Existence of equilibria with infinitely many consumers and infinitely many commodities

Journal

of Mathematical

Economics

12 (1983) 119-l 38. North-Holland

EXISTENCE

OF EQUILIBRIA WITH INFINITELY CONSUMERS AND INFINITELY MANY COMMODITIES

A Theorem Based on Models of Commodity

MANY

Differentiation*

Larry E. JONES Northwestern Received

August

University.

Evanston, IL 60201, USA

1982, revised version

accepted

July 1983

This paper presents a general result on the existence of competitive equilibria in exchange economies in which consumers and commodities arc both infinite in number. The result shows that - in this framework at least - the added assumptions necessary to handle models with infinitely many agents are remarkably similar to the additional restrictions needed when only finitely many commodities are available for trade. It is shown that the results apply, in a straightforward manner, to two of the common mqdels of consumer choice when commodity differentiation is an important consideration.

1. Introduction

The use of economic models featuring infinitely many commodities has become increasingly common in recent years. Examples include applications in capital theory [Balasko and Shell (1980)], the economics of uncertainty [Prescott and Townsend (1981)] and commodity differentiation [Dixit and Stiglitz (1977)]. In addition, many of these models feature price-taking consumption sectors with infinitely many households. All of the above references have this feature, at least in their standard interpretation. In fact, most of the well-known examples from the literature on commodity differentiation feature both infinitely many commodities and infinitely many price-taking consumers [Hotelling (1929), Prescott and Visscher (1977), and Novshek (1980) provide, additional examples]. In view of these facts, surprisingly few results of any generality have appeared concerning these models. This paper helps in rectifying this situation by presenting a general result on the existence of competitive equilibria for exchange economies with both a continuum of commodities and a continuum of households. Models with finitely many commodities and finitely many households are well studied [e.g., Debreu (1959)] as are those with finitely many *This work is based on the author’s dissertation. The patient assistance of Gerard Andreu Mas-Cole11 is gratefully acknowledged. The comments of an anonymous also very useful. Remaining errors are, however, mine. 03044068/83/$3.00

0

1983, Elsevier

Science

Publishers

B.V. (North-Holland)

Debreu’ and referee were

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L.E. Jones,

Equilibria

with infinitely

many consumers

and commodities

commodities and infinitely many households [see Hildenbrand (1974)]. In addition, several models with infinitely many commodities and finitely many households have been analyzed relatively completely [Bewley (1972) and Jones (1982) give two examples]. In contrast, general studies of the properties of models with both infinitely many commodities and households are few in number. Examples include Bewley (1970), Mas-Cole11 (1975) and the recent very general treatment by Ostroy (1982). All three papers present results on the existence of competitive equilibrium in large-square economies (this is the terminology introduced in Ostroy to describe models in which both the number of households and the number of commodities are infinite). In Bewley (1970), there are countably many commodities and the commodity space is taken to be I,, the space of bounded sequences. The treatment in Ostroy (1982), is more general, the commodity space is an abstract Banach Space (with some restrictions). No a priori specification of the collection of commodities is given (this would depend on the commodity space being considered). Of the three references given above, the presentation of Mas-Cole11 (1975) is most closely related to the model we will analyze here. The commodity spaces considered are similar in that both are subsets of the non-negative distributions on a compact metric space, T. A t in T is then interpreted as an individual commodity. The major distinction between the model analyzed in Mas-Cole11 and the one presented here is that here consumers are allowed to choose any non-negative distribution on T whereas in Mas-Colell, individual consumption sets feature significant indivisibilities. This is a key assumption in Mas-Cole11 as it is the basis of the continuity of equilibrium prices (as a function of t) which in turn is crucial for the success of the proof that equilibria exist. What the results of this paper show is that these indivisibilities are inessential as far as the existence of equilibrium is concerned. This is an important generalization for several reasons. First of all, it facilitates comparisons with the standard results on the existence of equilibrium with finitely many commodities. It is seen that the extra assumptions needed to handle the case with infinitely many commodities have natural interpretations as restrictions on the substitutability of commodities which have nearly the same characteristics (i.e., they are close in T). Second, since the treatment with convex consumption sets admits existence results with only finitely many households (this will not be true in general if there are indivisibilities), clean, direct comparisons between the assumptions with finitely many households and the assumptions with infinitely many households are available. One final point should be made concerning the issue of the appropriateness of convex and non-convex (i.e., with indivisibilities)

L.E. Jones, Equilibria with infinitely many consumers and commodities

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consumption sets in the modelling of commodity differentiation. Some authors have argued [Rosen (1974), Mas-Cole11 (1975)] that the nonconvexity of consumption sets is an essential aspect of any model of commodity differentiation. This statement is, I think, too strong. Both approaches have led to useful insights [see Hotelling (1929), Rosen (1974), Mas-Cole11 (1975), and d’Aspremont et al. (1979) for the case with indivisibilities; Spence (1976), Dixit and Stiglitz (1977), Hart (1979) and Novshek (198’0) for the case with convex consumption sets]. The approach with convex consumption sets has the added aIdvantage that the connections to the standard economic models [see Debreu (1959), Hildenbrand (1974)] are more transparent. Further, since existence of equilibria can be guaranteed with only finitely many consumers, limiting results should be easier in our framework. In section 2, we introduce the notation, definitions and assumptions we will need when considering the problem of the existence of equilibrium and state the main result of the paper. Section 3 is devoted to developing two examples of exchange economies satisfying all of the assumptions of the theorem. The examples are based on two of the standard models of consumer choice when commodity differentiation is important. Finally, section 4 contains the proof of the existence theorem and an example showing the necessity of our assumptions.

2. Definitions,

assumptions,

notation

In this section we provide the necessary background for the theorem on the existence of competitive equilibrium with infinitely many commodities and consumers. We will treat only the case of pure exchange. The formulation of commodities we will give here follows closely the approach of Mas-Colell. However, due to the assumed non-convexities of consumption sets in Mas-Colell’s model, our treatment of preferences is more closely related to the presentations in Hart (1979) and Jones (1982). The collection of commodities under consideration will be denoted by T with typical elements t and s. A t in T is to be thought of as a complete list of all economically relevant characteristics of the commodity in question. Although in most applications T will be a subset of R” for the appropriate choice of n, we will simply assume that T is a compact metric space. Whenever it is needed, d will denote a suitable metric for 7: Examples of T’s considered include both the location model, where T is either a line segment [Hotelling (1929), Prescott and Visscher (1977)] or a circle [Novshek (1980)], and the characteristics model of Lancaster (1975), where T is a subset of R” where n is the number of characteristics. These examples are considered in more detail in section 3. We will denote by GY(T) the Bore1 subsets of T and B will stand for a

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L.E. Jones, Equilibria with infinitely many consumers and commodities

typical element. Individual consumers will be modelled as choosing finite non-negative Bore1 measures (distributions) on ?: M will signify this collection and m will denote the typical element of M. M will be the consumption set for all agents in our economy. If an agent chooses an m in M as his consumption bundle, m(B) is the total amount of all commodities with characteristics in B which are available for consumption. It is natural to think of two measures in M as close if they assign similar quantities of commodities which are close under the metric on ?: This notion of closeness for M is captured by the topology of convergence in distribution. Formally, this notion of closeness is the weak* topology of the dual pairing of M with the continuous, real-valued functions on ?: All topological notions on M will be with respect to this topology. Note that the restriction to weak* continuous preferences (which we shall make below) does place meaningful economic restrictions on the substitutability relations among the various commodities. [See Brown and Lewis (1981) for another example of this type of phenomenon.] Further, we should note that the assumption of continuity of preferences convergence in distribution does place restrictions on the units in which commodities are measured - nearby commodities must be measured in comparable units. While this still leaves some flexibility (any change in units which is continuous on T will preserve the desired continuity property) we do not have the full freedom standard in general equilibrium theory. Of course, this is not a severe restriction; most economic models have ‘natural’ parameterizations in which close commodities are good substitutes. The examples of section 3 are in this category. Prices are bounded, non-negative, real-valued, %?(T)-measurable functions on ?: Prices will be denoted by p( .) where p(t) is the cost of buying one unit of commodity t. Thus, the usual interpretation of prices as commodity indexed lists will continue to hold here (we view this interpretation as essential). Since those price functions which are continuous (with respect to commodity description) will play a special role, they will be singled out as @‘CU. Value is calculated in the usual way. If p is a price function and m E M, the value of m at the prices p will be denoted by p-m and can be calculated as

p.m= JdWm(t). The budget set follows the normal definition; for e E M, fi(p;e)={mEMlp.mSp-e).

Note that if p is continuous, p is weak*-closed and if p is bounded below, B is bounded (i.e., {m(T)1 rnE/?} is bounded). Hence, since bounded subsets of

L.E. Jones, Equilibria with infinitely many co,wumers and commodities

123

M are conditionally compact, /-I is weak*-compact if p is both continuous and bounded below. One of the principal benefits of working with models having M as consumption sets is the fact that M has very appealing approximation properties. To see this, let 6, be the Dirac measure at t [i.e., 6,(B) = 1 if t EB, 0 otherwise). Then if T*={tl,. . .} is any dense subset of T,

u L wtl,.

. . ,4,),

n

is dense in M, where LS is the linear subspace of M spanned by the dti. That is, an economy with consumption sets equal to M and weak* continuous preference orderings can be very well approximated by an economy with a large but finite number of commodities. This fact will serve as the basis of the approximation on which the proof of existence will be based. Whenever it is needed, the closed subsets of a space S will be denoted by p(S). Topological notions on sets of subsets will always be under the topology of closed convergence. This is the topology on F(S) generated by the base [K;G,,..., G,]={FeF(S)IFnK=$3, FnGi#$J] as K ranges over the compact subsets of S and the Gi are open subsets of S. (See Hildenbrand (1974) for a more complete exposition of this topology.) Note that since T is a compact metric space, so is P(T). However, since M is not necessarily metrizeable (bounded subsets of M are metrizeable), we cannot conclude (e.g., through Theorem LB.2 of Hildenbrand) that F(M) and F(M x M) are compact and metrizeable. It will be shown in section 4 that compact subsets of F(M x M) are indeed metrizeable. Further, if M* is any compact subset of M, F(M* x M*) is compact and metrizeable. Preferences will be given in the usual representation, as elements of F(M x M). 9cF(M x M) will denote the collection of allowed preference relations and will be assumed compact. We will make the following assumptions concerning preferences: (i) For all k ~9, 2 is complete, transitive and reflective. (ii) For all 2 ~9, 2 is strictly monotone - m>m’ implies m>m' (m’, m) 4 21.

[i.e.,

since preferences are delinitionally weak* continuous Thus, nearby commodities are assumed to be uniformly (since 9 is compact) good substitutes. We will need to make one further assumption concerning the collection of preferences, B. The purpose of this assumption is to guarantee that the equilibria of our approximations tit together in a nice enough way to guarantee that passing to the limit is possible and successfully locates an equilibrium of the original economy. The intuition behind and the necessity of this assumption are more completely spelled out in Jones (1982). Note

P’cF(M

that

x M).

124

(UHS)

L.E. Jones,

Equilibria

with infinitely

many consumers

and commodities

For all y > 1, there is an p > 0 such thlat for all a>O, for all m E M, for all 2 E 9, and for all s, t E T with G!(s,t) < p,

That is, all individuals are willing to accept any trade in which the ‘terms’ (i.e., y) are strictly greater than one as long as the trade involves only commodities which have nearly the same characteristics. Roughly, (UHS) requires that marginal rates of substitutions between nearby commodities are uniformly (in m and 2) close to one. It was argued in Jones that conditions similar to (UHS) can be derived from restrictions to preferences having sufficiently smooth representations. Similar reasoning applies here. Proposition 1 (presented in section 4) gives a set of conditions based on smooth representations of preferences which are sufficient to guarantee that (UHS) is satisfied. In fact, the notation (UHS) stands for uniform household smoothness in the spirit of the assumptions of Jones (1982). LNote that a related condition is used in Hart (1979) as well.] That the existence of smooth utility functions is in fact stronger than necessary can be seen from the discussion, presented below, of the case where T is discrete. The force of (UHS) is to guarantee that in equilibrium, commodities with similar descriptions have similar prices. As noted above, this insures that the equilibria of approximating economies with large but finite numbers of commodities lit together in a sensible way. Note that our assumptions rule out some of the standard utility functions seen in economics when dealing with infinite dimensional commodity spaces. In particular, utility functions of the form

u(x(-1)=

f4x(W, 0

as seen in the continuous time capital theory literature, are ruled out. There are two problems with this type of utility function. First, this utility function cannot be extended to all of M such that it is ‘continuous. Second, (UHS) is not satisfied. We are now in a position to compare the assumptions on preferences needed in the model considered here with those used in other results on the existence of equilibrium. It is easy to see that if there are only finitely many commodities (i.e., T is discrete), assumption (UHS) is redundant. It is implied by the strict monotonicity of preferences. The other assumptions on 9 are all standard [see Hildenbrand (1974)]. Thus, the only additional assumptions involve restrictions on substitutability of neighboring commodities. [We should note,

L.E. Jones, Equilibria with infinitely many co,wumers and commodities

125

however, that due to technical considerations (e.g., the failure of Fatou’s Lemma in infinite dimensional spaces) the conclusion of the theorem is slightly weaker than is standard - see Mas-Cole11 (1975) for a more detailed discussion.] A comparison of the assumptions on preferences used here with those made in Jones reveals that the strengthening of assumptions needed here are very similar to strengthening used in Hildenbrand to go from economies with finitely many consumers to those with infinitely many consumers when only finitely many commodities are available. First, convexity of preferences is no longer necessary. This is standard. Second, the characteristics of the various consumers must be similar. This is manifested in our assumptions on preferences in two ways. (We will assume compactness of endowments below.) First, .Y must be compact. This is standard in treatments with infinitely many households. Second, the condition on smoothness of preferences is required to hold uniformly on 9’. This is, of course, a type of compactness of characteristics assumption as well. Notice that, in contrast to other approaches [Bewley (1970) Mas-Cole11 (1975)], no boundedness condition is imposed on consumption sets. In fact, an important step in the proof will be to show that the equilibrium prices of the approximating sequence are uniformly bounded below. This, coupled with a uniform bound on endowments, implies that allocations are uniformly bounded. Endowments will be elements of M. We will restrict the allowed endowments to a compact subset, E, of M. The typical element of E is e. Thus, for some sufficiently large c, e(T) 5 2 for all e in E. An economy is then a probability measure on 9 x E (relative to the Bore1 subsets under the product topology). We will denote the economy under consideration by 8. The marginals of d (and all other relevant distributions) will be denoted by subscripts. Thus, the marginal of & on 9 is &,, etc. We will follow Hart, Hildenbrand and Kohlberg (1974) and Mas-Cole11 (1975) in our definition of equilibrium. An equilibrium for 6’ is a pair (v,p) where p is a price function and v is a probability on B x E x M, such that V BXE-

-8,

&idB,=j,idv, v{(k,e,m)lm

where i:M+M maximizes

is the identity,

2 on P(p;e)}=l.

That is, the characteristics distribution supply equals consumption [eq. (2)], maximizing in the budget set [eq. (3)].

(3)

of v agrees with that of 6’ [eq. (l)], and all ‘assignments’ are utility

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L.E. Jones,

Equilibria

with infinitely

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and commodities

The interpretation is that if (k,e, m) is in the support of v, m is the allocation assigned to a consumer (or the appropriate percent of consumers) having characteristics (&e). Notice that there is a technical problem with this interpretation. Letting I = [0, l] and A be Lebesgue measure on I, if there is a Il/:I+BxE with d=lot+-’ ($ will be called a representation of d below) it need not be true that there is an f:I+M such that v =A o(I+G,~)-‘. Such an f would quite naturally be termed an assignment. However, it will be true that such an f does exist for some representation of b. The distinction here is primarily technical. Interested readers can refer to MasCole11 for further discussion. We can now state the main result of the paper, the existence of an equilibrium for 8. Theorem 1.

Zf supp(I,id 6,) = 7: an equilibrium, (v, p), exists for 8. Moreover,

peg(T).

Since SEid &‘,is the aggregate endowment are available We turn now to the presentation assumptions we have made. The proof of 4. T states that all commodities

of 8, the condition supp(J, id 6,) = in the aggregate. of two examples satisfying the Theorem 1 is contained in section

3. Two examples In this section, we will develop two examples of consumption sectors satisfying the assumptions listed in section 2. The examples are based on two common models of commodity differentiation. The first example is the location model of Hotelling (1929). Our treatment is similar to the discussion presented in Novshek (1980). We will let T be the unit circle in R2 (adding outside goods is a straightforward extension). For each t E 7: there is a household whose favorite commodity is t. Household t’s preferences, kt, are derived from a utility function U,, defined by u,(m) =JTfT(S)dm(s), where f,(s) is a continuous, non-negative, real-valued function on t which is maximized at s= t. Then, f,(s) is the marginal (and average) utility of consumption of commodity s by household t. Note that U, is linear on M and since f, is continuous, it follows that kt is weak* continuous. If, in addition, the ft’s are uniformly (in s and t) bounded above and below, and the map t+ft is continuous (uniform convergence on the ft’s), B = {kt 1t E T} is compact. With these preferences, it is straightforward to check that (UHS) reduces to:

LX. Jones, Equilibria with infinitely

many

For all y > 1, there is a p > 0 such that 1s l/y

co,wmers

and commodities

127

s’l< p implies

for all t.

It can be shown that this will also be satisfied under the conditions listed above (i.e., f’s bounded above and below and f, a continuous function of t). It is easy to see that the example of Novshek satisfies the necessary conditions since there f, is of the form

where f (which does not depend on t) is a continuous and decreasing function of 1x1. The conditions on endowments are even easier to verify. If t-+e, is any bounded (and measureable) assignment of endowments, E = {e, 1t E T} will be conditionally compact. Thus, all of the assumptions of the theorem will be satisfied as long as the support of the aggregate endowment distribution is ?: The second example is based on the characteristics model of Lancaster (1975). Here, there are L characteristics which the various commodities possess. The range of possible levels of the Zth characteristic that a commoditing can possess will be denoted by I,, a compact interval. Then, T=Z, x... x I, and a commodity is identified by its position in 7: Thus, carrots have high vitamin A coordinates and oranges have high vitamin C coordinates. If rnE M, the toal quantity of characteristic 1 available under the consumption bundle m is given by CI(4

= ST6 Wt).

Now, if a E I = [0, l] is a household,

a’s preferences

are of the form

U,(m)= YAc,(m),. . ., c,(m)), where V, is a standard utility function on L commodities. Again, if the map a+ V, is continuous, B = { ka 1a E I} is compact. Notice that for preferences to be strictly monotone, 0 cannot be in 7Y Finally, given any ;’ > 1, there is a p > 0 such that if d(s, t)
128

4. Proof of tbe theorem In this section, we give the details of the proof of the theorem presented in section 2. The strategy of the proof is relatively straightforward. First a representation, I++, is taken for B. From this, a series of approximating economies having only finitely many commodities is constructed. The proof of the theorem then follows an application of an existence theorem with finitely many commodities and a limiting argument combining the techniques of Mas-Cole11 with those used in Jones. The first step in this outline is accomplished through an application of Skorokhod’s Theorem [Hildenbrand (1974, I.D. (37)]. Unfortunately, since M x M is not a metric space, it is not immediate that LT(M x M) is a separable metric space (a necessary condition for Skorokhod’s Theorem to apply). However, as we will show below, compact subsets of Y(M x M) are separable and metrizeable. Before proceeding to the proof itself, we will need a few mathematical preliminaries. We begin by listing as facts a few results which we will find useful: F&t 1 (Hildenbrand, I.B. Theorem 2). If S is a locally compact metric space, F(S) is a compact metric space under the topology convergence.

separable of closed

Fact 2 (Hildenbrand I.D. (39)). Let (Z, &,p) be a probability space, and (S,p) be a metric space. If f,,f are measurable functions from Z into S and p o f;;’ converges to ~(f,(z)~f(z))+o a.e. --p, the sequence of distributions

/Loof-1.

If S is a metric Ls(F,)

space and F, is a sequence

of S, define

and s,,~E Fn4 with s,,~+s}.

Let S be a compact (Mas-Colell). subsets of S, p. is a sequence = 1, and p”+p, then ,u(Ls(F,)) = 1.

of closed p,(F,)

subsets

by Ls(F,)={s~S13,~

Fact 3

of closed

metric space. If F, is a sequence of probabilities on S such that

Let S be a topological space and let f, be a family of real-valued functions on S. The family {f,} is said to separate S if for all s, s’ ES, there is an n with

f,(s) z.Lw Fact

4 (Rudin, 3.8 (c)). If the compact topological space S can be separated by a countable family of continuous real-valued functions, S is metrizeable.

L.E. Jones, Equilibria with infinitely many consumers and commodities

129

Recall that P’cF(M x M) is the (compact) collection of preference relations considered and E, a bounded subset of M, is the collection of possible endowments. The economy 6 is a probability measure on (the Bore1 subsets of) 9 x E. Marginal distributions will be denoted by subscripts e.g., 8, is the marginal distribution of d on endowments, etc. Let T” be an increasing sequence of closed subsets of T converging to 7: If p”: T” +R continuous, we write (T”, 7: p) p E T) if T in convergence and for every nk and t”’ E T”‘, with P+t,

p”“(t”“)-p(t).

We have: Lemma 1 (Mas-Cole@ p”.m”+p.m.

If m”+m with suppm” c T” and (T”,p”)+(T,p),

then

If (T”,p”) is a sequence as above, we will say that it is equicontinuous, if for all p > 0 there is a 6 > 0, such that for all n, t, t’ E T” with d(t’, t) < 6 implies Ip”(t)-p”(t’)I
This is a natural extension of the normal functions with restricted domain. We have: Lemma subsets bounded peg(T)

definition

of equicontinuity

to

2 (Mas-Colell). Suppose T” is an increasing sequence of closed of T with T”+T and p”: T”+R+ are continuous and uniformly above: if (T”,p”) is equicontinuous, there is a subsequence nk and a with (Tnk,pnk)+(Tp).

We will also need the following technical result for our approximations: Lemma 3 (Mas-Colell). Let p be finite, non-negative measure on T For every p > 0, and every finite subset of 7: K = {t,, . . . , t,>, there are disjoint open sets Bi, i= 1,. . . , n, such that

6) (ii)

diam(B,)
(iii)

tiEBi

for

for all i, i=l,...,m.

L.E. Jones, Equilibria with infinitely many consumers and commodities

130

Let i be the identity

on M and definite

e* by

e* = sM id 8,. Then, e* is the aggregate social endowment. Note that e* EM and e*(T) s 2. Applying Lemma 3 successively with p” -+O, we can construct a sequence of finite subsets of 7: T” = {tl, . . . , tk,} and a sequence of families of disjoint open subsets By, i= 1,. . . , m,, such that

6)

T”cT”+l,

(ii)

T” + T

(iii)

t; E By:

64

diam(B1)

(9

e*

(in closed convergence),

< p”,

From this point forward, M” = LS(d,;, . . . ,6,:$.

T” and the By will be fixed. Further,

we will let

For y>O, define M,={mEMlm(T)Sy}, MF={mEMIm(T)
4.

The

map

g:~(MxM)+9(MfxM$:~-+~nM~xM~

is

continuous. to 2 [see Proof: Let R,, a E&, be a net in 9(M” x MO) converging Dugundji (1966) for a discussion of nets]. Let F,=g( R,), F=g(,)). If g is not continuous, we can assume that there is some U, open, with FE U and such that, for all BE A, there is an LX>~ with F,$ U. Without loss of generality (w.l.o.g.), we can assume that U is of the form U = [K;

G,,...,G,],

where K is a compact M x M. Thus, either: (i) (ii)

subset

of M x M and

the

Gi are open

for all /I, there is an a>p with F, n K #@, or for some i and all 8, there is an CL > p with Fa A G, = 0.

subsets

of

L.E. Jones, Equilibria with infinitely many comumers and commodities

Both of these give rise under assumption that k#+ 2.

Lemma 5.

Let X c R(M

simple

arguments

131

to contradictions

of the

x M) be compact, then 2” is metrizeable.

Proof: Take yn -+ co and define g,: X-+S(My” x My”) as in Lemma 4. Recall that 9(Mv” x My”) is compact, separable and metrizeable for all n. Let d, be a suitable metric for 9(My” x MVJ and let {Fy} be dense in 9(M:” x MFJ. Then, by Lemma 4, g;:X+R+:F+d,(g,(F),F;) is continuous.

Clearly,

gb separates

X. Thus, by Fact 4, X is metrizeable.

Recall that M” = LS(6,,, . . . , 6,,) 2 n M” x M”. Then, we have: Lemma 6.

and

define

h,,:Y-+B(M”

x M”): k-+

h” is continuous.

Proof: Take kk-+k and let Fk=h,(kk), F=h,(k). If K is a compact subset of M” x M” with F n K =& K is also a compact subset of M x M. Hence, for large k, Fk n K =fl as well. Suppose now that G is an open subset of M” x M” such that F n G#@ Let (m,,m,)~ F n G. Since 2 is strictly monotone and G is open, we can assume that m,>m,. Thus, K ={(m,,m,)} is a compact subset of M” x M” such that F n K = 0. It follows that, for large k, Fk n K = 0 and hence, by completeness, (m,, m2) E Fk as desired. above. Note that since Let 9” = h,(9), where h, is as defined M” x M”c M x M, Y’cF(M x M). Thus, we can define

We have: Lemma 7.

@ is compact, h, (2)

-+ 2.

Proof. Let U, be a covering of 6’. Take U,,.. . , U, such that .53q,J’;ui&J. We will show that there is an N such that for all n 2 N, 9’” c U. If this is not true, there are n, and Fk E Yk such that nk+ COand Fk 4 U for all k. Suppose Fk = h_( kk) where 2:” E 9. Without loss of generality, we can assume that k’+ 2. We need only show that Fk + 2. It is clear that if K is compact and 2 n K = 0, Fk n K = 0 for large k.

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and commodities

Suppose (m,,m,) E 2 A G, we can assume that m,>m,. Now, there are m:,mk,EMnk with mf+mi, i=l,2. Thus, for large k, m:>mk, and (m:,mk,)EG. That is, there is an (m:, rnz) E MnXx Mnx for all sufficiently large k such that (mf, mf) E G and m:>ml. An argument as in Lemma 6 shows that, for large k, (my, rnz) E Fk as desired. Thus, (UN” P) c U for some N. That 9’ is compact for i 5 N follows from Lemma 6. Hence, @ is compact. That h,,(k)-+2 was shown above. Define E” = E n M”. E” is the space of approximating e” E E” and p” is continuous on T”, for 2 E 9, define

endowments.

If

and @(k;p”;e”)={mEM”I Note that, under

the obvious

m maximizes notational

2 on p(p”; e’)}.

convention,

Finally, define 4( 2; p; e) = {m E M 1m maximizes 2 E 9, e E E. Then, we will need:

2

on fi(p; e)} for p E%?(T),

and e”+e where e”EE”, eeE. Suppose p*e>O Lemma 8. Let (T”,p”)+(lT;p) and m” E @( 2; p”, e”) for all n. Then, if p(t*) = 0 for some t*, m”(T) + 00. Proof: Suppose not, then by extracting subsequences, if necessary, we can assume that m”+m for some m. We claim that rnE @( k;p; e). To see this, suppose m’Ep(p;e) and m’>m. Since p.e:>O and T”-+7; there are ,LL” l/3”(p”; e”) with p’+m’ (recall that pn.e”+p.ej. Thus, for large n, pn>mn, a contradiction. Thus, m E b( 2; p; e). since p.(m + b,,) =p.m and Clearly this cannot be the case, however, m+d,,>m. If (Z, _cxZ, p) is a probability space and H: Z+M is a measurable function, the mean of H can be defined much as is done for functions taking values in finite dimensional vector spaces (see Rudin for details). That is, m* =Jz H(z)dp is that measure which assigns to each measurable subset B of T

m*(B)=JzfW(Wb

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L.E. Jones, Equilibria with infinitely many co,Psumers and commodities

The following

result will be of use in the proof of Theorem

Lemma 9 (Ma+Colell). Let H: M,-+M, probabilities on M, with ,u”+p. Then, jM, Hdp”-+JMy Hdp

1:

be continuous and let p”, p be

(weak*).

We turn now to: Proof of Theorem 1. Since 9 x E is a compact metric space, by Skorokhod’s Theorem, there is a lC/:I+YxE, where I=[O,l], with 6=%01,-~. $ will be fixed hereafter. We will denote the typical element of I by a and will let $(a)=(L,e,). For each

n, let r”:.P x E-+p

x E” be defined

by r”(& e) =(k”,

e”) where

?=~,(ZZ), e” =

Te(&)

6,:.

1

Clearly, r” is measurable (recall that h, is continuous and By is open). Define r+!P=r”otj:I+Y’ x E”. Then, $” is a representation for an economy, P, with finitely many commodities. Write (R:, ei) for $“(a). Since supp SEid 8, = IT: j1 eida>>O. (M” is isomorphic to RT, a fact which will be used implicitly hereafter the isomorphism being suppressed for notational convenience.) Further, by construction, e:(T) =e,(T) for a.e. a. By Theorem 2.11.2 of Hildenbrand (1974), there is an equilibrium for 8”. That is, there are prices p”: T”-+R and an allocation f”: I+ M” such that

(1)

(2) Since 2 o(V,f”)

f”(a) maximizes

2:

on p(p”;e”)

for a.e. a.

is strictly monotone on M” for all a, p”>>O. Defining on :P” x E” x M” such that - I, we see that vn is a probability

2;

v”{(~,e,m)EY” We will assume

v”=

x E” x M”Im~&‘(~;p”;e”)}=l.

(w.1.o.g.) that sup,n p”(t) = 1.

Note that v” can be interpreted as a probability on @x E x M. The remainder of the proof involves showing first, that the (T”,p”) are equicontinuous and second, that there is a v on @ x E x M with v”+v. From

134

L.E. Jones, Equilibria with infinitely many consumers and commodities

here, a subsequence is extracted so that (T”,p”)+( rp) with peg(T). The argument that (v,p) is in fact an equilibrium for d is then relatively straightforward. The proof that the (r”,p”) are equicontinuous follows a straightforward application of (UHS). The existence of a v satisfying our requirements is more difficult, however. First we will show that the p” are uniformly bounded below (using Lemma 8). From this, we will be able to conclude that, in fact, for some y, v” is a probability on .@x E x M,. Since this is a compact metric space, we will then be guaranteed that the v” do indeed posses a limit point. First, we will show that the (T”,p”) are equicontinuous. If the (T”,p”) are not equicontinuous, a simple argument shows the existence of sequences tk, sk, in

c”(a)=f”~(a)-f”k(a)(sk)G,, Clearly, ck is measurable

+ {p”“(sk)/p”“(tk))f”‘(a)(sk)b,k.

and ck(a) ~jP(p”~;e~~) for a.e. a [in fact,

L.E. Jones,

Equilibria

with infinitely

many consumers

and commodities

135

First, since I/J”-+ a.e. (use Lemma 7 and the definition of T”,By) and l, it follows by Fact 2 that v$~~+IZO$-~ =6. That is, v>.,$($n)v$xE. Further, since S,idv~=S,e::da=S,f”(a)da=S,idv~, and vl-tb,, vL-vM, we have (by Lemma 9)

Finally, we need only show that v(H) = 1, where H={(~;e;m)EPxXxMMy*Im

maximizes 2 on P(p;e)).

To see that this is so, let H” = {(2, e, m) E P x E” x M:, 1m maximizes 2 on /I”($‘; e”)}. It follows that v”(H”) = 1 for all n. A straightforward argument shows that Ls(H”) cH whence, by Fact 3, v(H) = 1. This completes the proof of the theorem. We will now fulfill an earlier promise and give a set of conditions sufficient to guarantee that assumption (UHS) is satisfied. To start, suppose that for each 2 ~9, there is a utility representation u i.e., m>m’ iff u?(m) > u,(m’). Consider the function of o! defined by

f(a) = uz(m+ ah), which is well-defined for all a 20. [If m(r) >O, it is defined for some a ~0 as well.] Denote by u&(m, t, a) =f’(a), uL(m, t, a) = f”(a),

if these exist.

These are, respectively, the first and second directional derivatives of the function u in the direction 6, at the point m+ aa,. [For more on differentiation in infinite dimensional spaces, see Hille (1948).] For notational convenience, we let uk(m, t) z ui(m, t, 0). Consider the family of functions of t, X’= {ui(m, r)} Then, we have: J Math

El

parameterized

by 2 and m.

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L.E. Jones, Equilibria with infinitely many consumers and commodities

Proposition 1. (1) (2) (3) (4)

Suppose that for all 2 ~9

a utility representation exists, and:

for all 2, m, t,u, u;(m, t,cz) and u;(m, t, a) exist; there exists yll and n2 >O such that nI 5 u;(m, t, ~1)5 nz for all 2, m, t, cr; there exists an n3 > 0 such that 1ug(m, t, IX)TSq3; the family offunctions on T given by A? above is equicontinuous.

In particular, for each fixed 2, m, u>(m, t) is a continuous function oft. Then, (UHS) is satisfied. The usefullness of this proposition is that it shows that we can interpret (UHS) as a condition on compactness of characteristics - i.e., since 2 is equicontinuous, it is conditionally compact. Thus, (UHS) says that marginal rates of substitution do not differ substantially across individuals. Before presenting the proof of the proposition we will derive a technical result we will need: Lemma 10. Under the conditions of Proposition 1, for all 6 > 0, there is a p >O such that for all m, 2,

ui(m, t)/ui(m, s) 2 1 - 6 Proof If the lemma is false, existence of a 6 >O and sequences u&(m’, t”)&(m”,

d(s, t) < p.

if

a straightforward argument implies k”, m”, s”, t” such that d(s”, t”)+O and

s”)< 1- 6

the

for all y1.

That is, uk.(m”, t”) < (1 - 6)u&(m”, s”). W.l.o.g., we can assume that s”, P-e*. Further, since X is equicontinuous, it is conditionally compact. Hence, by taking further subsequences if necessary, there is a continuous f on T with u&(m”, .) + f (.)

uniformly.

Thus, f (t*) =lim ui.(m”, t”) = lim u&.(m”,s”) which is impossible. is false, for some 6 > 1, there are Proof of Proposition 1. If the proposition LX” >O, mn, km, s” and t” with d(s”, t”)-+O, and

+ c?dSn for all n. mn + a”$,. <“mn _ We can assume

that

s”, t”+t*

and that

rx”+O. By a well known

version

of

L.E. Jones, Equilibria with infinitely many consumers and commodities

Taylor’s

remainder

u&f)

formula,

137

we have

+ tl”yu&(m”,s”) +(@.“)2ug.(m”, t”, e;)

5 up(m”) + ~“UL(m”,S”)+ (cC”)2u>.(m”,s”, f3;) for all

n,

where 0 < 0; < CI”y, 0 < t$ < cz”. Rearrangement

gives

v&.(m”, t”) <

1

+

ct”ugm”, s”, e;) -ct”y2ugm”, t”, f$)

t&&r”, s”) =

for all 12.

u&(m”, s”)

But, since a”+O, u> is bounded above and u> is bounded below, the right side of this inequality approaches 1 as n+co. Further, by Lemma 10, since y> 1, the left side is greater than 1+6 for any positive 6 if n is large enough. This contradiction completes the proof of the proposition. Combining Theorem 1 and Proposition 1, we see that competitive equilibria will exist as long as preferences are sufficiently smooth. We conclude with an example designed to demonstrate that the assumptions on compactness of characteristics are necessary for the theorem to hold. Example.

Let T={O,&$a ,... } with the normal Euclidean metric. There are 2-” consumers of type II described by (&, e,) with e, =8rin and 2” derived from the utility function

u,(m)=~d,(t)dm, where

&fin(t)=&

if

t=O

or

= 1

if

t= l/(n+

=l/(n-k+2)

if

t=l/k

t=l/k

for

kln+2,

l), for

ksn,

and O<&<$ is not compact (this would be equivalent to Then, 8={kDIn=l,...} equicontinuity of the f, in this case) and does not satisfy (UHS). Note that

138

L.E. Jones, Equilibria with infinitely many consumers and commodities

each k,, is weak* continuous and that each type of preferences does satisfy (UHS) individually. Suppose (7,~) is an equilibrium distribution for this economy. By the strict monotonicity of preferences, p(O) > 0, so (w.l.0.g.) we will assume p(0)= 1. Thus, for large n, p( l/n) zz 1. Using this fact, a tedious but straightforward argument shows that for large n, individuals of type n will only buy goods of type t= l/(n+ 1). This is impossible, however, since in this case, demand is 2” and supply is only 2-(“+ l). There is an equilibrium for this economy obtained by assigning each type his own endowment for consumption and announcing the prices p(l/n) = 2”, p(0) = co. References Balasko, Y. and K. Shell, 1980, The overlapping generations model, I: The case of pure exchange without money, Journal of Economic Theory 23, 281-306. Bewley, T., 1970, Equilibrium theory with an infinite dimensional commodity space, Unpublished Ph.D. dissertation (University of California, Berkeley, CA). Bewley, T., 1972, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory 4, 513-540. Brown, D. and L. Lewis, 1981, Myopic economic agents, Econometrica 49, 359-368. d’Aspremont, G., J. Gabszewicz and J. Thisse, 1979, On Hotelling’s ‘Stability in competition’, Econometrica 47, 1145-l 150. Debreu, G., 1959, Theory of value (Cowles Foundation, New Haven, CT). Dixit, A. and J. Stiglitz, 1977, Monopolistic competition and optimum product diversity, American Economic Review 67, 297-308. Dugundji, J., 1966, Topology (Allyn and Bacon, Boston, MA). Hart, O., 1979, Monopolistic competition in a large economy with differentiated commodities, Review of Economic Studies 46, l-30. Hart, S., W. Hildenbrand and E. Kohlberg, 1974, On equilibrium allocations as distributions on the commodity space, Journal of Mathematical Economics 1, 159-166. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Hille, E., 1948, Functional analysis and semigroups (Colloquium Publishing, New York). Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57. Jones, L., 1982, A competitive model of commodity differentiation, Econometrica, forthcoming. Lancaster, K., 1975, Socially optimal product differentiation, American Economic Review 65, 567-585. Mas-Colell, A., 1975, A model of equilibrium with differentiated commodities, Journal of Mathematical Economics 2, 263-295. Novshek, W., 1980, Equilibrium in simple spatial (or differentiated product) models, Journal of Economic Theory 22, 3133326. Ostroy, J., 1982, On the existence of Walrasian equilibrium in large-square economies, Mimeo. (University of California, Los Angeles, CA). Prescott, E. and R. Townsend, 1981, General competitive analysis in an economy with private information, Mimeo. (Carnegie-Mellon University, Pittsburgh, PA). Prescott, E. and M. Visscher, 1977, Sequential location among firms with foresight, Bell Journal of Economics 8, 378-393. Rosen, S., 1974, Hedonic prices and implicit markets: Product differentiation in pure competition, Journal of Political Economy 82, 3455. Rudin. W.. 1973. Functional analvsis (McGraw Hill, New York.) Spenck, M., 1976, Product selecfion, ‘fixed costs and monopohstic competition, Review of Economic Studies 43, 217-236.