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Interdependent preferences with a continuum of agents
Journal of Mathematical Economics 41 (2005) 665–686
Interdependent preferences with a continuum of agents Mitsunori Noguchi∗ Department of Economics, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan Received 18 January 2002; received in revised form 31 October 2002; accepted 8 June 2004 Available online 15 December 2004
Abstract We prove the existence of an equilibrium in exchange economies in which agents t form a finite measure space (T, T, µ), and the agents’ preferences are interdependent. We introduce the notion of reference coalition C(t, p) for each t and for each price system p, which is simply the coalition of the agents who belong to a certain income range specified by t when p prevails in the market. The interdependent preferences of each agent t operate through the so-called “reference consumption vector η”, which represents the aggregate effect on t’s preferences, resulting from varying consumption choices made by other agents. We choose for η the consumption vectors of the form 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s), where x is an assignment of commodity bundles and where the measure 1/(µ[C(t, p)])µ represents the relative frequencies of t coming into contact with each member of C(t, p). In view of the law of large numbers, the above integral can be interpreted as the “consumption trend” of the agents in C(t, p). © 2004 Elsevier B.V. All rights reserved. JEL classification: D51 Keywords: Interdependent preferences; Measure space of agents; Equilibrium
1. Introduction In this paper, we prove the existence of an equilibrium in exchange economies E in which participating agents t form a finite measure space (T, T, µ) and their preferences are ∗
Tel.: +81 5613 88514; fax: +81 5283 34767. E-mail address: [email protected] (M. Noguchi).
0304-4068/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2004.06.003
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interdependent. Each t ∈ T is characterized by his initial endowment e(t), interdependent preference relation, and what we call the “reference coalition,” which is simply the coalition consisting of the agents who belong to a certain income range specified by t. The reference coalition may vary among different agents, and even for the same agent it may still vary, depending on what price system p prevails, since the defining income range varies according to p. We denote the reference coalition of agent t at price system p by C(t, p). By “interdependent preference relations,” we mean preference relations in the usual sense but with an additional feature of being allowed to depend on other agents’ consumption. In our formulation, the interdependent preferences of each t operate through a single consumption vector η, called the “reference consumption vector,” which is a representative consumption vector summarizing the various influences on t’s preferences, resulting from the varying consumption choices made by other agents. We denote by R(t, η) the interdependent preference relation of agent t under the influence of reference consumption vector η. It would be unrealistic to assume that each agent can directly observe or get accurate information about the consumption choices made by all the agents in his reference coalition. It seems more natural to assume that each participating agent t comes into contact, at random, with finitely many agents s in C(t, p), and then t modifies his preferences according to the average consumption behavior of these agents s. In view of the law of large numbers, such average behavior would tend to the expectation of x over C(t, p), as the frequency of the “random contact” of t increases, where x denotes the assignment of commodity bundles to each agent. Therefore, it would be reasonable to assume that η equals the expectation of x over C(t, p), which would appear to agent t as the “consumption trend” of the agents in C(t, p) . Before we can proceed this way, however, we must specify a probability measure on the “events” in C(t, p) so that we can sensibly discuss the “random contact” of agent t to the agents in C(t, p). For an arbitrary coalition A, the number µ(A) can be interpreted as the fraction of the totality of agents contained in A. Thus the relative frequency for a given agent t to encounter a certain agent in C(t, p) must be compatible with µ, and this account gives a rationale for the use of 1/(µ[C(t, p)])µ for the required probability measure on C(t, p), where µ[C(t, p)] > 0 is warranted for the agents t with positive income. Consequently, the interdependent preference relations take the form R(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s)) for all agents t with positive income. There may be an issue of whether agent t himself should be included in his own reference coalition or not. However, as apparent from our formulation of interdependence, if (T, T, µ) is atomless, it does not matter one way or the other since each individual agent is negligible. Our notion of interdependence is simultaneous in the sense that all individual consumption choices are compatible with the reference consumption vectors η which are determined by the individual choices themselves, and hence, there is no reason to exclude agents from their own reference coalitions even if some of the agents have positive measure. In the above setting, we prove the existence of a competitive equilibrium in E in which the commodity space is the l-dimensional Euclidean space Rl , the consumption possibility sets are all identical and equal to the positive orthant of Rl , the preference relations satisfy a certain convexity condition, and most distinctively, the incomes of the agents sufficiently vary. We list the possible difficulties one might encounter in dealing with interdependent preferences. The standard method in the literature often constructs a sequence of sub-economies
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Ek from the original economy E, usually by restricting the consumption possibility sets to compact subsets whose sizes relate to the indices k. We then manage to obtain a set of equilibria (pk , xk ), one in each sub-economy Ek , utilizing the compactness of the consumption possibility sets. In the next step, we hope the sequence (pk , xk ) will converge to some limit (p, ¯ x¯ ), which hopefully relates to an honest equilibrium in the original economy E. If preferences depend on both price systems p and assignments x, as in the case at hand, preference relations take the general form R(t, p, x), and somewhere along the way, we frequently end up in a situation in which R(t, p, ¯ x¯ )¯x(t) needs to be approximated by R(t, pk , xk )xk (t). Here are some of the possible difficulties: 1. Although {pk } can be assumed to lie in a price simplex and hence admits a convergent subsequence, it may be difficult to find a convergent subsequence of {xk }. If preferences were independent of x, we would not need to face this problem, and we could make use of Fatou’s lemma to obtain a limit x¯ with the property that x¯ (t) ∈ Ls{xk (t)}. This is all we need to approximate R(t)¯x(t) by R(t)xk (t). This is exactly the case handled by Schmeidler (1969). A similar argument works even with price dependent preferences, as shown by Greenberg et al. (1979). The most significant advantage of the preferences without dependence on x would, of course, lie in the fact that the Lyapunov Convexity Theorem becomes applicable with a non-atomic measure space of agents, and consequently the convexity assumption on the preference relations can be disposed of. One possibility to resolve this difficulty is to factor R(t, p, x) as R(t, ξ), ξ = f (t, p, x) – our formulation is a special case of this – and manage to make {f (t, pk , xk )} converge to some limit, say, η¯ . 2. Even if we obtain a limit η¯ by the above method, there is, in general, no guarantee that η¯ coincides with f (t, p, ¯ x¯ ) where x¯ is the limit obtained by the use of Fatou’s lemma. We overcome the first difficulty by showing that the sequence {f (t, pk , xk )} belongs to an order interval, which is surely compact in Rl , and the second difficulty by following Schmeidler’ s limiting argument: His argument does not require the limit to be an equilibrium in E, but by showing that the limit price system is strictly positive, it deduces that the elements in the sequence {(pk , xk )} eventually become equilibria in E. The amount of literature on interdependent preferences is enormous. Duesenberry’s classical work (Duesenberry, 1949) concerning the so-called consumption function problem attempted to explain the statistical discrepancy between the Kuznets data (the data on aggregate savings and income in the period 1869–1929), and budget study data for 1935– 1936 and 1941–1942, by objecting to one of the fundamental assumptions of aggregate demand theory at the time: Every individual’s consumption behavior is independent of that of every other individual. Duesenberry, indeed, proposed the interdependent utility function of the i-th individual with consumption expenditure Ci to be of the form Ui = Ui [Ci / αij Cj ] where Cj is the consumption of the j-th individual and αij the weight applied by the ith individual to the expenditure of the jth (Duesenberry, p.32, 1949). Observe that the “support” of αij can be thought of as, in our terminology, a reference coalition of the i th individual, and αij precisely corresponds to the probability distribution of what we call the random contact of the i-th individual to the j-th individual in the support of αij . There are some notable differences between Duesenberry’s construction and ours: firstly,
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we consider each commodity separately, and secondly, it is inevitable for us to take the effect of price changes on reference coalitions into consideration in our microeconomic analysis of consumer behavior. Johnson (1952) investigated the macroeconomic aspects of income redistribution with interdependent preferences: he introduced “interdependence effect” as well as “income effect” in his analysis of how income transfers among individuals affect the aggregate propensity to consume. Pollak (1976, 1977, 1978) published a series of papers concerning endogenous taste, interdependent preferences, price dependent preferences, etc. His ideas of simultaneous (long-run) interdependence and steady-state (long-run) demand function are somewhat related to our equilibrium concept. Leibenstein (1950) incorporated interdependent preferences into his microeconomic analysis of consumer’s demand theory: He divided the external effects on utility into the bandwagon effect, the snob effect, and the Veblen effect. In his analysis of the bandwagon effect, each consumer’s demand function d, given a price system p, depends on the total demand D, and hence D must be obtained as a fixed point of the total demand function with p fixed. This line of argument is essential in our fixed-point theorem approach. Finally, it might seem possible to treat, at least with no price dependence, more general non-ordered interdependent preferences with a measure space of agents by the abstract economy setting. However, as Balder (1998) pointed out, this approach involves extremely delicate mathematical complications.
2. The model and the theorem In what follows (T, T, µ) denotes a complete, finite, separable measure space, i.e., µ is a real-valued non-negative, countably additive separable measure defined on a complete σ-algebra T of subsets of a set T such that µ(T ) < ∞. Rl denotes the l-dimensional Euclidean space. The i-th coordinate of the point ξ in Rl is denoted by ξ i (i = 1, . . . , l). For any two elements ξ, η in Rl , ξ ≥ η means ξ i ≥ ηi i i for every i = 1, . . . , l, ξ > η means lξ ≥ iη iand ξ = η, and ξ η means ξ > η for every i = 1, . . . , l. The scalar product i=1 ξ η of ξ and η is denoted by ξ · η and the norm √ ξ · ξ of ξ by |ξ|. Rl+ denotes the positive orthant {ξ ∈ Rl |ξ ≥ 0} of Rl , and Rl++ the subset {ξ ∈ Rl |ξ 0} of Rl . 0 denotes the zero element of Rl , 1 the vector (1, . . . , 1) in Rl , and ei the i-th coordinate vector in Rl . For any two element ξ, η in Rl , [ξ, η] denotes the order interval (ξ + Rl+ ) ∩ (η − Rl+ ). For any subset A of Rl , coA denotes its convex hull, ¯ its closure. For any two subsets A, B of Rl , A \ B denotes setAc its complement, and A theoretic subtraction. B(Rl+ ) denotes the Borel subsets of Rl+ , and T ⊗ B(Rl+ ) the obvious product. For a µ-integrable function f, T f (t)dµ(t) denotes its integral. For brevity we often omit the domain T, the variable t, and the measure dµ, and simply write f unless confusion is expected to result. L1 (µ, Rl ) denotes the space of equivalence classes of Rl -valued µ l integrable functions f : T → R normed by f = |f |. L1 (µ, Rl )∗ denotes the dual of L1 (µ, Rl ), and ·, · denotes the pairing of these spaces. For any correspondence φ : T → Rl , define L1 (µ, φ) ≡ {f ∈ L1 (µ, Rl )|f (t) ∈ φ(t) µ − a.e.}. Gφ denotes the graph of φ.
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All agents are assumed to have the same consumption set Rl+ , and an element ξ in is a consumption vector or commodity bundle. An element p of Rl+ \ {0} is a price system. (T, T, µ) is a measure space of agents. A µ-integrable function x : T → Rl+ is an assignment. An assignment e : T → Rl+ is an initial assignment where e(t) represents the initial endowment of agent t. B(t, p) = {ξ ∈ Rl+ |p · ξ ≤ p · e(t)} is the budget set of agent t at price system p. An assignment x such that x = e is an allocation. In our model, preferences of agent t are influenced by the consumption behavior of other agents, and we assume that the interdependent preferences of each t operate through a single consumption vector η ∈ Rl+ , called the reference consumption vector which is a representative consumption vector summarizing the various influences on t’s preferences, resulting from the varying consumption choices made by other agents. Define I ≡{open sets U in R such that U¯ \ U is countable}. Note that for some open subset U in R, U¯ \ U may not be countable: for example, take U = the complement of the Cantor ternary set. We consider functions I : Rl+ × Rl+ × Rl+ \ {0} → I having the following properties: Rl+
1. for any positive number λ, I(e, ", λp) = λI(e, ", p) holds; 2. I(e, ", p) ⊂ (0, ∞); 3. for any w ∈ Rl+ satisfying p · w ∈ I(e, ", p) and for any convergent sequence pn → p in Rl+ \ {0}, there is a natural number N such that n ≥ N implies pn · w ∈ I(e, ", pn ); 4. for any w ∈ Rl+ and for any convergent sequence pn → p in Rl+ \ {0}, pn · w ∈ I(e, ", pn ) implies p · w ∈ I(e, ", p); 5. for any w ∈ Rl+ satisfying p · w ∈ I(e, ", p) and for any convergent sequence (en , "n ) → (e, ") in Rl+ × Rl+ , there is a natural number N such that n ≥ N implies p · w ∈ I(en , "n , p); 6. for any w ∈ Rl+ and for any convergent sequence (en , "n ) → (e, ") in Rl+ × Rl+ , p · w ∈ I(en , "n , p) implies p · w ∈ I(e, ", p). Intuitively speaking, I(e, ", p) represents an income range in the income-scale, relative to income p · e and with magnitude p · ". Example 1. We list some examples of such I which are of importance in practical applications. In what follows, we adhere to the convention that (0, 0) ≡ ∅. 1. I(e, ", p) = (max{0, p · e − p · "}, p · e + p · "), 2. I(e, ", p) = (p · e + p · ", ∞), 3. I(e, ", p) = (0, max{0, p · e − p · "}). Observing that max{0, p · e − p · "} is a continuous function of (e, ", p), it is a simple matter to justify that the above examples all possess the required properties. Let δ : T → Rl+ be a fixed measurable function. Define C(t, p) ≡ {s ∈ T |p · e(s) ∈ I(e(t), δ(t), p)}. We require that µ[C(t, p)] > 0 for all (t, p) ∈ T × Rl+ \ {0} satisfying
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p · e(t) > 0. C(t, p) is said to be the reference coalition of agent t at price system p. Observe that: 1. for any positive number λ, C(t, λp) = C(t, p) holds for each (t, p); 2. C(t, p) ∈ T holds for each (t, p). Remark 1. As a typical example, we can take I(e, ", p) = (max{0, p · e − p · "}, p · e + p · "). With this choice of I and with appropriately chosen e : T → Rl+ and δ, C(t, p) becomes non-empty for all (t, p) ∈ T × Rl+ \ {0} satisfying p · e(t) > 0 and can be expressed as C(t, p) = {s ∈ T | max{0, p · e(t) − p · δ(t)} < p · e(s) < p · e(t) + p · δ(t)}. In this example, C(t, p) consists of those agents whose income in the income-scale is positive and fall within a certain range with respect to the income of agent t. The value of p · δ(t) simply determines the size of such range. Also note that if one wishes C(t, p) to consist of those agents whose income in the income-scale falls between 70% and 130% of the income of agent t, simply take δ(t) = 0.3e(t). Note that this example is compatible with Duesenberry’s observation about interdependence of preference systems. Duesenberry (1949, p.30) indeed writes: “In general, however, it appears that income is one of the principal status criteria.” But since social status rankings in our society form a continuous series rather than a set of clearly defined group rankings, every individual must associate with some people of higher or lower status than his own. We can also define C(t, p) in terms of I(e, ", p) = (p · e + p · ", ∞) or I(e, ", p) = (0, max{0, p · e − p · "}). These may be viewed as the reference coalition of an agent t in the extreme case of “Emulation” or “Competition” in the sense of Johnson (1952). R(t, η) ⊂ Rl+ × Rl+ is the preference relation of agent t (under the influence of reference consumption vector η). Definition 1. An exchange economy E is a pair [(T, T, µ), ((C(t, p))p∈Rl
+ \{0}
, (R(t, η))η∈Rl , e(t))t∈T ]. +
Definition 2. An equilibrium for the exchange economy E is a price system p¯ and an allocation x¯ such that for almost all t in T, if p¯ · e(t) > 0, then x¯ (t) is maximal with respect to 1 R(t, x¯ (s)dµ(s)) µ[C(t, p)] ¯ C(t,p) ¯ in the budget set B(t, p), ¯ and if p¯ · e(t) = 0, then x¯ (t) = 0.
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We can now state our assumptions and our main theorem. (A.1) Transitivity: for each (t, η) ∈ T × Rl+ , aR(t, η)b and bR(t, η)c implies aR(t, η)c; (A.2) Irreflexivity: for each (t, η) ∈ T × Rl+ , aR(t, η)a does not hold; (A.3) Continuity: for each (t, η) ∈ T × Rl+ and for each b ∈ Rl+ , the set {a|aR(t, η)b} is open relative to Rl+ ; for each t ∈ T and for each a ∈ Rl+ , the set {(b, η)|aR(t, η)b} is open relative to Rl+ × Rl+ ; (A.4) Convexity: for each (t, η) ∈ T × Rl+ and for each a ∈ Rl+ , the set {b|aR(t, η)b}c is convex; (A.5) Desirability: for each (t, η) ∈ T × Rl+ , a > b implies aR(t, η)b; (A.6) Measurability: for each a ∈ Rl+ and for each b ∈ Rl+ , {(t, η)|aR(t, η)b} ∈ T ⊗ B(Rl+ ); (A.7) Resource Availability: e 0; (A.8) Income Diversity: for each (t, p) ∈ T × Rl+ \ {0} such that p · e(t) > 0, c ∈ I(e(t), δ(t), p) \ I(e(t), δ(t), p) implies µ[{s ∈ T |p · e(s) = c}] = 0; (A.8) Income Diversity (stronger version): for each p ∈ Rl+ \ {0}, µ[{s ∈ T |p · e(s) = const}] = 0. Remark 2. The assumptions listed above are standard in the literature, except for (A.8) and (A.8) . (A.8) asserts in words that if an income c is realized as a “borderline income” of the agents in the reference coalition of some agent t with positive income at some price system p, the set of agents with income equal to c forms a null set. (A.8) is certainly stronger than (A.8) and implies (A.8). Note that (A.8) clearly implies µ[{s ∈ T |e(s) = const}] = 0. However, the converse may not hold: to see this, let (T, T, µ) be the unit interval [0, 1] with Lebesgue measure, l = 2, and e(t) = te2 , t ∈ [0, 1]. Then µ[{s ∈ T |e(s) = const}] = 0 but µ[{s ∈ T |p · e(s) = 0}] = 1 for p = e1 ∈ R2+ \ {0}. Observe that e(t) = t(e1 + e2 ), t ∈ [0, 1] is an example satisfying (A.8) . Although (A.8) appears to be a good sufficient condition and more concise than (A.8), it exhibits many shortcomings. For example, we see from the above argument that (A.8) significantly restrict the cardinality of T: if µ(T ) > 0, T must be more than countable. A more serious shortcoming is that (A.8) turns (A.7) vacuous: (A.8) implies µ[{s ∈ T |ei (s) = 0}] = 0 for i = 1, . . . , l, and hence e(t) 0 µ − a.e. This is quite unsatisfactory because proving existence of an equilibrium with (A.7) instead of with the stronger condition e(t) 0 µ − a.e. is one of the main accomplishments in Schmeidler (1969). Remark 3. Note that the reference consumption vectors η of the form 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s) are defined only for t at price system p such that µ[C(t, p)] > 0, which is guaranteed, by the definition of C(t, p), for t at price system p such that p · e(t) > 0. For t at price system p such that p · e(t) = 0, we simply let any element η ∈ Rl+ be the reference consumption vector. In what follows, this does not cause any problems since we carefully avoid discussing maximality with respect to R(t, η) for t at price system p such that p · e(t) = 0. In our main theorem, the equilibrium price systems p¯ turn out to be strictly positive, which reduces the budget set of agent t with p · e(t) = 0 to {0}, and as a consequence, the equilibrium consumption vector x(t) = 0 will be maximal
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in {0} with respect to R(t, η) for “any” η. We next present a concrete example that satisfies (A.1)–(A.8). Example 2. Let (T, T, µ) be the unit interval [0, 1] with Lebesgue measure, and l = 2. Let u : T × R2+ × R2+ → R be a function satisfying the following conditions: 1. 2. 3. 4.
u(t, a, η) is jointly continuous in (a, η); u(t, a, η) is quasi-concave in a; u(t, a, η) is strictly monotonic in a; u(t, a, η) is jointly measurable in (t, η).
We now define aR(t, η)b ⇐⇒ u(t, a, η) > u(t, b, η). Then (A.1)–(A.6) are easily seen to be satisfied. Let te1 if 0 ≤ t < 21 e(t) = te2 if 21 ≤ t ≤ 1. Then e = 18 e1 + 38 e2 0, and hence (A.7) is satisfied even though it is not true that e(t) 0 µ − a.e. Let I(e, ", p) = (max{0, p · e − p · "}, p · e + p · ") and δ(t) = 21 e(t). Then I(e(t), δ(t), p) = ( 21 p · e(t), 23 p · e(t)). Thus I(e(t), δ(t), p) \ I(e(t), δ(t), p) = { 21 p · e(t), 23 p · e(t)}. Note that p · e(t) =
tp1
if 0 ≤ t <
tp2
if 21 ≤ t ≤ 1,
1 2
We must show that µ[C(t, p)] > 0 for all (t, p) ∈ T × Rl+ \ {0} satisfying p · e(t) > 0. Observe that p · e(t) > 0 ⇐⇒ p1 > 0, t ∈ (0, 21 ) or p2 > 0, t ∈ [ 21 , 1]. Assume the first case. Then I(e(t), δ(t), p) = ( 21 tp1 , 23 tp1 ), and s ∈ 0, 21 | 21 tp1 < p · e(s) < 23 tp1 = s ∈ 0, 21 | 21 tp1 < sp1 < 23 tp1 = s ∈ 0, 21 | 21 t < s < 23 t ⊃ 21 t, t . Thus µ[C(t, p)] ≥ 21 t > 0. We next assume the second case. Then I(e(t), δ(t), p) = ( 21 tp2 , 23 tp2 ), and
s ∈ 21 , 1 | 21 tp2 < p · e(s) < 23 tp2 = s ∈ 21 , 1 | 21 tp2 < sp2 < 23 tp2 = s ∈ [ 21 , 1]| 21 t < s < 23 t ⊃ 21 , 43 . Thus µ[C(t, p)] ≥ 41 . These establish our claim. We next show that for each p > 0, µ[{s ∈ T |p · e(s) = c}] = 0 holds for each c > 0. To this end, note that for each p > 0, {s ∈ T |p ·
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e(s) = c} (c > 0) consists of at most two elements. Thus (A.8) is satisfied. Observe that this example does not satisfy (A.8) since for p = e1 , µ[{s ∈ T |p · e(s) = 0}] = µ[{s ∈ T |s = 0 or 21 ≤ s ≤ 1}] = 21 > 0. Furthermore, for u(t, a, η) =
a · (η + 1)
if 0 ≤ t <
1·a+1·η
if 21 ≤ t ≤ 1,
1 2
we can show that p = ( 21 , 21 ) and (0, t) x(t) = t 2 , t 3 3
if 0 ≤ t <
1 2
if 21 ≤ t ≤ 1
constitute an equilibrium. To see this, note that
= C t, 21 , 21
∅ t
, 23 t 2
t ,1 2
if t = 0 if 0 < t <
2 3
if 23 ≤ t ≤ 1,
and (0, t) if 0 < t < 13 2 9t − 1 15t 2 + 1 1 , if 13 ≤ t < 23 x(s)dµ(s) = 24t 24t 1 1 C t, 2 , 2 µ C t, 21 , 21 1 3 − t2 , if 23 ≤ t ≤ 1. 8 − 4t 8 − 4t For t ∈ [ 21 , 1], any vector a satisfying the budget equality is maximal with respect to u(t, ·, η) for any η, and hence a = ( 3t , 23 t) is maximal in B(t, ( 21 , 21 )) with respect to u(t, ·, 1/(µ[C(t, ( 21 , 21 ))]) C(t,( 1 , 1 )) x(s)dµ(s)). For t ∈ (0, 13 ), we 2 2 have u(t, a, 1/(µ[C(t, ( 21 , 21 ))]) C(t,( 1 , 1 )) x(s)dµ(s)) = a · (1, 1 + t), which is maximal 2 2
at a = (0, t) in B(t, ( 21 , 21 )), and for t ∈ [ 13 , 21 ), we have u(t, a, 1/(µ[C(t, ( 21 , 21 ))]) 2 2 C(t,( 21 , 21 )) x(s)dµ(s)) = a · ((9t − 1)/24t + 1, (15t + 1)/24t + 1), which is maximal at a = (0, t) in B(t, ( 21 , 21 )). Finally, we compute x = ( 18 , 38 ) = e as required. Main Theorem. If assumptions (A.1)–(A.8) are satisfied, an exchange economy E has an equilibrium (p, ¯ x¯ ).
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3. An auxiliary theorem We proceed as in Schmeidler (1969, p.579) with a slight alteration in notation. For a ˜ p) = B(t, p) ∩ {ξ ∈ Rl+ |ξ ≤ k[1 · e(t)]1}. positive real number k > 1, define B(t, Definition 3. A k-bounded partial equilibrium for an exchange economy E is a price system p and an allocation x such that for almost all t ∈ T , if p · e(t) > 0, then x(t) is ˜ p), and if p · e(t) = 0, maximal with respect to R(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s)) in B(t, ˜ p). then x(t) ∈ B(t, Auxiliary Theorem. For k > 1, if assumptions (A.1)–(A.8) are satisfied, an exchange economy E has a k-bounded partial equilibrium (p, x) with p · x(t) = p · e(t) holding outside the same exceptional set as appeared in Definition 3. Proof of auxiliary theorem. Let ) = {p ∈ Rl+ |1 · p = 1} and define a correspondence K : T → Rl+ by K(t) = {ξ ∈ Rl+ |ξ ≤ k[1 · e(t)]1}. For (t, p, x) ∈ T × ) × L1 (µ, K), let H(t, p, x) = 1 l ˜ ˜ {b ∈ B(t, p)| a ∈ R+ |aR t, µ[C(t,p)] C(t,p) x(s)dµ(s) b ∩ B(t, p) = ∅} if p · e(t) > 0 ˜ B(t, p) if p · e(t) = 0. It is well known that L1 (µ, K) is weakly compact in L1 (µ, Rl ) [see, for example, Yannelis (1991, p. 7)] and that the weak topology on L1 (µ, K) is metrizable [see Dunford and Schwartz (1958, Theorem 3, p.434) and Kolmogorov and Fomin (1970, p. 381)]. From this point on, we adhere to the convention that L1 (µ, K) is endowed with the weak topology unless otherwise stated. Since K admits a measurable selection [x(t) = 0, for example], L1 (µ, K) is clearly non-empty and also a convex subset of L1 (µ, Rl ). We define a correspondence γ1 : ) × L1 (µ, K) → ) by γ1 (p, x) = ArgMax q · x − e |q ∈ ) and γ2 : ) × L1 (µ, K) :→ L1 (µ, K) by γ2 (p, x) = {y ∈ L1 (µ, K)|y(t) ∈ H(t, p, x) µ–a.e.}. We proceed with the following theorem whose proof is now standard and is sketched in the Appendix A. Theorem 1. γ1 × γ2 : ) × L1 (µ, K) → ) × L1 (µ, K) is a non-empty, convex-valued correspondence with a closed graph.
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We can now appeal to the Fan–Glicksberg fixed point theorem and obtain a fixed point (p, x) with the following properties: 1. p · ( x − e) ≥ q · ( x − e) for all q ∈ ), 2. x(t) ∈ H(t, p, x) µ − a.e. Denote the exceptional null set appearing above by Tk so that x(t) ∈ H(t, p, x) holds for all t ∈ T \ Tk . The remainder of the proof is almost verbatim to the arguments given in Schmeidler (1969) for establishing the similar claim. We only need to make minor modifications in Schmeidler (1969, p. 580) so that his arguments extend to cover our case as well. Recall from the definition of H(t, p, x) that statement 2 above can be rewritten as ˜ p), and x(t) is maximal in B(t, ˜ p) x(t) ∈ B(t, 1 with respect to R(t, x(s)dµ(s)) µ[C(t, p)] C(t,p) 2. if t ∈ T \ Tk and p · e(t) > 0 ˜ p) x(t) ∈ B(t, if t ∈ T \ Tk and p · e(t) = 0. We must show that x is an allocation, i.e., x = e. We show this by constructing an appropriate modification y of x. To this end, let t ∈ T \ Tk . Since x(t) ∈ H(t, p, x) ⊂ ˜ p) ⊂ B(t, p), we have B(t, p · x(t) ≤ p · e(t). We claim that p · x(t) = p · e(t). If p · e(t) = 0, the claim clearly holds. Assume p · e(t) > 0. If x(t) = k[1 · e(t)]1, it follows that x(t) e(t) since k > 1. This would imply p · x(t) > p · e(t), which is a contradiction. Thus, x(t) < k[1 · e(t)]1. We can, then, find a coordinate vector ei and a real number " > 0, such that x(t) + "ei < k[1 · e(t)]1. If p · e(t) > p · x(t) ≥ 0, there exists " , 0 < " < ", such that p · (x(t) + " ei ) = p · x(t) + ˜ p), and by " pi ≤ p · x(t) + " < p · e(t) since pi ≤ 1. Consequently, x(t) + " ei ∈ B(t, (A.5), x(t) + " ei becomes more desirable than x(t), which contradicts the maximality of x(t). We have established that for each t ∈ T \ Tk , p · x(t) = p · e(t). Define b≡
x−
e.
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From the above equality, we obtain p · b = 0. Upon substituting this in statement 1 above, we have 0 ≥ q · b for all q ∈ ). By setting q = ei , i = 1, . . . , l, we get bi ≤ 0 for all i. We define J ≡ {i|bi = 0} and J ≡ {i|bi < 0}. If J = ∅, we get x = e, and we are done. Note that for i ∈ J , we have pi = 0. For each i ∈ J , define Si ≡ {t ∈ T |xi (t) < ei (t)}. Observe that for each i ∈ J , we have µ(Si ) > 0, since µ(Si ) = 0 would entail xi (t) ≥ ei (t) µ − a.e, and hence bi = 0. We claim that for each i ∈ J , |bi | ≤ Si ei − xi , and Si ei − xi > 0. Since bi < 0, we compute i
|b | = −b
i
We clearly have i ∈ J so that
= Si
i
Si
i
e −x +
T \Si
i
e −x
i
≤
Si
ei − x i .
ei − xi > 0, hence the claim holds. We set ci ≡
Si
ei − xi for each
ci > 0, |bi | ≤ ci for each i ∈ J . We next define a modification y of x in the following manner: yi (t) ≡ xi (t) for all t ∈ T if i ∈ J, and i
y (t) ≡
xi (t) + xi (t)
|bi | i (e (t) − xi (t)) ci
t ∈ Si t ∈ T \ Si
if i ∈ J . Recalling that pi = 0 for each i ∈ J , we obtain p · y(t) =
pi yi (t)
i∈J
=
pi xi (t)
= p · x(t).
i∈J
Combining this with the previous equality, we deduce that for each t ∈ T \ Tk , p · y(t) = p · e(t). This shows, in particular, that for each t ∈ T \ Tk , y(t) ∈ B(t, p). ˜ p). It suffices to show this for t ∈ Si , i ∈ J . Recall that for i ∈ J , We must show y(t) ∈ B(t, i i we have |b | ≤ c . Hence, from the definition of yi (t) on Si , we obtain yi (t) ≤ ei (t). Since
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k > 1, we compute yi (t) ≤ ei (t) ≤ k[1 · e(t)] for each i ∈ J such that t ∈ Si . For the rest of i, we have yi (t) = xi (t). Therefore, y(t) ≤ k[1 · e(t)]1, and as a consequence, ˜ p) y(t) ∈ B(t, for each t ∈ T \ Tk . In particular, we have established that for each t ∈ T \ Tk , if p · e(t) = 0, ˜ p). then y(t) ∈ B(t, ˜ We need to show that for each t ∈ T \ Tk , if p · e(t) > 0, y(t) is maximal in B(t, p) with respect to R(t, 1/(µ[C(t, p)]) C(t,p) y(s)dµ(s)). To this end, note that if i ∈ J , we have |bi |/ci (ei (t) − xi (t)) > 0 on Si . Thus, if t ∈ T \ Tk satisfies p · e(t) > 0, t ∈ i∈J Si , then by (A.5), we obtain y(t)R(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s))x(t),
which contradicts the maximality of x(t). Therefore, we have
Si ⊂ {s ∈ T |p · e(s) = 0} ∪ Tk .
i∈J
Since {s ∈ T |x(s) = y(s)} = i∈J Si , x and y agree on {s ∈ T |p · e(s) > 0} ∩ (T \ Tk ). Note also that by the definition of C(t, p), p · e(s) > 0 for each s ∈ C(t, p). Thus, x and y agree on C(t, p) ∩ (T \ Tk ), and hence C(t,p) x(s)dµ(s) = C(t,p) y(s)dµ(s). Finally, we show that y = e. For i ∈ J , we compute |bi | i |bi | i i i i i i y = x + i (e − x ) + x = x + i c = xi + |bi | c c Si T \Si i i i i i = ei . = x − x − e = x −b Therefore, y is an allocation. This completes our proof. We now prove our main theorem. Proof of Main Theorem. For each integer k > 1 there is a k-bounded partial equilibrium (pk , xk ). Let Tk be the exceptional null set for each k. Without any loss of generality, we can assume that {pk } converges to some element p¯ ∈ ). Then by the Fatou lemma in several dimension [see Hildenbrand (1974, p. 69)], there is a µ-integrable function x¯ such that x¯ (t) ∈ Ls(xk (t)) µ − a.e. As in Schmeidler (1969, p. 583), µ[{t ∈ T |p¯ · e(t) > 0}] > 0, and hence we can safely choose t ∈ T such that p¯ · e(t) > 0, t ∈ ∪k>1 Tk , and x¯ (t) ∈ Ls(xk (t)). In the proof of Corollary 1 in Appendix A, we see that for pk → p, ¯ µ[C(t, pk )] → that the order inµ[C(t, p)] ¯ >0, and hence 1/(µ[C(t, pk )]) < M for large k. Observe terval [0, M e] is non-empty compact and that 0 ≤ 1/(µ[C(t, pk )]) C(t,pk ) xk (s)dµ(s) < M C(t,pk ) xk (s)dµ(s) ≤ M xk = M e. Passing to a subsequence if necessary, we may assume that pk → p, ¯ 1/(µ[C(t, pk )]) C(t,pk ) xk (s)dµ(s) → η¯ , and xk (t) → x¯ (t). We pro-
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ceed as in Schmeidler (1969, p. 583) for establishing p¯ 0. We have, by the budget equality for each k, p¯ · x¯ (t) = lim pk · xk (t) = lim pk · e(t) = p¯ · e(t) > 0. k
k
Consequently, there is an index i such that p¯ i > 0 and x¯ i (t) > 0. Suppose p¯ j = 0 for some index j, and let z = x¯ (t) + ej . Then (A.5) implies zR(t, η¯ )¯x(t). Note that p¯ · z = p¯ · x¯ (t) > 0. Thus there is " > 0 such that z = z − "ei ∈ Rl+ and z R(t, η¯ )¯x(t). Since p¯ · z = p¯ · z − "p¯ i < p¯ · e(t), there is k1 such that k > k1 implies pk · z < pk · e(t). Note that there is k2 such that k > k2 implies pk · e(t) > 0, and there is k3 such that k > k3 implies z ≤ k[1 · e(t)]1. By (A.3) we can also find k4 such that k > k4 implies z R(t, 1/(µ[C(t, pk )]) C(t,pk ) xk (s)dµ(s))xk (t). We can now choose k > max{k1 , k2 , k3 , k4 } ˜ pk ), pk · e(t) > 0, but this contradicts to the maximality of xk (t) in such that z ∈ B(t, ˜ pk ) with respect to R(t, 1/(µ[C(t, pk )]) B(t, ¯ 0. C(t,pk ) xk (s)dµ(s)). This establishes p For the rest of our proof, we proceed as in Schmeidler (1969, p. 584). Since p¯ 0, we can choose δ > 0 and k0 such that for each k > k0 , pik > δ holds for all i, i = 1, . . . , l. Assume k > k0 . Then, for each z ∈ B(t, pk ), we have δzi ≤ pik zi ≤ pk · z ≤ pk · e(t) ≤ 1 · e(t), which implies zi ≤ 1/δ[1 · e(t)]. Choose k ≥ max{1/δ, k0 + 1}. Then, for each z ∈ B(t, pk ), we have zi ≤ k[1 · e(t)], and hence z ≤ k[1 · e(t)]1. Thus, for such k, we obtain B(t, pk ) ⊂ {z ∈ Rl+ |z ≤ k[1 · e(t)]1}. We claim that (pk , xk ) is a desired competitive equilibrium. Let t ∈ T \ Tk . If pk · e(t) > ˜ pk ) with respect to R(t, 1/(µ[C(t, pk )]) 0, xk (t) is maximal in B(t, C(t,pk ) xk (s)dµ(s)). As ˜ we see above, we have B(t, pk ) = B(t, pk ). Assume pk · e(t) = 0. Since pik > δ > 0(i = 1, . . . , l), we obtain e(t) = 0. Thus, B(t, pk ) = {0}, and xk (t) = 0. Appendix A. Proof of Theorem 1 For each e ∈ Rl+ and for a convergent sequence pn → p in Rl+ \ {0}, define S = {z ∈ Rl+ |p · z ∈ I(e, ", p)} and Sn = {z ∈ Rl+ |pn · z ∈ I(e, ", pn )}. We prove the following lemmas: Lemma 1. S¯ = {z ∈ Rl+ |p · z ∈ I(e, ", p)} ¯ Then there is a sequence zi → y such that zi ∈ S, i.e., p · zi ∈ I(e, ", p). Proof. Let y ∈ S. Since taking the inner product with p is continuous, we obtain p · y ∈ I(e, ", p), or S¯ ⊂ {z ∈ Rl+ |p · z ∈ I(e, ", p)}. For proving the converse, assume p · z ∈ I(e, ", p). Let B 1 (z) be the open ball centered i
at z with radius 1i . Then, since taking the inner product with p is an open mapping, there is zi such that zi ∈ B 1 (z) and p · zi ∈ I(e, ", p), i.e., zi ∈ S. Since |zi − z| < 1/ i → 0, we i ¯ This completes the proof. have zi → z. Thus z ∈ S. ¯ Then τ(Sn Lemma 2. Let τ be a finite Borel measure on Rl+ such that τ(S) = τ(S). 0.
S) →
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679
Proof. We first show that τ(S \ Sn ∩ S) → 0. From statement 3 in the definition of ∞ I, it follows that z ∈ S implies z ∈ Sn for all n ≥ N, and hence z ∈ ∪∞ j=1 ∩n=j (Sn ∩ S) = lim inf n (Sn ∩ S). Thus we have S ⊂ lim inf n (Sn ∩ S). Since lim supn (Sn ∩ S) = ∞ ∩∞ j=1 ∪n=j (Sn ∩ S) ⊂ S, Fatou’s lemma yields τ(S) ≤ τ[lim inf (Sn ∩ S)] ≤ lim inf τ(Sn ∩ S) n
n
≤ lim sup τ(Sn ∩ S) ≤ τ[lim sup(Sn ∩ S)] ≤ τ(S), n
n
and hence limn→∞ τ(Sn ∩ S) = τ(S). It is now immediate that τ(S \ Sn ∩ S) = τ(S) − τ(Sn ∩ S) → 0. We next show that τ(Sn \ Sn ∩ S) → 0. To this end, note that S ⊂ lim inf n Sn clearly ¯ This can be seen as follows: let z ∈ ∩∞ ∪∞ Sn . holds. We claim that lim supn Sn ⊂ S. j=1 n=j Then we can choose n1 < n2 < n3 < . . . such that pni · z ∈ I(e, ", pni ) for all i. Note that from statement 4 in the definition of I, we obtain p · z ∈ I(e, ", p), which implies, by Lemma ¯ We can now appeal to Fatou’s lemma again and obtain 1, that z ∈ S. τ(S) ≤ τ[lim inf Sn ] ≤ lim inf τ(Sn ) n
n
¯ ≤ lim sup τ(Sn ) ≤ τ[lim sup(Sn )] ≤ τ(S). n
n
¯ we have limn→∞ τ(Sn ) = τ(S) = τ(S). ¯ From this, it is immediThus if τ(S) = τ(S), ate that τ(Sn \ Sn ∩ S) = τ(Sn ) − τ(Sn ∩ S) → τ(S) − τ(S) = 0. Since τ(Sn S) = τ(S \ Sn ∩ S) + τ(Sn \ Sn ∩ S), τ(Sn S) → 0 as desired. Proposition 1. For each t ∈ T , (p, x) → C(t,p) x(s)dµ(s) is continuous on {p ∈ )|p · e(t) > 0} × L1 (µ, K), where L1 (µ, K) is endowed with the weak topology. Proof. Let (pn , xn ) → (p, x) bean arbitrary convergent sequence in {p ∈ )|p · e(t) > 0} × L1 (µ, K). We need to show C(t,pn ) xn (s)dµ(s) → C(t,p) x(s)dµ(s). We have the following inequalities: xn (s)dµ(s) − x(s)dµ(s) C(t,pn )
C(t,p)
xn (s)dµ(s) − xn (s)dµ(s) C(t,pn ) C(t,p) + xn (s)dµ(s) − x(s)dµ(s) .
≤
C(t,p)
C(t,p)
Let χC(t,p) be the characteristic function of C(t, p). For the convergence of the second term on the right hand side of the above inequality, it suffices to show that each component converges, i.e., χC(t,p) (s)xi (s)dµ(s) − χC(t,p) (s)xi (s)dµ(s) → 0, n
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which amounts to showing χC(t,p) ei , xn → χC(t,p) ei , x, but this follows trivially from the fact that χC(t,p) ei ∈ L1 (µ, Rl )∗ and xn → x weakly. We next show that the first term on the right hand side of the above inequality converges √ as well. Since xn is a measurable selection of K, we have 0 ≤ |xn (s)| ≤ k l[1 · e(s)], and hence xn (s)dµ(s) − xn (s)dµ(s) C(t,pn ) C(t,p) ≤ |χC(t,pn ) (s) − χC(t,p) (s)xn (s)|dµ(s) ≤
√ |χC(t,pn ) (s) − χC(t,p) (s)|k l[1 · e(s)]dµ(s)
√ =k l
C(t,pn ) C(t,p)
1 · e(s)dµ(s).
Since A → A 1 · e(s)dµ(s), A ∈ T, is absolutely continuous with respect to µ, it suffices to show µ[C(t, pn ) C(t, p)] → 0. Note that C(t, p) = e−1 (S) and C(t, pn ) = e−1 (Sn ), where we let e = e(t) and " = δ(t) in the definitions of S and Sn , respectively. Thus we can write µ[C(t, pn ) C(t, p)] = µ[e−1 (Sn ) e−1 (S)] = µ[e−1 (Sn S)] = τ(Sn S), where τ ≡ µ ◦ e−1 is a finite Borel measure on Rl+ . From Lemma 1, we have S¯ \ S = {z ∈ Rl+ |p · z ∈ I(e(t), δ(t), p) \ I(e(t), δ(t), p)}, and since I(e(t), δ(t), p) \ I(e(t), δ(t), p) is countable by the definition of I, S¯ \ S equals the countable union of the sets having the form {z ∈ Rl+ |p · z = c}, where c ∈ I(e(t), δ(t), p) \ I(e(t), δ(t), p). Observe that by (A.8), we have τ[{z ∈ Rl+ |p · z = c}] = µ[{s ∈ T |p · e(s) = c}] = 0, and hence τ(S¯ \ S) = 0 or ¯ = τ(S). We can now appeal to Lemma 2 in order to establish our claim. τ(S) 1 Corollary 1. For each t ∈ T , (p, x) → µ[C(t,p)] C(t,p) x(s)dµ(s) is defined and continuous on {p ∈ )|p · e(t) > 0} × L1 (µ, K), where L1 (µ, K) is endowed with the weak topology. Proof. Let t ∈ T . If {p ∈ )|p · e(t) > 0} = ∅, the statement in the corollary above is trivially true. So, assume {p ∈ )|p · e(t) > 0} = ∅. It suffices to show p → µ[C(t, p)] > 0 is continuous on {p ∈ )|p · e(t) > 0}. To see this , let pn → p be an arbitrary convergent sequence in {p ∈ )|p · e(t) > 0}. Then, as in the last part of the proof of Proposition 1, dµ(s) − dµ(s) |µ[C(t, pn )] − µ[C(t, p)]| = C(t,pn )
C(t,p)
= χC(t,pn ) (s)dµ(s) − χC(t,p) (s)dµ(s) ≤ |χC(t,pn ) (s) − χC(t,p) (s)|dµ(s) = µ[C(t, pn ) This establishes our claim.
C(t, p)] → 0.
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Proposition 2. For each (p, x) ∈ ) × L1 (µ, K), t → {t ∈ T |p · e(t) > 0}.
C(t,p) x(s)dµ(s)
681
is measurable in
Proof. Define a function G : Rl+ × Rl+ → Rl+ by G(a, b) = {s∈T |p·e(s)∈I(a,b,p)} x(s)dµ(s). Since C(t,p) x(s)dµ(s) = G(e(t), δ(t)), it suffices to show that G(a, b) is continuous in the image of {t ∈ T |p · e(t) > 0} under e × δ : T → Rl+ × Rl+ . To this end, let (an , bn ) → (a, b) be an arbitrary convergent sequence in e × δ({t ∈ T |p · e(t) > 0}). Define V ≡ {z ∈ Rl+ |p · z ∈ I(a, b, p)} and Vn ≡ {z ∈ Rl+ |p · z ∈ I(an , bn , p)}. As in the proof for Proposition 1, we obtain the following results. • V¯ = {z ∈ Rl+ |p · z ∈ I(a, b, p)}; we simply repeat the argument in the proof of Lemma 1, with a, b substituted for e, ", respectively. • V ⊂ liminfn (Vn ∩ V ); this is a direct consequence of statement 5 in the definition of I. • limsupn Vn ⊂ V¯ ; this can be seen as follows: let z ∈ limsupn Vn . Then we can choose n1 < n2 < n3 < . . . such that p · z ∈ I(ani , bni , p) for all i. From statement 6 in the definition of I, we deduce that p · z ∈ I(a, b, p), or z ∈ V¯ . ¯ we Repeating the argument in the proof of Lemma 2 with V, Vn , V¯ instead of S, Sn , S, l ¯ conclude that for any finite Borel measure τ on R+ , if τ(V ) = τ(V ), then τ(Vn V ) → 0. Note that {s ∈ T |p · e(s) ∈ I(an , bn , p)} = e−1 (Vn ) and {s ∈ T |p · e(s) ∈ I(a, b, p)} = e−1 (V ). So, as in the proof of Proposition 1, we have |G(an , bn ) − G(a, b)| = x(s)dµ(s) − x(s)dµ(s) {s∈T |p·e(s)∈I(an ,bn ,p)} {s∈T |p·e(s)∈I(a,b,p)} x(s)dµ(s) − x(s)dµ(s) = e−1 (Vn )
e−1 (V )
= χe−1 (Vn ) (s)x(s)dµ(s) − χe−1 (V ) (s)x(s)dµ(s) ≤ |χe−1 (Vn ) (s) − χe−1 (V ) (s)x(s)|dµ(s) √ ≤ k l |χe−1 (Vn ) (s) − χe−1 (V ) (s)|1 · e(s)dµ(s) √ =k l
e−1 (Vn ) e−1 (V )
1 · e(s)dµ(s).
Defining τ as in the proof of Proposition 1, it suffices to show that µ[e−1 (Vn ) e−1 (V )] = τ(Vn V ) → 0. Since V¯ \ V is the countable union of the sets in the form {z ∈ Rl+ |p · z = c}, c ∈ I(a, b, p) \ I(a, b, p), and I(a, b, p) = I(e(t), δ(t), p) for some t satisfying p · e(t) > 0, (A.8) implies τ[{z ∈ Rl+ |p · z = c}] = µ[{s ∈ T |p · e(s) = c}] = 0 and hence τ(V¯ ) = τ(V ), and this establishes our claim.
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Corollary 2. For each (p, x) ∈ ) × L1 (µ, K), t → 1/(µ[C(t, p)]) surable in {t ∈ T |p · e(t) > 0}.
C(t,p) x(s)dµ(s) is mea-
Proof. Since µ[C(t, p)] > 0 on {t ∈ T |p · e(t) > 0}, it suffices to show that t → µ[C(t, p)] is measurable in {t ∈ T |p · e(t) > 0}. To this end, define a function H : Rl+ × Rl+ → Rl+ by H(a, b) = µ[{s ∈ T |p · e(s) ∈ I(a, b, p)}]. We must show that H is continuous in the image of {t ∈ T |p · e(t) > 0} under e × δ : T → Rl+ × Rl+ . We have |H(an , bn ) − H(a, b)| = |µ[e−1 (Vn )] − µ[e−1 (V )]| = τ(Vn
≤ µ[e−1 (Vn
V )]
V ),
where V, Vn , τ are defined as in the proof of Proposition 2. As in the proof of Proposition 2, the last term converges to 0. This completes our proof. Proof of Theorem 1. Since the continuity of (p, x) → p · ( x − e) follows from Lemma 1 in (Noguchi, 1977, P. 8), Berge’s maximality theorem (1963) implies that γ1 is a nonempty, convex, compact valued, upper semi-continuous correspondence. We next show that t → H(t, p, x) admits a measurable graph. Define Tp = {t ∈ T |p · e(t) > 0} ∈ T. Following Schmeidler (1969, p. 582), we have GH(·,p,x) = [(Tpc × Rl+ ) ∩ GB(·,p) ] ∪ [(Tp × Rl+ ) ∩ GH(·,p,x) ]. ˜ Define g(t, ξ) = p · ξ − p · e(t). By Castaing and Valadier (1977, Theorem III.14, p. 70), g is jointly measurable in (t, ξ). Consequently, we have GB(·,p) = {(t, ξ) ∈ T × Rl+ |g(t, ξ) ≤ 0} ∈ T ⊗ B(Rl+ ), and since the correspondence t → K(t) clearly admits a measurable graph, so does the ˜ p). Thus, (Tpc × Rl+ ) ∩ GB(·,p) correspondence t → B(t, ∈ T ⊗ B(Rl+ ). We must show that ˜ (Tp × Rl+ ) ∩ GH(·,p,x) ∈ T ⊗ B(Rl+ ). To this end, proceeding as in Schmeidler (1969, p. 582), note that (Tp × Rl+ ) ∩ GH(·,p,x) ˜ p)} \ (t, b) ∈ Tp × Rl+ |∃a ∈ B(t, ˜ p) such that = {(t, b) ∈ Tp × Rl+ |b ∈ B(t, aR t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s) b
By (A.3)(Continuity), the statement ˜ p) such that aR t, ∃a ∈ B(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s) b
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683
is equivalent to ˜ ∃ rational r ∈ B(t, p) such that rR t,
1 µ[C(t, p)]
Thus, l ˜ (t, b) ∈ Tp × R+ |∃a ∈ B(t, p) such that aR t, =
C(t,p)
x(s)dµ(s) b.
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s) b
˜ p) such that (t, b) ∈ Tp × Rl+ |∃ rational r ∈ B(t,
1 x(s)dµ(s) b rR t, µ[C(t, p)] C(t,p) ˜ p)} {(t, b) ∈ Tp × Rl+ |r ∈ B(t, =
rational r∈Rl+
∩ (t, b) ∈
Tp × Rl+ |rR(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s))b
.
Note that ˜ p)} {(t, b) ∈ Tp × Rl+ |r ∈ B(t, ˜ p)} ∩ (Tp × Rl+ ) = {(t, b) ∈ T × Rl+ |r ∈ B(t, ˜ p)} × Rl+ ] ∩ (Tp × Rl+ ) = [{t ∈ T |r ∈ B(t, = [{t ∈ T |p · r ≤ p · e(t), r ≤ k[1 · e(t)]1} × Rl+ ] ∩ (Tp × Rl+ ) ∈ T ⊗ B(Rl+ ). In what follows, we abbreviate R(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s)) as R. Note that by (A.3) (Continuity), if rRb, then there exists an open ball Ub centered at b, such that for all b ∈ Ub , rRb holds. It follows that we can safely choose a rational r ∈ Rl+ such that rRr and r > b. On the other hand, if there exists a rational r ∈ Rl+ such that rRr and r > b hold, then by (A.5) (Desirability) and (A.1) (Transitivity), we get rRb. Thus, the statement “rRb” is equivalent to “∃ rational r ∈ Rl+ such that rRr and r > b. Consequently, {(t, b) ∈ Tp × Rl+ |rRb} = {(t, b) ∈ Tp × Rl+ |∃ rational r ∈ Rl+ such that rRr and r > b} [{(t, b) ∈ Tp × Rl+ |rRr } ∩ {(t, b) ∈ Tp × Rl+ |r > b}] = rational r ∈Rl+
Note that for each rational r , we have {(t, b) ∈ Tp × Rl+ |r > b} = Tp × {b ∈ Rl+ |r > b} ∈ T ⊗ B(Rl+ ),
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and {(t, b) ∈ Tp × Rl+ |rRr } = {t ∈ Tp |rRr } × Rl+ . Thus, it suffices to prove the following claim in order to complete the proof of the fact that the correspondence t → H(t, p, x) admits a measurable graph. Claim. For each (a, b) ∈ Rl+ × Rl+ , {t ∈ Tp |aR(t, 1/(µ[C(t, p)]) where p and x are fixed.
C(t,p) x(s)dµ(s))b}
∈ T,
As a matter of fact, this is the only essential alteration needed in order to apply Lemma 3 in Schmeidler (1969, p. 582) to the case at hand. We verify the above claim. To this l l end, recall that by (A.6), we have {(t, η) ∈ T × R+ |aR(t, η)b} ∈ T ⊗ B(R+ ), and also by Corollary 2, h(t) ≡ 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s) is measurable in Tp . We must show that Gh ∈ T ⊗ B(Rl+ ). Note that Gh = {(t, η) ∈ Tp × Rl+ |η = h(t)}. Since Tp ∈ T, we can choose a measurable extension h¯ : T → Rl+ of h. Then, we have ¯ Gh = {(t, η) ∈ T × Rl+ |η = h(t)} ∩ (Tp × Rl+ ). ¯ is jointly meaBy Castaing and Valadier (1977, Theorem III.14, p. 70), q(t, η) = η − h(t) l ). surable in T × Rl+ . Consequently, Gh ∈ T ⊗ B(R + Since {t ∈ Tp |aR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))b} = projT [{(t, η) ∈ T × Rl+ |aR(t, η)b} ∩ Gh ], where projT denotes the projection onto T, we can appeal to Castaing and Valadier (1977, Theorem III.23, p. 75)to establish our claim. By Lemma 2 Schmeidler (1969, p. 581), we see that for each t ∈ T , H(t, p, x) = ∅. Thus, by Castaing and Valadier (1977, Theorem III.22, p. 74), t → H(t, p, x) admits a measurable selection, and hence γ2 (p, x) = ∅. We also need to show that H(t, p, x) is convex. For thispurpose, it suffices to consider the ˜ p) = ∅ case p · e(t) > 0. Note that {a ∈ Rl+ |aR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))ξ} ∩ B(t, is equivalent to ξ ∈ ∩a∈B(t,p) {b ∈ Rl+ |aR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))b}c , and the last ˜ set is convex by (A.4), which yields that H(t, p, x) is convex, and hence γ2 (p, x) is convex as well. We still need to show Gγ2 is closed, but this is simply Theorem 5.5 in Yannelis (1991, p. 19) or claim 6 in Khan and Vohra (1984, p. 139). Thus we only need to check that the assumptions are all satisfied. ˜ p) is comWe need to show that H(t, p, x) is compact. If p · e(t) = 0, H(t, p, x) = B(t, ˜ pact. For p · e(t) > 0, we show that H(t, p, x) is closed in B(t, p). To see this, let ξn → ξ¯ ˜ p) such that ξn ∈ H(t, p, x). Then, be a convergent sequence in B(t, a ∈ Rl+ |aR(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s))ξn
˜ p) = ∅. ∩ B(t,
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¯ ∈ B(t, ˜ p) such that If ξ¯ ∈ H(t, p, x), there exists an element A ¯ AR(t,
1 µ[C(t, p)]
C(t,p)
¯ x(s)dµ(s))ξ.
¯ Then, (A.3) implies that AR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))ξn for large n, but this is a contradiction. We next show that (p, x) → H(t, p, x) is upper semi-continuous. To this end, we sim¯ p, ply mimic the proof in Schmeidler (1969, Lemma 6, p. 582). Let (ξn , pn , xn ) → (ξ, ¯ x¯ ) be a convergent sequence in K(t) × ) × L1 (µ, K) such that ξn ∈ H(t, pn , xn ). Since pn · ˜ p), ξn ≤ pn · e(t), ξn ≤ k[e(t) · 1]1, if p¯ · e(t) = 0, then ξ¯ ∈ B(t, ¯ and hence ξ¯ ∈ H(t, p, ¯ x¯ ). Now assume p¯ · e(t) > 0. Then pn · e(t) > 0 for n > n0 for some n0 . Suppose there is ¯ Then we can find a ∈ Rl ˜ p) ¯ (s)dµ(s))ξ. ¯ such that a R(t, 1/(µ[C(t, p)]) ¯ a ∈ B(t, + C(t,p) ¯ x ¯ and p¯ · a < p¯ · e(t), which implies that ¯ such that a R(t, 1/(µ[C(t, p)]) ¯ x (s)dµ(s)) ξ C(t,p) ¯ pn · a < pn · e(t) for n > n1 for some n1 . Since (p, x) → 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s) is continuous on {p ∈ )|p · e(t) > 0} × L1 (µ, K) by Corollary 1, (A.3) implies that a R(t, 1/(µ[C(t, pn )]) C(t,pn ) xn (s)dµ(s))ξn for n > n2 for some n2 . These imply that for large enough n, we obtain {a ∈ Rl+ |aR(t, 1/(µ[C(t, pn )]) C(t,pn ) xn (s)dµ(s))ξn } ∩ ˜ pn ) = ∅, pn · e(t) > 0, but this is a contradiction. B(t, Final remark. As we mentioned in the introduction, one of the major advantages in considering atomless economies is the fact that it allows us to drop the convexity assumption on the preferences by the use of the Lyapunov convexity theorem, as demonstrated by Aumann (1966), Schmeidler (1969), Greenberg et al. (1979), and other authors. The dependence of the preferences on assignments x makes this known approach extremely difficult, hence any attempt to drop the convexity assumption on the preferences in our framework seems to be very challenging. The commodity space considered in this paper is finite dimensional. It may be possible to extend our results to cover infinite dimensional commodity spaces with the property that order intervals can be made compact with an appropriately chosen topology. The author is indebt to the anonymous referee of JME for carefully checking the original draft and providing helpful comments and suggestions. Needless to say, the author is solely responsible for any remaining errors.
References Aumann, R.J., 1966. Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34, 1–17. Balder, E., 1998. Incompatibility of Usual Conditions for Equilibrium Existence in Continuum Economies Without Ordered Preferences (Revision). University of Utrecht, Mimeo. Berge, C., 1963. Topological Spaces. Macmillan, New York. Castaing, C., Valadier, M., 1977. Complex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 480. Springer-Verlag, New York. Duesenberry, J.S., 1949. Income, Savings, and the Theory of Consumer Behavior. Cambridge, Massachusetts.
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Dunford, N., Schwartz, J.T., 1958. Linear Operator, Part II. Interscience, New York. Greenberg, J., Shitovitz, B., Wieczorek, A., 1979. Existence of equilibria in atomless production economies with price dependent preferences. Journal of Mathematical Economics 6, 31–41. Hildenbrand, W., 1974. Core and Equilibria of a Large Economy. Princeton University Press, New York. Johnson, H.G., 1952. The effects of income-redistribution on aggregate consumption with interdependence of consumer’s preferences. Economica XIX, 131–147. Khan, M.A., Vohra, R., 1984. Equilibrium in abstract economies without ordered preferences and with a measure space of agents. Journal of Mathematical Economics 13, 133–142. Kolmogorov, A., Fomin, S., 1970. Introductory Real Analysis. Dover, New York. Leibenstein, H., 1950. Bandwagon, Snob, and Veblen effects in the theory of consumers’ demand. Quarterly Journal of Economics 64, 183–207. Noguchi, M., 1997. Economies with a continuum of consumers, a continuum of suppliers and an infinite dimensional commodity space. Journal of Mathematical Economics 27, 1–21. Pollak, R., 1976. Interdependent preferences. American Economic Review 66, 309–320. Pollak, R., 1977. Price dependent preferences. American Economic Review 67, 64–75. Pollak, R., 1978. Endogenous tastes in demand and welfare analysis. American Economic Review, Papers and Proceedings 68, 374–379. Schmeidler, D., 1969. Competitive equilibria in markets with a continuum of traders and incomplete preferences. Econometrica 37, 578–585. Yannelis, N.C., 1991. Integration of Banach-valued correspondences. In: Ali Khan, M., Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces. Springer-Verlag, pp. 2–35.
Further reading Bourbaki, N., 1989. General Topology. Springer-Verlag, New York.
Interdependent preferences with a continuum of agents Mitsunori Noguchi∗ Department of Economics, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan Received 18 January 2002; received in revised form 31 October 2002; accepted 8 June 2004 Available online 15 December 2004
Abstract We prove the existence of an equilibrium in exchange economies in which agents t form a finite measure space (T, T, µ), and the agents’ preferences are interdependent. We introduce the notion of reference coalition C(t, p) for each t and for each price system p, which is simply the coalition of the agents who belong to a certain income range specified by t when p prevails in the market. The interdependent preferences of each agent t operate through the so-called “reference consumption vector η”, which represents the aggregate effect on t’s preferences, resulting from varying consumption choices made by other agents. We choose for η the consumption vectors of the form 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s), where x is an assignment of commodity bundles and where the measure 1/(µ[C(t, p)])µ represents the relative frequencies of t coming into contact with each member of C(t, p). In view of the law of large numbers, the above integral can be interpreted as the “consumption trend” of the agents in C(t, p). © 2004 Elsevier B.V. All rights reserved. JEL classification: D51 Keywords: Interdependent preferences; Measure space of agents; Equilibrium
1. Introduction In this paper, we prove the existence of an equilibrium in exchange economies E in which participating agents t form a finite measure space (T, T, µ) and their preferences are ∗
Tel.: +81 5613 88514; fax: +81 5283 34767. E-mail address: [email protected] (M. Noguchi).
0304-4068/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2004.06.003
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interdependent. Each t ∈ T is characterized by his initial endowment e(t), interdependent preference relation, and what we call the “reference coalition,” which is simply the coalition consisting of the agents who belong to a certain income range specified by t. The reference coalition may vary among different agents, and even for the same agent it may still vary, depending on what price system p prevails, since the defining income range varies according to p. We denote the reference coalition of agent t at price system p by C(t, p). By “interdependent preference relations,” we mean preference relations in the usual sense but with an additional feature of being allowed to depend on other agents’ consumption. In our formulation, the interdependent preferences of each t operate through a single consumption vector η, called the “reference consumption vector,” which is a representative consumption vector summarizing the various influences on t’s preferences, resulting from the varying consumption choices made by other agents. We denote by R(t, η) the interdependent preference relation of agent t under the influence of reference consumption vector η. It would be unrealistic to assume that each agent can directly observe or get accurate information about the consumption choices made by all the agents in his reference coalition. It seems more natural to assume that each participating agent t comes into contact, at random, with finitely many agents s in C(t, p), and then t modifies his preferences according to the average consumption behavior of these agents s. In view of the law of large numbers, such average behavior would tend to the expectation of x over C(t, p), as the frequency of the “random contact” of t increases, where x denotes the assignment of commodity bundles to each agent. Therefore, it would be reasonable to assume that η equals the expectation of x over C(t, p), which would appear to agent t as the “consumption trend” of the agents in C(t, p) . Before we can proceed this way, however, we must specify a probability measure on the “events” in C(t, p) so that we can sensibly discuss the “random contact” of agent t to the agents in C(t, p). For an arbitrary coalition A, the number µ(A) can be interpreted as the fraction of the totality of agents contained in A. Thus the relative frequency for a given agent t to encounter a certain agent in C(t, p) must be compatible with µ, and this account gives a rationale for the use of 1/(µ[C(t, p)])µ for the required probability measure on C(t, p), where µ[C(t, p)] > 0 is warranted for the agents t with positive income. Consequently, the interdependent preference relations take the form R(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s)) for all agents t with positive income. There may be an issue of whether agent t himself should be included in his own reference coalition or not. However, as apparent from our formulation of interdependence, if (T, T, µ) is atomless, it does not matter one way or the other since each individual agent is negligible. Our notion of interdependence is simultaneous in the sense that all individual consumption choices are compatible with the reference consumption vectors η which are determined by the individual choices themselves, and hence, there is no reason to exclude agents from their own reference coalitions even if some of the agents have positive measure. In the above setting, we prove the existence of a competitive equilibrium in E in which the commodity space is the l-dimensional Euclidean space Rl , the consumption possibility sets are all identical and equal to the positive orthant of Rl , the preference relations satisfy a certain convexity condition, and most distinctively, the incomes of the agents sufficiently vary. We list the possible difficulties one might encounter in dealing with interdependent preferences. The standard method in the literature often constructs a sequence of sub-economies
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Ek from the original economy E, usually by restricting the consumption possibility sets to compact subsets whose sizes relate to the indices k. We then manage to obtain a set of equilibria (pk , xk ), one in each sub-economy Ek , utilizing the compactness of the consumption possibility sets. In the next step, we hope the sequence (pk , xk ) will converge to some limit (p, ¯ x¯ ), which hopefully relates to an honest equilibrium in the original economy E. If preferences depend on both price systems p and assignments x, as in the case at hand, preference relations take the general form R(t, p, x), and somewhere along the way, we frequently end up in a situation in which R(t, p, ¯ x¯ )¯x(t) needs to be approximated by R(t, pk , xk )xk (t). Here are some of the possible difficulties: 1. Although {pk } can be assumed to lie in a price simplex and hence admits a convergent subsequence, it may be difficult to find a convergent subsequence of {xk }. If preferences were independent of x, we would not need to face this problem, and we could make use of Fatou’s lemma to obtain a limit x¯ with the property that x¯ (t) ∈ Ls{xk (t)}. This is all we need to approximate R(t)¯x(t) by R(t)xk (t). This is exactly the case handled by Schmeidler (1969). A similar argument works even with price dependent preferences, as shown by Greenberg et al. (1979). The most significant advantage of the preferences without dependence on x would, of course, lie in the fact that the Lyapunov Convexity Theorem becomes applicable with a non-atomic measure space of agents, and consequently the convexity assumption on the preference relations can be disposed of. One possibility to resolve this difficulty is to factor R(t, p, x) as R(t, ξ), ξ = f (t, p, x) – our formulation is a special case of this – and manage to make {f (t, pk , xk )} converge to some limit, say, η¯ . 2. Even if we obtain a limit η¯ by the above method, there is, in general, no guarantee that η¯ coincides with f (t, p, ¯ x¯ ) where x¯ is the limit obtained by the use of Fatou’s lemma. We overcome the first difficulty by showing that the sequence {f (t, pk , xk )} belongs to an order interval, which is surely compact in Rl , and the second difficulty by following Schmeidler’ s limiting argument: His argument does not require the limit to be an equilibrium in E, but by showing that the limit price system is strictly positive, it deduces that the elements in the sequence {(pk , xk )} eventually become equilibria in E. The amount of literature on interdependent preferences is enormous. Duesenberry’s classical work (Duesenberry, 1949) concerning the so-called consumption function problem attempted to explain the statistical discrepancy between the Kuznets data (the data on aggregate savings and income in the period 1869–1929), and budget study data for 1935– 1936 and 1941–1942, by objecting to one of the fundamental assumptions of aggregate demand theory at the time: Every individual’s consumption behavior is independent of that of every other individual. Duesenberry, indeed, proposed the interdependent utility function of the i-th individual with consumption expenditure Ci to be of the form Ui = Ui [Ci / αij Cj ] where Cj is the consumption of the j-th individual and αij the weight applied by the ith individual to the expenditure of the jth (Duesenberry, p.32, 1949). Observe that the “support” of αij can be thought of as, in our terminology, a reference coalition of the i th individual, and αij precisely corresponds to the probability distribution of what we call the random contact of the i-th individual to the j-th individual in the support of αij . There are some notable differences between Duesenberry’s construction and ours: firstly,
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we consider each commodity separately, and secondly, it is inevitable for us to take the effect of price changes on reference coalitions into consideration in our microeconomic analysis of consumer behavior. Johnson (1952) investigated the macroeconomic aspects of income redistribution with interdependent preferences: he introduced “interdependence effect” as well as “income effect” in his analysis of how income transfers among individuals affect the aggregate propensity to consume. Pollak (1976, 1977, 1978) published a series of papers concerning endogenous taste, interdependent preferences, price dependent preferences, etc. His ideas of simultaneous (long-run) interdependence and steady-state (long-run) demand function are somewhat related to our equilibrium concept. Leibenstein (1950) incorporated interdependent preferences into his microeconomic analysis of consumer’s demand theory: He divided the external effects on utility into the bandwagon effect, the snob effect, and the Veblen effect. In his analysis of the bandwagon effect, each consumer’s demand function d, given a price system p, depends on the total demand D, and hence D must be obtained as a fixed point of the total demand function with p fixed. This line of argument is essential in our fixed-point theorem approach. Finally, it might seem possible to treat, at least with no price dependence, more general non-ordered interdependent preferences with a measure space of agents by the abstract economy setting. However, as Balder (1998) pointed out, this approach involves extremely delicate mathematical complications.
2. The model and the theorem In what follows (T, T, µ) denotes a complete, finite, separable measure space, i.e., µ is a real-valued non-negative, countably additive separable measure defined on a complete σ-algebra T of subsets of a set T such that µ(T ) < ∞. Rl denotes the l-dimensional Euclidean space. The i-th coordinate of the point ξ in Rl is denoted by ξ i (i = 1, . . . , l). For any two elements ξ, η in Rl , ξ ≥ η means ξ i ≥ ηi i i for every i = 1, . . . , l, ξ > η means lξ ≥ iη iand ξ = η, and ξ η means ξ > η for every i = 1, . . . , l. The scalar product i=1 ξ η of ξ and η is denoted by ξ · η and the norm √ ξ · ξ of ξ by |ξ|. Rl+ denotes the positive orthant {ξ ∈ Rl |ξ ≥ 0} of Rl , and Rl++ the subset {ξ ∈ Rl |ξ 0} of Rl . 0 denotes the zero element of Rl , 1 the vector (1, . . . , 1) in Rl , and ei the i-th coordinate vector in Rl . For any two element ξ, η in Rl , [ξ, η] denotes the order interval (ξ + Rl+ ) ∩ (η − Rl+ ). For any subset A of Rl , coA denotes its convex hull, ¯ its closure. For any two subsets A, B of Rl , A \ B denotes setAc its complement, and A theoretic subtraction. B(Rl+ ) denotes the Borel subsets of Rl+ , and T ⊗ B(Rl+ ) the obvious product. For a µ-integrable function f, T f (t)dµ(t) denotes its integral. For brevity we often omit the domain T, the variable t, and the measure dµ, and simply write f unless confusion is expected to result. L1 (µ, Rl ) denotes the space of equivalence classes of Rl -valued µ l integrable functions f : T → R normed by f = |f |. L1 (µ, Rl )∗ denotes the dual of L1 (µ, Rl ), and ·, · denotes the pairing of these spaces. For any correspondence φ : T → Rl , define L1 (µ, φ) ≡ {f ∈ L1 (µ, Rl )|f (t) ∈ φ(t) µ − a.e.}. Gφ denotes the graph of φ.
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All agents are assumed to have the same consumption set Rl+ , and an element ξ in is a consumption vector or commodity bundle. An element p of Rl+ \ {0} is a price system. (T, T, µ) is a measure space of agents. A µ-integrable function x : T → Rl+ is an assignment. An assignment e : T → Rl+ is an initial assignment where e(t) represents the initial endowment of agent t. B(t, p) = {ξ ∈ Rl+ |p · ξ ≤ p · e(t)} is the budget set of agent t at price system p. An assignment x such that x = e is an allocation. In our model, preferences of agent t are influenced by the consumption behavior of other agents, and we assume that the interdependent preferences of each t operate through a single consumption vector η ∈ Rl+ , called the reference consumption vector which is a representative consumption vector summarizing the various influences on t’s preferences, resulting from the varying consumption choices made by other agents. Define I ≡{open sets U in R such that U¯ \ U is countable}. Note that for some open subset U in R, U¯ \ U may not be countable: for example, take U = the complement of the Cantor ternary set. We consider functions I : Rl+ × Rl+ × Rl+ \ {0} → I having the following properties: Rl+
1. for any positive number λ, I(e, ", λp) = λI(e, ", p) holds; 2. I(e, ", p) ⊂ (0, ∞); 3. for any w ∈ Rl+ satisfying p · w ∈ I(e, ", p) and for any convergent sequence pn → p in Rl+ \ {0}, there is a natural number N such that n ≥ N implies pn · w ∈ I(e, ", pn ); 4. for any w ∈ Rl+ and for any convergent sequence pn → p in Rl+ \ {0}, pn · w ∈ I(e, ", pn ) implies p · w ∈ I(e, ", p); 5. for any w ∈ Rl+ satisfying p · w ∈ I(e, ", p) and for any convergent sequence (en , "n ) → (e, ") in Rl+ × Rl+ , there is a natural number N such that n ≥ N implies p · w ∈ I(en , "n , p); 6. for any w ∈ Rl+ and for any convergent sequence (en , "n ) → (e, ") in Rl+ × Rl+ , p · w ∈ I(en , "n , p) implies p · w ∈ I(e, ", p). Intuitively speaking, I(e, ", p) represents an income range in the income-scale, relative to income p · e and with magnitude p · ". Example 1. We list some examples of such I which are of importance in practical applications. In what follows, we adhere to the convention that (0, 0) ≡ ∅. 1. I(e, ", p) = (max{0, p · e − p · "}, p · e + p · "), 2. I(e, ", p) = (p · e + p · ", ∞), 3. I(e, ", p) = (0, max{0, p · e − p · "}). Observing that max{0, p · e − p · "} is a continuous function of (e, ", p), it is a simple matter to justify that the above examples all possess the required properties. Let δ : T → Rl+ be a fixed measurable function. Define C(t, p) ≡ {s ∈ T |p · e(s) ∈ I(e(t), δ(t), p)}. We require that µ[C(t, p)] > 0 for all (t, p) ∈ T × Rl+ \ {0} satisfying
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p · e(t) > 0. C(t, p) is said to be the reference coalition of agent t at price system p. Observe that: 1. for any positive number λ, C(t, λp) = C(t, p) holds for each (t, p); 2. C(t, p) ∈ T holds for each (t, p). Remark 1. As a typical example, we can take I(e, ", p) = (max{0, p · e − p · "}, p · e + p · "). With this choice of I and with appropriately chosen e : T → Rl+ and δ, C(t, p) becomes non-empty for all (t, p) ∈ T × Rl+ \ {0} satisfying p · e(t) > 0 and can be expressed as C(t, p) = {s ∈ T | max{0, p · e(t) − p · δ(t)} < p · e(s) < p · e(t) + p · δ(t)}. In this example, C(t, p) consists of those agents whose income in the income-scale is positive and fall within a certain range with respect to the income of agent t. The value of p · δ(t) simply determines the size of such range. Also note that if one wishes C(t, p) to consist of those agents whose income in the income-scale falls between 70% and 130% of the income of agent t, simply take δ(t) = 0.3e(t). Note that this example is compatible with Duesenberry’s observation about interdependence of preference systems. Duesenberry (1949, p.30) indeed writes: “In general, however, it appears that income is one of the principal status criteria.” But since social status rankings in our society form a continuous series rather than a set of clearly defined group rankings, every individual must associate with some people of higher or lower status than his own. We can also define C(t, p) in terms of I(e, ", p) = (p · e + p · ", ∞) or I(e, ", p) = (0, max{0, p · e − p · "}). These may be viewed as the reference coalition of an agent t in the extreme case of “Emulation” or “Competition” in the sense of Johnson (1952). R(t, η) ⊂ Rl+ × Rl+ is the preference relation of agent t (under the influence of reference consumption vector η). Definition 1. An exchange economy E is a pair [(T, T, µ), ((C(t, p))p∈Rl
+ \{0}
, (R(t, η))η∈Rl , e(t))t∈T ]. +
Definition 2. An equilibrium for the exchange economy E is a price system p¯ and an allocation x¯ such that for almost all t in T, if p¯ · e(t) > 0, then x¯ (t) is maximal with respect to 1 R(t, x¯ (s)dµ(s)) µ[C(t, p)] ¯ C(t,p) ¯ in the budget set B(t, p), ¯ and if p¯ · e(t) = 0, then x¯ (t) = 0.
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We can now state our assumptions and our main theorem. (A.1) Transitivity: for each (t, η) ∈ T × Rl+ , aR(t, η)b and bR(t, η)c implies aR(t, η)c; (A.2) Irreflexivity: for each (t, η) ∈ T × Rl+ , aR(t, η)a does not hold; (A.3) Continuity: for each (t, η) ∈ T × Rl+ and for each b ∈ Rl+ , the set {a|aR(t, η)b} is open relative to Rl+ ; for each t ∈ T and for each a ∈ Rl+ , the set {(b, η)|aR(t, η)b} is open relative to Rl+ × Rl+ ; (A.4) Convexity: for each (t, η) ∈ T × Rl+ and for each a ∈ Rl+ , the set {b|aR(t, η)b}c is convex; (A.5) Desirability: for each (t, η) ∈ T × Rl+ , a > b implies aR(t, η)b; (A.6) Measurability: for each a ∈ Rl+ and for each b ∈ Rl+ , {(t, η)|aR(t, η)b} ∈ T ⊗ B(Rl+ ); (A.7) Resource Availability: e 0; (A.8) Income Diversity: for each (t, p) ∈ T × Rl+ \ {0} such that p · e(t) > 0, c ∈ I(e(t), δ(t), p) \ I(e(t), δ(t), p) implies µ[{s ∈ T |p · e(s) = c}] = 0; (A.8) Income Diversity (stronger version): for each p ∈ Rl+ \ {0}, µ[{s ∈ T |p · e(s) = const}] = 0. Remark 2. The assumptions listed above are standard in the literature, except for (A.8) and (A.8) . (A.8) asserts in words that if an income c is realized as a “borderline income” of the agents in the reference coalition of some agent t with positive income at some price system p, the set of agents with income equal to c forms a null set. (A.8) is certainly stronger than (A.8) and implies (A.8). Note that (A.8) clearly implies µ[{s ∈ T |e(s) = const}] = 0. However, the converse may not hold: to see this, let (T, T, µ) be the unit interval [0, 1] with Lebesgue measure, l = 2, and e(t) = te2 , t ∈ [0, 1]. Then µ[{s ∈ T |e(s) = const}] = 0 but µ[{s ∈ T |p · e(s) = 0}] = 1 for p = e1 ∈ R2+ \ {0}. Observe that e(t) = t(e1 + e2 ), t ∈ [0, 1] is an example satisfying (A.8) . Although (A.8) appears to be a good sufficient condition and more concise than (A.8), it exhibits many shortcomings. For example, we see from the above argument that (A.8) significantly restrict the cardinality of T: if µ(T ) > 0, T must be more than countable. A more serious shortcoming is that (A.8) turns (A.7) vacuous: (A.8) implies µ[{s ∈ T |ei (s) = 0}] = 0 for i = 1, . . . , l, and hence e(t) 0 µ − a.e. This is quite unsatisfactory because proving existence of an equilibrium with (A.7) instead of with the stronger condition e(t) 0 µ − a.e. is one of the main accomplishments in Schmeidler (1969). Remark 3. Note that the reference consumption vectors η of the form 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s) are defined only for t at price system p such that µ[C(t, p)] > 0, which is guaranteed, by the definition of C(t, p), for t at price system p such that p · e(t) > 0. For t at price system p such that p · e(t) = 0, we simply let any element η ∈ Rl+ be the reference consumption vector. In what follows, this does not cause any problems since we carefully avoid discussing maximality with respect to R(t, η) for t at price system p such that p · e(t) = 0. In our main theorem, the equilibrium price systems p¯ turn out to be strictly positive, which reduces the budget set of agent t with p · e(t) = 0 to {0}, and as a consequence, the equilibrium consumption vector x(t) = 0 will be maximal
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in {0} with respect to R(t, η) for “any” η. We next present a concrete example that satisfies (A.1)–(A.8). Example 2. Let (T, T, µ) be the unit interval [0, 1] with Lebesgue measure, and l = 2. Let u : T × R2+ × R2+ → R be a function satisfying the following conditions: 1. 2. 3. 4.
u(t, a, η) is jointly continuous in (a, η); u(t, a, η) is quasi-concave in a; u(t, a, η) is strictly monotonic in a; u(t, a, η) is jointly measurable in (t, η).
We now define aR(t, η)b ⇐⇒ u(t, a, η) > u(t, b, η). Then (A.1)–(A.6) are easily seen to be satisfied. Let te1 if 0 ≤ t < 21 e(t) = te2 if 21 ≤ t ≤ 1. Then e = 18 e1 + 38 e2 0, and hence (A.7) is satisfied even though it is not true that e(t) 0 µ − a.e. Let I(e, ", p) = (max{0, p · e − p · "}, p · e + p · ") and δ(t) = 21 e(t). Then I(e(t), δ(t), p) = ( 21 p · e(t), 23 p · e(t)). Thus I(e(t), δ(t), p) \ I(e(t), δ(t), p) = { 21 p · e(t), 23 p · e(t)}. Note that p · e(t) =
tp1
if 0 ≤ t <
tp2
if 21 ≤ t ≤ 1,
1 2
We must show that µ[C(t, p)] > 0 for all (t, p) ∈ T × Rl+ \ {0} satisfying p · e(t) > 0. Observe that p · e(t) > 0 ⇐⇒ p1 > 0, t ∈ (0, 21 ) or p2 > 0, t ∈ [ 21 , 1]. Assume the first case. Then I(e(t), δ(t), p) = ( 21 tp1 , 23 tp1 ), and s ∈ 0, 21 | 21 tp1 < p · e(s) < 23 tp1 = s ∈ 0, 21 | 21 tp1 < sp1 < 23 tp1 = s ∈ 0, 21 | 21 t < s < 23 t ⊃ 21 t, t . Thus µ[C(t, p)] ≥ 21 t > 0. We next assume the second case. Then I(e(t), δ(t), p) = ( 21 tp2 , 23 tp2 ), and
s ∈ 21 , 1 | 21 tp2 < p · e(s) < 23 tp2 = s ∈ 21 , 1 | 21 tp2 < sp2 < 23 tp2 = s ∈ [ 21 , 1]| 21 t < s < 23 t ⊃ 21 , 43 . Thus µ[C(t, p)] ≥ 41 . These establish our claim. We next show that for each p > 0, µ[{s ∈ T |p · e(s) = c}] = 0 holds for each c > 0. To this end, note that for each p > 0, {s ∈ T |p ·
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e(s) = c} (c > 0) consists of at most two elements. Thus (A.8) is satisfied. Observe that this example does not satisfy (A.8) since for p = e1 , µ[{s ∈ T |p · e(s) = 0}] = µ[{s ∈ T |s = 0 or 21 ≤ s ≤ 1}] = 21 > 0. Furthermore, for u(t, a, η) =
a · (η + 1)
if 0 ≤ t <
1·a+1·η
if 21 ≤ t ≤ 1,
1 2
we can show that p = ( 21 , 21 ) and (0, t) x(t) = t 2 , t 3 3
if 0 ≤ t <
1 2
if 21 ≤ t ≤ 1
constitute an equilibrium. To see this, note that
= C t, 21 , 21
∅ t
, 23 t 2
t ,1 2
if t = 0 if 0 < t <
2 3
if 23 ≤ t ≤ 1,
and (0, t) if 0 < t < 13 2 9t − 1 15t 2 + 1 1 , if 13 ≤ t < 23 x(s)dµ(s) = 24t 24t 1 1 C t, 2 , 2 µ C t, 21 , 21 1 3 − t2 , if 23 ≤ t ≤ 1. 8 − 4t 8 − 4t For t ∈ [ 21 , 1], any vector a satisfying the budget equality is maximal with respect to u(t, ·, η) for any η, and hence a = ( 3t , 23 t) is maximal in B(t, ( 21 , 21 )) with respect to u(t, ·, 1/(µ[C(t, ( 21 , 21 ))]) C(t,( 1 , 1 )) x(s)dµ(s)). For t ∈ (0, 13 ), we 2 2 have u(t, a, 1/(µ[C(t, ( 21 , 21 ))]) C(t,( 1 , 1 )) x(s)dµ(s)) = a · (1, 1 + t), which is maximal 2 2
at a = (0, t) in B(t, ( 21 , 21 )), and for t ∈ [ 13 , 21 ), we have u(t, a, 1/(µ[C(t, ( 21 , 21 ))]) 2 2 C(t,( 21 , 21 )) x(s)dµ(s)) = a · ((9t − 1)/24t + 1, (15t + 1)/24t + 1), which is maximal at a = (0, t) in B(t, ( 21 , 21 )). Finally, we compute x = ( 18 , 38 ) = e as required. Main Theorem. If assumptions (A.1)–(A.8) are satisfied, an exchange economy E has an equilibrium (p, ¯ x¯ ).
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3. An auxiliary theorem We proceed as in Schmeidler (1969, p.579) with a slight alteration in notation. For a ˜ p) = B(t, p) ∩ {ξ ∈ Rl+ |ξ ≤ k[1 · e(t)]1}. positive real number k > 1, define B(t, Definition 3. A k-bounded partial equilibrium for an exchange economy E is a price system p and an allocation x such that for almost all t ∈ T , if p · e(t) > 0, then x(t) is ˜ p), and if p · e(t) = 0, maximal with respect to R(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s)) in B(t, ˜ p). then x(t) ∈ B(t, Auxiliary Theorem. For k > 1, if assumptions (A.1)–(A.8) are satisfied, an exchange economy E has a k-bounded partial equilibrium (p, x) with p · x(t) = p · e(t) holding outside the same exceptional set as appeared in Definition 3. Proof of auxiliary theorem. Let ) = {p ∈ Rl+ |1 · p = 1} and define a correspondence K : T → Rl+ by K(t) = {ξ ∈ Rl+ |ξ ≤ k[1 · e(t)]1}. For (t, p, x) ∈ T × ) × L1 (µ, K), let H(t, p, x) = 1 l ˜ ˜ {b ∈ B(t, p)| a ∈ R+ |aR t, µ[C(t,p)] C(t,p) x(s)dµ(s) b ∩ B(t, p) = ∅} if p · e(t) > 0 ˜ B(t, p) if p · e(t) = 0. It is well known that L1 (µ, K) is weakly compact in L1 (µ, Rl ) [see, for example, Yannelis (1991, p. 7)] and that the weak topology on L1 (µ, K) is metrizable [see Dunford and Schwartz (1958, Theorem 3, p.434) and Kolmogorov and Fomin (1970, p. 381)]. From this point on, we adhere to the convention that L1 (µ, K) is endowed with the weak topology unless otherwise stated. Since K admits a measurable selection [x(t) = 0, for example], L1 (µ, K) is clearly non-empty and also a convex subset of L1 (µ, Rl ). We define a correspondence γ1 : ) × L1 (µ, K) → ) by γ1 (p, x) = ArgMax q · x − e |q ∈ ) and γ2 : ) × L1 (µ, K) :→ L1 (µ, K) by γ2 (p, x) = {y ∈ L1 (µ, K)|y(t) ∈ H(t, p, x) µ–a.e.}. We proceed with the following theorem whose proof is now standard and is sketched in the Appendix A. Theorem 1. γ1 × γ2 : ) × L1 (µ, K) → ) × L1 (µ, K) is a non-empty, convex-valued correspondence with a closed graph.
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We can now appeal to the Fan–Glicksberg fixed point theorem and obtain a fixed point (p, x) with the following properties: 1. p · ( x − e) ≥ q · ( x − e) for all q ∈ ), 2. x(t) ∈ H(t, p, x) µ − a.e. Denote the exceptional null set appearing above by Tk so that x(t) ∈ H(t, p, x) holds for all t ∈ T \ Tk . The remainder of the proof is almost verbatim to the arguments given in Schmeidler (1969) for establishing the similar claim. We only need to make minor modifications in Schmeidler (1969, p. 580) so that his arguments extend to cover our case as well. Recall from the definition of H(t, p, x) that statement 2 above can be rewritten as ˜ p), and x(t) is maximal in B(t, ˜ p) x(t) ∈ B(t, 1 with respect to R(t, x(s)dµ(s)) µ[C(t, p)] C(t,p) 2. if t ∈ T \ Tk and p · e(t) > 0 ˜ p) x(t) ∈ B(t, if t ∈ T \ Tk and p · e(t) = 0. We must show that x is an allocation, i.e., x = e. We show this by constructing an appropriate modification y of x. To this end, let t ∈ T \ Tk . Since x(t) ∈ H(t, p, x) ⊂ ˜ p) ⊂ B(t, p), we have B(t, p · x(t) ≤ p · e(t). We claim that p · x(t) = p · e(t). If p · e(t) = 0, the claim clearly holds. Assume p · e(t) > 0. If x(t) = k[1 · e(t)]1, it follows that x(t) e(t) since k > 1. This would imply p · x(t) > p · e(t), which is a contradiction. Thus, x(t) < k[1 · e(t)]1. We can, then, find a coordinate vector ei and a real number " > 0, such that x(t) + "ei < k[1 · e(t)]1. If p · e(t) > p · x(t) ≥ 0, there exists " , 0 < " < ", such that p · (x(t) + " ei ) = p · x(t) + ˜ p), and by " pi ≤ p · x(t) + " < p · e(t) since pi ≤ 1. Consequently, x(t) + " ei ∈ B(t, (A.5), x(t) + " ei becomes more desirable than x(t), which contradicts the maximality of x(t). We have established that for each t ∈ T \ Tk , p · x(t) = p · e(t). Define b≡
x−
e.
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From the above equality, we obtain p · b = 0. Upon substituting this in statement 1 above, we have 0 ≥ q · b for all q ∈ ). By setting q = ei , i = 1, . . . , l, we get bi ≤ 0 for all i. We define J ≡ {i|bi = 0} and J ≡ {i|bi < 0}. If J = ∅, we get x = e, and we are done. Note that for i ∈ J , we have pi = 0. For each i ∈ J , define Si ≡ {t ∈ T |xi (t) < ei (t)}. Observe that for each i ∈ J , we have µ(Si ) > 0, since µ(Si ) = 0 would entail xi (t) ≥ ei (t) µ − a.e, and hence bi = 0. We claim that for each i ∈ J , |bi | ≤ Si ei − xi , and Si ei − xi > 0. Since bi < 0, we compute i
|b | = −b
i
We clearly have i ∈ J so that
= Si
i
Si
i
e −x +
T \Si
i
e −x
i
≤
Si
ei − x i .
ei − xi > 0, hence the claim holds. We set ci ≡
Si
ei − xi for each
ci > 0, |bi | ≤ ci for each i ∈ J . We next define a modification y of x in the following manner: yi (t) ≡ xi (t) for all t ∈ T if i ∈ J, and i
y (t) ≡
xi (t) + xi (t)
|bi | i (e (t) − xi (t)) ci
t ∈ Si t ∈ T \ Si
if i ∈ J . Recalling that pi = 0 for each i ∈ J , we obtain p · y(t) =
pi yi (t)
i∈J
=
pi xi (t)
= p · x(t).
i∈J
Combining this with the previous equality, we deduce that for each t ∈ T \ Tk , p · y(t) = p · e(t). This shows, in particular, that for each t ∈ T \ Tk , y(t) ∈ B(t, p). ˜ p). It suffices to show this for t ∈ Si , i ∈ J . Recall that for i ∈ J , We must show y(t) ∈ B(t, i i we have |b | ≤ c . Hence, from the definition of yi (t) on Si , we obtain yi (t) ≤ ei (t). Since
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k > 1, we compute yi (t) ≤ ei (t) ≤ k[1 · e(t)] for each i ∈ J such that t ∈ Si . For the rest of i, we have yi (t) = xi (t). Therefore, y(t) ≤ k[1 · e(t)]1, and as a consequence, ˜ p) y(t) ∈ B(t, for each t ∈ T \ Tk . In particular, we have established that for each t ∈ T \ Tk , if p · e(t) = 0, ˜ p). then y(t) ∈ B(t, ˜ We need to show that for each t ∈ T \ Tk , if p · e(t) > 0, y(t) is maximal in B(t, p) with respect to R(t, 1/(µ[C(t, p)]) C(t,p) y(s)dµ(s)). To this end, note that if i ∈ J , we have |bi |/ci (ei (t) − xi (t)) > 0 on Si . Thus, if t ∈ T \ Tk satisfies p · e(t) > 0, t ∈ i∈J Si , then by (A.5), we obtain y(t)R(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s))x(t),
which contradicts the maximality of x(t). Therefore, we have
Si ⊂ {s ∈ T |p · e(s) = 0} ∪ Tk .
i∈J
Since {s ∈ T |x(s) = y(s)} = i∈J Si , x and y agree on {s ∈ T |p · e(s) > 0} ∩ (T \ Tk ). Note also that by the definition of C(t, p), p · e(s) > 0 for each s ∈ C(t, p). Thus, x and y agree on C(t, p) ∩ (T \ Tk ), and hence C(t,p) x(s)dµ(s) = C(t,p) y(s)dµ(s). Finally, we show that y = e. For i ∈ J , we compute |bi | i |bi | i i i i i i y = x + i (e − x ) + x = x + i c = xi + |bi | c c Si T \Si i i i i i = ei . = x − x − e = x −b Therefore, y is an allocation. This completes our proof. We now prove our main theorem. Proof of Main Theorem. For each integer k > 1 there is a k-bounded partial equilibrium (pk , xk ). Let Tk be the exceptional null set for each k. Without any loss of generality, we can assume that {pk } converges to some element p¯ ∈ ). Then by the Fatou lemma in several dimension [see Hildenbrand (1974, p. 69)], there is a µ-integrable function x¯ such that x¯ (t) ∈ Ls(xk (t)) µ − a.e. As in Schmeidler (1969, p. 583), µ[{t ∈ T |p¯ · e(t) > 0}] > 0, and hence we can safely choose t ∈ T such that p¯ · e(t) > 0, t ∈ ∪k>1 Tk , and x¯ (t) ∈ Ls(xk (t)). In the proof of Corollary 1 in Appendix A, we see that for pk → p, ¯ µ[C(t, pk )] → that the order inµ[C(t, p)] ¯ >0, and hence 1/(µ[C(t, pk )]) < M for large k. Observe terval [0, M e] is non-empty compact and that 0 ≤ 1/(µ[C(t, pk )]) C(t,pk ) xk (s)dµ(s) < M C(t,pk ) xk (s)dµ(s) ≤ M xk = M e. Passing to a subsequence if necessary, we may assume that pk → p, ¯ 1/(µ[C(t, pk )]) C(t,pk ) xk (s)dµ(s) → η¯ , and xk (t) → x¯ (t). We pro-
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ceed as in Schmeidler (1969, p. 583) for establishing p¯ 0. We have, by the budget equality for each k, p¯ · x¯ (t) = lim pk · xk (t) = lim pk · e(t) = p¯ · e(t) > 0. k
k
Consequently, there is an index i such that p¯ i > 0 and x¯ i (t) > 0. Suppose p¯ j = 0 for some index j, and let z = x¯ (t) + ej . Then (A.5) implies zR(t, η¯ )¯x(t). Note that p¯ · z = p¯ · x¯ (t) > 0. Thus there is " > 0 such that z = z − "ei ∈ Rl+ and z R(t, η¯ )¯x(t). Since p¯ · z = p¯ · z − "p¯ i < p¯ · e(t), there is k1 such that k > k1 implies pk · z < pk · e(t). Note that there is k2 such that k > k2 implies pk · e(t) > 0, and there is k3 such that k > k3 implies z ≤ k[1 · e(t)]1. By (A.3) we can also find k4 such that k > k4 implies z R(t, 1/(µ[C(t, pk )]) C(t,pk ) xk (s)dµ(s))xk (t). We can now choose k > max{k1 , k2 , k3 , k4 } ˜ pk ), pk · e(t) > 0, but this contradicts to the maximality of xk (t) in such that z ∈ B(t, ˜ pk ) with respect to R(t, 1/(µ[C(t, pk )]) B(t, ¯ 0. C(t,pk ) xk (s)dµ(s)). This establishes p For the rest of our proof, we proceed as in Schmeidler (1969, p. 584). Since p¯ 0, we can choose δ > 0 and k0 such that for each k > k0 , pik > δ holds for all i, i = 1, . . . , l. Assume k > k0 . Then, for each z ∈ B(t, pk ), we have δzi ≤ pik zi ≤ pk · z ≤ pk · e(t) ≤ 1 · e(t), which implies zi ≤ 1/δ[1 · e(t)]. Choose k ≥ max{1/δ, k0 + 1}. Then, for each z ∈ B(t, pk ), we have zi ≤ k[1 · e(t)], and hence z ≤ k[1 · e(t)]1. Thus, for such k, we obtain B(t, pk ) ⊂ {z ∈ Rl+ |z ≤ k[1 · e(t)]1}. We claim that (pk , xk ) is a desired competitive equilibrium. Let t ∈ T \ Tk . If pk · e(t) > ˜ pk ) with respect to R(t, 1/(µ[C(t, pk )]) 0, xk (t) is maximal in B(t, C(t,pk ) xk (s)dµ(s)). As ˜ we see above, we have B(t, pk ) = B(t, pk ). Assume pk · e(t) = 0. Since pik > δ > 0(i = 1, . . . , l), we obtain e(t) = 0. Thus, B(t, pk ) = {0}, and xk (t) = 0. Appendix A. Proof of Theorem 1 For each e ∈ Rl+ and for a convergent sequence pn → p in Rl+ \ {0}, define S = {z ∈ Rl+ |p · z ∈ I(e, ", p)} and Sn = {z ∈ Rl+ |pn · z ∈ I(e, ", pn )}. We prove the following lemmas: Lemma 1. S¯ = {z ∈ Rl+ |p · z ∈ I(e, ", p)} ¯ Then there is a sequence zi → y such that zi ∈ S, i.e., p · zi ∈ I(e, ", p). Proof. Let y ∈ S. Since taking the inner product with p is continuous, we obtain p · y ∈ I(e, ", p), or S¯ ⊂ {z ∈ Rl+ |p · z ∈ I(e, ", p)}. For proving the converse, assume p · z ∈ I(e, ", p). Let B 1 (z) be the open ball centered i
at z with radius 1i . Then, since taking the inner product with p is an open mapping, there is zi such that zi ∈ B 1 (z) and p · zi ∈ I(e, ", p), i.e., zi ∈ S. Since |zi − z| < 1/ i → 0, we i ¯ This completes the proof. have zi → z. Thus z ∈ S. ¯ Then τ(Sn Lemma 2. Let τ be a finite Borel measure on Rl+ such that τ(S) = τ(S). 0.
S) →
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Proof. We first show that τ(S \ Sn ∩ S) → 0. From statement 3 in the definition of ∞ I, it follows that z ∈ S implies z ∈ Sn for all n ≥ N, and hence z ∈ ∪∞ j=1 ∩n=j (Sn ∩ S) = lim inf n (Sn ∩ S). Thus we have S ⊂ lim inf n (Sn ∩ S). Since lim supn (Sn ∩ S) = ∞ ∩∞ j=1 ∪n=j (Sn ∩ S) ⊂ S, Fatou’s lemma yields τ(S) ≤ τ[lim inf (Sn ∩ S)] ≤ lim inf τ(Sn ∩ S) n
n
≤ lim sup τ(Sn ∩ S) ≤ τ[lim sup(Sn ∩ S)] ≤ τ(S), n
n
and hence limn→∞ τ(Sn ∩ S) = τ(S). It is now immediate that τ(S \ Sn ∩ S) = τ(S) − τ(Sn ∩ S) → 0. We next show that τ(Sn \ Sn ∩ S) → 0. To this end, note that S ⊂ lim inf n Sn clearly ¯ This can be seen as follows: let z ∈ ∩∞ ∪∞ Sn . holds. We claim that lim supn Sn ⊂ S. j=1 n=j Then we can choose n1 < n2 < n3 < . . . such that pni · z ∈ I(e, ", pni ) for all i. Note that from statement 4 in the definition of I, we obtain p · z ∈ I(e, ", p), which implies, by Lemma ¯ We can now appeal to Fatou’s lemma again and obtain 1, that z ∈ S. τ(S) ≤ τ[lim inf Sn ] ≤ lim inf τ(Sn ) n
n
¯ ≤ lim sup τ(Sn ) ≤ τ[lim sup(Sn )] ≤ τ(S). n
n
¯ we have limn→∞ τ(Sn ) = τ(S) = τ(S). ¯ From this, it is immediThus if τ(S) = τ(S), ate that τ(Sn \ Sn ∩ S) = τ(Sn ) − τ(Sn ∩ S) → τ(S) − τ(S) = 0. Since τ(Sn S) = τ(S \ Sn ∩ S) + τ(Sn \ Sn ∩ S), τ(Sn S) → 0 as desired. Proposition 1. For each t ∈ T , (p, x) → C(t,p) x(s)dµ(s) is continuous on {p ∈ )|p · e(t) > 0} × L1 (µ, K), where L1 (µ, K) is endowed with the weak topology. Proof. Let (pn , xn ) → (p, x) bean arbitrary convergent sequence in {p ∈ )|p · e(t) > 0} × L1 (µ, K). We need to show C(t,pn ) xn (s)dµ(s) → C(t,p) x(s)dµ(s). We have the following inequalities: xn (s)dµ(s) − x(s)dµ(s) C(t,pn )
C(t,p)
xn (s)dµ(s) − xn (s)dµ(s) C(t,pn ) C(t,p) + xn (s)dµ(s) − x(s)dµ(s) .
≤
C(t,p)
C(t,p)
Let χC(t,p) be the characteristic function of C(t, p). For the convergence of the second term on the right hand side of the above inequality, it suffices to show that each component converges, i.e., χC(t,p) (s)xi (s)dµ(s) − χC(t,p) (s)xi (s)dµ(s) → 0, n
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which amounts to showing χC(t,p) ei , xn → χC(t,p) ei , x, but this follows trivially from the fact that χC(t,p) ei ∈ L1 (µ, Rl )∗ and xn → x weakly. We next show that the first term on the right hand side of the above inequality converges √ as well. Since xn is a measurable selection of K, we have 0 ≤ |xn (s)| ≤ k l[1 · e(s)], and hence xn (s)dµ(s) − xn (s)dµ(s) C(t,pn ) C(t,p) ≤ |χC(t,pn ) (s) − χC(t,p) (s)xn (s)|dµ(s) ≤
√ |χC(t,pn ) (s) − χC(t,p) (s)|k l[1 · e(s)]dµ(s)
√ =k l
C(t,pn ) C(t,p)
1 · e(s)dµ(s).
Since A → A 1 · e(s)dµ(s), A ∈ T, is absolutely continuous with respect to µ, it suffices to show µ[C(t, pn ) C(t, p)] → 0. Note that C(t, p) = e−1 (S) and C(t, pn ) = e−1 (Sn ), where we let e = e(t) and " = δ(t) in the definitions of S and Sn , respectively. Thus we can write µ[C(t, pn ) C(t, p)] = µ[e−1 (Sn ) e−1 (S)] = µ[e−1 (Sn S)] = τ(Sn S), where τ ≡ µ ◦ e−1 is a finite Borel measure on Rl+ . From Lemma 1, we have S¯ \ S = {z ∈ Rl+ |p · z ∈ I(e(t), δ(t), p) \ I(e(t), δ(t), p)}, and since I(e(t), δ(t), p) \ I(e(t), δ(t), p) is countable by the definition of I, S¯ \ S equals the countable union of the sets having the form {z ∈ Rl+ |p · z = c}, where c ∈ I(e(t), δ(t), p) \ I(e(t), δ(t), p). Observe that by (A.8), we have τ[{z ∈ Rl+ |p · z = c}] = µ[{s ∈ T |p · e(s) = c}] = 0, and hence τ(S¯ \ S) = 0 or ¯ = τ(S). We can now appeal to Lemma 2 in order to establish our claim. τ(S) 1 Corollary 1. For each t ∈ T , (p, x) → µ[C(t,p)] C(t,p) x(s)dµ(s) is defined and continuous on {p ∈ )|p · e(t) > 0} × L1 (µ, K), where L1 (µ, K) is endowed with the weak topology. Proof. Let t ∈ T . If {p ∈ )|p · e(t) > 0} = ∅, the statement in the corollary above is trivially true. So, assume {p ∈ )|p · e(t) > 0} = ∅. It suffices to show p → µ[C(t, p)] > 0 is continuous on {p ∈ )|p · e(t) > 0}. To see this , let pn → p be an arbitrary convergent sequence in {p ∈ )|p · e(t) > 0}. Then, as in the last part of the proof of Proposition 1, dµ(s) − dµ(s) |µ[C(t, pn )] − µ[C(t, p)]| = C(t,pn )
C(t,p)
= χC(t,pn ) (s)dµ(s) − χC(t,p) (s)dµ(s) ≤ |χC(t,pn ) (s) − χC(t,p) (s)|dµ(s) = µ[C(t, pn ) This establishes our claim.
C(t, p)] → 0.
M. Noguchi / Journal of Mathematical Economics 41 (2005) 665–686
Proposition 2. For each (p, x) ∈ ) × L1 (µ, K), t → {t ∈ T |p · e(t) > 0}.
C(t,p) x(s)dµ(s)
681
is measurable in
Proof. Define a function G : Rl+ × Rl+ → Rl+ by G(a, b) = {s∈T |p·e(s)∈I(a,b,p)} x(s)dµ(s). Since C(t,p) x(s)dµ(s) = G(e(t), δ(t)), it suffices to show that G(a, b) is continuous in the image of {t ∈ T |p · e(t) > 0} under e × δ : T → Rl+ × Rl+ . To this end, let (an , bn ) → (a, b) be an arbitrary convergent sequence in e × δ({t ∈ T |p · e(t) > 0}). Define V ≡ {z ∈ Rl+ |p · z ∈ I(a, b, p)} and Vn ≡ {z ∈ Rl+ |p · z ∈ I(an , bn , p)}. As in the proof for Proposition 1, we obtain the following results. • V¯ = {z ∈ Rl+ |p · z ∈ I(a, b, p)}; we simply repeat the argument in the proof of Lemma 1, with a, b substituted for e, ", respectively. • V ⊂ liminfn (Vn ∩ V ); this is a direct consequence of statement 5 in the definition of I. • limsupn Vn ⊂ V¯ ; this can be seen as follows: let z ∈ limsupn Vn . Then we can choose n1 < n2 < n3 < . . . such that p · z ∈ I(ani , bni , p) for all i. From statement 6 in the definition of I, we deduce that p · z ∈ I(a, b, p), or z ∈ V¯ . ¯ we Repeating the argument in the proof of Lemma 2 with V, Vn , V¯ instead of S, Sn , S, l ¯ conclude that for any finite Borel measure τ on R+ , if τ(V ) = τ(V ), then τ(Vn V ) → 0. Note that {s ∈ T |p · e(s) ∈ I(an , bn , p)} = e−1 (Vn ) and {s ∈ T |p · e(s) ∈ I(a, b, p)} = e−1 (V ). So, as in the proof of Proposition 1, we have |G(an , bn ) − G(a, b)| = x(s)dµ(s) − x(s)dµ(s) {s∈T |p·e(s)∈I(an ,bn ,p)} {s∈T |p·e(s)∈I(a,b,p)} x(s)dµ(s) − x(s)dµ(s) = e−1 (Vn )
e−1 (V )
= χe−1 (Vn ) (s)x(s)dµ(s) − χe−1 (V ) (s)x(s)dµ(s) ≤ |χe−1 (Vn ) (s) − χe−1 (V ) (s)x(s)|dµ(s) √ ≤ k l |χe−1 (Vn ) (s) − χe−1 (V ) (s)|1 · e(s)dµ(s) √ =k l
e−1 (Vn ) e−1 (V )
1 · e(s)dµ(s).
Defining τ as in the proof of Proposition 1, it suffices to show that µ[e−1 (Vn ) e−1 (V )] = τ(Vn V ) → 0. Since V¯ \ V is the countable union of the sets in the form {z ∈ Rl+ |p · z = c}, c ∈ I(a, b, p) \ I(a, b, p), and I(a, b, p) = I(e(t), δ(t), p) for some t satisfying p · e(t) > 0, (A.8) implies τ[{z ∈ Rl+ |p · z = c}] = µ[{s ∈ T |p · e(s) = c}] = 0 and hence τ(V¯ ) = τ(V ), and this establishes our claim.
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Corollary 2. For each (p, x) ∈ ) × L1 (µ, K), t → 1/(µ[C(t, p)]) surable in {t ∈ T |p · e(t) > 0}.
C(t,p) x(s)dµ(s) is mea-
Proof. Since µ[C(t, p)] > 0 on {t ∈ T |p · e(t) > 0}, it suffices to show that t → µ[C(t, p)] is measurable in {t ∈ T |p · e(t) > 0}. To this end, define a function H : Rl+ × Rl+ → Rl+ by H(a, b) = µ[{s ∈ T |p · e(s) ∈ I(a, b, p)}]. We must show that H is continuous in the image of {t ∈ T |p · e(t) > 0} under e × δ : T → Rl+ × Rl+ . We have |H(an , bn ) − H(a, b)| = |µ[e−1 (Vn )] − µ[e−1 (V )]| = τ(Vn
≤ µ[e−1 (Vn
V )]
V ),
where V, Vn , τ are defined as in the proof of Proposition 2. As in the proof of Proposition 2, the last term converges to 0. This completes our proof. Proof of Theorem 1. Since the continuity of (p, x) → p · ( x − e) follows from Lemma 1 in (Noguchi, 1977, P. 8), Berge’s maximality theorem (1963) implies that γ1 is a nonempty, convex, compact valued, upper semi-continuous correspondence. We next show that t → H(t, p, x) admits a measurable graph. Define Tp = {t ∈ T |p · e(t) > 0} ∈ T. Following Schmeidler (1969, p. 582), we have GH(·,p,x) = [(Tpc × Rl+ ) ∩ GB(·,p) ] ∪ [(Tp × Rl+ ) ∩ GH(·,p,x) ]. ˜ Define g(t, ξ) = p · ξ − p · e(t). By Castaing and Valadier (1977, Theorem III.14, p. 70), g is jointly measurable in (t, ξ). Consequently, we have GB(·,p) = {(t, ξ) ∈ T × Rl+ |g(t, ξ) ≤ 0} ∈ T ⊗ B(Rl+ ), and since the correspondence t → K(t) clearly admits a measurable graph, so does the ˜ p). Thus, (Tpc × Rl+ ) ∩ GB(·,p) correspondence t → B(t, ∈ T ⊗ B(Rl+ ). We must show that ˜ (Tp × Rl+ ) ∩ GH(·,p,x) ∈ T ⊗ B(Rl+ ). To this end, proceeding as in Schmeidler (1969, p. 582), note that (Tp × Rl+ ) ∩ GH(·,p,x) ˜ p)} \ (t, b) ∈ Tp × Rl+ |∃a ∈ B(t, ˜ p) such that = {(t, b) ∈ Tp × Rl+ |b ∈ B(t, aR t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s) b
By (A.3)(Continuity), the statement ˜ p) such that aR t, ∃a ∈ B(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s) b
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683
is equivalent to ˜ ∃ rational r ∈ B(t, p) such that rR t,
1 µ[C(t, p)]
Thus, l ˜ (t, b) ∈ Tp × R+ |∃a ∈ B(t, p) such that aR t, =
C(t,p)
x(s)dµ(s) b.
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s) b
˜ p) such that (t, b) ∈ Tp × Rl+ |∃ rational r ∈ B(t,
1 x(s)dµ(s) b rR t, µ[C(t, p)] C(t,p) ˜ p)} {(t, b) ∈ Tp × Rl+ |r ∈ B(t, =
rational r∈Rl+
∩ (t, b) ∈
Tp × Rl+ |rR(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s))b
.
Note that ˜ p)} {(t, b) ∈ Tp × Rl+ |r ∈ B(t, ˜ p)} ∩ (Tp × Rl+ ) = {(t, b) ∈ T × Rl+ |r ∈ B(t, ˜ p)} × Rl+ ] ∩ (Tp × Rl+ ) = [{t ∈ T |r ∈ B(t, = [{t ∈ T |p · r ≤ p · e(t), r ≤ k[1 · e(t)]1} × Rl+ ] ∩ (Tp × Rl+ ) ∈ T ⊗ B(Rl+ ). In what follows, we abbreviate R(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s)) as R. Note that by (A.3) (Continuity), if rRb, then there exists an open ball Ub centered at b, such that for all b ∈ Ub , rRb holds. It follows that we can safely choose a rational r ∈ Rl+ such that rRr and r > b. On the other hand, if there exists a rational r ∈ Rl+ such that rRr and r > b hold, then by (A.5) (Desirability) and (A.1) (Transitivity), we get rRb. Thus, the statement “rRb” is equivalent to “∃ rational r ∈ Rl+ such that rRr and r > b. Consequently, {(t, b) ∈ Tp × Rl+ |rRb} = {(t, b) ∈ Tp × Rl+ |∃ rational r ∈ Rl+ such that rRr and r > b} [{(t, b) ∈ Tp × Rl+ |rRr } ∩ {(t, b) ∈ Tp × Rl+ |r > b}] = rational r ∈Rl+
Note that for each rational r , we have {(t, b) ∈ Tp × Rl+ |r > b} = Tp × {b ∈ Rl+ |r > b} ∈ T ⊗ B(Rl+ ),
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and {(t, b) ∈ Tp × Rl+ |rRr } = {t ∈ Tp |rRr } × Rl+ . Thus, it suffices to prove the following claim in order to complete the proof of the fact that the correspondence t → H(t, p, x) admits a measurable graph. Claim. For each (a, b) ∈ Rl+ × Rl+ , {t ∈ Tp |aR(t, 1/(µ[C(t, p)]) where p and x are fixed.
C(t,p) x(s)dµ(s))b}
∈ T,
As a matter of fact, this is the only essential alteration needed in order to apply Lemma 3 in Schmeidler (1969, p. 582) to the case at hand. We verify the above claim. To this l l end, recall that by (A.6), we have {(t, η) ∈ T × R+ |aR(t, η)b} ∈ T ⊗ B(R+ ), and also by Corollary 2, h(t) ≡ 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s) is measurable in Tp . We must show that Gh ∈ T ⊗ B(Rl+ ). Note that Gh = {(t, η) ∈ Tp × Rl+ |η = h(t)}. Since Tp ∈ T, we can choose a measurable extension h¯ : T → Rl+ of h. Then, we have ¯ Gh = {(t, η) ∈ T × Rl+ |η = h(t)} ∩ (Tp × Rl+ ). ¯ is jointly meaBy Castaing and Valadier (1977, Theorem III.14, p. 70), q(t, η) = η − h(t) l ). surable in T × Rl+ . Consequently, Gh ∈ T ⊗ B(R + Since {t ∈ Tp |aR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))b} = projT [{(t, η) ∈ T × Rl+ |aR(t, η)b} ∩ Gh ], where projT denotes the projection onto T, we can appeal to Castaing and Valadier (1977, Theorem III.23, p. 75)to establish our claim. By Lemma 2 Schmeidler (1969, p. 581), we see that for each t ∈ T , H(t, p, x) = ∅. Thus, by Castaing and Valadier (1977, Theorem III.22, p. 74), t → H(t, p, x) admits a measurable selection, and hence γ2 (p, x) = ∅. We also need to show that H(t, p, x) is convex. For thispurpose, it suffices to consider the ˜ p) = ∅ case p · e(t) > 0. Note that {a ∈ Rl+ |aR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))ξ} ∩ B(t, is equivalent to ξ ∈ ∩a∈B(t,p) {b ∈ Rl+ |aR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))b}c , and the last ˜ set is convex by (A.4), which yields that H(t, p, x) is convex, and hence γ2 (p, x) is convex as well. We still need to show Gγ2 is closed, but this is simply Theorem 5.5 in Yannelis (1991, p. 19) or claim 6 in Khan and Vohra (1984, p. 139). Thus we only need to check that the assumptions are all satisfied. ˜ p) is comWe need to show that H(t, p, x) is compact. If p · e(t) = 0, H(t, p, x) = B(t, ˜ pact. For p · e(t) > 0, we show that H(t, p, x) is closed in B(t, p). To see this, let ξn → ξ¯ ˜ p) such that ξn ∈ H(t, p, x). Then, be a convergent sequence in B(t, a ∈ Rl+ |aR(t,
1 µ[C(t, p)]
C(t,p)
x(s)dµ(s))ξn
˜ p) = ∅. ∩ B(t,
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¯ ∈ B(t, ˜ p) such that If ξ¯ ∈ H(t, p, x), there exists an element A ¯ AR(t,
1 µ[C(t, p)]
C(t,p)
¯ x(s)dµ(s))ξ.
¯ Then, (A.3) implies that AR(t, 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s))ξn for large n, but this is a contradiction. We next show that (p, x) → H(t, p, x) is upper semi-continuous. To this end, we sim¯ p, ply mimic the proof in Schmeidler (1969, Lemma 6, p. 582). Let (ξn , pn , xn ) → (ξ, ¯ x¯ ) be a convergent sequence in K(t) × ) × L1 (µ, K) such that ξn ∈ H(t, pn , xn ). Since pn · ˜ p), ξn ≤ pn · e(t), ξn ≤ k[e(t) · 1]1, if p¯ · e(t) = 0, then ξ¯ ∈ B(t, ¯ and hence ξ¯ ∈ H(t, p, ¯ x¯ ). Now assume p¯ · e(t) > 0. Then pn · e(t) > 0 for n > n0 for some n0 . Suppose there is ¯ Then we can find a ∈ Rl ˜ p) ¯ (s)dµ(s))ξ. ¯ such that a R(t, 1/(µ[C(t, p)]) ¯ a ∈ B(t, + C(t,p) ¯ x ¯ and p¯ · a < p¯ · e(t), which implies that ¯ such that a R(t, 1/(µ[C(t, p)]) ¯ x (s)dµ(s)) ξ C(t,p) ¯ pn · a < pn · e(t) for n > n1 for some n1 . Since (p, x) → 1/(µ[C(t, p)]) C(t,p) x(s)dµ(s) is continuous on {p ∈ )|p · e(t) > 0} × L1 (µ, K) by Corollary 1, (A.3) implies that a R(t, 1/(µ[C(t, pn )]) C(t,pn ) xn (s)dµ(s))ξn for n > n2 for some n2 . These imply that for large enough n, we obtain {a ∈ Rl+ |aR(t, 1/(µ[C(t, pn )]) C(t,pn ) xn (s)dµ(s))ξn } ∩ ˜ pn ) = ∅, pn · e(t) > 0, but this is a contradiction. B(t, Final remark. As we mentioned in the introduction, one of the major advantages in considering atomless economies is the fact that it allows us to drop the convexity assumption on the preferences by the use of the Lyapunov convexity theorem, as demonstrated by Aumann (1966), Schmeidler (1969), Greenberg et al. (1979), and other authors. The dependence of the preferences on assignments x makes this known approach extremely difficult, hence any attempt to drop the convexity assumption on the preferences in our framework seems to be very challenging. The commodity space considered in this paper is finite dimensional. It may be possible to extend our results to cover infinite dimensional commodity spaces with the property that order intervals can be made compact with an appropriately chosen topology. The author is indebt to the anonymous referee of JME for carefully checking the original draft and providing helpful comments and suggestions. Needless to say, the author is solely responsible for any remaining errors.
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Further reading Bourbaki, N., 1989. General Topology. Springer-Verlag, New York.