The Weak Weak Axiom (WWA) for the aggregate excess demand function ensures uniqueness of equilibrium in regular economies. Jerison (1999) shows that the WWA holds if the excess demand satisfies the hypothesis of Nondecreasing Dispersion of Excess Demand (NDED). This note offers a new hypothesis guaranteeing that the WWA holds and suggests a new way for obtaining supporting empirical evidence

BRIGHI, Luigi

R. John

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A Hypothesis Guaranteeing the Weak WeakAxiomLuigi Brighi and Reinhard John∗January 2005Abstract: The Weak Weak Axiom (WWA) for the aggregate excess demand func-tion ensures uniqueness of equilibrium in regular economies. Jerison (1999) showsthat the WWA holds if the excess demand satisfies the hypothesis of NondecreasingDispersion of Excess Demand (NDED). This note offers a new hypothesis guar-anteeing that the WWA holds and suggests a new way for obtaining supportingempirical evidence.JEL classification: D5Keywords: Weak axiom of revealed preference, Nondecreasing dispersion of excessdemand, Uniqueness of equilibrium∗Dipartimento di Economia Politica, Universita` di Modena e Reggio Emilia, vialeBerengario 51, 41100 Modena, Italy (brighi@unimo.it) and Wirtschaftstheorie I, Univer-sita¨t Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany (rjohn@uni-bonn.de). We wishto thank Mario Forni for helpful discussions and suggestions.11 IntroductionThe Weak Weak Axiom (WWA), introduced in the economic literature byKihlstrom,Mas-Colell and Sonnenschein (1976), is a milder version of Samuel-son’s Weak Axiom of Revealed Preference. The WWA is a very useful prop-erty for the excess demand of competitive general equilibrium models sinceit ensures uniqueness of equilibrium in regular economies and nice compara-tive statics results as those shown, for example, in Nachbar (2002) and Quah(2003).The aggregate excess demand does not generally possess any relevantstructure. In large economies, however, if individual behavior is sufficientlyheterogeneous some good properties are likely to emerge.1 Jerison (1999) putforward a hypothesis concerning the joint distribution of individual charac-teristics, i.e. endowments and demand behavior, guaranteeing that the excessdemand satisfies the WWA. The hypothesis is called Nondecreasing Disper-sion of Excess Demand (NDED) and conveys the idea that the patterns ofconsumption of the population tend to be more dispersed as wealth increases.In the present note we propose a new hypothesis ensuring that the excessdemand of a competitive general equilibrium model satisfies the WWA. Thehypothesis is related to the effects on demand of ad hoc income redistribu-tions and suggests a new way to check whether the WWA is consistent withempirical evidence.2 Notation and definitionsLet us consider an exchange economy with n goods and denote by A thefinite set of agents. Each agent a ∈ A is characterized by a vector of initialendowments ωa ∈ Rn+ and a demand function fa : Rn++×R+ → Rn+. We assumethat fa satisfies the budget identity, i.e. pTfa(p,w) = w, is homogeneous of1See, for instance, the contribution by Hildenbrand (1983, 1994), Jerison (1982), andMarhuenda (1995) among others.2degree zero and is differentiable. The individual excess demand is za(p) =fa(p, pTωa)− ωa, so that the function Z : Rn++ → Rn given byZ(p) = E[za(p)] :=1#A∑a∈Aza(p)is the (mean) excess demand function of the exchange economy. The decom-position of the Jacobian matrix of the excess demand, obtained by using theSlutsky equation, is∂Z(p) = S(p) −M(p) (1)where S(p) is the average of individual Slutsky matrices and M(p) is theaverage of individual wealth effect matrices, ma(p)za(p)T , where ma(p) is thevector of marginal propensities to consume of agent a, i.e. ∂wfa(p, pTωa).The Weak Weak Axiom (WWA) amounts to pseudomonotonicity of theexcess demand function Z, i.e. for all prices p and q(q − p)TZ(p) ≤ 0 implies (q − p)TZ(q) ≤ 0.For regular excess demand, the WWA is characterized by the following first-order condition.2 For all v ∈ RnvTp = vTZ(p) = 0 implies vT∂Z(p)v ≤ 0. (2)Under mild rationality assumptions S(p) is negative semidefinite, thus by (1)and (2), the WWA holds ifvTp = vTZ(p) = 0 implies vTM(p)v ≥ 0 (3)i.e. when the wealth effect matrixM(p) = E[ma(p)za(p)T ] is positive semidef-inite on a suitable subspace.Let us consider the function za(p, λ) = fa(p, pTωa + λ) − ωa, i.e. theexcess demand of agent a at price p when nominal income is increased bythe positive amount λ. Moreover, let Z(p, λ) = E[za(p, λ)] and, for anyv ∈ Rn, consider the variance of the projection of za on v, i.e. σ2v(λ) =2Recall that an excess demand function is regular if at any zeroes its Jacobian matrixhas rank n − 1. See John (2000) and Brighi (2004) for a more general characterization ofpseudomonotonicity and the weak axioms.3E[vTza(p, λ)−vTZ(p, λ)]2. Jerison (1999) introduced the following hypothesison the joint distribution of individual characteristics called NondecreasingDispersion of Excess Demand (NDED).• For all p ∈ Rn++ and for all v ∈ RnvTp = vTZ(p) = 0 implies ∂λσ2v(0) ≥ 0. (4)It can be seen3 that (4) and (3) are equivalent, therefore if S(p) is negativesemidefinite NDED is a sufficient condition for the excess demand to satisfythe WWA.Formally, NDED requires a nondecreasing variance of the projections ofza on v. From the economic point of view, this hypothesis is meant to capturethe idea that the consumption patterns of the population tend to be moredispersed after a generalized increase in income. Let us illustrate NDED bymeans of a simple example that will also be used in the subsequent section...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........ .............................................................................................................................................. ....................................................................................................................................................................................................................................................pvECaECbx1x2ωbωa••ϕb(p)ϕb(p, λ)ϕa(p)ϕa(p, λ)Fig. 1. A two by two exampleExample. There are only two goods and two agents. The demand functionsof agents a and b are respectivelyfa(p,w) =(2w2p1 + p2,w2p1 + p2)and fb(p,w) =(wp1 + 2p2,2wp1 + 2p2)3See Lemma 3 in Jerison (1999)4................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................. .................................................................................................................................................. ...........................................................z1z2••pvza(p)zb(p)za(p, λ)zb(p, λ)Fig. 2. The hypothesis of NDEDand the vectors of initial endowments are ωa = (0, 10) and ωb = (10, 0). IfZ(p) 6= 0, conditions (2) or (3) are trivially satisfied in the two goods case.Therefore, let us focus on a price vector at which Z vanishes, e.g. p = (1, 1),and a vector v = (1,−1) so that vTp = 0. This case is depicted in Fig. 1 whereECa and ECb are the Engel curves of the two agents and the dashed linerepresents the budget line of each agent after a generalized increase in incomeby the amount λ > 0. ϕi(p) = fi(p, pTωi) and ϕi(p, λ) = fi(p, pTωi + λ) fori = a, b are the demanded bundles before and after the increase in income.Fig. 2 translates the above example in terms of excess demand and showsthat NDED is satisfied. Indeed, the black dots representing the projectionsof z on v are further away from the origin than the starting vectors of theexcess demand and this means that the variance σ2v(λ) is increased.53 A new hypothesisAs we have seen in Section 2, the hypothesis of NDED refers to the behaviorof individual excess demand in response to a generalized increase in income.On the other hand, the hypothesis that we are going to introduce will beconcerned with consumers behavior in response to ad hoc income redistribu-tions.Given the price p, the vector v = q− p is a ‘redistributive’ price change ifvTZ(p) = 0. The price change v is called redistributive because if any agenta is given the amount of incomeβa = −vTza(p), (5)then∑a βa = −(#A)vTZ(p) = 0, i.e. the βa’s are simply a redistribution ofincome among the population. In addition, by imposing the condition vTp =0 we are taking a specific price normalization for q. Specifically, the priceq is normalized by the vector (1/pT p)p, since vTp = 0 yields qT (1/pT p)p =1. Let us thus consider only normalized, redistributive price changes andthe associated income redistributions. Formally, given p and v such thatvTZ(p) = 0 and vTp = 0, the associated income redistribution is the set{βa | βa = −vTza(p), a ∈ A}.This kind of ad hoc income redistributions approximates the effect of pricechanges on the distribution of wealth. More precisely, βa is a compensatingvariation of income, i.e. the income that, after the price change, can be takenaway from an agent so that he can buy exactly the old bundle of goods atthe new prices. Indeed, the compensating variation of income ya is definedbyqTωa − ya = qTfa(p, pTωa).which yieldsya = qTωa − qTfa(p, pTωa) = −qTza(p) == −(p + v)Tza(p) = −vTza(p) = βa.6In order to introduce the new hypothesis let us consider the Examplein Section 2. As we know, this is a case where NDED holds so that thewealth effect matrix M(p) is positive semidefinite on the orthogonal space toZ(p). From Fig. 2 it is easily seen that βa < 0 and βb > 0. Therefore, thehypothetical income redistribution consists of a tax for agent a and an incometransfer for agent b. Let us suppose that this hypothetical tax/transfer policyis actually implemented. The budget line of agent b will shift upward andthat of agent a inward as shown in Fig. 3. The new demanded bundles are,respectively,ϕa(p, βa) = fa(p, pTωa+βa) and ϕb(p, βb) = fb(p, pTωb+βb). Themean demand will be Φ(p, βa, βb) = E[ϕi(p, βi)] and the mean endowmentsω = E(ωi), with i = a, b. Fig. 3 shows that the (mean) excess demand,Z(p, βa, βb) = Φ(p, βa, βb)−ω, and the excess demand of agent b, zb(p, βb) =ϕb(p, βb) − ωb, point in the same direction so that the sign of vTZ(p, βa, βb)is the same as the sign of vTzb(p, βb). Loosely speaking, the excess demandchange resulting from the hypothetical income redistribution is driven by theagent receiving a positive income transfer......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................. .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................pvECaECbx1x2ωbωaω•Φ(p,βa,βb)ϕb(p,βb)ϕa(p,βa)zb(p,βb)Z(p,βa,βb) •Fig. 3 The hypothesis of NER.In order to generalize the above idea, let us consider any redistributionpolicy providing the generic agent a with the tax/transfer tβa, where t > 0.Accordingly, the excess demand of agent a will beza(p, tβa) = fa(p, pTωa + tβa)− ωa.7The (mean) excess demand bundle associated with the tax/transfer policy isgiven byZ(p, t) = E[za(p, tβa)].The group of agents receiving an income transfer will be denoted by A+, i.e.A+ = {a ∈ A |βa ≥ 0}, and their (mean) excess demand will beZ+(p, t) = E[za(p, tβa)|A+] := 1#A+∑a∈A+za(p, tβa).If we generalize the intuition behind the example we ought to require thatsign [vTZ(p, t)] = sign [vTZ+(p, t)]for all t > 0 sufficiently small. However, by (5) and the definition of A+, thesign on the right hand side must be non positive so that we remain with thefollowing hypothesis:• Negative effect of redistributions (NER). Let v ∈ Rn be a normalized,redistributive price change at p, i.e vTp = 0 and vTZ(p) = 0, and{βa |βa = −vTza(p), a ∈ A} the associated redistribution of income.Then for all t > 0 sufficiently smallvTZ(p, t) ≤ 0.The hypothesis of NER has an interpretation in terms of real wealth.After the redistribution the new excess demand is Z(p, t) and the overallcompensating income variation is vTZ(p, t). If vTZ(p, t) is negative, as thehypothesis of NER requires, the net effect on total real wealth is positive.In other words, after any ad hoc income redistribution the population ofconsumers will be better off (in the aggregate) at the new prices q ratherthan at the old ones p, since qTZ(p, t) ≤ pTZ(p, t).Let us show that the hypothesis of NER has a desirable implication onthe aggregate wealth effect matrix M(p).Lemma. If NER is satisfied then, for any p and v such that vTp = 0 andvTZ(p) = 0, it holds vTM(p)v ≥ 0.8Proof. Since vTp = 0 and vTZ(p) = 0, by NER, we have1tvT [Z(p, t)− Z(p)] ≤ 0for all small t > 0. Taking the limit as t goes to zero and using (5) yields0 ≥ limt→0vT{1t[Z(p, t)− Z(p)]}== vT{E limt→01t[fa(p, pTωa + tβa) − fa(p, pTωa)]}== vT {E[ma(p)βa]} == −vT {E[ma(p)za(p)T ]} v == −vTM(p)v.Assuming that the aggregate substitution matrix in the Slutsky decom-position is negative semidefinite, the Lemma and (3) yield immediately thefollowing result.Proposition. If Z is a regular excess demand and NER holds then Z satisfiesthe WWA.Another consequence of the Lemma is that NER has the same implica-tion as NDED on the wealth effect matrix M(p). Therefore, not only thehypothesis of NDED but also NER could be tested, in principle, by meansof panel or cross-section data as indicated in Jerison (1999, 2001).As a final remark, we notice that the hypothesis of NER suggests a quitestraightforward way to verify whether the WWA is consistent with empir-ical evidence. Indeed, with cross-section data on households’ demand andendowments, the suggested procedure consists of the following steps brieflysketched below.1. Subdivide the population of households A into K classes of wealth, sothat Ak is the group of households in wealth class k, with k = 1, ...,Kand A = ∪Kk=1Ak. Denote by αk = #Ak/#A the relative frequency ofhouseholds in wealth class k.92. For each wealth class compute the (mean) excess demand bundle4 Zk =∑a∈Ak za/#Ak so that the (mean) excess demand of the whole sampleis Z =∑Kk=1 αkZk.3. Choose a redistributive price change v for the population A, i.e. vTZ =0 and compute the tax/transfer policy βa = −vTza, for all a ∈ A.Compute the new distribution of wealth, where individual wealth isnow given by pTωa+βa. The new distribution of households into wealthclasses is denoted by A′k and relative frequencies by α′k = #A′k/#A.4. Compute the excess demand of the whole sample by using the newfrequencies α′k, i.e. compute Z′ =∑Kk=1 α′kZk and apply statisticaltechniques to check whether vTZ ′ is negative.Since NER is concerned with the behavior of demand when households’wealth changes, we had to make supplementary assumptions in order touse cross-section data to asses the empirical plausibility of our hypothesis.Specifically, in Step 4, the implicit assumption is that even if the relativefrequency of households in one wealth class has changed, the excess demandbehavior of the households in the class, on average, remains the same. Thisassumption is obviously justified if demand behavior and endowments areindependently distributed. Less severe conditions on the joint distributionof individual characteristics may probably be sufficient.4Since prices remain fixed, we omit p as an argument of the excess demand functions.10ReferencesBrighi, L. (2004) “A Stronger Criterion for the Weak Weak Axiom”, Journalof Mathematical Economics, 40, 93-103.Hildenbrand, W. (1983) “On the Law of Demand,” Econometrica, 51, 997-1019.Hildenbrand, W. (1994) Market Demand: Theory and Empirical Evidence.Princeton, NJ: Princeton University Press.Jerison, M. (1982) “The Representative Consumer and the Weak Axiomwhen the Distribution of Income is Fixed”, Report DP 150, SUNY Albany.Jerison, M. (1999) “Dispersed Excess Demands, the Weak Axiom and Unique-ness of Equilibrium”, Journal of Mathematical Economics, 31, 15-48.Jerison, M. (2001) “Demand Dispersion, Metonymy and Ideal Panel Data”in G. Debreu, W. Neuefeind and W.Trockel (eds.) Economics essays: AFestschrift for Werner Hildenbrand, Berlin, SpringerJohn, R. (2000) “A First Order Characterization of Generalized Monotonic-ity”, Mathematical Programming, 88, 147-155.Kihlstrom, R., Mas-Colell, A. and H. Sonnenschein (1976) “The DemandTheory of the Weak Axiom of Revealed Preference”, Econometrica, 44,971-978.Marhuenda, F. (1995) “Distribution of Income and Aggregation of Demand,”Econometrica, 63, 647-666.Nachbar, J. H. (2002) “General Equilibrium Comparative Statics” Econo-metrica, 70, 2065-2074.Quah, J. K.-H. (2003) “Market Demand and Comparative Statics whenGoods are Normal” Journal of Mathematical Economics, 39, 317-333.11

A Hypothesis Guaranteeing the Weak Weak Axiom

Abstract: The Weak Weak Axiom (WWA) for the aggregate excess demand func-tion ensures uniqueness of equilibrium in regular economies. Jerison (1999) shows that the WWA holds if the excess demand satisfies the hypothesis of Nondecreasing Dispersion of Excess Demand (NDED). This note offers a new hypothesis guar-anteeing that the WWA holds and suggests a new way for obtaining supporting empirical evidence. JEL classification: D

Luigi Brighi

Reinhard John

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A Hypothesis Guaranteeing the Weak Weak Axiom

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