Volume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households

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1 Volume 29, Issue 3 Existence of competitive equilibrium in economies wit multi-member ouseolds Noriisa Sato Graduate Scool of Economics, Waseda University Abstract Tis paper focuses on te existence of a competitive equilibrium in te general equilibrium model wit multi-member ouseolds introduced by Haller (2000). Two main results are obtained: te first is based on te assumption of ``budget exaustion,'' wic is also used in te existence teorem by Gersbac and Haller (1999). Te second is based on te assumption tat every ouseold as at least one member wose preference is nonsatiated wit respect to feasible ouseold consumption, and tis is applicable to te case in wic budget exaustion does not old. I am grateful to te anonymous referee for te valuable comments and suggestions. Citation: Noriisa Sato, (2009) ''Existence of competitive equilibrium in economies wit multi-member ouseolds'', Economics Bulletin, Vol. 29 no.3 pp Submitted: Apr Publised: July 24, 2009.

2 1 Introduction As an alternative to te traditional general equilibrium model, in wic ouseolds are treated as if tey were single consumers, Haller (2000) introduces a general equilibrium model in wic eac ouseold may consist of several (eterogeneous) members. In is model, eac member of a ouseold as is or er own utility function, wic may depend on te oter members consumption as well as is or er own consumption (in oter words, consumption externalities may exist between members of te same ouseold). Te excange of goods and services takes place only among ouseolds and not among individuals. However, te beavior of a ouseold in markets is collectively decided by its members: taking a price system as given, te ouseold members collectively decide teir demand for goods and services under teir common budget constraint. Haller (2000) assumes tat as a result of suc collective decision-making, every ouseold as a wole cooses a consumption plan tat is efficient in te sense tat tere is no oter possible coice tat satisfies te budget constraint, and makes at least one member better off witout making any oter members worse off. 1 Ten, a competitive equilibrium among ouseolds is defined as a state in wic every ouseold makes an efficient coice and all markets clear. Haller (2000) provides several fundamental results for te properties of competitive equilibrium among ouseolds. For example, e establises te first welfare teorem and generalizes it to te core inclusion statement. He also examines conditions under wic a competitive equilibrium allocation among ouseolds can be realized troug competitive excange among individuals. Furter implications of competition among multi-member ouseolds are investigated in Gersbac and Haller (2001, 2005), wic extend Haller s (2000) model (in wic te ouseold structure is given) to te case of a variable ouseold structure. In tis paper, we focus on te existence of a competitive equilibrium among ouseolds and provide two main results. Our first result is based on te assumption of budget exaustion. Budget exaustion asserts tat to acieve an efficient consumption coice, ouseolds need to exaust teir budget. Under tis assumption, Gersbac and Haller (1999) prove te existence of a competitive equilibrium among ouseolds wit free disposal. 2 However, in teir proof tey implicitly assume tat budget exaustion olds in a truncated economy rater tan in an original economy. In contrast, we assume budget exaustion in te original economy and derive te existence of a competitive equilibrium among ouseolds. Our existence result also differs from te result of Gersbac and Haller (1999), in tat we discard te free disposal assumption by allowing negative equilibrium prices. 3 Next we consider an alternative to budget exaustion. As Haller (2000) illustrates, wen we take negative intra-ouseold externalities into account, budget exaustion may not old even if every consumer as a monotonic preference wit respect to is or er individual consumption. However, we prove tat even if budget exaustion fails, te existence of a competitive equilibrium among ouseolds is still ensured, provided tat eac ouseold as at least one member wose utility function is nonsatiated 1 Tis beavioral ypotesis is originated by Ciappori (1988, 1992), wo intends to develop a collective decision-making model suc tat testable implications of ouseold beavior can be derived, and ouseold members individual preferences and te collective decision-making process can be recovered from te observation of te ouseold beavior. 2 Budget exaustion is also used in Haller (2000) to establis te first welfare teorem. 3 Wit some additional assumptions, we can obtain te existence of a strictly positive equilibrium price. 1

3 wit respect to te feasible ouseold consumption. Tis paper is organized as follows. In Section 2, we introduce te model and definitions. In Section 3, we provide a preliminary result to be used in te proofs of our main existence results. In Section 4, we prove te existence of a competitive equilibrium among ouseolds under budget exaustion. In Section 5, we provide anoter existence result by replacing budget exaustion wit te nonsatiation assumption. In Section 6, we make concluding remarks. Finally, all te proofs are provided in te Appendix. 2 Model and Definitions Following Haller (2000), we define a pure excange economy wit multi-member ouseolds as follows. In te economy, tere exist l commodities and H ouseolds (l, H N, 1 l <, 1 H < ). 4 Eac ouseold = 1,, H consists of finitely many members i = m wit m = 1,, M() and M() 1. Let I denote te set of all consumers in te economy, tat is, I = {m : = 1,, H and m = 1,, M()}. Eac consumer i I as a consumption set X i = R l +. Let X = m X m for eac, and let X = i X i. We call an element x = (x m ) m of X a ouseold consumption bundle of, and x = (x i ) i of X an allocation. For a given allocation x X, we denote ouseold s consumption bundle in x by x. Let U i : X R be te utility function of consumer i = m. We allow consumption externalities only between members of te same ouseolds (intra-ouseold externalities). Eac ouseold is endowed wit a commodity bundle ω R l +. A price system p is a l-dimensional vector in R l. Note tat tis model coincides wit te standard Arrow Debreu economy if M() = 1 for all. m x m = ω (free disposal is not allowed). An allocation x X is feasible if For eac, a ouseold consumption bundle x X is feasible for if tere exists (x k ) k k X k suc tat x = (x, (x k ) k ) X is feasible. Let F denote te set of ouseold consumption bundles of tat are feasible for. Since X i = R l + for all i I, it is easy to ceck tat F = {x X : m x m k ω k } for all. Note tat F is compact for all, and if x = (x ) X is a feasible allocation, ten x F for all. Along wit te standard continuity and concavity, we use te following properties of consumers utility functions. Strict Monotonicity. U m is strictly monotonic if U m (x m, (x n ) n m ) is strictly increasing in x m. 5 4 We use te following matematical notations. Let N denote te set of natural numbers. For k N, let R k denote te k-dimensional Euclidean space For x, y R k, by x y, we mean x j y j for all j = 1,, k, and by x y, we mean x j > y j for all j. Furter, by x > y, we mean x y and x y. Let R k + = {x R k : x 0}. For x, y R k, we denote by x y = k j=1 x jy j te inner product, by x = x x te Euclidean norm, and by B(x 0, r) = {x R k : x x 0 < r} te open ball centered at x 0 wit radius r. For a set A R k, we denote by int A, cl A and co A te interior of A, closure of A and convex ull of A, respectively. 5 For X R l, a function f : X R is strictly increasing if f(y) > f(x) for all x, y X wit y > x. 2

4 Nonsatiation. U m is nonsatiated on F if for all x F, tere exists y X suc tat U m (y ) > U m (x ). Local Nonsatiation. U m is locally nonsatiated on F if for all x F and r > 0, tere exists y B(x, r) X suc tat U m (y ) > U m (x ). Non-Negative Externalities. U m exibits non-negative externalities if U m is nondecreasing in (x n ) n m. 6 It is clear tat te strict-monotonicity of U m implies te local nonsatiation. Note also tat under te concavity of U m, te nonsatiation implies te local nonsatiation. For x X and p R l, we denote Ten, we define s budget set B (p) by p x = p x m. B (p) = {x X : p x p ω }. Furter, we define s efficient budget set EB (p) as follows. m Definition 1. Houseold s consumption bundle x X belongs to EB (p) if and only if x B (p) and tere is no y B (p) suc tat U m (y ) U m (x ) for all m, and U m (y ) > U m (x ) for some m. We call an element x of EB (p) an efficient coice or efficient consumption bundle of. Note tat Definition 1 does not specify te exact decision mecanism used in a ouseold to acieve its efficient coices. Definition 2. An element (x, p ) X R l is a competitive equilibrium among ouseolds if (i) x EB (p ) for all, and (ii) m x m = ω. In a competitive equilibrium, eac ouseold makes an efficient coice under its budget constraint and markets clear. Note tat we allow negative equilibrium prices. 3 Preliminary Result In tis paper, we prove te existence of a competitive equilibrium among ouseolds under two different sets of assumptions. In te existence analysis, we suppose tat a specific decision mecanism is used in every ouseold to acieve its efficient coices: every ouseold maximizes a weigted sum of its members utility functions subject to te budget constraint. 6 For X R l, a function f : X R is nondecreasing if f(y) f(x) for all x, y X wit y x. 3

5 For eac, let = {λ = (λ 1,, λ M() ) R M() + : m λ m = 1}. Let W : X R be te function defined by W (x, λ ) = m λ m U m (x ). Moreover, for eac λ, we define a function W λ : X R by W λ (x ) = W (x, λ ). Te function W λ is a weigted sum of te ouseold members utility functions, were λ m is te weigt of te m-t member s utility. We call tis W λ ouseold s welfare function wit weigt λ. In a ouseold, a maximization of te welfare function leads to an efficient consumption coice. Indeed, it is easy to see tat for λ wit λ 0, if x X maximizes W λ ( ) over B (p), ten, x EB (p). Note tat welfare functions wit different weigts may yield different efficient consumption bundles. Let = wit a generic element λ = (λ ). Eac λ defines te list of welfare functions (W λ ) of te ouseolds. Te following lemma is a preliminary result to be used in te proofs of our main existence results. Lemma 1. Let λ. Suppose tat ω 0 for all, W λ is continuous on X for all, and W λ is concave on X for all. Ten, tere exist an element (x, p ) X R l and a non-negative real number α R + suc tat (i) p x p ω + α for all, (ii) For eac, if W λ (iii) m x m = ω. (y ) > W λ (x ), ten p y > p ω + α, 4 Existence under budget exaustion In tis section, we address te existence of a competitive equilibrium under te following assumption. (BE). Budget Exaustion For all and p R l, if x EB (p), ten p x = p ω. 4

6 Tis assumption is introduced in Gersbac and Haller (1999) as one of te sufficient conditions for te existence of a competitive equilibrium. 7 Under (BE) and some standard assumptions, tey first prove te existence of a competitive equilibrium among ouseolds wit free disposal (terefore wit nonnegative equilibrium price) and ten prove te existence of a strictly positive equilibrium price wit exact market clearing by adding several conditions (Gersbac and Haller 1999, Proposition 1). However, it sould be noted tat in Gersbac and Haller s proof, tey implicitly assume tat (BE) olds in te truncated economy rater tan in te original economy (tey employ te truncation tecnique in teir proof, as we do in te proof of Lemma 1). Indeed, (BE) may not old in te truncated economy even if (BE) olds in te original economy (and vice versa). 8 In contrast, in tis paper, we derive te existence of a competitive equilibrium among ouseolds (wit exact market clearing) by assuming (BE) in te original economy: Teorem 1. Suppose tat ω 0 for all, U m is continuous and concave on X for all and m, and, (BE) olds. Ten, tere exists a competitive equilibrium among ouseolds (x, p ) X R l wit p 0. Te first two assumptions in Teorem 1 are also used in te first part of Proposition 1 in Gersbac and Haller (1999). Wit some additional assumptions, we can obtain te existence of a strictly positive equilibrium price, wic is te same conclusion as (te corrected version of) te second part of Proposition 1 in Gersbac and Haller (1999). Corollary 1. Suppose tat all te assumptions of Teorem 1 old. Suppose furter tat tere exists at least one suc tat U m is strictly monotonic and exibits non-negative externalities for all m. Ten, tere exists a competitive equilibrium among ouseolds (x, p ) X R l wit p 0. Corollary 1 differs tecnically from te second part of Proposition 1 in Gersbac and Haller (1999) since, as noted above, Corollary 1 is based on (BE) for te original economy, wereas Gersbac and Haller s result is based on (BE) for te truncated economy. 5 Existence under (local) nonsatiation In tis section, we provide anoter existence result by replacing (BE) wit te following assumption. (PNSM). Presence of nonsatiated members For all, tere exists at least one member m wose utility function U m is nonsatiated on F. 7 Tis assumption is also used in Haller (2000) to establis te first welfare teorem. 8 Te autor is grateful to an anonymous referee for providing tese observations. 5

7 Te next lemma asserts tat if a ouseold as at least one member wose utility function is nonsatiated on F, te nonsatiation property is inerited by s welfare function wit a certain weigt. Lemma 2. For a ouseold, suppose tat U m is continuous on X for all m, and U m is nonsatiated on F for at least one m. Ten, tere exists λ suc tat λ 0 and W λ is nonsatiated on F. Te proof is provided in te Appendix. We now state te existence of a competitive equilibrium among ouseolds. Teorem 2. Suppose tat ω 0 for all, U m is continuous and concave on X for all m, and, (PNSM) olds. Ten, tere exists a competitive equilibrium among ouseolds (x, p ) X R l wit p 0. 6 Concluding Remarks Te assumption (BE) asserts tat every efficient consumption bundle of a ouseold lies on te budget line. In a one-person ouseolds model, i.e., te standard Arrow Debreu model, (BE) is implied, for example, by te local nonsatiation assumption. In contrast, in te multi-member ouseolds model, as sown in Haller (2000), 9 (BE) may not old even if every consumer as a monotonic preference wit respect to is or er individual consumption. However, Teorem 2 demonstrates tat even if (BE) fails, te existence of a competitive equilibrium is still ensured, provided tat every ouseold as at least one member wose preference is nonsatiated on te feasible consumption set. Unfortunately, unlike (BE), te assumption (PNSM) does not ensure strong Pareto optimality (in te usual sense) of a competitive equilibrium allocation (see Example 1 below). However, it is wort noting tat every competitive equilibrium allocation among ouseolds is weakly Pareto optimal witout any furter assumptions. Moreover, one can easily verify tat every equilibrium allocation x satisfies te following welfare property, wic is weaker tan weak Pareto optimality, but stronger tan strong Pareto optimality: Tere is no y X suc tat U m (y ) U m (x ) for all m I, and U m (y ) > U m (x ) for at least one m for eac. In oter words, for an equilibrium allocation x, tere is no allocation y in wic every individual acieves at least te same utility as in x, and at least one member in eac ouseold acieves a strictly iger utility. 9 Haller 2000, Example 3.3, p

8 Example 1. Let l = 1, H = 2. Tere exist tree consumers, simply labeled i = 1, 2, 3, wit generic consumption bundles x i R 2. Consumers 1 and 2 form a ouseold denoted by. Consumer 3 forms a second ouseold denoted by k. Let ω = 2 and ω k = 1. Te utility functions of consumers are U 1 (x ) = x 1 x 2, U 2 (x ) = x 2 x 1, U 3 (x k ) = x 3. Note tat every U i satisfies te local nonsatiation property. Ten, it is easy to ceck tat x = (x, x k) = [(1, 1), 1] and p = 1 constitute a competitive equilibrium among ouseolds, and x is strongly Pareto dominated by te allocation y = [(1/2, 1/2), 2]. Finally, te following observation also clarifies te difference between te existence result under (BE) and tat under (PNSM): If (BE) olds, ten any coice of utilitarian welfare weigts λ = (λ ) wit λ 0 leads to te existence of a competitive equilibrium among ouseolds (x, p ) (in wic eac ouseold maximizes W λ on B (p )). In contrast, if (PNSM) olds, ten some utilitarian welfare weigts λ = (λ ) wit λ 0 lead to te existence of a competitive equilibrium among ouseolds, but oters may not. More specifically, from te proof of Lemma 2, eac ouseold must assign a weigt tat is sufficiently close to 1 to a nonsatiated member. References P.-A. Ciappori. Rational ouseold labor supply. Econometrica, 56:63 89, P.-A. Ciappori. Collective labor supply and welfare. Te Journal of Political Economy, 100: , M. Geistdoerfer-Florenzano. Te Gale Nikaido Debreu lemma and te existence of transitive equilibrium wit or witout te free-disposal assumption. Journal of Matematical Economics, 9: , H. Gersbac and H. Haller. Allocation among multi-member ouseolds: issues, cores and equilibria. In A. Alkan, C. D. Aliprantis, and N. C. Yannelis, editors, Current trends in economics, pages New York, Springer, H. Gersbac and H. Haller. Collective decisions and competitive markets. Review of Economic Studies, 68: , H. Gersbac and H. Haller. Wen inefficiency begets efficiency. Economic Teory, 25: , H. Haller. Houseold decisions and equilibrium efficiency. International Economic Review, 41: ,

9 Appendix In te proof of Lemma 1, we use te following teorem by Geistdoerfer-Florenzano (1982). Teorem 3. Let P be a closed convex cone (wit vertex 0) of R l, and B = {p R l : p 1}, S = {p R l : p = 1} be respectively te (closed) unit ball and te unit-spere of R l. Let ζ : B P R l be an upper semicontinuous correspondence wit nonempty compact convex values. Assume tat p S P, z ζ(p), p z 0. Ten, tere exists p B P suc tat ζ(p) P 0, were is te polar cone of P. P 0 = {z R l : z q 0, q P } Proof of Lemma 1. Let B = {p R l : p 1} and S = {p R l : p = 1}. Suppose tat λ is te element suc tat W λ is continuous and concave on X for all. For eac, we define s relaxed budget correspondence β : B X by β (p) = {x X r : p x p ω + (1 p )}. Let r = 2 ω and X r = X cl B(0, r), were B(0, r) = {x R lm() : x < r}. From te definition, F X B(0, r) for all. Let β r : B X r be te correspondence defined by β r (p) = β (p) X r. It is easy to see tat β r is nonempty convex valued. Moreover, since ω 0 and X r is compact, te correspondence β r is continuous on B. 10 For eac, we define s demand correspondence φ : B X r by φ (p) = {x X r : x β r (p) and W λ (x ) W λ (y ) for all y β(p)}. r By te concavity of W λ, te correspondence φ is convex valued for all. Moreover, by Berge s Maximum Teorem, φ is upper semicontinuous wit nonempty compact values. For eac, we define te correspondence φ : B R l by φ (p) = {x R l : x = m x m, x φ (p)}. Ten, it is readily verified tat φ is upper semicontinuous wit nonempty compact convex values. Finally, we define te market excess demand correspondence ζ : B R l by ζ(p) = φ (p) ω. 10 It is readily verified tat β r is upper semicontinuous on B. To see tat βr on B, note first tat te correspondence γ : B X r defined by is lower semicontinuous γ (p) = {x X r : p x < p ω + (1 p )} as an open grap in B X r. Moreover, since ω 0, we ave β r(p) = cl(γ (p)) for all p B, were cl(γ (p)) denotes te closure of γ (p) in X r. Since te closure of a correspondence wit an open grap is lower semicontinuous, β r is lower semicontinuous. 8

10 Te correspondence ζ is clearly upper semicontinuous wit nonempty compact convex values, and satisfies p z 0 for all p S and z ζ(p). Applying Teorem 3 as P = R l, we obtain p B wit 0 ζ(p ). From te definition of ζ, tere exists x = (x ) X r suc tat and x φ (p ) for all, (1) 0 = x m m ω. (2) Note tat equation (2) implies tat x F for all. Let α = (1 p ) 0. Ten, it is clear tat (x, p ) X r B and α R + satisfy conditions (i) and (iii) in te statement of te lemma. Tus, it remains to sow tat x maximizes W λ on β (p ) (not just on β(p r )) for all. Suppose tat for some, tere exists y β (p ) wit W λ (y ) > W λ (x ). Since x β(p r ) F X B(0, r), tere exists t (0, 1) wit t x +(1 t) y β(p r ). Moreover, by te concavity of W λ wic contradicts (1). W λ, we ave (t x +(1 t) y ) > W λ (x ), Proof of Teorem 1. Take an arbitrary λ = (λ ) wit λ 0. For eac, since U m is continuous and concave on X for all m, te function W λ is continuous and concave on X. Applying Lemma 1 for λ = (λ ), we obtain an element (x, p ) X R l and a non-negative real number α R + suc tat (i) p x p ω + α for all, (ii) For eac, if W λ (iii) m x m = ω. (y ) > W λ (x ), ten p y > p ω + α, We divide te proof into two cases according to te value of p. Case 1. p = 0. Note first tat (ii) implies tat W λ (x ) W λ (y ) for all y X and. Ten, we must ave m x m = ω for eac. Indeed, if m x m ω for some, ten tere exists p 0 suc tat p m x m < p ω, wic contradicts (BE). It is now clear tat for any p 0, te element (x, p) X R l \ {0} constitutes a competitive equilibrium among ouseolds. Case 2. p 0. 9

11 We prove tat (x, p ) is a competitive equilibrium among ouseolds. In view of (iii), it suffices to sow tat x EB (p ) for all. We first prove p x p ω for all. (3) Suppose tat p x < p ω for some. Since λ 0 and (ii) olds, we ave x EB (p ). Ten, (BE) implies tat p x = p ω, wic is a contradiction. We next prove tat p x = p ω for all. Suppose tat p x > p ω for some. Ten, summing up te equations in (3) for, we obtain p x m > p ω, m wic contradicts (iii). Ten, since λ 0 for all, (ii) implies tat x EB (p ) for all. Terefore, we conclude tat (x, p ) X R l is a competitive equilibrium among ouseolds. Proof of Corollary 1. From Teorem 1, tere exists a competitive equilibrium among ouseolds (x, p ) X R l. Suppose tat p j 0 for some j = 1,, l. Let be te ouseold suc tat for all m, te utility function U m is strictly monotonic in x m and exibits non-negative externalities. For eac m, define m s consumption bundle y m = (y m1,, y ml ) X m by x ms + 1 if s = j, y ms = x ms if s j. Let y = (y m ) m X. Since eac U m is strictly monotonic and exibits non-negative externalities, we ave U m (y ) > U m (x ) for all m. Moreover, since p j 0, we ave p y < p x p ω. However, tis contradicts te fact tat x EB (p ). Proof of Lemma 2. Let 1 be te consumer wose utility function U 1 is nonsatiated on F. Note tat W : X R is continuous on X since U m is continuous for all m. Let f : R be te function defined by f(λ ) = max x F W (x, λ ). By Berge s Maximum Teorem, f is continuous on. Let x = argmax x F U 1 (x ). Since U 1 is nonsatiated on F, tere exists y X \ F suc tat U 1 (y ) > U 1 (x ). Tus, for (1, 0,, 0), W (y, (1, 0,, 0)) = U 1 (y ) > U 1 (x ) = f(1, 0,, 0). 10

12 Since W and f are continuous in λ, tere exists λ suc tat λ 0 and W (y, λ ) > f(λ ). Ten, W λ is nonsatiated on F. Indeed, for all x F, W λ wic completes te proof. (y ) = W (y, λ ) > f(λ ) W λ (x ), Proof of Teorem 2. For eac, by Lemma 2, tere exists λ suc tat λ 0 and W λ is continuous and concave on X, and nonsatiated on F. Note tat togeter wit te concavity, te nonsatiation property of W λ implies te local nonsatiation property of W λ. Applying Lemma 1 for λ = (λ ), we obtain an element (x, p ) X R l and a non-negative real number α R + suc tat (i) p x p ω + α for all, (ii) For eac, if W λ (iii) m x m = ω. (y ) > W λ (x ), ten p y > p ω + α, We prove tat (x, p ) is a competitive equilibrium among ouseolds. In view of (iii), it suffices to sow tat x EB (p ) for all. Note first tat since W λ is (locally) nonsatiated on F for all, we must ave p 0. We next prove tat x B (p ) for all. For all, since x F and W λ is locally nonsatiated on F, we ave Summing up te equations in (4) for, we obtain p x m = p ω + α. (4) m p x m = p ω + Hα. m However, since (iii) implies p x m = p ω, m we must ave α = 0. Tus, x B (p ) for all. Since λ 0 and (ii) olds, it is clear tat x EB (p ) for all. 11