derived from the consumers’ demands for produced goods and the producers’ demands for inputs in the original economy Žsee Section 3, below.. The problem analyzed by Hildenbrand Ž1989. arises because the households’ shares of aggregate wealth vary with prices. Freixas and Mas-Colell Ž1987. consider the case in which the wealth shares are fixed, e.g., when endowments are collinear. They consider the class of consumption sectors in which all households have the same arbitrary Engel curve at some price vector, and they ask what restrictions on the Engel curve are necessary for mean demand to satisfy the weak axiom in eÕery consumption sector in the class. They show that the Engel curve must lie in a plane, and for each commodity the demand must be a convex or concave function of wealth. Engel curves of this form are not robust since perturbations cause them to leave the plane. These Engel curve restrictions are very strong, but so is the hypothesis. The class of allowed consumption sectors is very large. In particular, individual substitution effects are allowed to be arbitrarily small, so they cannot contribute to the weak axiom for mean demand. In addition, the wealth distribution is unrestricted, so, in particular, mean demand is required to satisfy the weak axiom in every two-consumer economy with the given Engel curve Žall the other consumers’ wealth levels can be set equal to zero.. As we show in Proposition 1X , below, nondecreasing dispersion of excess demands ŽNDED. is the weakest joint restriction on the consumers’ Engel curves and endowments leading to a weak version of the weak axiom in the aggregate. The restrictions derived by Freixas and Mas-Colell are implications of IDED when IDED is required to hold for eÕery collinear endowment distribution, including distributions concentrated at two points. One might ask if these same restrictions are implied when the distribution of wealth is required to be unimodal so that there must be a continuum of consumers. We show in Proposition 3, below, that they are.

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Next, we ask if the restrictions obtained by Freixas and Mas-Colell follow from the assumption that the consumers’ Engel curves are identical. Allowing for heterogeneous Engel curves broadens the class of communities satisfying the weak axiom in the aggregate for every wealth distribution. But we show that the communities in the class are still very special. The consumers’ Engel curves at a given price vector must all be linear or else they must all lie in a single plane Žin which case they must satisfy other curvature restrictions described in Proposition 4, below.. This conclusion follows because we allow the entire aggregate wealth to be held by just two consumers. These results suggest that IDED can fail because of insufficient consumer heterogeneity. If the consumers’ Engel curves are identical but their wealth levels are not, then there must be enough diÕersity of wealth in order for IDED to hold. It is generally not sufficient to have wealth concentrated at two distinct levels, however far apart they are. On the other hand, IDED can hold with a concentrated wealth distribution if the consumers’ demand functions have the right kind of diversity. The issue of consumer heterogeneity is discussed further in Section 7. Section 2 presents notation and differential characterizations of demand and excess demand functions satisfying a weak version of the weak axiom. In Section 3, we define and characterize increasing dispersion of demand and IDED. In Fig. 1 we show that these conditions must be violated if mean demand or excess demand violates the weak axiom. We also state the main proposition, that IDED implies uniqueness of equilibrium in competitive economies with constant returns production. Section 4 describes a nonparametric test of IDED when demand functions and endowments are independently distributed Žor under weaker assumptions that relate wealth effects to cross-section data.. Sections 5 and 6 extend the main theorem of Freixas and Mas-Colell Ž1987., first by restricting the allowed consumption sectors to ones with unimodal wealth distribution, then by allowing heterogeneous consumer demand functions. Section 7 discusses related literature and remaining open problems. The longer proofs are in Section 8. 2. Notation and preliminary results l l be a convex set containing Rqq and not containing 0 g R l. A Let P ; Rq l demand function in a l-good economy is a function F: P = Rq™ Rq that is homogeneous of degree zero and satisfies the budget identity pF Ž p,w . s w for each price vector p g P and wealth w G 0. 1 We will restrict attention to the set of

1

Unless otherwise noted, vectors are columns. For vectors u and z in R l we write uz in place of uPz when there is no chance of confusion. We write u4 z wrespectively, uG z x when every component of the vector u is strictly greater than wresp. at least as great asx the corresponding component of vector z. The set of vectors orthogonal to z is denoted z H '  u:uPzs 04. We use capital letters to represent matrix and vector-valued functions, and let small letters represent scalar-valued functions.

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C 1 demand functions, F , endowed with the weak C 1 topology, cf. Hirsch Ž1976.. An excess demand function on Q ; P is a function Z:Q ™ R l that is homogeneous of degree zero and satisfies Walras’ Law: pZ Ž p . s 0 for each p g Q. A consumption sector is a joint distribution of households’ demand functions and l endowments, formally, a Borel probability measure m on F = Rq . A household a l of type a has the endowment vector v g Rq , the demand function F a g F , with ith component Fia , and the excess demand function Z a Ž p . ' F a Ž p, pv a . y v a. In the private ownership economies studied below, the demand vector F a Ž p, pv a . includes amounts of goods that are not purchased, but were initially owned and not sold during the period under consideration. For most of the results in this paper, the consumption sector can be finite. The measure m can be discrete. The mean endowment vector for the consumption sector m is v ' v a d m. A

H

household of type a is competitiÕe if it is a utility maximizer, i.e., if there is a l function u:Rq ™ R such that for each Ž p,w . g P = Rq the vector F a Ž p,w . is the l unique maximizer of u over the budget set b Ž p,w . '  x g Rq : px F w4 . A consumption sector is competitiÕe if every household type in its support is competitive. We treat the households’ demand functions as primitive in order to allow for household types that are not competitive. The marginal propensity to consume of household type a with wealth w at price vector p is the vector M a Ž p,w . ' Ew F a Ž p,w . of derivatives of the demands with respect to wealth. The aÕerage propensity to consume of household type a at Ž p,w . g P = Rqq is A a Ž p,w . ' Ž1rw . F a Ž p,w .. The Slutsky matrix of household type a at price and wealth vector Ž p,w . is S a Ž p,w . ' Ep F a Ž p,w . q M a Ž p,w . F a Ž p,w .

T

ŽThe superscript T denotes the transpose. The ij element of the l = l matrix Ep F a Ž p,w . is EFia Ž p,w .rEpj .. A l = l matrix M is positiÕe semidefinite wresp. positiÕe definite x on V ; R l if T z M z G w)x0 for each nonzero z g V. We omit the reference to V if it is R l. Note that the matrix M need not be symmetric. The matrix M is negatiÕe semidefinite wresp. negatiÕe definite x on V if yM is positive semidefinite wresp. positive definitex on V. A function G: X ; R l ™ R l is pseudomonotone on Y ; X if for each p and q in Y, GŽ q .Ž p y q . F 0 implies GŽ p .Ž p y q . F 0. We omit reference to Y if it is the domain of G. We say that a demand function F satisfies the weak weak axiom of reÕealed preference ŽWWA. if F ŽP,1. is pseudomonotone. We say that an excess demand function satisfies WWA if it is pseudomonotone. A demand function F satisfies the weak axiom if qF Ž p,1. F 1 and F Ž p,1. / F Ž q,1. imply pF Ž q,1. ) 1. An excess demand function Z satisfies the weak axiom if qZ Ž p . F 0 and Z Ž p . / Z Ž q . imply pZ Ž q . ) 0. It is easy to show that the demand and excess demand functions F a and Z a satisfy the weak axiom if household type a is

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competitive. The converse is true in the case of two goods Ž l s 2. but not otherwise. The WWA for demand functions has the following differential characterization. Lemma 1. (Kihlstrom et al., 1976) A C 1 demand function F a satisfies the WWA if l and only if for each Ž p,w . g Rqq = Rqq the Slutsky matrix S a Ž p,w . is negatiÕe semidefinite. We restrict attention to two classes of economies: pure exchange economies without disposal, and production economies with free disposal and convex, constant returns to scale technology. All of these economies are characterized by a consumption sector m and a production set, which is a closed convex cone Y ; R l. We refer to the pure exchange case as the case of no production. In that case Y s  04 . In the production case, we assume that z F y g Y implies z g Y Žfree l disposal. and Y l Rq s  04 Žno free lunch.. The latter condition states that inactivity is possible and that it is impossible to produce output without using any input. With production or without it, equilibrium profit must be zero, so it is unnecessary to specify the households’ profit shares. The mean demand and mean excess demand for the consumption sector m at price vector p are, respectively, F Ž p . ' F a Ž p, pv a . d m and Z Ž p . '

H

H

F a Ž p, pv a . y v a d m ,

defined for p g P. These functions depend on m , but as long as the community is fixed, we omit reference to it. In the economy determined by m and Y a Ž competitiÕe. equilibrium price vector is a vector p g P such that Ži. Z Ž p . g Y and Žii. py F 0 for all y g Y. Condition Ži. implies that the net consumer demand can be satisfied by the production sector. Condition Žii. states that profit is maximized by the production vector Z Ž p . since pZ Ž p . s 0. The homogeneity of Z implies that for any equilibrium price vector p and any scalar l ) 0, l p is also an equilibrium. Thus, when we speak of uniqueness of equilibrium, we mean uniqueness up to scalar multiplication. The definition of equilibrium requires supply to equal demand for all goods when there is no production. But in the production case there can be excess supply in equilibrium. It is possible that Z Ž p . is not in the boundary of the production set at an equilibrium price vector p. Then the price of each good in excess supply is zero. ŽIf not, condition Žii. is violated.. This is allowed since the set of prices P can contain vectors with components equal to zero. We will be concerned with conditions under which the mean excess demand function Z satisfies WWA Ži.e., is pseudomonotone. and conditions under which the competitive equilibrium is unique. Pseudomonotonic excess demand functions have a differential characterization similar to the one in Lemma 1, but this characterization requires an additional hypothesis. We call an excess demand

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function Z on an open set Q ; P regular if its Jacobian matrix EZŽ p . has rank l y 1 for each p g Q such that ZŽ p . s 0. Note that l y 1 is the maximal possible rank of the Jacobian matrix since the homogeneity of Z implies EZŽ p . p s 0. John Ž1995. proves the following. Lemma 2. A regular C 1 excess demand function Z on a conÕex set Q ; P is pseudomonotone if for each p g Q, the Jacobian EZŽ p . is negatiÕe semidefinite on ZŽ p . H . The conÕerse is true if Q is open in R l. In Sections 5 and 6 we will show that in order for IDED to be satisfied for every endowment distribution, the households’ Engel functions must have special forms, which we now define. Let W be an interval in Rq and fix p g P. A function G:W ™ R l with pGŽ w . s w for each w g W is called a p-Engel curÕe on W. A p-Engel curve on an interval has no torsion ŽNT. if its image is contained in a plane Ža two-dimensional affine space in R l .. A real valued function defined on an interval of R has uniform curÕature ŽUC. if it is convex or is concave. A p-Engel curve on W has UC if it can be extended to a p-Engel curve G s Ž G 1, . . . ,G l . on w0, ax > W for which each component function G j has UC. In other words, a p-Engel curve has UC if it can be extended to the origin in such a way that the demand for each good is a convex or concave function of wealth. We say that household type a has UC at p on an interval W if the p-Engel function F a Ž p,P . on W has UC. An Engel curve with inferior goods can have UC if it is defined on a bounded interval. But if the Engel curve has UC on all of Rq then all the goods must be normal.

3. Increasing dispersion of excess demands, the weak axiom and unique equilibrium In this section we define IDED and show that, under standard hypotheses, it implies that the mean excess demand is pseudomonotone Žsatisfies WWA. and that equilibrium is unique. Before doing so, we illustrate the relationship between dispersion and the WWA of revealed preference for mean excess demand. Fig. 1 shows how the mean excess demand can violate the WWA even though all the consumers satisfy the axiom. ŽWith two goods and smooth demands, the consumers are indistinguishable from utility maximizers.. In Fig. 1, both consumers have the endowment vector at the intersection of the two budget lines. The black dots represent the mean demand vectors at the price vectors p and q. Consumer 1 has demand vectors marked with x’s and consumer 2 with o’s. These individual demands satisfy the WWA, but the mean demands and excess demands do not. For this to happen there must be a region in which increasing wealth makes the consumers’ demand vectors wresp. excess demand vectorsx approach each other. In order for the consumers to satisfy the WWA, their Engel curves

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Fig. 1. Violation of IDED and of the weak axiom for mean excess demand.

associated with prices p must pass to the left of their demand vectors at the prices q. It is easy to see that if the consumers’ demand vectors are moved along the budget line while holding the mean demands fixed, there is no way to avoid having the Engel curves approach each other as they do in the figure. This shows that in the two commodity case, some form of increasing dispersion must be violated if mean demand or mean excess demand are to violate the WWA. In order to define increasing wresp. nondecreasingx dispersion of excess demands, we consider Zla Ž p . ' F a Ž p, l q pv a . y v a , which is what the excess demand of household type a would be at price vector p if the household were to receive l units of additional wealth. IDED wresp. nondecreasing dispersionx at p requires, roughly, that a slight increase in l, starting from zero, increases wresp. does not reducex the variance of the households’ excess demands in any direction orthogonal to p and to the total excess demand vector Z Ž p .. The variance of the projections of the households’ excess demand vectors in the direction z g R l is Varm  Õ P Zla Ž p . 4 '

H

2

Õ P Ž Zla Ž p . y Zl Ž p . . d m ,

where ZlŽ p . ' HZla Ž p .d m. Formally, the consumption sector m has IDED at p wresp. NDED at p x if for each nonzero z in ZŽ p . H jp H

ElVarm  Õ P Zla Ž p . 4 < ls0 ) w G x 0.

Ž 1.

If either IDED or NDED holds at every p g P then we omit reference to the price vector. It is important to note that the IDED hypothesis only restricts the households’ excess demands. It does not restrict the endowment distribution or the

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distribution of household demand functions separately; it only restricts the joint distribution of endowments and demand functions. In particular, IDED does not require the households’ endowment vectors to be collinear and does not require the demand functions to have special forms. This follows from Remark 1, below. It will be useful to have alternative characterizations of these hypotheses about dispersion. When the households receive additional wealth l, the covariance matrix of their excess demands is T

T

Covm Zla Ž p . ' Zla Ž p . Zla Ž p . d m y Zla Ž p . d m Zla Ž p . d m .

H

H

H

Lemma 3. The following conditions are equiÕalent. (a) Consumption sector m has IDED [resp. NDED] at p. (b) The matrix El Covm Zla Ž p .< ls0 is positiÕe definite on ZŽ p . H lp H w resp. positiÕe semidefinite on ZŽ p . Hx . (c) The matrix H M a Ž p, pv a . Z a Ž p . T d m is positiÕe definite on ZŽ p . H lp H w resp. positiÕe semidefinite on ZŽ p . Hx . Proof. Fix p, let M ' HM a d m , with M a evaluated at Ž p, pv a .. Define C ' HŽ M a y M .Ž Z a Ž p . y ZŽ p .. T d m s HM a Z a Ž p . T d m y MZŽ p . T. Since pM a s 1 for every a , Cp s C T p s 0. IDED wresp. NDEDx at p requires 0 - wFxEl H w z P Ž Zla Ž p . y ZlŽ p ..x2 d m < ls0 s 2H w z P Ž M a y M .xwŽ Z a Ž p . y ZŽ p .. P z xd m s zT Ž C q C T . z whenever 0 / z g ZŽ p . H lp H . Since El Covm Zla Ž p .< ls0 s C q C T, this shows that Ža. and Žb. are equivalent. Equivalence of Žb. and Žc. follows from the fact that zT Ž C q C T . z s 2 zTHM a Z a Ž p . T zd m if z P Z Ž p . s 0.I Remark 1. IDED is a robust Žstable. property of a consumption sector. To see this, suppose that m has IDED at p g P and consider a sequence of consumption sectors m n converging setwise to m , cf. the work of Royden Ž1988 p. 269.. Suppose that all these consumption sectors have compact support in the endowment distribution. Let Z wresp. Z n x be the mean excess demand function in m wresp. m n x. By Lemma 3, IDED at p implies appropriate signs for principal minors of the matrix HM a Ž p, pv a . Z a Ž p . T d m bordered by the vector p Žand also ZŽ p . if it is nonzero.. Since each F a is C 1, Proposition 18, p. 270 of Royden Ž1988. implies that each of these principal minors is approached by the corresponding principal minor for the consumption sector m n as n approaches infinity. Therefore m n has IDED for sufficiently large n. Remark 2. The m cannot have IDED if the number of household types in the support of m is less than l y 1, where l is the number of goods. If the number of household types is less than l y 1 then, for fixed p, there is a nonzero vector that is orthogonal to p and to every Z a Ž p . with a in the support of m. Then by Lemma 3, IDED does not hold at p.

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When the households’ endowment vectors are collinear, IDED is closely related to IDD, a property referred to as ‘spreading Engel curves’ by Jerison Ž1982., cf. Hildenbrand Ž1994.. The definition of IDD is like that of IDED, with the households’ excess demands replaced by their demands. To make this precise, let F Ž p . ' HF a Ž p, pv a .d m be the mean demand of the consumption sector m. The consumption sector has IDD increasing demand dispersion at p wresp. NDD nondecreasing demand dispersion at p x if for each nonzero vector z g F Ž p . H

ElVarm  Õ P F a Ž p, l q pv a . 4 < ls0 ) w G x 0. We omit reference to p if the property holds for all p g P. IDD wresp. NDDx requires that the variance of the households’ demands in directions orthogonal to the mean demand increases wresp. does not decreasex when the households’ wealth levels rise by the same small amount. IDD has empirical support when the households’ total expenditures are see. Haerdle et al. Ž1991., Hildenbrand and Kneip Ž1993. and Hildenbrand Ž1994. who work with up to 14 aggregated commodities. In Section 7 we discuss the significance of commodity aggregation for tests of IDED and IDD. Lemma 4. Consider a consumption sector m in which the households’ endowment Õectors are collinear, and let v a ' ua v for each a in the support of m . (a) m has IDD [resp. NDD] at p if and only if HM a Ž F a . T d m y Hua M aHŽ F a . T d m is positiÕe definite on p H w resp. positiÕe semidefinite x when M a and F a are eÕaluated at Ž p, pv a .. (b) IDD [resp. NDD] at p implies IDED [resp. NDED] at p. (c) The conÕerse of (b) is false: IDED at p does not imply NDD at p. (d) If Z(p) s 0 then NDED at p implies NDD at p. We now turn to the economic implications of NDED or IDED. We will need an assumption about the consistency of the individual demands. We say that the consumption sector has ŽNAS. nonpositive average substitution wresp. strict NASx at p if the mean of the households’ Slutsky matrices, HS a Ž p, pv a .d m is negative semidefinite wresp. negative definite on p Hx . If either NAS or strict NAS is satisfied for every p g P, we omit the reference to p. Competitive consumption sectors have nonpositive average substitution. But NAS is considerably weaker then the assumption that the consumption sector is competitive. It does not require that the households maximize utility. It holds if they merely satisfy the weak version of the weak axiom. Moreover, it is a property of the average of the households’ Slutsky matrices. If some households violate the weak axiom, the substitution effects of other households can make up for that fact. Strict NAS is only slightly stronger than NAS. It holds if NAS holds and a set of households with positive measure have Slutsky matrices of maximal rank l y 1. ŽIt is sufficient that a set of households with positive measure maximize differentiable utility functions.. Even strict NAS is weak. It does not imply that the mean

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excess demand function Z has any useful properties. Even in a competitive consumption sector the mean excess demand function Z is essentially arbitrary, cf. the work of Shafer and Sonnenschein Ž1982.. On the other hand, as shown below, when we add IDED, we obtain the weak axiom for mean excess demand and unique competitive equilibrium. Empirical demand studies have found that some consumers or households violate the weak axiom. But the violations for any particular consumer or household are typically small, and the violations of different consumers can cancel each other out, cf. the works of Battalio et al. Ž1973. and Sippel Ž1997.. As long as the violations are unrelated across households, the mean of the Slutsky substitution effects can be negative, in which case, NAS is satisfied. IDED is important because of the following. Proposition 1. Consider an economy as specified aboÕe, in which the consumption sector m has NAS and a regular mean excess demand function Z. (a) If m has NDED, then Z satisfies the WWA (is pseudomonotone), and the set of equilibrium price Õectors is conÕex. (b) If, in addition, there is no production or if m has strict NAS or IDED, then there is at most one equilibrium price Õector, up to scalar multiple. Remark 3. The first part of Proposition 1Ža. follows easily from Lemmas 2 and 3. To see this, consider the Jacobian matrix of mean excess demand,

E ZŽ p.

s

H

Ep F a Ž p,w a . q M a Ž p,w a . Ž v a .

T

dm

Ž 2. T

s S a Ž p,w a . d m y M a Ž p,w a . Z a Ž p . d m ,

H

H

where w a ' pv a . NAS implies that HS a Ž p,w a .d m is negative semidefinite, and NDED implies, by Lemma 3, that HM a Ž p,w a . Z a Ž p . T d m is positive semidefinite on ZŽ p . H . So together, NAS and NDED imply that EZŽ p . is negative semidefinite on ZŽ p . H , and by Lemma 2, Z satisfies WWA if it is regular. Eq. Ž2. suggests that NDED is the weakest restriction on the joint distribution of endowments and Engel curves that, together with NAS, implies WWA for Žregular. mean excess demand. Any hypothesis that implies WWA for Z without restricting the households’ substitution effects also implies NDED. To see this, let us restrict attention to the class of consumption sectors with NAS and with regular mean excess demand. Now consider any additional hypothesis that implies WWA for Z without further restricting the households’ substitution effects. For each p g P, the hypothesis must apply to cases in which the Slutsky matrix S a Ž p,w a . is arbitrarily close to 0 for every a in the support of the consumption sector. Then, by Eq. Ž2., negative semidefiniteness of EZ Ž p . on Z Ž p . H requires HM a Ž p,w a . Z a Ž p . T d m to be positive semidefinite on ZŽ p . H . By Lemma 3,

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NDED must hold, so the hypothesis is at least as strong as NDED. Proposition 1X , below, gives a precise version of this result. It depends on the following terminology. A priÕate ownership consumption sector is a measure on the Cartesian product l Ž l of Rq the space of endowments. and a set of preference preorderings on Rq .A l l l preference preordering on Rq is a subset R of Rq = Rq satisfying for each xX , x l and y in Rq , Ži. reflexivity: Ž x, x . g R; and Žii. completeness: Ž x, y . g R or Ž y, x . g R; and Žiii. transitivity: Ž xX , x . g R and Ž x, y . g R imply Ž xX , y . g R. The preferred set at x, RˆŽ x . '  y:Ž y, x . g R4 , is interpreted as the set of vectors at least as desirable as x according to R. The preference relation R is said to be closed l wresp. conÕex x if RˆŽ x . is closed wresp. convexx for each x g Rq . A closed preference preordering R generates a demand correspondence x R Ž p,w . '  x g b Ž p,w .:Ž x, y . g R,; y g b Ž p,w .4 / 0, where b Ž p,w . '  x g l Rq : px F w4 . The closed preordering R and an endowment vector v generate an excess demand correspondence zv R Ž p . ' x R Ž p, pv . y v . The mean excess demand correspondence Z˜ of a private ownership consumption sector at price vector p is the mean of the excess demand sets zv R Ž p . with respect to the distribution over v and R if this mean exists; otherwise Z˜Ž p . is empty. An excess demand correspondence Z˜ satisfies WWA if for each p and q in the domain of Z˜ and every x g Z˜Ž p . and y g Z˜Ž q ., py F 0 implies qx G 0. We can now give a precise formulation for the proposition that NDED is the weakest restriction on the distribution of endowments and Engel curves leading to the WWA for mean excess demand. Proposition 1X . If a consumption sector m Õiolates NDED at p then for each scalar k ) 0 there is a priÕate ownership consumption sector with closed, conÕex preferences and with (a) the same joint distribution of endowments and p-Engel curÕes on [0, k] as in m , and (b) mean excess demand that Õiolates WWA. In the proof we construct a private ownership consumption sector with properties Ža. and Žb.. The preferences are satiated, but are locally nonsatiated in the budget set b Ž p, k .. The preferred sets are not necessarily strictly convex. However, they have smooth strictly convex boundaries with positive Gaussian curvature in a neighborhood of the p-Engel curve. They generate continuously differentiable demand functions for all households at prices near p. Under the hypothesis of Proposition 1X it is probably also possible to construct a private ownership consumption sector with properties Ža. and Žb. in which the consumers have globally defined smooth demand functions. We conclude this section by discussing two possible difficulties in applying Proposition 1 to private ownership economies. In the definition in Section 2, equilibrium price vectors are elements of P. But the set P is to some extent l arbitrary. In principle, it could be taken to Rqq , but this might not be satisfactory. If the households can be satiated there might be free goods in equilibrium. But the

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sufficient conditions for uniqueness of equilibrium given in Proposition 1 imply only that there is at most one equilibrium price vector in P. The proposition would not rule out other equilibria with some price equal to zero. In a private ownership economy an equilibrium price vector must be in the intersection of the domains of all the consumers’ excess demand correspondences. ˜ If it is convex, it can be taken to be P. Otherwise, P can Call this intersection P. l be chosen to be any convex subset of P˜ that contains Rqq . If P is different from P˜ and if P contains an equilibrium, then to prove uniqueness of equilibrium a separate argument Ždifferent from the proof of Proposition 1. must be used to l eliminate the price vectors in the boundary of Rq that are in P˜ but not in P. A second difficulty arises in the commonly applied class of general equilibrium models in which there are fixed supplies of primary Žnonproduced. factors. Hildenbrand Ž1989. shows that in such models, mean excess demand generically violates the weak axiom. It can also be shown that in these models IDED is generically violated. The problem comes from the fact that the households do not demand the primary inputs directly. But they do so indirectly through their demands for produced goods. Under weak hypotheses, with constant returns technology, such a model can be reduced to a pure exchange model representing the households’ indirect excess demands for the primary factors. In this ‘reduced’ model, IDED can hold. Suppose, for example, that each produced good is produced in a competitive industry with an input–output technology. Let the jth columns of the matrices A and B be, respectively, the vectors of produced and primary inputs required to produce one unit of good j. Assume that the economy is capable of simultaneously producing strictly positive quantities of all the produced goods. This means that there is a nonnegative vector z such that all components of the vector Ž I y A . z are strictly positive. Then Ž I y A . is nonsingular with a nonnegative inverse. In equilibrium, profits are zero, so the price vectors p˜ for produced goods and q for primary inputs satisfy p˜ T Ž I y A . s q T B. Therefore the entire price vector can be written as pŽ q ., a function of the vector of primary input prices. Let F˜ a be the demand function of type a for produced goods. Then the demand of type a in the reduced model is BŽ I y A .y1 Ž F˜ a Ž pŽ q ., pŽ q . v a . y v ˜ a ., a where v ˜ is the vector of endowments of produced goods owned initially by type a . If NAS holds in the original model then it does in the reduced model too, and it is possible for the consumption sector in the reduced model to have IDED so that Proposition 1 applies.

4. Testing for increasing dispersion of excess demands In this section we show that under certain conditions NDED or IDED can be tested using cross-section data on households’ consumption and endowments. The test uses the statistical method of average derivatives without any need to specify

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the functional forms of the households’ demands. IDED is a restriction on the effect of an increase in wealth on the dispersion of households’ excess demands. Cross-section data cannot tell us the effect of change in wealth on a given set of households. Rather they show the consumption of different sets of households at different wealth levels. In order to test for NDED or IDED using cross-section data, we must make an assumption connecting the cross-section to the effects of wealth changes. In this section Žbut not in any others., we will assume that the households’ demand functions and their endowments are independently distributed. It follows that the distribution of demands by richer households represent what poorer households would demand if their wealth levels were higher. It is not necessary for this independence to hold for the entire population. It is enough if the consumption sector can be partitioned into groups within which the independence holds. The test also applies under weaker conditions described below. Under this independence assumption, NDED and IDED can be tested using household consumption and endowment data that are thought of as randomly drawn from the distribution over vectors Ž F a Ž p, pv a ., v a . induced by a consumption sector m. The distribution m must represent a large population so that the expression for DŽ p . in Eq. Ž3. is well-defined. In that expression, m p < w denotes the distribution of a conditional on p P v a s w, and m p Ž w . is the p-wealth distribution induced by m at price vector p: m p Ž w . ' m Ž a : p P v a F w4.. Define the cross-section Engel function for m by F Ž p,w . s HF a Ž p,w .d m p < w, and the cross-section conditional endowment function, v Ž p,w . ' Hv a d m p < w. Proposition 2. Let m be a consumption sector with independently distributed demands and endowments in which the matrix DŽ p . '

H E HF w

H

y

a

T Ž p,w . F a Ž p,w . d m p < w d m p Ž w . T

Ew F Ž p,w . v Ž p,w . d m p Ž w . T

y v Ž p,w . Ew F Ž p,w . d m p Ž w .

H

Ž 3.

is well-defined. Then m has NDED [resp. IDED] at p if and only if DŽ p . is positiÕe semidefinite on Z Ž p . H w resp. positiÕe definite on Z Ž p . H lp H x. Proof. If demand functions and endowments are independently distributed then m p < w is independent of w. Therefore DŽ p .

T

T

T

T

s M aŽ F a . dmq F aŽ M a . dmy M aŽ v a . dmy v aŽ M a . dm

H

H

H

H

T

s M a Z a Ž p . Td m q Z a Ž p .Ž M a . d m ,

H

H

Ž 4.

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where F a and M a are evaluated at Ž p, pv a .. By Lemma 3, m has NDED wresp. IDEDx if and only if DŽ p . is positive semidefinite on Z Ž p . H wresp. positive definite on Z Ž p . H lp H x.I The matrix DŽ p . in Eq. Ž3. is the sum of matrices of average derivatives that can be estimated nonparametrically using demands and endowment vectors from a single cross-section, as described by Kneip Ž1993.. To test for NDED the matrix DŽ p . can be projected on Z Ž p . H and the projected matrix tested for positive semidefiniteness. This is equivalent to partitioning DŽ p . into

ž

D1

D2

DT2

D3

/

,

where D 2 is Ž l y 1. = 1, and testing for nonnegativity of the eigenvalues of the matrix D1 q D 3 zz T y z DT2 y D 2 z T, where z ' Ž1rZl Ž p ..Ž Z1Ž p ., . . . ,Zly1Ž p ... Alternatively, one could test whether the principal minors of order three or more in the bordered Hessian matrix

ž

0 ZŽ p.

ZŽ p. T

DŽ p .

/

are nonpositive. The distributions of these statistics can be estimated by computing the statistics from each of several ‘bootstrap’ samples, created by sampling with replacement from the original data. It is unlikely that in a large heterogeneous community the households’ demand functions and endowment vectors are independently distributed. Endowment vectors are determined in part by past purchases, and therefore by past demand functions which are likely to be similar to the current demand function. But the independence assumption in Proposition 2 is unnecessarily strong. Eq. Ž4. holds under the metonymy assumption of Marhuenda Ž1995 p. 654., which is weaker than independence. In fact it is enough if the identity Ž4. holds only approximately. It is more reasonable to assume independence, or Marhuenda’s metonymy, for smaller more homogeneous groups of households. If the community can be partitioned into subgroups for which the identity Ž4. holds approximately, or if the difference between the left and right-hand sides is uncorrelated across subgroups, then the matrix DŽ p . can be estimated for each of these subgroups, and the average of the estimates, weighted in proportion to the sizes of the subgroups, is an estimate of HM a Ž p, pv a . Z a Ž p, pv a . T d m for the whole population. If this weighted average matrix is positive semidefinite on Z Ž p . H then NDED holds at p. The first matrix on the right-hand side of Eq. Ž3.,

˜' A

H E HF w

a

T Ž p,w . F a Ž p,w . d m p < w d m p Ž w . ,

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has the same formal expression as the matrix A used by Haerdle et al. Ž1991. to estimate the symmetrized mean income effect matrix. However the matrices A and ˜ are not the same. In A, w is the total expenditure during a fixed period and A F a Ž p,w . represents amounts of goods bought during the period. Data on these ˜ taken from Eq. variables are collected in consumer expenditure surveys. But in A, Ž3., w is the value of the entire endowment, and F a Ž p,w . includes amounts of goods purchased, but also amounts of goods initially owned and not sold during the period. It might be difficult to obtain accurate data on the goods that are not traded. Another problem raised by estimation of the matrix DŽ p . is that we have taken the household to be the consumption unit because that is the way that expenditure data are typically collected. But the consistency argument used to justify the hypothesis that a single consumer satisfies the WWA does not extend to households. The weak axiom does not necessarily aggregate, as seen in Fig. 1. ŽThis is the motivation for the present paper.. 5. Collinear endowments and identical household demand functions The main result of Freixas and Mas-Colell Ž1987. was described Section 1. They obtain strong restrictions on an Engel curve that are necessary in order for the mean demand in every competitive consumption sector in a particular class to satisfy the weak axiom. The households in these consumption sectors are all required to have the given Engel curve, but there is no restriction on the distribution of wealth. This means that the weak axiom is required to hold in two-household consumption sectors, since all the other households can be assigned zero wealth. In Proposition 3, below, we show that the same conclusion applies when the weak axiom is only required to hold in all competitive consumption sectors with a given Engel curve and unimodal wealth distribution. Actually, the statement of Proposition 3 is slightly different and somewhat stronger than this. We use the fact that NDED is the weakest Engel curve restriction leading to the WWA in the aggregate when endowments are collinear. We find restrictions on a p-Engel curve that are implied if it satisfies NDED for every collinear wealth distribution with sufficiently concentrated unimodal wealth distribution. Note that we are proving a stronger theorem if we can obtain the same conclusions as before when additional restrictions are placed on the class of allowed consumption sectors. Our proof uses a local characterization of the NT and UC Engel curve restrictions obtained by Freixas and Mas-Colell. When the Engel curve is C 2 these two restrictions can be combined into the following l condition. A C 2 p-Engel curve G:W ; Rq™ Rq satisfies NTUC at w if T 2 Ž .w E G w wEGŽ w . y GŽ w .x is positive semidefinite. The p-Engel curve G satisfies NTUC if it satisfies NTUC at every point in its domain. Recall that the p-wealth distribution for the consumption sector m is defined by m p Ž w . ' m Ž a : p P v a F w4..

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Proposition 3. Let F(p,P ) be a C 2 p-Engel curÕe on a bounded interÕal W. The following conditions, (a), (b), (c) and (d) are equiÕalent. (a) NDED holds at p in eÕery consumption sector such that: (a1) all endowment Õectors are collinear; and (a2) the support of the p-wealth distribution is contained in W; and (a3) all household types in the support haÕe the p-Engel function F(p,P ) on W. (b) There exists e ) 0 such that NDED holds at p in eÕery consumption sector that satisfies (a1), (a2), (a3) and (b1) the p-wealth distribution is unimodal with support of length less than e . (c) F(p,P ) satisfies NTUC on W. (d) F(p,P ) satisfies NT and UC on W. Conditions (a), (b), (c) and (d) are satisfied if for some e ) 0 mean excess demand satisfies WWA in eÕery priÕate ownership consumption sector with identical household preferences satisfying (a1), (a2), (a3) and (b1). This proposition extends the main result of Freixas and Mas-Colell Ž1987. since it shows that it is enough to require the WWA for mean excess demand to hold in consumption sectors with unimodal wealth distributions and arbitrarily little wealth inequality. 2 The strong implications come partly from the assumption that the households’ Engel functions are identical. However in the next section we allow for heterogeneous household demand functions and obtain implications almost as strong.

6. Collinear endowments and heterogeneous household demand functions Proposition 3 above describes the weakest Engel curve restrictions leading to the WWA in the aggregate Žfor every smooth wealth density. if the households’ preferences are identical. In this section we allow household Engel curves to differ. But we allow the aggregate wealth to be held by two households, as do Freixas and Mas-Colell. The implications still turn out to be very restrictive. At any given price vector, the households’ Engel curves must be linear or else must lie in a single plane. In the latter case, the demand vectors must spread out as wealth increases, and this imposes additional restrictions described below. The specification of a consumption sector in Section 2 is not adequate for the analysis in this section. We want to consider the effects of varying the distribution of wealth in a fixed community of households. To do this, we need to be able to distinguish between Ži. a change in the wealth levels of two households that 2

In addition, the proof of the main theorem ŽTheorem 3. provided by Freixas and Mas-Colell Ž1987. applies only if all the goods are normal.

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initially had the same preferences and endowments, and Žii. the splitting of a single household into two households. The second type of change is ruled out if the community of households is fixed. Thus we need to identify the households, not just their demand functions. The simplest way to do this is to restrict attention to a finite community. The conclusions can be extended to consumption sectors represented by continuous distributions. In this section, a community with I members is a set of C 1 demand functions i I  F 4is1 . The corresponding marginal and average propensities to consume are denoted M i and Ai. A Ždiscrete. consumption sector is a community with an endowment vector v i for each household i. According to Proposition 4, below, if the households’ Engel curves are not linear, then the households can be partitioned into ‘Engel families.’ These families can be uniformly ranked according to the values of their marginal propensities to consume for each good. To make this notion of ranking precise, consider two sets of households, K and H. We say that these sets are uniformly ranked at p if there is a set of goods J such that for every pair of households i g K and h g H, inf  M ji Ž p,w . :w ) 0 4 G sup  M jh Ž p,w . :w ) 0 4 ; j g J and inf  M jh Ž p,w . :w ) 0 4 G sup  M ji Ž p,w . :w ) 0 4 ; j f J. ŽThe superscripts represent households and subscripts represent commodities.. The households in the set K all have marginal propensities to consume of the goods in set J that are at least as high as the corresponding marginal propensities of the households in set H at all wealth levels. For the other goods the households in K never have higher marginal propensities to consume than the households in H. By the mean value theorem, the same statements are true with ‘marginal’ replaced by ‘average.’ To define the Engel families, call Ei Ž p . '  l F i Ž p,w . : Ž l ,w . 4 0 4 the Engel cone of household i at p. A p-Engel family is a set of households with the same Engel cone at p. Two households, i and h are replicas at p if there is a scalar k ) 0 satisfying Ai Ž p,w . s A h Ž p,kw . for all w ) 0. A household satisfies UC at p if the household’s p-Engel curve satisfies UC on Rq Žsee Section 2.. Proposition 4. A community {F i } with at least three households has NDED at p for eÕery collinear endowment distribution if and only if condition (a) or (b) holds. (a) Linear Engel curÕes: F i (p,w) s wF i (p,1), for each household i and wealth w G 0. (b) Coplanar Engel curÕes: (b1) The set j i {F i (p,w):w G 0} is contained in a plane; and

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(b2) any two households in the same p-Engel family are replicas and satisfy (UC) at p; and (b3) all pairs of p-Engel families are uniformly ranked. Case Ža. holds if the households’ have homothetic preferences that are not necessarily identical. In case Žb., the households’ demand vectors lie in the same Žtwo-dimensional. plane. If two households have the same vector of budget shares at price vector p for some assignment of endowments, then their p-Engel curves are replicas of each other and have UC. If the households’ average propensity to consume vectors are different no matter what their wealth levels are, then for each good j and each wealth distribution, the household with the higher average propensity to consume j also has a higher marginal propensity to consume j.

7. Conclusion This paper has introduced an interpretable and potentially testable condition implying that the consumers’ mean excess demand satisfies the weak axiom and that competitive equilibrium is unique. The condition, IDED, is related to Increasing Demand Dispersion, a restriction introduced by Jerison Ž1982. and studied empirically by Hildenbrand and Kneip Ž1993. and Hildenbrand Ž1994.. IDD implies IDED when the consumers’ shares of aggregate wealth do not vary with prices Že.g., when their endowments are collinear, assuming that profits are zero.. But IDD implies uniqueness of general equilibrium only in that case. IDED is important because it implies uniqueness whether or not the consumers’ wealth shares vary with prices. There is an alternative approach to uniqueness of equilibrium in the case of price-dependent wealth shares. Grandmont Ž1992. and Quah Ž1997. show how sufficient dispersion in consumers’ characteristics Ždefined in various ways. implies that equilibrium is unique and mean demand F satisfies the Law of Demand: Ž p y q . P Ž F Ž p,w . y F Ž q,w . . F 0

Ž 5.

for price vectors p and q such that mean wealth w s p P v s q P v is constant. Marhuenda Ž1995. obtains the same conclusion when the density function for the consumers’ endowments rises sufficiently slowly where the population density is high. It can be shown that as the consumer heterogeneity increases, as specified in the papers cited above, violations of IDED shrink to zero when consumer endowments are collinear. In that sense, IDED can be considered a more general sufficient condition for uniqueness of equilibrium. Consumer heterogeneity Žas specified by Quah. that is sufficient for the Law of Demand also implies near linearity of the mean demand with respect to wealth. As specified by Grandmont

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Ž1992. and Kneip Ž1998., it implies that mean budget shares do not respond much to price changes. IDED is weaker Žfor a given the distribution of endowments. since it does not imply any restriction on the functional form of the mean demand. As shown in Sections 5 and 6, above, the households’ Engel curves must have special forms in order for IDED to hold for every distribution of collinear endowments. There must be weaker restrictions implied if IDED is required to hold for some more restricted class of endowment distributions. The forms of such intermediate restrictions remain an open question. One way to approach this problem would be to try to characterize communities by the endowment distributions for which they satisfy IDED. IDED can be tested using cross-section data on household expenditures and endowments under assumptions linking wealth effects to cross-sectional variation—assumptions that are also made by Grandmont Ž1992., Marhuenda Ž1995. and Quah Ž1997.. The issue of empirical testing raises the question why we do not simply test the Law of Demand directly. For example it is possible to test for negative semidefiniteness of the Jacobian of mean excess demand. However this would require estimating derivatives at a point using discrete data. Small measurement errors can lead to large estimation errors. The tests of IDED proposed here involve estimation of derivatives averaged over the wealth distribution, and this reduces the sensitivity to measurement errors. The monotonicity inequality Ž5. that defines the Law of Demand, above, can in principle be tested directly, but in practice the necessary data are not readily available. The test should be made using demand vectors observed at different prices when the value of the mean endowment vector is fixed. Such data can be obtained using the clever experimental method of Sippel Ž1997. for eliciting points of a consumer’s demand function, cf. Heid and Orth-Schwall Ž1995.. However, it would be very expensive to obtain data for large numbers of commodities. The alternative is to use time series data on prices and demands. But if the prices are observed over a short period then there are few observations. And if the period is longer, and then the price changes are associated not only with changes in policy or technology, but also with changes in population, preferences and endowments. In that case, the data do not provide a real test of the Law of Demand or the weak axiom for the mean excess demand function of a fixed consumption sector. Returning to the definition of IDED, we see that this property cannot be expected to hold for highly disaggregated commodities. With finely defined commodities there are likely to be inferior goods for which the dispersion of excess demands decreases as the households get richer. ŽThe dispersion approaches zero as wealth increases since both the endowments and the purchases of these goods fall to zero for most households.. In unpublished work, J.-U. Scheer has found support for IDD using French data with total expenditures divided among 60 commodity aggregates, but violations of IDD when total expenditures are divided among 410 aggregates.

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It is important to note that IDD and IDED are not necessary for uniqueness of equilibrium. They are restrictions on the joint distribution of Engel functions and endowments, and thus omit the contribution of substitution effects. ŽSee the decomposition Ž2. in Remark 3, above.. Statistical tests of IDED must use data in which close substitutes are treated as a single commodity. If IDED is satisfied when goods are aggregated in this way, it can be considered as evidence in favor of uniqueness of equilibrium. The aggregation may hide inferior goods for which IDED is violated. But within a single commodity aggregate, the large substitution effects could take over precisely where the wealth effects in IDED fail. Mas-Colell Ž1991. shows that in a pure exchange economy Žpossibly the reduced economy from a production economy with primary factors in fixed supply. sufficiently flat indifference surfaces, at least as flat as Cobb–Douglas, are sufficient for unique equilibrium. It seems unlikely that all goods would be sufficiently close substitutes according to that criterion, but the substitution effects are complemented by the wealth effects in the definition of IDED. In order to deal with highly disaggregated commodities one must allow for the fact that most goods are traded in discrete units. The differential approach taken in this paper does not apply to the demand for such commodities. Extending the analysis to this case is a topic for further research. The concept of uniqueness of equilibrium must be generalized. But once that is done, it seems quite plausible that in large economies the presence of goods consumed in either one unit or not at all would contribute to approximate uniqueness, appropriately defined.

8. Proofs

Proof of Lemma 4. IDD wresp. NDDx at p is equivalent to the requirement that for each nonzero z orthogonal to F ' HF a Ž p, pv a .d m , 0 - w Fx E lVarm  z P F a Ž p, l q pv a . 4 < ls0

T s 2 zT Ž M a y M .Ž F a y F . zd m

H

T

˜ s 2 zT M a Ž F a . zd m s 2 zT Cz,

H

Ž 6. where M a and F a are evaluated at Ž p, pv a . and where M ' HM a d m and C˜ '; M a Ž F a . T d mHua M a d m F T. This shows that NDD holds at p if C˜ is positive semidefinite. The converse is true because every l-vector u can be written ˜ s ˜ s C˜ T p s 0, uTCu as u s z q l p with l s Ž uF .rŽ pF . and z g F H . Since Cp T ˜ T ˜ ˜ Ž . z Cz, so by Eq. 6 , u Cu G 0 if NDD holds at p. This shows that C is positive semidefinite. To prove the other equivalence in Ža., note that IDD holds at p if

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and only if the inequality in Eq. Ž6. is strict for every nonzero z g F H . This is ˜ ) 0 whenever u s z q l p g p H , with equivalent to the requirement that uT Cu H 0 / z g F . But this is equivalent to C˜ being positive definite on p H because z s 0 if and only if u s l p g p H , which holds if and only if l s 0 and u s 0. This completes the proof of Ža.. To prove Žb., suppose that NDD holds at p. Then HM a Ž F a . T d m is positive semidefinite on F H . Given a nonzero z g Z Ž p . H , let u ' z y l p, where l s Ž zF .rŽ pF .. Then uF s 0 and uZŽ p . s 0, so u v s 0, where v ' Hv a d m. Therefore, 0 F uTHM a Ž F a . T ud m s uTHM a Ž Z a q ua v . T ud m s uTHM a Ž Z a . T ud m s Ž z y l p .HM a Ž Z a . T zd m s zTHM a Ž Z a . T zd m. The last equation holds since p P M a s 1 and zZ Ž p . s 0. By Lemma 3, this shows that m has NDED at p. If IDD holds at p then HM a Ž F a . T d m is positive definite on F H . Using the notation in the previous paragraph, if 0 / z g Z Ž p . H lp H then u / 0 since u s 0 implies z s l p g p H , which implies z s 0. Therefore 0 uTHM a Ž Z a . T ud m s zTHM a Ž Z a . T zd m , so HM a Ž Z a . T d m is positive definite on Z Ž p . H lp H . By Lemma 3, IDED holds at p. To prove Žd., suppose that Z Ž p . s 0 and that NDED holds at p. Consider an arbitrary z g F H s v H . Then z P Z Ž p . s 0, so Lemma 3 implies 0 F zTHM a Z a Ž p . T zd m s zTHM a Ž Z a Ž p . q ua v . T zd m s zTHM a Ž F a . T zd m , so from the proof of Ža. above, NDD holds at p. To prove Žc. we must show that IDED at p does not imply NDD at p. Suppose that there are only two goods and that Z Ž p . / 0. Then IDED holds automatically at p. To see this, note that p P Z Ž p . s 0, so z g Z Ž p . H implies z s l p for some scalar l, and z g p H implies z s 0. Thus by giving an example in which NDD is violated at p, we show that IDED does not imply NDD at p. Consider a consumption sector with only two types of households, 1 and 2. The demand of household 1 is generated by the utility function u1 Ž x 1 , x 2 . s 6 x 11r2 q 4 x 23r4 and that of household 2 by u 2 Ž x 1 , x 2 . s x 1g x 21y g . Let p s Ž1,1.. Then x s Ž x 1 , x 2 . 4 0 is in the image of the p-Engel curve of household 1 whenever u11 Ž x .ru12 Ž x . s 1, which holds whenever x 2 s x 12 . The budget identity implies that the demand for good 1 by household 1 is x 1 when that household has wealth w 1Ž x 1 . ' x 1 q x 12 . Differentiating implicitly, we find that household 1 with wealth 2 has marginal propensity to consume M11 Ž p,2. s 1rwX1Ž1. s 1r3 for good 1, since w 1Ž1. s 2. Household 1 has demand F11 Ž p,2. s 1 for good 1 when its wealth is 2. Household 2 with wealth 2 has marginal propensity to consume M12 Ž p,w 2 . s g and demand F12 Ž p,2. s 2g for good 1 when its wealth is 2. By part Ža. of Lemma 4, NDD at p implies 0 F M11 F11 q M12 F12 y Ž 1r2 . Ž M11 q M12 .Ž F11 q F12 . s Ž 1r2 . Ž M11 y M12 .Ž F11 y F12 . , where all functions are evaluated at Ž p,2.. But M11 y M12 s Ž1r3. y g and F11 y F12 s 1 y 2g , so if 1r3 - g - 1r2 then NDD does not hold.I

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37

Proof of Proposition 1. Fix p and let w a s pv a. The Jacobian matrix of the mean excess demand at p is

E ZŽ p .

s

H

Ep F a Ž p,w a . q M a Ž p,w a . Ž v a .

T

dm

Ž 7. a

a

a

a

a

T

s S Ž p,w . d m y M Ž p,w . Z Ž p . d m .

H

H

NAS and NDED, with Lemma 3, imply that EZŽ p . is negative semidefinite on ZŽ p . H . By Lemma 2, Z is pseudomonotone. If IDED holds or if NAS holds strictly, then EZŽ p . is negative definite on ZŽ p . H lp H . Assume that NDED holds. To prove that the equilibrium set is convex, define D '  p g R l : py F 0, ; y g Y 4 , P ) ' D l P and E '  p g P ) : pZ Ž q . G 0, ;q g P ) 4 . Note that D, P ) and E are intersections of convex sets, and so are convex. We will show that E is the set of equilibrium price vectors. Suppose that p is an equilibrium price vector. Then p g P ) and Z Ž p . g Y. Thus, for all q g P ) , qZ Ž p . F 0. Since Z is pseudomonotone, pZ Ž q . G 0 for all q g P ) , so p g E. Next, we will show that every element of E is an equilibrium price vector. But first we consider certain properties of D. If there is no production, then Y s  04 l and D s R l. In the production case, D is a nonempty subset of Rq . Nonemptiness follows from the fact that 0 is in the boundary of Y. If D is not contained in l Rq then q j - 0 for some q g D and some j. But then ye j g Y and q P Žye j . ) 0, where e j is the l-vector of unit length with jth component equal to 1. This contradicts the definition of D. Whether there is production or not, D contains a strictly positive vector p. To see this in the production case, suppose first that there is some j such that q j s 0 for all q g D. Then q P e j s 0 for all q g D, so since Y s  y g R l : py F 0, l ; p g D4 , e j g Y. But this contradicts the ‘no free lunch’ hypothesis Y l Rq s  04 . j j It follows that for each j there is some q g D with q j ) 0. Let p be a convex combination of the vectors q j with strictly positive weights for each q j. Then p is strictly positive and in D since D is convex. Now consider an arbitrary p g E ; P ) . For any q g P ) and l g Ž0,1., pl ' l q q Ž1 y l. p is in P ) since P ) is convex. Since p g E, we have pZ Ž pl . G 0. This implies qZ Ž pl . F 0 since 0 s pl Z Ž pl . s l qZ Ž pl . q Ž1 y l. pZ Ž pl .. By continuity of Z, qZ Ž p . F 0, and this holds for every q g P ) . Since the production set Y is a closed, convex cone, Y s  y g R l : py F 0, ; p g D4 ; cf. the work of Valentine Ž1964 Theorem 5.3, p. 59.. Suppose that p is not an equilibrium. Then Z Ž p . f Y so with or without production there is some nonnegative q g D with qZ Ž p . ) 0. Let ql ' l p q Ž1 y l. q. Then ql g P ) since D is convex and p is in D and is strictly positive. For l ) 0 sufficiently small ql Z Ž p . ) 0, which contradicts the hypothesis that p g E. This proves that p is an equilibrium and therefore that the set of equilibrium price vectors is the convex set E. Now, we turn to the proof of uniqueness of equilibrium. Suppose again that NDED holds. Let p and q be distinct equilibrium price vectors. Let z s q y p

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and define pl s p q lz for l g w0,1x, as above. Suppose first that there is no production. Since the set of equilibria is convex, Z Ž p q lz . s 0 for l g w0,1x. Differentiating with respect to l, we obtain EZ Ž p . z s 0. Since Z is homogeneous, EZ Ž p . p s 0. Since Z is regular EZ Ž p . has rank l y 1, so p and z are linearly dependent. This proves that p is the unique equilibrium up to scalar multiple. Next, consider the case of production and suppose that IDED holds or that NAS holds strictly. Since p and q are equilibria, pZ Ž q . F 0 and qZ Ž p . F 0. Since Z is pseudomonotone, pZ Ž q . s qZ Ž p . s 0. Since the equilibrium set is convex, pZ Ž pl . F 0 and 0 F Ž pl y p . Z Ž pl . s lzZ Ž pl .. Therefore, zZ Ž pl . G 0 and zZ Ž p . s 0, and differentiating with respect to l we obtain zTEZ Ž p . z G 0. Define u s z y l p, where l s Ž pz .rŽ p P p .. Then pu s 0 and uZ Ž p . s 0, since zZ Ž p . s 0. Note that Walras’ Law Ž pZ Ž p . s 0. implies pTEZ Ž p . s yZ Ž p . T. Therefore, using the homogeneity of Z, we obtain uTEZ Ž p . u s Ž z y l p . TEZ Ž p .Ž z y l p . s zTEZ Ž p . z q l Z Ž p . T z s zTEZ Ž p . z G 0. This implies u s 0 since EZ Ž p . is negative definite on Z Ž p . H lp H . So z s l p and again p is the unique equilibrium up to scalar multiple.I Proof of Proposition 1X . Suppose that m violates NDED at some p. We will specify a private ownership consumption sector with convex preferences and with properties Ža. and Žb. of Proposition 1X . To do this, we specify for each household type a a convex preference relation that generates a C 1 excess demand function on a neighborhood of Ž p, pv a . and a ’s p-Engel curve on w0,k x. In a neighborhood of F a Ž p, pv a ., the constructed indifference surface through F a Ž p, pv a . coincides with a sphere that is tangent to the budget frontier orthogonal to p. By making the radius of the sphere small we make elements of the household’s Slutsky matrix small. Then by Eq. Ž7., violation of NDED implies that the Jacobian of mean excess demand fails to be negative semidefinite on the appropriate hyperplane, so mean excess demand violates WWA. Consider a household of type a with wealth w a ' pv a ) 0 at p, and with p-Engel curve G. Fix z ) max k,w a 4 . Let r:w0, z x ™ Rq be a strictly decreasing, continuous function, with r Ž z . s 0. For each w G 0 and z g w0,zx, let B Ž w, z . be the closed ball of radius r Ž z . and center GŽ w . q Ž r Ž z .r< p <. p. This ball is tangent to the budget frontier  x G 0: px s w4 at the point GŽ w . on the p-Engel curve. For l z g w0, z x, define QŽ z . to be the intersection of Rq with the convex hull of j w g w z, z x B Ž w, z .. For z - 0, define QŽ z . to be the set of nonnegative vectors that are a distance yz or less from some point in QŽ0.. Finally, define a preference l relation on Rq with preferred set P Ž x . equal to the intersection of the QŽ z . sets l that contain x. This is well-defined since, for each x g Rq , there is some z such X that x g QŽ z .. By construction, z - z implies QŽ z . ; QŽ zX .. l For each x g Rq , P Ž x . is compact and convex. To see this, note that for Ž z G 0, j w g w z, z x B w, z . is the set of points within the distance r Ž z . from the compact image of G on w z, z x translated by Ž r Ž z .r< p <. p. This set is compact, so

M. Jerisonr Journal of Mathematical Economics 31 (1999) 15–48

39

its convex hull QŽ z . is compact and convex. For z - 0, QŽ z . is the set of points within the distance yz from a compact, convex set, so for all z g R, QŽ z . is compact and convex. Thus P Ž x . is an intersection of compact, convex sets. Next, we show that the preference relation defined by preferred sets P Ž x . is a preordering. By definition x g P Ž x .. If y f P Ž x . then there is some z such that x g QŽ z . and y f QŽ z .. If y g QŽ zX . then zX - z and QŽ z . ; QŽ zX .. It follows that x g P Ž y .. Therefore, the preference relation is complete. If xX g P Ž x . and x g P Ž y ., then every QŽ z . containing y also contains x and hence xX , so xX g P Ž x ., and the preference relation is transitive. Thus the preference relation is a preordering. For a fixed z g w0, z . we will show that in a neighborhood of GŽ z ., the indifference surface for the preferences P through GŽ z . coincides with the boundary ball B Ž z, z .. We begin by showing that for w g Ž z, z x, the angle between p and GŽ w . y GŽ z . is bounded below pr2. For this, it suffices to show that the cosine of that angle is bounded above 0. The cosine equals p P Ž GŽ w . y GŽ z ..rw < p < < GŽ w . y GŽ z .
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M. Jerisonr Journal of Mathematical Economics 31 (1999) 15–48

Consider a private ownership consumption sector with consumer preferences as defined above for each a . The measure m then determines a measure on the space of these preferences. If m violates NDED at p then by Lemma 3, there exists u g Z Ž p . H such that uTHM a Ž p,w a . Z a Ž p . T ud m - 0. Each consumer in the private ownership consumption sector has the same marginal propensity to consume and excess demand vectors as the corresponding a type in the m consumption sector at Ž p,w a . and p. By Eq. Ž7., the matrix uTEZ Ž p . u in the private ownership consumption sector can be made arbitrarily close to uTHM a Ž p,w a . Z a Ž p . T ud m - 0 by making the Slutsky matrices in the private ownership consumption sector sufficiently small. By Lemma 2, the mean excess demand in the private ownership consumption sector violates WWA.I The next lemma will be used in the proofs of Propositions 3 and 4. Lemma 5. A rank one square matrix uzT is positive semidefinite if and only if z s 0 or u s lz for some scalar l G 0. Proof. The if part is immediate. To prove the only if part, note that if u and z are linearly independent then there is some x with x P z s 0 and x P u / 0. If x P u ) wresp. -x0 then for sufficiently small l ) 0 and x s Ž x y lz .wresp. x q lz x we have x P z - w)x 0 and x P u ) w-x0. Then x T uzT x - 0. So if uzT is positive semidefinite and z / 0 then u s lz and zŽ lz . zT z G 0, so l G 0.I Proof of Proposition 3. We will prove Ža. ´ Žb. ´ Žd. ´ Ža.. In specifying a consumption sector, we drop the parameter a since the demand functions F a are all identical. Ža. ´ Žb. is trivial. To prove Žb. ´ Žc., consider the delta function sequence r t Ž w . s Ž1rt . r wŽ w y y .rt x, where r is a continuous density with compact support, mean 0 and variance s 2 ) 0. Each density r t has mean y, and approaches a point mass as t approaches 0. We consider consumption sectors in which all households have the demand function F, and we omit p as a function argument since it is fixed. Let y be an arbitrary point in the interior of W, the domain of the p-Engel function F ŽP.. Let t be small enough so that the density r t has support of length less then e , contained in W. Consider a consumption sector with mean endowment vector v ' HF Ž w . r t Ž w .dw and a collinear endowment distribution such that the p-wealth distribution has density r t . Under the hypothesis Žb. of Proposition 3, NDED holds at p, so by Lemma 3, HM Ž w .Ž F Ž w . y Ž wry . v . Tr t Ž w .dw is positive semidefinite on Z Ž p . H . By construction, Z Ž p . s 0. Using the definition of v , it follows that for sufficiently small t ) 0 the matrix T

T

E Ž t . s y M Ž w . F Ž w . r t Ž w . dw y wM Ž w . r t Ž w . dw F Ž w . r t Ž w . dw

H

H

H

M. Jerisonr Journal of Mathematical Economics 31 (1999) 15–48

41

is positive semidefinite. Using the change of variable w s tu q y, and the definition M s EF, we have T

E Ž t . s y E F Ž tu q y . F Ž t u q y . r Ž u . d u

H

T

y Ž tu q y . E F Ž t u q y . r Ž u . d u F Ž tu q y . r Ž u . d u .

H

H

Note that E Ž0. s 0, and that EE Ž t . s H Ž t . q J Ž t ., where H Ž t . s y u Ž E 2 F . F Tr Ž u . d u y u Ž tu q y . E 2 Fr Ž u . d u F Tr Ž u . d u ,

H

H

H

and J Ž t . s y uE FE F Tr Ž u . d u y uE Fr Ž u . d u F Tr Ž u . d u

H

H

H

y Ž tu q y . E Fr Ž u . d u uE F Tr Ž u . d u ,

H

H

where all the missing arguments are tu q y. Is it easy verify that EE Ž0. s 0, so E 2 E Ž0. must be positive semidefinite. Since Hur Ž u .d u s 0, we have T

T

E J Ž 0 . s y u 2E 2 F Ž y . E F Ž y . r Ž u . d u q y u 2E F Ž y . E 2 F Ž y . r Ž u . d u

H

H

T

y u 2E 2 F Ž y . r Ž u . d u F Ž y . r Ž u . d u

H

H

T

y yE F Ž y . r Ž u . d u u 2E 2 F Ž y . r Ž u . d u

H

H

s s 2E 2 F Ž y . yE F Ž y . y F Ž y . 3

T

.

Since F is not necessarily C , H is not necessarily differentiable. However, straightforward calculation shows that H Ž0. s 0 and lim t ™ 0 wŽ1rt . H Ž t . y E J Ž0.x s 0. So EH Ž0. exists and is equal to E J Ž0.. This proves that E 2 E Ž0. s 2 s 2 E 2 F Ž y .w yEF Ž y . y F Ž y .xT. Since E 2 EŽ0. is positive semidefinite, F satisfies ŽNTUC. at y. Žc. ´ Žd.: Define GŽ y . ' eyy F Ž e y . when ln y g Ž a,b .. Then F Ž w . s wGŽln w .,EF Ž w . s EGŽln w . q GŽln w . and wE 2 F Ž w . s E 2 GŽln w . q EGŽln w . for w ) 0. So NTUC implies that the matrix ŽE 2 G q EG .ŽEG . T evaluated at each point in the interval Žln a, ln b . is positive semidefinite. By Lemma 5, the vectors E 2 G Ž y . and EG Ž y . must be collinear and point in the same direction. This implies that the length of EG Ž y . is nondecreasing in y. If EG Ž y . / 0 for some y, then there exists a scalar valued function t with E 2 GŽ z . s t Ž z .EGŽ z . for z in a neighborhood of y. Solving this differential equation shows that there is a scalar valued function h and a vector z such that EGŽ z . s hX Ž z . z for z in the neighborhood of y. If EGŽ y . s 0 then EGŽ z . s 0 for z in w a, y x. Therefore, EG Ž y . s hX Ž y . z for all y in w a,b x. Integrating, we obtain G Ž y . s hŽ y . z q u and F Ž w . s k Ž w . z q wu for a vector u

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M. Jerisonr Journal of Mathematical Economics 31 (1999) 15–48

and a scalar valued function k. Thus, the image of F is contained in the plane spanned by the vectors z and u, so F satisfies NT. To prove ŽUC., note that ŽNTUC. requires that if F Ž w . s k Ž w . z q wu then kY Ž w .w wkX Ž w . y k Ž w .x zzT is positive semidefinite. If z s 0 then F is linear in w and satisfies ŽUC., so we consider the case with z / 0. Then, kY Ž w . mŽ w . G 0 where mŽ w . ' wkX Ž w . y k Ž w .. Note that mX Ž w . s wkY Ž w .. Suppose that for some w we have mŽ w . ) 0. Then kY Ž w . G 0 and mX Ž w . G 0. We can conclude that mŽ y . ) 0 for all y ) w. To see this, note that if z is the infimum of the set of zX ) w such that mŽ zX . s 0, then by the mean value theorem mX Ž y . - 0 and mŽ y . ) 0 for some y g Ž w, z .. Hence, kY Ž y . mŽ y . - 0, which contradicts our hypothesis. If for some w we have kY Ž w . ) 0 then mŽ w . G 0 and mX Ž w . ) 0. Therefore, mŽ y . ) 0 and kY Ž y . G 0 for y ) w. A similar argument shows that if kY Ž w . - 0 then kY F 0 on w w,b x. This shows that k is convex or concave on the interval w a,b x. Finally, if a ) 0 and mŽ a. ) wresp.-x0 then kY G wresp.Fx0 on w a,b x. This completes the proof that F satisfies ŽUC.. Žd. ´ Ža.: Suppose that F satisfies NT and UC at p. Then there is a convex function f on Rq and vectors B and C such that F Ž w . s f Ž w . B q wC for w g W. Given a consumption sector m satisfying Ža1., Ža2. and Ža3., we must show that NDED holds at p. We will prove that NDD holds at p and apply Lemma 4. For this we must show that HEF Ž w .Ž F Ž w . y F . T d m w is positive semidefinite on F H , where F ' HF Ž w .d m w . Given z orthogonal to F, it is enough to show that HzT ŽEF Ž w .. F Ž w . T zd m w G 0. Define g Ž w . ' zF Ž w . for w G 0 and aŽ w . ' g Ž w .rw for w ) 0. Suppose that zB G 0 so that g is convex. Since g is C 2 and g X is nondecreasing, dw wg X Ž w . y g Ž w .xrdw s wg Y Ž w . G 0. Since g Ž0. s 0, we have g X G a and a X G 0. Since Hg Ž w .d m w s 0, there is some w with g Ž w . s 0. It follows that aŽ w . G wFx0 if w G wFx w, and that wg X Ž w .aŽ w . G wg X Ž w . aŽ w . for all w ) 0. Integrating we obtain HzT ŽEF Ž w .. F Ž w . T zd m w s Hg X Ž w . g Ž w .d m w G Hwg X Ž w . aŽ w .d m w s 0. This proves that HŽEF Ž w .. F Ž w . T d m w is positive semidefinite on F H , so the consumption sector satisfies NDD and NDED at p. If zB - 0 then the above argument applies with g replaced by yg. The last part of Proposition 3 follows from Proposition 1X .I Consider C 1 functions f i :Rq™ Rq for i s 1,2, and let a i Ž w . ' f i Ž w .rw for w ) 0. The next three lemmas characterize Engel curves that satisfy uniformly increasing dispersion. We say that f 1 and f 2 spread if for all positive w and z, Ž f 1X Ž w . y f 2X Ž z ..Ž a1Ž w . y a 2 Ž z .. G 0. Lemma 6. Suppose that f 1 and f 2 spread. If a1Ž w . s a1Ž z . then f 1X Ž w . s f 2X Ž z .. Proof. Suppose that for some w and z, a1Ž w . s a 2 Ž z . and f 1X Ž w . ) f 2X Ž z .. If f 1X Ž w . F a1Ž w . then f 2X Ž z . - a2 Ž z . and a 2X Ž z . - 0. For y - z sufficiently close to z, a2 Ž y . ) a2 Ž z . s a1Ž w . and f 2X Ž y . - f 1X Ž w ., which contradicts the hypothesis that f 1 and f 2 spread. If instead f 1X Ž w . ) a1Ž w . then a1X Ž w . ) 0. For y - w sufficiently

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close to w, a1Ž y . - a1Ž w . s a 2 Ž z . and f 1X Ž y . ) f 2X Ž z ., again contradicting the hypothesis. Therefore a1Ž w . s a2 Ž z . implies f 1X Ž w . F f 2X Ž z .. Interchanging w and z we obtain f 1X Ž w . s f 2X Ž z ..I Lemma 7. Suppose that f 1 and f 2 spread. If for some w and z, a1Ž w . ) a2 Ž z . and a2X Ž z . G 0 then a1Ž y . ) a 2 Ž z . for all y ) w. Proof. Define a ' a2 Ž z . and suppose that a1Ž yX . F a for some yX ) w. Since a1 is continuous, the set  wX ) w:a1Ž wX . F a4 contains a minimum value, y, with a1Ž y . s a. Since f 1Ž y . s ay and f 1Ž w . ) aw, we have f 1Ž y . y f 1Ž w . - aŽ y y w .. By the mean value theorem, there is some t g Ž w, y . with f 1X Ž t . - a. The definition of y implies that a1Ž t . ) a s a2 Ž z .. Since a2X Ž z . G 0, we have f 2X Ž z . G a 2 Ž z . s a ) f 1X Ž t ., contradicting the spread hypothesis. Therefore no such yX exists.I Lemma 8. Suppose that f 1 and f 2 spread and that for some y and z, a1X Ž y . 0wresp.) 0x and a1Ž y . s a 2 Ž z .. Then f 1 and f 2 are concave wresp. convexx and there is a unique constant k ) 0 such that a1Ž w . s a2 Ž kw . for all w ) 0. Proof. Under the hypotheses of the lemma, let K be the largest interval containing y such that for each w g K, a1X Ž w . - 0 and a1Ž w . is in the image of a 2 . Fix w g K and note that a 2 Ž t . s a1Ž w . for some t. Since a1X Ž w . - 0, Lemma 5 implies a 2 Ž t . s a1Ž w . ) f 1X Ž w . s f 2X Ž t ., and hence a 2X Ž t . - 0. This shows that a2X Ž t . - 0 for each t with a2 Ž t . s a1Ž w .. Therefore, there is only one such t Žgiven w ., and we denote it k Ž w . w. By the same argument, if w g K and if wX satisfies a1Ž wX . s a1Ž w . then a1X Ž wX . - 0, so there can only be one such wX , and it equals w. It follows that w g K for every w with a1Ž w . g a 2 Ž K .. Furthermore, a 2X Ž y . - 0 implies that K is a neighborhood of y. By the implicit function theorem, the function k solving a 2 Ž k Ž w . w . s a1Ž w . is 1 C on K. By definition of a1 , f 1Ž w . k Ž w . s f 2 Ž k Ž w . w . for all w g K. Differentiating yields f 1X Ž w . k Ž w . q f 1 Ž w . kX Ž w . s f 2X Ž k Ž w . w . wkX Ž w . q k Ž w . . Ž 8. X X By Lemma 6, f 1 Ž w . s f 2 Ž k Ž w . w ., and substituting this into Eq. Ž8. yields g Ž w . kX Ž w . s 0, where g Ž w . ' f 1Ž w . y wf X1Ž w .. Since a1X Ž w . - 0 for w g K, we have g Ž w . ) 0 and kX Ž w . s 0 on K. Thus we can reinterpret k as a scalar satisfying a1Ž w . s a2 Ž kw . for all w g K. For w and y in K with y ) w, we have a1Ž y . - a1Ž w . s a 2 Ž kw .. Since f 1 and f 2 spread, f 1X Ž y . F f 2X Ž kw . s f 1X Ž w ., where the last equality follows from Lemma 6. Thus f 1 is concave on K. To complete the proof we must extend these results from K to Ž0,`.. Suppose that K has a finite upper endpoint b. We noted above that g Ž w . ) 0 on K. Since f 1 is concave on K, g is differentiable almost everywhere on K, and the derivative g X Ž w . s ywf Y1 Ž w . is nonnegative wherever it exists. Thus g is nondecreasing on K, and g Ž b . ) 0. This implies that a1X Ž kb . - 0. But then b must be in the interior of K, a contradiction. So no such upper endpoint exists.

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Suppose that K has a lower endpoint t ) 0. If for some w - t we have a1Ž w . - a1Ž t ., then there exists wX g Ž w,t . with a1Ž wX . g a1Ž K .. As noted in the first paragraph of this proof, we then have wX g K, which contradicts the definition of t. It follows that a1Ž w . G a1Ž t . for w - t. If a1X Ž t . - 0 then a1X Ž t . s kaX2 Ž kt . - 0. Since a1Ž t . s a2 Ž kt ., this implies that a1Ž t . is in the interior of the image of a2 , which contradicts the definitions of K and t. Therefore a1X Ž t . s 0, a2 Ž kt . s a1Ž t . s f 1X Ž t . s f 2X Ž kt . and a 2X Ž kt . s 0. If for some w - t, a1Ž w . ) a1Ž t . s a 2 Ž kt ., then Lemma 7 implies a1Ž t . ) a2 Ž kt ., a contradiction. We can conclude that a1Ž w . s a1Ž t . for w - t. By a symmetric argument, a 2 Ž y . s a 2 Ž kt . for y - kt. So a1Ž w . s a 2 Ž kw . on Ž0,`.. Since a1 is constant on Ž0,t ., f 1X is also constant, and f 1 is concave on Ž0,`.. Since f 2 Ž kw . s kf 1Ž w ., f 2 is also concave. The proof for the case of a1X Ž y . ) 0 is similar. In that case each f i is convex.I Lemma 9. Let p be fixed and suppose that for some vector A and for every w g Rqq the matrix Ž M i Ž p,w . y A .Ž A i Ž p,w . y A . T is positive semidefinite. Then there is a nonnegative real-valued function f and a vector B Žboth depending on p . such that F i Ž p,w . s f Ž w . B q wA for all w G 0. Proof. Define GŽ w . ' F i Ž p,w . y wA. Then GXi Ž w . s M i Ž p,w . y A and so, under the hypothesis of the lemma, GX Ž w .GŽ w . T is positive semidefinite for all w ) 0. Suppose that GŽ w . / 0, and let W be the largest interval containing w such that G / 0 on the interior of W. Since G is C 1, Lemma 5 implies that there is a scalar valued function g G 0 satisfying GX Ž w . s g Ž w .GŽ w . for w g W. The solution to this differential equation has the form GŽ w . s f Ž w . B on W for some vector B / 0, with f Ž w . s expŽ Hbw g Ž s .d s . ) 0, where b is the lower bound of W. Since g G 0, f is nondecreasing. Since w was arbitrary, this shows that if GŽ w . / 0 then GŽ y . / 0 for every y ) w. Therefore, GŽ w . s 0 for w below the lower bound of W. It follows that f can be extended to satisfy GŽ w . s f Ž w . B on Rqq with f nonnegative and f Ž0. s 0. Then F i Ž p,w . s f Ž w . B q wA for w G 0.I Proof of Proposition 4. Since p is fixed we omit it as an argument of all functions. To prove the if part of the proposition, note that

Ž 1r2. Ýwi wh Ž M i Ž wi . y M h Ž wi . .Ž Ai Ž wi . y A h Ž wh . .

T

i,h T

s Ý w h M i Ž wi . F i Ž wi . y wi M i Ž wi . F h Ž w h .

T

Ž 9.

i,h T

T

s wÝM i Ž wi . F i Ž wi . y Ýwi M i Ž wi . ÝF i Ž wi . . i

i

i

If condition Ža. of Proposition 4 is satisfied, the households’ p-Engel curves are linear. Then M i Ž wi . s Ai Ž wi . for each household i, so the matrix in Eq. Ž9. is positive semidefinite. This implies that the consumption sector has NDD, and, by Lemma 4, that the sector has NDED when the endowment vectors are collinear.

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Suppose instead that Žb. of Proposition 4 holds. Since F i satisfies NT, there is a function f i and vectors B and C Ždepending on p . such that F i Ž w . s f i Ž w . B q wC for w G 0. Then M i Ž w . s f iX Ž w . B q C and A i Ž w . s a i Ž w . B q C, where a i Ž w . ' f i Ž w .rw. If households i and h belong to different p-Engel families then Žpossibly after reversing the labels i and h. f iX Ž wi . G f hX Ž wh . for all wi ) 0 and wh ) 0. Since f i Ž0. s 0 for all i, a i Ž wi . - a hŽ w h . would contradict the mean value theorem. Therefore, a i Ž wi . G a hŽ w h .. If households i and h are p-replicas and satisfy UC, then without loss of generality f i can be assumed to be convex and a i nondecreasing. Since there is some k satisfying f hŽ w . s f i Ž kw .rk for all w G 0, we have f hX Ž w . f iX Ž kw . and a i Ž w . s a hŽ w . for all w ) 0. Then f iX Ž wi . y f hX Ž wh . G wFx0 and a i Ž wi . y a hŽ w h . G wFx0 whenever wi G wFx kwh . It follows that in all possible cases, Ž M i Ž wi . y M h Ž w h ..Ž Ai Ž wi . y A h Ž w h .. T s X Ž f i Ž wi . y f hX Ž wh ..Ž a i Ž wi . y a hŽ w h .. BB T is positive semidefinite, since the scalar valued function on the right-hand side is nonnegative. Since this holds for all i and h, the expression in Eq. Ž9. is positive semidefinite, so the consumption sector satisfies NDD and NDED when the endowments are collinear. This completes the if part of the proof. For the only if part, we begin by proving the following claim: if the community satisfies NDED for every collinear endowment distribution then for every pair of households i and h and pair of positive scalars wi and w h the matrix ŽM i Ž wi . y M h Ž w h ..ŽAi Ž wi . y Ai Ž w h .. T is positive semidefinite. Suppose that this matrix is not positive semidefinite. We will show that there is a collinear endowment distribution for which NDED is violated. Let household k have wealth w k ) 0. By choosing w k sufficiently small for k different from i and h we can make the expression in Eq. Ž9. arbitrarily close to wi w hŽM i Ž wi . y M h Ž w h ..ŽAi Ž wi . y Ah Ž wh .. T, hence not positive semidefinite. For each k s 1, . . . , I, define v ' ÝF k Ž w k . and v k ' Ž w krw . v , where w ' Ýw k . Then the right-hand side of Eq. Ž9. is wÝ M k Ž F k . T y wÝ M kv k , so this matrix is not positive semidefinite. But by Lemma 3, it follows that NDED is violated. This completes the proof of the claim. By Lemma 5, ŽM i Ž wi . y M h Ž wh ..ŽAi Ž wi . y Ah Ž wh .. T is symmetric for every i and h and positive wi and w h . In the terminology of Jerison Ž1984., the community has pairwise symmetry. By Theorem 3X of Jerison Ž1984., at least one of the following conditions must hold for vectors B, C, Bi , C i and scalar e , all depending on p: Ž a . F i Ž w . s w Žln w . B q wC i ; Ž b . F i Ž w . s w e Bi q wC; I Žg . for each Ž w 1 , w 2 , . . . ,wi . 4 0, the points  M i Ž wi ., Ai Ž i .4is1 are collinear; k k they are contained in the line  l M Ž w k . q Ž1 y l. A Ž w k .: l g R4 if M k Ž wk . / A k Ž w k .. Since the demand vectors F i Ž w i . are nonnegative, cases Ž a . and Ž b . above can hold only if B s Bi s 0. In that case, Ža. of Proposition 4 is satisfied. Suppose instead that Žg . holds. We first prove Žb1. of the proposition. Suppose that there are three households with wealth levels such that their average propensity to

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consume vectors are distinct. Then every household’s p-Engel curve is contained in the span of at least two of the three average propensity to consume vectors, so Žb1. holds. Suppose instead that for each vector of wealth levels, Ž w 1 , w 2 , . . . ,wi . the set of corresponding average propensity vectors  Ai Ž wi .4i consists of two distinct points. If two households have identical average propensity vectors no matter what their wealth levels are then by varying the wealth of one of these households, holding that of the other constant, we see that each household’s average propensity to consume vector does not vary with wealth. Thus if Ai varies with wealth for some household i then that average propensity vector must be different from the average propensity of the other households, and the other households must have the same constant average propensity to consume vector, A hŽ w h . s A for every h / i. By Lemma 9, F i Ž w . s f i Ž w . B q wA for all w G 0, so Žb1. holds. Finally, Žb1. holds if all the households’ average propensity to consume vectors are equal at every wealth distribution, since in that case they are all constant and identical. We have shown that there are nonzero vectors B and C and functions f i for each household i satisfying F i Ž w . s f i Ž w . B q wC for all w G 0. Define a i Ž w . ' f i Ž w .rw for each household i and w ) 0. For every pair of households i and h and pair of wealth levels wi and w h the matrix ŽM i Ž wi . y M h Ž w h ..ŽAi Ž wi . y Ah Ž wh .. T s Ž f iX Ž wi . y f hX Ž w h ..Ž a i . y a hŽ wh .. BB T is positive semidefinite. Therefore Ž f iX Ž w i . y f hX Ž wh )..Ž a i Ž wi . y a hŽ wh .. is nonnegative, and the functions f i and f h spread. ŽSee the definition before Lemma 6.. We will prove Žb2. and Žb3. together. Suppose that f iX Ž w . ) f hX Ž y . and f iX Ž wX . f hŽ yX .for some strictly positive w, y, wX and yX . By Lemma 6, a i Ž w . ) a h Ž y . and a i Ž wX . - a h Ž yX . .

Ž 10 .

Suppose that a i Ž wX . ) a i Ž w .. Then the interval w a i Ž w .,a i Ž wX .x is contained in the image of a h , and by Sard’s theorem ŽGuillemin and Pollack, 1974. there exists z with a hŽ z . in the interval and a hX Ž z . / 0. If a hX Ž z . - 0wresp.) 0x then Lemma 8 implies that f i and f h are concave wresp. convexx and that there is a scalar k satisfying a i Ž zX . s a hŽ kzX . for all zX ) 0. This implies that i and h are in the same p-Engel family and are replicas and satisfy UC at p. Suppose instead that a i Ž wX . - a i Ž w .. Since a hŽ y . and a hŽ yX . cannot both be below a i Ž wX . or both above a i Ž w ., the interval w a i Ž wX .,a i Ž w .x intersects the image of a h . If the intersection has a nonempty interior then the above argument implies that i and h are UC replicas at p. If the intersection has empty interior then a h is constant, and A h s a h B q C does not vary with wealth. Then F i Ž z . s a i Ž z . B q C s A h q Ž a i Ž z . y a h . B s A h q f Ž z .B for some nonnegative function f, where the last equality follows from Lemma 9. Therefore, a i Ž z . G a h for all z or a i Ž z . F a h for all z. This contradicts the inequalities in Eq. Ž10.. If a i Ž wX . s a i Ž w . then by Eq. Ž10., this value of a i is in the interior of the image of a h . By the argument in the previous paragraph Žinterchanging i and h., a i Ž z . cannot be constant for all z, so the images of a i and a h have nonempty

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interior and i and h are UC replicas at p. We have shown that if f iX Ž w . ) f hX Ž y . and f iX Ž wX . - f hX Ž yX . then households i and h are UC replicas at p and hence are in the same p-Engel family. It follows that if households i and h are in different p-Engel families, and if f iX Ž w . ) f hX Ž y . for some w ) 0 and y ) 0 then f iX Ž wX . G f hX Ž yX . for all wX and yX . Suppose that households i and h are in the same p-Engel family. If Ai is constant then A h Ž w . s Ai Ž y . for all positive w and y. So i and h are replicas with linear p-Engel curves, hence satisfy UC at p. Suppose instead that Ai and A h vary with wealth. Then there exist w, y, wX and yX satisfying Eq. Ž10.. The argument above shows that i and h are replicas and satisfy UC at p. This proves Žb2.. Next, consider two different p-Engel families, and household i in one family and h in the other. Then Žpossibly relabeling the households. f iX Ž w . ) f hX Ž y . for some w ) 0 and y ) 0, which from above implies f iX Ž wX . G f hX Ž yX . for all wX and yX . Each household j in the p-Engel family of i is a replica of i and satisfies f jX Ž z . s f iX Ž kz . for some k and all z. Therefore f jX Ž wX . G f XmŽ yX . for every wX G 0 and yX G 0 and every household m in the p-Engel family of h. From the identity F i Ž w . s f i Ž w . B q wC it follows that the p-Engel families are uniformly ranked. This completes the proof of Žb3..I

Acknowledgements Several of the results in this paper were presented under the title ‘Dispersed Demands and the Weak Axiom in the Aggregate,’ at the 1990 Bonn Workshop on Mathematical Economics. The author thanks the participants, and especially J.-M. Grandmont, W. Hildenbrand, A. Kneip, J.-U. Scheer and R. John for helpful comments and discussions. The research was supported by the Deutsche Forschungsgemeinschaft, SFB 303.

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