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Equilibria and core of large economies
Journal of Mathematical Economics 2 (1975) 155-169. 0 North-Holland
EQUILIBRIA
Publishing Company
AND CORE OF LARGE ECONOMIES* Hildegard DIERKER University of Bonn, Bonn, Germany
In this paper we apply results on regular economies to study equilibria and core in a nondifferentiable framework. We show that the distributions of agents’ characteristics of regular economies form a dense subset of all distributions of agents’ characteristics. Therefore ‘most’ economies have equilibria which are contained in fmitely many e-balls. And the core of ‘most’ sufficiently large economies is contained in finitely many c-balls centered at equilibrium allocations of these economies.
1. Introduction In this paper we study equilibria and core of exchange economies in a nondifferentiable framework. We describe an economy by a measurable mapping from the space of agents into the space of preferences and endowments. We assume that preferences are continuous, strictly monotone and convex. The convexity of preferences implies that mean demand as well as the set of equilibrium prices of an economy depend only on the distribution of agents’ preferences and initial endowments. This distribution is called the characteristic distribution of the economy [Debreu (1974)]. We call an economy regular if its characteristic distribution is regular. A regular economy has locally unique equilibria. Using a theorem on the approximation of preferences [Mas-Cole11 (1974)] we show that every economy can be approximated by simple, regular economies (Theorem 1). In Proposition 4 we show that the equilibrium price correspondence is continuous at regular characteristic distributions. This implies that the equilibrium price correspondence is e-continuous on an .open dense set (Theorem 2). For the study of the core we restrict agents’ characteristics to a compact set containing only strictly convex preferences. We know that for atomless economies the core coincides with the set of equilibrium .allocations [Aumann (1964)]. Bewley (1973) and W. Hildenbrand (1974) have *Presented at the Mathematical Social Science Board Colloquium on Mathematical Bconomics in August I974 at the University of California, Berkeley. The author would like to thank .E. Dierker and K. and W. Hildenbrand for many helpful discussions and comments.
H. Dierker, Equilibria and core of large economies
156
shown that the same result holds approximately in large competitive economies : for every core allocation of a sufficiently large competitive economy there is a price system at which all the consumers demand nearly the same commodity bundles they get in the core allocation. But this price system need not be close to an equilibrium price system for this economy. We show: for every E > 0 there is a ‘large class’ of economies in which every core allocation can be decentralized by an equilibrium price system, i.e., the core of such an economy is contained in a finite number of s-balls centered at equilibrium allocations of this economy. Our result uses the notion of a purely competitive sequence of economies and a limit theorem from W. Hildenbrand (1974). But the idea is essentially due to Bewley (1973). 2. The space of economies We assume that the commodity space is the Euclidean 1 space R’. The norm of a commodity bundle x = (x1, . . . , x,) E R’ is
The price of the hth commodity is denoted p,, . We will assume that all commodities are desirable; therefore, we consider positive prices only. Let S=(p~R*/lpI=l
and
p,,>O,
h=l,...,Z)
denote the price space, s its closure in R’. As we are only studying exchange, all agents in our model are consumers who are characterized by their consumption sets, their initial endowments and their preference relations. We assume that each consumer has the consumption set P={xER~Ix~>O, an initial endowment preference
h=l,...,
Z},
COE P
relation 5 on P.
and a complete, transitive and reflexive binary As usual < denotes strict preference and - denotes
indifference. Let B0 denote the set of continuous preferences on P which satisfy: (i)
x I y
and
x # y
(ii) Convexity condition.
Monotony condition.
x5
and
tE[O, l]
(iii) Boundary condition.
The closure of every indifference set is contained in P.
Remark.
y
A preference relation in 8,
implies implies
x < y. x5
tx+(l-t)y.
is defined only on the open positive
H. Dierker, Equilibria and core of large economies
orthant P of R’. One can extend it continuously following way :
157
onto the closure of P in the
ifxEP\PandyEP,definexiy; if x, y EP\P, define x N y. We endow g0 with the topology of closed convergence. Thus gc, becomes a separable metric space [cf. W. Hildenbrand (1974, ch. l)]. The space of consumers’ characteristics is B0 x P endowed with the product topology. Remark. Let /?(a, p) denote the budget set of consumer a E 9, x P at price ~. p E S and ~(0, p) its closure in R J. In our framework the demand cp(a,p) of consumer a at pricep is
_rp(a, P) = {x E B(a, P) I Y 5 x for all y E /?(a, p)] . As in W. Hildenbrand (1974, ch. 1) one can show that individual demand q:BOxPxS-+R’ (59 09 P)
H qP(5, o4 P)
has the following properties : Proposition I o,p) E B,, x P x S. Then ~(5, o,p) is non-empty, compact and convex. (ii) cpis upper hemi-continuous (u.h.c.). (iii) Let (s,, 0,) E go x P be convergent to (5, o) E 9”. x P and (p,J s S be convergent top E s\S. Then
(i) Let (5,
inf{lxlIxE~(~,,On,pn)}-)CO
for n-a.
We consider economies of the following kind: Definition 1. An economy is a measurable mapping d of a probability space (A, &, v) into (8, x P, S?(S, x P)) such that the distribution v 0 8-l of B is a measure with compact support.
Here g(8, x P) denotes the Bore1 a-algebra of Y,, x P. The distribution v 0 b-l of 6’ is called the characteristic distribution of the economy [Debreu (1974)]. Let JZ*(~~ x P) denote the set of characteristic distributions. An economy is called simple, if A consists of finitely many points Notation. and v is the counting measure. An economy is called atomless, if v is atomless. Let e : PO x P -+ P denote the projection. It describes the initial endowment of every agent, se o &’dv describes mean endowment. Since supp v o 8- ’ is compact in P, we have
158
H. Dierker, Equilibria and core of large economies
Definition 2.
An allocation for
the economy B is an integrable function
f:A+P. An allocation f is attainable tf jf dv = j e 0 8 dv.
We consider economies as similar if they have similar characteristic distributions and similar mean endowments. Definition 3. A sequence 8” : (A,, &,,, v,) + 8, gent to the economy d : (A, d, v) -+ B,, x P if
x P of economies is conver-
(1) (v, o ~5’;‘) is weakly convergent to v 0 b- I; (2) J e o 8” dv, is convergent to 5 e 0 d dv .
Let p denote the Prohorov-metric [Billingsley (1968)] which induces the weak topology and define a metric p* on A*(ge x P) as follows : Definition 4.
Let p 1, p2 E A*(PO
P*~cL~)
= P~~cL~)+
x
P).
1j4v-kb2
1.
Clearly, p* induces the convergence of economies. The set A*(gO x P) of characteristic distributions, endowed with the metric p*, is a separable metric space. frhis follows from Parthasarathy (1967, pp. 43-44).] The mean demand of an economy d at price p E S equals s cp(&(m),p) dv. It depends on the characteristic distribution of 8’ only. Lemma 1.
Let d : (A, d, v) + Bo x P be an economy,p E S. Then
~,&W),
P) dv = j e,,xr CP(*,P) d(v 0 8-l).
The proof follows from the change of variable formula in W. Hildenbrand (1974, D. II). For the study of equilibrium prices it is therefore sufficient to know the mean demand correspondence
@:.M*(B,xP)xS+R’ 01&-+(.,P)d~. Proposition 2 Let p E &*(9,, x P) and p E S. Then @(p,p) convex. (ii) @is upper hemi-continuous.
(i)
is non-empty, compact and
H. Dierker, Equilibriaand core of large economies
159
(iii) Let (p,,) c JT*(cY~ x P) be convergent to p E &Y*(L~‘~x P) and (p,,) E S be convergent top E s\S. Then
inf {1x11x e cl+, , p,)] + 00 for
12--, co.
The proof goes along the lines of the proofs of Theorem 3 and Proposition 6 in W. Hildenbrand (1974, ch. 1.3). Since mean demand exhibits the boundary behavior described in Proposition 2, equilibrium prices must be positive. Definition 5. Let 6’ : (A, ~4, v) -+ PO x P be an economy. A price system p E S is an equilibrium price system for 8, if there is an allocation f : A + P for 8 such that (0
f(a) E so(S(a),p)
(ii) jfdv
a.e. on
A;
= Jeobdv.
Let II(&) denote the set of equilibriumprice systemsfor 8. The properties of mean demand imply that mean excess demand,
has non-empty, compact, convex values, is bounded from below, upper hemicontinuous and shows the following boundary behavior: Let (p,J c A*(gO x P) be convergent to ,u E A’*(pO xP) and (p,,) c S be convergent top E S\S. Then inf{IxlER(xEZ(p,,,pn)}+~ These properties
guarantee
for
n-+co.
existence of equilibria.
equilibrium price correspondence,
n:
.4z*(9’,xP)
--) s w-HPESIOEZG4P)l>
has the following properties : Proposition 3 n(p) is non-empty and compactfor all p E: .J%?*(~~x P), (ii) n is upper hemi-continuous. (i)
For a proof see W. Hildenbrand (1974, ch. 2.2).
More precisely:
the
160
H. Dierker, Equilibriaand core of large economies
3. contimlity of n
Consider the set of smooth, convex preferences on P, i.e., the set of those preferences 5 E Be for which ((x, y) E P x P 1x N y} is a C2 submanifold in Px P. A smooth, convex preference relation can be represented by a continuously differentiable (C’) function g : P + S which is locally integrable and locally convex [Debreu (1972)]. Let G denote the set of C1 functions g : P + S representing smooth preferences in B,, . We endow G with the topology of uniform C’ convergence on compact sets. In Appendix I we show that this topology is finer than the topology of closed convergence. The set G contains preferences which give rise to C’ demand functions. From Debreu (1972) we know that a preference relation has a C’ demand function, if its indifference surfaces have non-vanishing Gaussian curvature. Therefore an economy has a C’ mean demand function, if its characteristic distribution is concentrated on a compact set in G such that the indifference surfaces of all preferences have non-vanishing Gaussian curvature. Definition 6.
~1E &Y*(POx P) is called regular, tf
(i) supp ,u is compact in G; (ii) there is an open set B in S with n(p) c B E B -C S such that cp(a, *)IE is a C’ function for all a E Supp p. (iii) 2(~, *)la is transverse regular to zero. (Here z(p, a) denotes mean excess demand of p for the first I - 1 commodities.) An economy is called regular tf its characteristic distribution is regular. Let &!g(G x P) denote the set of regular characteristic distributions.
For a detailed description of &z(GxP), see H. Dierker (1975). Obviously, regular economies have locally unique equilibria. Since the set of equilibria is compact, regular economies have only a finite number of equilibria. We shall show that the set of equilibria of a regular economy depends continuously on the parameters of the economy. Moreover, we show that every economy can be approximated by regular economies. Let 6 denote the Hausdorff distance of compact sets of B, x P. Theorem 1. For every economy d : (A, &, v) -+ 9”. x P there is a sequence of simple, regular economies 8k:(Ak,~k,~k)+90~P, such that
(i) (gk) is convergent to I;
k=
1,2 ,...,
H. Dierker, Equilibria and core of large economies
(ii) G(supp~,o~~~,suppv06-‘)+0
for
161
k-co.
The proof relies on the following approximation theorem for preferences which is a slight modification of MasColell(l974, Theorem 2 and Remark 4) : Approximation Theorem.
Let
every indifference surface of g has non-vanishing Gaussian curvature
’
Then G,, is dense in 9”. . Proof of Theorem I. Let E G supp (v o d- ‘) be a dense subset of supp (v 0 b- ‘). From Parthasarathy (1967, p. 44) we know that v 0 b- r can be approximated in the weak topology by measures which are concentrated on finite subsets of E. Let (~3 G &*(S, x P) be a sequence of measures giving rational weights to finitely many points from E, such that p(v,, v o b- ‘) + 0 fork + 00. Clearly, Ls (supp v& = supp (v 0 S-l), where Ls (supp v& denotes the set of x E go x P such that every neighborhood of x intersects infinitely many supp vk. Since the set of all closed subsets of the compact space supp v 0 6-r, endowed with the Hausdorff distance, is sequentially compact [Hausdorff (1957, p. 172)], we can assume that (supp vk) iS COnVergent to supp v 0 d- ‘. As (vk) + v 0 b- ’ weakly and supp vk E supp v 0 d- I, we conclude that p*(vk, v 0 b- ‘) + 0 for k + CO.Since G, is dense in 8,) it is no restriction to assume that every preference relation in supp vk has a C’ demand function. Also, it is no restriction to assume that vk is a regular distribution, because we can achieve this by an arbitrarily small change of the endowments in supp vk [Debreu (1970)]. Thus vk is a regular distribution. It is easy to construct a simple economy with characteristic distribution vk: take a set A, with cardinality #A, equal to the least common denominator of the values of vk . If a, E supp vk has value s,/# A,, map exactly s, points from Ak onto a,, . Thus we get a sequence of simple, regular economies approximating 8. Q.E.D. Corollary 1.
_&!z(Gx P) is dense in &%‘*(Y, x P).
The next proposition shows that the equilibrium price correspondence continuous on the set of regular characteristic distributions. Proposition 4. Let p E Jlg(G x P). The III : (.M*(gO x P), p*) + S is continuous at p.
equiIibrium price
Il is
correspondence
Proof. Let B,a, respectively Uea, denote the closed, respectively the open, s-neighborhood of a E R'. We have to show that n is lower hemi-continuous at p. As n(p) = {ql, . . . ,
162
H. Dierker, Equilibria and core of large economies
qk}, it is sufficient to show that for E > 0 there exists a neighborhood U of p such that IT($) n B,qj # C$for all ,u’ E U and i = 1, . . . , k. Choose 8 < E such that
the closed S-neighborhoods of the qi are disjoint. As Ii’ is u.h.c. we can choose a neighborhood U’ of p such that
Let @L, *) denote mean excess demand of ,u for the first I- 1 commodities. Since p is regular, there is 6 < 8 such that &, .)lean(,,) is transverse regular to zero. We can find some cc > 0 such that every continuous function f: S + R' satisfying Max.
If(P)- %~9]
< a
~sBaWt)
has a zero in B,q,, i = 1, . . . , k. From the upper hemi-continuity that there is a neighborhood U c U’ of ,u such that
&P’,P)
C U&h
P)
for
P’ E U
and
of 2 follows,
p E B&W.
Let p’ E U. We have to show that lI(p’) n B,qi # 4. Approximate p’ by a sequence p; of simple, regular distributions for which mean excess demand is a function. For n sufficiently large,
This means there is pni E It@;) n B,q,, i = 1, . . . , k. Letp, be a cluster point of (pai). Since ll is u.h.c., Q.E.D.
pi E IT&‘) n B&q,, i = 1, . . ., k.
Theorem 1 and Proposition 4 together imply that II is s-continuous on an open dense set, i.e., if we look at two characteristic distributions in this set which are close enough, then the Hausdorff distance of their sets of equilibria is Eat most. Theorem 2. For every E > 0 there is an open dense set in (A*(gO which the equilibrium price correspondence II is e-continuous.
x
P), p*) on
Proof. Since 17 is continuous at regular distributions, we can find for ~12 > 0 and every regular distribution p, an open neighborhood U(E/~, p) such that
WP’) = 4, &W for every $ E U(s/2, p).
and &4
E
4,JW),
H. Dierker,
Equilibria and core of large economies
163
Let 46,
P*) =
U W/2, PO. p regular
This set is open dense, owing to Theorem 1; and on it Il is s-continuous.
Q.E.D.
Remark. Of course, Theorem 2 follows directly from the upper hemicontinuity of II [see Fort (1949) or E. Dierker (1974)]. But the construction of the open dense set enables us to understand the structure of the set of equilibrium price systems of a large class of economies: Take a characteristic distribution p from this open dense set. Arbitrarily close to p we can find a regular characteristic distribution CL,,such that n@) is contained in the s-balls centered at the equilibrium price systems of pa. In order to know n(p) approximately, it is sufficient to know the finitely many price systems of U&J. In order to make a similar statement for the core, it is useful to consider a notion of convergence of economies which is finer than the one we have been using. This is the convergence of economies including convergence of supports. It is induced by the following metric rl on .M*(Y,, x P). Definition 7.
Let p 1, p2 G Af*(Po
1~1,/J2)
x P).
= P~1,cLz)+G(suPP~~,suPP~z).
Remark. (&!*(9,, x P), q) is a separable metric space. The metric tl induces a tier topology than p*. Therefore mean demand, mean excess demand and the equilibrium price correspondence are u.h.c. with respect to rl. Theorem 1 implies that &$(G x P) is dense in (&*(Pe x P), q). Therefore we can prove the following analogue of Theorem 2 : Proposition 5. For every E > 0 there is an open dense set in (A?*@~ x P), q) on which ll is e-continuous.
4. The core Consider the economy 8 : (A, d, v) --f 9, x P. An allocation f for d can be improved upon, if there is a coalition S E JZ?of positive weight and an allocation h, attainable for S, assigning to every person in S a commodity bundle which he prefers to the bundle he gets atJ Definition 8. The core of b, denoted C(g), is the set of all attainable allocationsfor I which cannot be improved upon.
It is easy to show that every equilibrium allocation is in the core. Also we know that in an atomless economy every core allocation can be decentralized by
H. Dierker, EquiIibriaand core of large economies
164
an equilibrium price system [Aumann (1964)]. In this case the study of the set of equilibrium allocations coincides with the study of the core. Now we wonder what the study of the equilibrium price correspondence l7 can contribute to the study of the core of a more general economy. We constrain ourselves to purely competitive sequences of economies of the following kind: Let B, = {5 E B,, 1 5 is strictly convex} and restrict consumers’ characteristics to a compact setKE B,xP. Definition 9. competitive’ if
A sequence ~5’”: (A,, &“, v,) + K of simple economies is purely
(1) the sequence of distributions (v, 0 8; ‘) is weakly convergent to a measure p on K; (2) G(supp v, 0 8; I, supp Jo) + 0 for (3) #A,+cx, for n+co.
n + co.
Here #A,, denotes the cardinality of A,. Obviously, (1) implies
For purely competitive sequences one can prove the following theorem which is a slight modification of Theorem 2 from W. Hildenbrand (1974, ch. 3.2). Theorem. Let K be a compact set in B, x P. Let the sequence (8,) of simple economies with characteristics in K be purely competitive. Then for every E > 0 there is an integer ii such that for every n 2 n and every allocation f E C(&,,) there is aprice vectorp E lT(p) with theproperty (f(a)-cp(g,,(a),
p>I 5 E for every
a E A,, .
Here p is an equilibrium price system for the limit distribution ,u. It may differ considerably from the equilibrium prices of &‘”[see Bewley (1973)]. But assume for a moment that 17 is continuous at ~1.Then we can find an equilibrium price system p,, for c?,,which is sufficiently close to p. This means that the core allocation f can approximately be decentralized by p.. As long as we stay within the framework of smooth preferences and K is sufficiently diversified, the set of regular characteristic distributions on K is an open dense set (with respect to q) on which l7 is continuous [H. Dierker (1975)]. Thus, one is led to suppose that ‘in general’ a purely competitive sequence has a regular limit distribution with a finite number of equilibrium price systems. The equilibria of a sufficiently large economy from this sequence are close to those of ‘This definition is more restrictive than the definition of a purely competitive sequence given by W. Hildenbrand (1974), because he does not require (2).
H. Dierker,
Equilibria and core of large economies
165
the limit distribution, Therefore the core of such an economy is contained in a finite number of s-balls centered at the equilibrium allocations of this economy. When we admit preferences from P’,, then ll is s-continuous and the ‘equivalence theorem’ for the core will depend on E. Let us make this precise. If the equilibrium price correspondence n is to be s-continuous on an open dense set, the compact set K c Ys x P must display a certain variety of agents’ characteristics. The proof of Theorem 1 shows that it is sufficient to postulate : (i) Every preference relation 5 E pr, K can be approximated by a sequence of preferences from (pr 1 K) n G,, which give rise to C’ demand functions. (ii) The interior points of pr, K E P are dense in pr, K, (pr, and pr, denote the projections on 8, and P, respectively). Let 4*(K) denote the set of characteristic distributions on K and A’;(K) the set of regular characteristic distributions on K. Obviously, A*(K) is the set of all probability distributions on K. Theorem 3. Let K c Y’s x P be a compact set satisfying (i) and (ii). For every E > 0 there is an open dense subset A!(&, K) of (A*(K), n) which contains A%$@) and is of the folIowing kind: For every purely competitive sequence g,,: (4,
dn, v,> + K
n = 1,2,...,
with limit distribution u = lim v, 0 8; ’ E A’(&,K) n there is an integer no such that for every n 2 n, and every core allocation f, E C(R,,) there exists an equilibriumprice systemp, E II(8,) with theproperty: If,(a) - ~(~Aa), p,)I 5 e for every
a E 4.
The proof of Theorem 3 shows that it is sufficient to consider finitely many p,, E II( in order to ‘decentralize’ the core allocations of 8”. This means that the core of 8” is contained in finitely many s-balls centered at equilibrium allocations of 8”. Proof.
(A*(K), p*) is a compact space. Since II is u.h.c.,
is a compact subset of S. Individual demand rp : 8, x P x S + R'is a continuous function. It is uniformly continuous on K x So. Thus, for E > 0 there is B(E) > 0 such that Iq(a, p)- q(a, p)] I e/3 for all a E K and p, p E S,, with/p-PI
I 8(s).
166
H. Dierker, Equilibria and core of large economies
The theorem on purely competitive sequences says that there is n(.s/3) such that for n 2 n(s/3) and every core allocation f, E C(&“), there exists a price systemp E II(,u) with the property P>I 5
~_G)-~(~&),
e/3
for every
a E A,.
As in Proposition 4 one can show that ll is continuous on &g(K). Properties (i) and (ii) of Kguarantee, that &g(K) is dense in k’*(K). Choose for every p,, E &g(K) an open neighborhood U&J on which II is G(s)-continuous. Define
This set is open dense. Let p = limv,o&;lE_M(.s,K). ” Then p is contained in some U(p,J with p. E k’:(K). This means, there is p E II&) with Ij-pl I 8(e). As ~(v,, 0 6’; I, p) --f 0, there is no 2 n(.e/3) such that v, 0 &; 1 E U&J for n 2 n,. Consequently, there is pn E 17(gn) with IF-P.1
5 %&I.
Summing up - for n 2 n, and every a E A,:
Theorem 3 yields a sharper result for purely competitive sequences with a regular limit distribution. Corollary 2. Let K E 8, x P be a compact set satisfying (i) and (ii). Assume, thepurely competitive sequence
8”: (A,, d,,
4
+ K,
n=
1,2,...,
has a regular limit distribution p = limv,o&;l.
n
Then, for every E > 0 there is an integer no such that for every n 2 n, and every core allocution f, E C(8”) there exists an equilibrium price system p,, E li’(&,,) with the property If,@) - 40(~“(4, P&l 5 E forevery
UEA,.
H. Dierker, Equilibria and core of large economies
167
Remark. Theorem 3 gives for every agent in A, the same bound for the difference between the core allocation and the equilibrium allocation. To get such a result one has to restrict agents’ characteristics to a compact set and to assume the convergence of the supports of the characteristic distributions. Without these assumptions one can still prove an analogue of Theorem 3 on the distributions of the core allocations in the commodity space R’. This analogue makes use of Theorem 1 from W. Hildenbrand (1974, ch. 3.2) and our Theorem 2. Appendix I
We want to show that the uniform C’ convergence on compact sets induces a topology on the set G of smooth, convex preferences which is finer than the topology of closed convergence. Proposition 6.2 Let (g,) c G be convergent to g E G, C1 uniformly on compact sets. Then the sequence of preferences, represented by g,,, is convergent in the topology of closed convergence to the preference relation represented by g. Proof. Let x0 E P. Denote the indifference set of x0 with respect to g, by Z,(X,) and the indifference set of x0 with respect to g by I&,). Let d, be a metric for the closed convergence of closed subsets of R’. Lemma.
d,(l,(x,,),
1(x,)) + 0
for
n + co.
Suppose the lemma is true. Let A denote the diagonal of P and u,(xO), respectively u(x,J, the distance from zero to Z,(x,) n A, respectively Z(xO) n A. This defines utility functions un(-), respectively u( *), for g,, respectively g. Clearly, Max
XEP
*
l%W4X)\ +O 1+1x1”
for
n-+03.
This implies that the sequence of preferences, represented by u,,, is convergent to the preference represented by u in the topology of closed convergence [see Kannai (1970) and W. Hildenbrand (1974, ch. 1.2)]. It remains to prove the lemma. Proof of the lemma. It is no restriction to assume that the closed limit of (Zn(x,& exists [W. Hildenbrand (1974, B. II)]. Let
lim Z,(x,) = IO. n
Obviously, x0 E I,. Suppose IO # Z(xo). 2A similar result is contained in Mas-Colell(l972).
168
H. Dierker, Equilibriaand core of large economies
Let H denote the hyperplane through 0 which is orthogonal to the diagonal d. Let pr, denote the projection onto H. Because preferences are monotone, we know that pr, : P + H maps 1(x0) and I&J, n = 1, 2, . . .) onto H. Therefore, pr, maps I0 = lirn 1,(x0) onto H as well. As I, # 1(x,), there exist y E I, and z E 1(x,) such that pr, y = pr, z and /y-z1 > 0. (Seefig. 1,) Consider the plane E determined by x,,, y, z. Let (aI, uz) be a Cartesian coordinate system for E n P, where v2 is taken in the direction of A and let t E R denote time. Commodity
2
Fig. 1
Because of monotony, the plane E cuts every indifference surface of g,,, respectively g, transversely. Therefore we define smooth vector-fields fn, respectivelyf, on E n P in the following way : f, assigns to every x E E n P that unit tangent vector of I,(x) n E at x which has positive first coordinate. Similarly, f assigns to every x E E n P that unit tangent vector of I(x) n Eat x which has positive first coordinate. Since (g,) -_)g C’ uniformly on compact sets, also cf,) --f f C1 uniformly on compact sets. Let v”(t) = (q(t), v;(t)) be the integral curve off, starting at time t, in x0 and v(t) = (vi(t), vz(t)) the integral curve off starting at time to in x0. Consider some compact cube K c E n P containing x0, y and z in its interior. Since f,, respectivelyf, are normalized, it follows from monotony that we can find some compact time interval To G R, to E To, such that v”(t), respectively v(t), are outside the compact Kif t $To . We have VGE Iv”(t)-v(t)] + 0 [see Hartman (1964, p. 4)]. This implies &(1,(x,) n K, 1(x,) n K) + 0 for n + co, a contradiction to the choice of y and z. Thus I, = 1(x0). Q.E.D.
H. Dierker, Equilibria and core of large economies
169
Remark. The proof does not make use either of the convexity of g,, respectively g, or the strictness of monotony. It uses the following monotony assumption : x i,, x+P, respectively x 4, x+P for all x E P.
It is easy to find examples showing that the C’ topology on G is strictly finer than the topology of closed convergence. References Aumann, R.J., 1964, Markets with a continuum of traders, Econometrica 32,39-50. Bewley, T., 1973, Edgeworth’s conjecture, Econometrica 41,425454. Billingsley, P., 1968, Convergence of probability measures (Wiley, New York). Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38,387-392. Debreu, G., 1972, Smooth preferences, Econometrica 40,603-615. Debreu, G., 1974, Four aspects of the mathematical theory of economic equilibrium, Working Paper IP-211 (Institute of Business and Economic Research, University of California, Berkeley, Calif.). Dierker, E., 1974, Topological methods in Walrasian economics (Springer, Heidelberg). Dierker, H., 1975, Smooth preferences and the regularity of equilibria, to appear in Journal of Mathematical Economics. Fort, M.K., 1949, A unified theory of semi-continuity, Duke Mathematical Journal 16, 237246. Hartman, Ph., 1964, Ordinary differential equations (Wiley, New York). Hausdorff, F., 1957, Set theory (Chelsea Publishing Company, New York). Hicks, N., 1965, Notes on differential geometry (Van Nostrand Reinhold, New York). Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton). Kannai, Y., 1970, Continuity properties of the core of a market, Econometrica 38,791-815. Mas-Colell, A., 1972, Smooth preferences and differentiable demand functions, Working Paper IP-175 (Institute of Business and Economic Research, University of California, Berkeley, Calif.). Mas-Colell, A., 1974, Continuous and smooth consumers: Approximation theorems, Journal of Economic Theory 8,305-336. Parthasarathy, K.R., 1967, Probability measures on metric spaces (Academic Press, New York).
EQUILIBRIA
Publishing Company
AND CORE OF LARGE ECONOMIES* Hildegard DIERKER University of Bonn, Bonn, Germany
In this paper we apply results on regular economies to study equilibria and core in a nondifferentiable framework. We show that the distributions of agents’ characteristics of regular economies form a dense subset of all distributions of agents’ characteristics. Therefore ‘most’ economies have equilibria which are contained in fmitely many e-balls. And the core of ‘most’ sufficiently large economies is contained in finitely many c-balls centered at equilibrium allocations of these economies.
1. Introduction In this paper we study equilibria and core of exchange economies in a nondifferentiable framework. We describe an economy by a measurable mapping from the space of agents into the space of preferences and endowments. We assume that preferences are continuous, strictly monotone and convex. The convexity of preferences implies that mean demand as well as the set of equilibrium prices of an economy depend only on the distribution of agents’ preferences and initial endowments. This distribution is called the characteristic distribution of the economy [Debreu (1974)]. We call an economy regular if its characteristic distribution is regular. A regular economy has locally unique equilibria. Using a theorem on the approximation of preferences [Mas-Cole11 (1974)] we show that every economy can be approximated by simple, regular economies (Theorem 1). In Proposition 4 we show that the equilibrium price correspondence is continuous at regular characteristic distributions. This implies that the equilibrium price correspondence is e-continuous on an .open dense set (Theorem 2). For the study of the core we restrict agents’ characteristics to a compact set containing only strictly convex preferences. We know that for atomless economies the core coincides with the set of equilibrium .allocations [Aumann (1964)]. Bewley (1973) and W. Hildenbrand (1974) have *Presented at the Mathematical Social Science Board Colloquium on Mathematical Bconomics in August I974 at the University of California, Berkeley. The author would like to thank .E. Dierker and K. and W. Hildenbrand for many helpful discussions and comments.
H. Dierker, Equilibria and core of large economies
156
shown that the same result holds approximately in large competitive economies : for every core allocation of a sufficiently large competitive economy there is a price system at which all the consumers demand nearly the same commodity bundles they get in the core allocation. But this price system need not be close to an equilibrium price system for this economy. We show: for every E > 0 there is a ‘large class’ of economies in which every core allocation can be decentralized by an equilibrium price system, i.e., the core of such an economy is contained in a finite number of s-balls centered at equilibrium allocations of this economy. Our result uses the notion of a purely competitive sequence of economies and a limit theorem from W. Hildenbrand (1974). But the idea is essentially due to Bewley (1973). 2. The space of economies We assume that the commodity space is the Euclidean 1 space R’. The norm of a commodity bundle x = (x1, . . . , x,) E R’ is
The price of the hth commodity is denoted p,, . We will assume that all commodities are desirable; therefore, we consider positive prices only. Let S=(p~R*/lpI=l
and
p,,>O,
h=l,...,Z)
denote the price space, s its closure in R’. As we are only studying exchange, all agents in our model are consumers who are characterized by their consumption sets, their initial endowments and their preference relations. We assume that each consumer has the consumption set P={xER~Ix~>O, an initial endowment preference
h=l,...,
Z},
COE P
relation 5 on P.
and a complete, transitive and reflexive binary As usual < denotes strict preference and - denotes
indifference. Let B0 denote the set of continuous preferences on P which satisfy: (i)
x I y
and
x # y
(ii) Convexity condition.
Monotony condition.
x5
and
tE[O, l]
(iii) Boundary condition.
The closure of every indifference set is contained in P.
Remark.
y
A preference relation in 8,
implies implies
x < y. x5
tx+(l-t)y.
is defined only on the open positive
H. Dierker, Equilibria and core of large economies
orthant P of R’. One can extend it continuously following way :
157
onto the closure of P in the
ifxEP\PandyEP,definexiy; if x, y EP\P, define x N y. We endow g0 with the topology of closed convergence. Thus gc, becomes a separable metric space [cf. W. Hildenbrand (1974, ch. l)]. The space of consumers’ characteristics is B0 x P endowed with the product topology. Remark. Let /?(a, p) denote the budget set of consumer a E 9, x P at price ~. p E S and ~(0, p) its closure in R J. In our framework the demand cp(a,p) of consumer a at pricep is
_rp(a, P) = {x E B(a, P) I Y 5 x for all y E /?(a, p)] . As in W. Hildenbrand (1974, ch. 1) one can show that individual demand q:BOxPxS-+R’ (59 09 P)
H qP(5, o4 P)
has the following properties : Proposition I o,p) E B,, x P x S. Then ~(5, o,p) is non-empty, compact and convex. (ii) cpis upper hemi-continuous (u.h.c.). (iii) Let (s,, 0,) E go x P be convergent to (5, o) E 9”. x P and (p,J s S be convergent top E s\S. Then
(i) Let (5,
inf{lxlIxE~(~,,On,pn)}-)CO
for n-a.
We consider economies of the following kind: Definition 1. An economy is a measurable mapping d of a probability space (A, &, v) into (8, x P, S?(S, x P)) such that the distribution v 0 8-l of B is a measure with compact support.
Here g(8, x P) denotes the Bore1 a-algebra of Y,, x P. The distribution v 0 b-l of 6’ is called the characteristic distribution of the economy [Debreu (1974)]. Let JZ*(~~ x P) denote the set of characteristic distributions. An economy is called simple, if A consists of finitely many points Notation. and v is the counting measure. An economy is called atomless, if v is atomless. Let e : PO x P -+ P denote the projection. It describes the initial endowment of every agent, se o &’dv describes mean endowment. Since supp v o 8- ’ is compact in P, we have
158
H. Dierker, Equilibria and core of large economies
Definition 2.
An allocation for
the economy B is an integrable function
f:A+P. An allocation f is attainable tf jf dv = j e 0 8 dv.
We consider economies as similar if they have similar characteristic distributions and similar mean endowments. Definition 3. A sequence 8” : (A,, &,,, v,) + 8, gent to the economy d : (A, d, v) -+ B,, x P if
x P of economies is conver-
(1) (v, o ~5’;‘) is weakly convergent to v 0 b- I; (2) J e o 8” dv, is convergent to 5 e 0 d dv .
Let p denote the Prohorov-metric [Billingsley (1968)] which induces the weak topology and define a metric p* on A*(ge x P) as follows : Definition 4.
Let p 1, p2 E A*(PO
P*~cL~)
= P~~cL~)+
x
P).
1j4v-kb2
1.
Clearly, p* induces the convergence of economies. The set A*(gO x P) of characteristic distributions, endowed with the metric p*, is a separable metric space. frhis follows from Parthasarathy (1967, pp. 43-44).] The mean demand of an economy d at price p E S equals s cp(&(m),p) dv. It depends on the characteristic distribution of 8’ only. Lemma 1.
Let d : (A, d, v) + Bo x P be an economy,p E S. Then
~,&W),
P) dv = j e,,xr CP(*,P) d(v 0 8-l).
The proof follows from the change of variable formula in W. Hildenbrand (1974, D. II). For the study of equilibrium prices it is therefore sufficient to know the mean demand correspondence
@:.M*(B,xP)xS+R’ 01&-+(.,P)d~. Proposition 2 Let p E &*(9,, x P) and p E S. Then @(p,p) convex. (ii) @is upper hemi-continuous.
(i)
is non-empty, compact and
H. Dierker, Equilibriaand core of large economies
159
(iii) Let (p,,) c JT*(cY~ x P) be convergent to p E &Y*(L~‘~x P) and (p,,) E S be convergent top E s\S. Then
inf {1x11x e cl+, , p,)] + 00 for
12--, co.
The proof goes along the lines of the proofs of Theorem 3 and Proposition 6 in W. Hildenbrand (1974, ch. 1.3). Since mean demand exhibits the boundary behavior described in Proposition 2, equilibrium prices must be positive. Definition 5. Let 6’ : (A, ~4, v) -+ PO x P be an economy. A price system p E S is an equilibrium price system for 8, if there is an allocation f : A + P for 8 such that (0
f(a) E so(S(a),p)
(ii) jfdv
a.e. on
A;
= Jeobdv.
Let II(&) denote the set of equilibriumprice systemsfor 8. The properties of mean demand imply that mean excess demand,
has non-empty, compact, convex values, is bounded from below, upper hemicontinuous and shows the following boundary behavior: Let (p,J c A*(gO x P) be convergent to ,u E A’*(pO xP) and (p,,) c S be convergent top E S\S. Then inf{IxlER(xEZ(p,,,pn)}+~ These properties
guarantee
for
n-+co.
existence of equilibria.
equilibrium price correspondence,
n:
.4z*(9’,xP)
--) s w-HPESIOEZG4P)l>
has the following properties : Proposition 3 n(p) is non-empty and compactfor all p E: .J%?*(~~x P), (ii) n is upper hemi-continuous. (i)
For a proof see W. Hildenbrand (1974, ch. 2.2).
More precisely:
the
160
H. Dierker, Equilibriaand core of large economies
3. contimlity of n
Consider the set of smooth, convex preferences on P, i.e., the set of those preferences 5 E Be for which ((x, y) E P x P 1x N y} is a C2 submanifold in Px P. A smooth, convex preference relation can be represented by a continuously differentiable (C’) function g : P + S which is locally integrable and locally convex [Debreu (1972)]. Let G denote the set of C1 functions g : P + S representing smooth preferences in B,, . We endow G with the topology of uniform C’ convergence on compact sets. In Appendix I we show that this topology is finer than the topology of closed convergence. The set G contains preferences which give rise to C’ demand functions. From Debreu (1972) we know that a preference relation has a C’ demand function, if its indifference surfaces have non-vanishing Gaussian curvature. Therefore an economy has a C’ mean demand function, if its characteristic distribution is concentrated on a compact set in G such that the indifference surfaces of all preferences have non-vanishing Gaussian curvature. Definition 6.
~1E &Y*(POx P) is called regular, tf
(i) supp ,u is compact in G; (ii) there is an open set B in S with n(p) c B E B -C S such that cp(a, *)IE is a C’ function for all a E Supp p. (iii) 2(~, *)la is transverse regular to zero. (Here z(p, a) denotes mean excess demand of p for the first I - 1 commodities.) An economy is called regular tf its characteristic distribution is regular. Let &!g(G x P) denote the set of regular characteristic distributions.
For a detailed description of &z(GxP), see H. Dierker (1975). Obviously, regular economies have locally unique equilibria. Since the set of equilibria is compact, regular economies have only a finite number of equilibria. We shall show that the set of equilibria of a regular economy depends continuously on the parameters of the economy. Moreover, we show that every economy can be approximated by regular economies. Let 6 denote the Hausdorff distance of compact sets of B, x P. Theorem 1. For every economy d : (A, &, v) -+ 9”. x P there is a sequence of simple, regular economies 8k:(Ak,~k,~k)+90~P, such that
(i) (gk) is convergent to I;
k=
1,2 ,...,
H. Dierker, Equilibria and core of large economies
(ii) G(supp~,o~~~,suppv06-‘)+0
for
161
k-co.
The proof relies on the following approximation theorem for preferences which is a slight modification of MasColell(l974, Theorem 2 and Remark 4) : Approximation Theorem.
Let
every indifference surface of g has non-vanishing Gaussian curvature
’
Then G,, is dense in 9”. . Proof of Theorem I. Let E G supp (v o d- ‘) be a dense subset of supp (v 0 b- ‘). From Parthasarathy (1967, p. 44) we know that v 0 b- r can be approximated in the weak topology by measures which are concentrated on finite subsets of E. Let (~3 G &*(S, x P) be a sequence of measures giving rational weights to finitely many points from E, such that p(v,, v o b- ‘) + 0 fork + 00. Clearly, Ls (supp v& = supp (v 0 S-l), where Ls (supp v& denotes the set of x E go x P such that every neighborhood of x intersects infinitely many supp vk. Since the set of all closed subsets of the compact space supp v 0 6-r, endowed with the Hausdorff distance, is sequentially compact [Hausdorff (1957, p. 172)], we can assume that (supp vk) iS COnVergent to supp v 0 d- ‘. As (vk) + v 0 b- ’ weakly and supp vk E supp v 0 d- I, we conclude that p*(vk, v 0 b- ‘) + 0 for k + CO.Since G, is dense in 8,) it is no restriction to assume that every preference relation in supp vk has a C’ demand function. Also, it is no restriction to assume that vk is a regular distribution, because we can achieve this by an arbitrarily small change of the endowments in supp vk [Debreu (1970)]. Thus vk is a regular distribution. It is easy to construct a simple economy with characteristic distribution vk: take a set A, with cardinality #A, equal to the least common denominator of the values of vk . If a, E supp vk has value s,/# A,, map exactly s, points from Ak onto a,, . Thus we get a sequence of simple, regular economies approximating 8. Q.E.D. Corollary 1.
_&!z(Gx P) is dense in &%‘*(Y, x P).
The next proposition shows that the equilibrium price correspondence continuous on the set of regular characteristic distributions. Proposition 4. Let p E Jlg(G x P). The III : (.M*(gO x P), p*) + S is continuous at p.
equiIibrium price
Il is
correspondence
Proof. Let B,a, respectively Uea, denote the closed, respectively the open, s-neighborhood of a E R'. We have to show that n is lower hemi-continuous at p. As n(p) = {ql, . . . ,
162
H. Dierker, Equilibria and core of large economies
qk}, it is sufficient to show that for E > 0 there exists a neighborhood U of p such that IT($) n B,qj # C$for all ,u’ E U and i = 1, . . . , k. Choose 8 < E such that
the closed S-neighborhoods of the qi are disjoint. As Ii’ is u.h.c. we can choose a neighborhood U’ of p such that
Let @L, *) denote mean excess demand of ,u for the first I- 1 commodities. Since p is regular, there is 6 < 8 such that &, .)lean(,,) is transverse regular to zero. We can find some cc > 0 such that every continuous function f: S + R' satisfying Max.
If(P)- %~9]
< a
~sBaWt)
has a zero in B,q,, i = 1, . . . , k. From the upper hemi-continuity that there is a neighborhood U c U’ of ,u such that
&P’,P)
C U&h
P)
for
P’ E U
and
of 2 follows,
p E B&W.
Let p’ E U. We have to show that lI(p’) n B,qi # 4. Approximate p’ by a sequence p; of simple, regular distributions for which mean excess demand is a function. For n sufficiently large,
This means there is pni E It@;) n B,q,, i = 1, . . . , k. Letp, be a cluster point of (pai). Since ll is u.h.c., Q.E.D.
pi E IT&‘) n B&q,, i = 1, . . ., k.
Theorem 1 and Proposition 4 together imply that II is s-continuous on an open dense set, i.e., if we look at two characteristic distributions in this set which are close enough, then the Hausdorff distance of their sets of equilibria is Eat most. Theorem 2. For every E > 0 there is an open dense set in (A*(gO which the equilibrium price correspondence II is e-continuous.
x
P), p*) on
Proof. Since 17 is continuous at regular distributions, we can find for ~12 > 0 and every regular distribution p, an open neighborhood U(E/~, p) such that
WP’) = 4, &W for every $ E U(s/2, p).
and &4
E
4,JW),
H. Dierker,
Equilibria and core of large economies
163
Let 46,
P*) =
U W/2, PO. p regular
This set is open dense, owing to Theorem 1; and on it Il is s-continuous.
Q.E.D.
Remark. Of course, Theorem 2 follows directly from the upper hemicontinuity of II [see Fort (1949) or E. Dierker (1974)]. But the construction of the open dense set enables us to understand the structure of the set of equilibrium price systems of a large class of economies: Take a characteristic distribution p from this open dense set. Arbitrarily close to p we can find a regular characteristic distribution CL,,such that n@) is contained in the s-balls centered at the equilibrium price systems of pa. In order to know n(p) approximately, it is sufficient to know the finitely many price systems of U&J. In order to make a similar statement for the core, it is useful to consider a notion of convergence of economies which is finer than the one we have been using. This is the convergence of economies including convergence of supports. It is induced by the following metric rl on .M*(Y,, x P). Definition 7.
Let p 1, p2 G Af*(Po
1~1,/J2)
x P).
= P~1,cLz)+G(suPP~~,suPP~z).
Remark. (&!*(9,, x P), q) is a separable metric space. The metric tl induces a tier topology than p*. Therefore mean demand, mean excess demand and the equilibrium price correspondence are u.h.c. with respect to rl. Theorem 1 implies that &$(G x P) is dense in (&*(Pe x P), q). Therefore we can prove the following analogue of Theorem 2 : Proposition 5. For every E > 0 there is an open dense set in (A?*@~ x P), q) on which ll is e-continuous.
4. The core Consider the economy 8 : (A, d, v) --f 9, x P. An allocation f for d can be improved upon, if there is a coalition S E JZ?of positive weight and an allocation h, attainable for S, assigning to every person in S a commodity bundle which he prefers to the bundle he gets atJ Definition 8. The core of b, denoted C(g), is the set of all attainable allocationsfor I which cannot be improved upon.
It is easy to show that every equilibrium allocation is in the core. Also we know that in an atomless economy every core allocation can be decentralized by
H. Dierker, EquiIibriaand core of large economies
164
an equilibrium price system [Aumann (1964)]. In this case the study of the set of equilibrium allocations coincides with the study of the core. Now we wonder what the study of the equilibrium price correspondence l7 can contribute to the study of the core of a more general economy. We constrain ourselves to purely competitive sequences of economies of the following kind: Let B, = {5 E B,, 1 5 is strictly convex} and restrict consumers’ characteristics to a compact setKE B,xP. Definition 9. competitive’ if
A sequence ~5’”: (A,, &“, v,) + K of simple economies is purely
(1) the sequence of distributions (v, 0 8; ‘) is weakly convergent to a measure p on K; (2) G(supp v, 0 8; I, supp Jo) + 0 for (3) #A,+cx, for n+co.
n + co.
Here #A,, denotes the cardinality of A,. Obviously, (1) implies
For purely competitive sequences one can prove the following theorem which is a slight modification of Theorem 2 from W. Hildenbrand (1974, ch. 3.2). Theorem. Let K be a compact set in B, x P. Let the sequence (8,) of simple economies with characteristics in K be purely competitive. Then for every E > 0 there is an integer ii such that for every n 2 n and every allocation f E C(&,,) there is aprice vectorp E lT(p) with theproperty (f(a)-cp(g,,(a),
p>I 5 E for every
a E A,, .
Here p is an equilibrium price system for the limit distribution ,u. It may differ considerably from the equilibrium prices of &‘”[see Bewley (1973)]. But assume for a moment that 17 is continuous at ~1.Then we can find an equilibrium price system p,, for c?,,which is sufficiently close to p. This means that the core allocation f can approximately be decentralized by p.. As long as we stay within the framework of smooth preferences and K is sufficiently diversified, the set of regular characteristic distributions on K is an open dense set (with respect to q) on which l7 is continuous [H. Dierker (1975)]. Thus, one is led to suppose that ‘in general’ a purely competitive sequence has a regular limit distribution with a finite number of equilibrium price systems. The equilibria of a sufficiently large economy from this sequence are close to those of ‘This definition is more restrictive than the definition of a purely competitive sequence given by W. Hildenbrand (1974), because he does not require (2).
H. Dierker,
Equilibria and core of large economies
165
the limit distribution, Therefore the core of such an economy is contained in a finite number of s-balls centered at the equilibrium allocations of this economy. When we admit preferences from P’,, then ll is s-continuous and the ‘equivalence theorem’ for the core will depend on E. Let us make this precise. If the equilibrium price correspondence n is to be s-continuous on an open dense set, the compact set K c Ys x P must display a certain variety of agents’ characteristics. The proof of Theorem 1 shows that it is sufficient to postulate : (i) Every preference relation 5 E pr, K can be approximated by a sequence of preferences from (pr 1 K) n G,, which give rise to C’ demand functions. (ii) The interior points of pr, K E P are dense in pr, K, (pr, and pr, denote the projections on 8, and P, respectively). Let 4*(K) denote the set of characteristic distributions on K and A’;(K) the set of regular characteristic distributions on K. Obviously, A*(K) is the set of all probability distributions on K. Theorem 3. Let K c Y’s x P be a compact set satisfying (i) and (ii). For every E > 0 there is an open dense subset A!(&, K) of (A*(K), n) which contains A%$@) and is of the folIowing kind: For every purely competitive sequence g,,: (4,
dn, v,> + K
n = 1,2,...,
with limit distribution u = lim v, 0 8; ’ E A’(&,K) n there is an integer no such that for every n 2 n, and every core allocation f, E C(R,,) there exists an equilibriumprice systemp, E II(8,) with theproperty: If,(a) - ~(~Aa), p,)I 5 e for every
a E 4.
The proof of Theorem 3 shows that it is sufficient to consider finitely many p,, E II( in order to ‘decentralize’ the core allocations of 8”. This means that the core of 8” is contained in finitely many s-balls centered at equilibrium allocations of 8”. Proof.
(A*(K), p*) is a compact space. Since II is u.h.c.,
is a compact subset of S. Individual demand rp : 8, x P x S + R'is a continuous function. It is uniformly continuous on K x So. Thus, for E > 0 there is B(E) > 0 such that Iq(a, p)- q(a, p)] I e/3 for all a E K and p, p E S,, with/p-PI
I 8(s).
166
H. Dierker, Equilibria and core of large economies
The theorem on purely competitive sequences says that there is n(.s/3) such that for n 2 n(s/3) and every core allocation f, E C(&“), there exists a price systemp E II(,u) with the property P>I 5
~_G)-~(~&),
e/3
for every
a E A,.
As in Proposition 4 one can show that ll is continuous on &g(K). Properties (i) and (ii) of Kguarantee, that &g(K) is dense in k’*(K). Choose for every p,, E &g(K) an open neighborhood U&J on which II is G(s)-continuous. Define
This set is open dense. Let p = limv,o&;lE_M(.s,K). ” Then p is contained in some U(p,J with p. E k’:(K). This means, there is p E II&) with Ij-pl I 8(e). As ~(v,, 0 6’; I, p) --f 0, there is no 2 n(.e/3) such that v, 0 &; 1 E U&J for n 2 n,. Consequently, there is pn E 17(gn) with IF-P.1
5 %&I.
Summing up - for n 2 n, and every a E A,:
Theorem 3 yields a sharper result for purely competitive sequences with a regular limit distribution. Corollary 2. Let K E 8, x P be a compact set satisfying (i) and (ii). Assume, thepurely competitive sequence
8”: (A,, d,,
4
+ K,
n=
1,2,...,
has a regular limit distribution p = limv,o&;l.
n
Then, for every E > 0 there is an integer no such that for every n 2 n, and every core allocution f, E C(8”) there exists an equilibrium price system p,, E li’(&,,) with the property If,@) - 40(~“(4, P&l 5 E forevery
UEA,.
H. Dierker, Equilibria and core of large economies
167
Remark. Theorem 3 gives for every agent in A, the same bound for the difference between the core allocation and the equilibrium allocation. To get such a result one has to restrict agents’ characteristics to a compact set and to assume the convergence of the supports of the characteristic distributions. Without these assumptions one can still prove an analogue of Theorem 3 on the distributions of the core allocations in the commodity space R’. This analogue makes use of Theorem 1 from W. Hildenbrand (1974, ch. 3.2) and our Theorem 2. Appendix I
We want to show that the uniform C’ convergence on compact sets induces a topology on the set G of smooth, convex preferences which is finer than the topology of closed convergence. Proposition 6.2 Let (g,) c G be convergent to g E G, C1 uniformly on compact sets. Then the sequence of preferences, represented by g,,, is convergent in the topology of closed convergence to the preference relation represented by g. Proof. Let x0 E P. Denote the indifference set of x0 with respect to g, by Z,(X,) and the indifference set of x0 with respect to g by I&,). Let d, be a metric for the closed convergence of closed subsets of R’. Lemma.
d,(l,(x,,),
1(x,)) + 0
for
n + co.
Suppose the lemma is true. Let A denote the diagonal of P and u,(xO), respectively u(x,J, the distance from zero to Z,(x,) n A, respectively Z(xO) n A. This defines utility functions un(-), respectively u( *), for g,, respectively g. Clearly, Max
XEP
*
l%W4X)\ +O 1+1x1”
for
n-+03.
This implies that the sequence of preferences, represented by u,,, is convergent to the preference represented by u in the topology of closed convergence [see Kannai (1970) and W. Hildenbrand (1974, ch. 1.2)]. It remains to prove the lemma. Proof of the lemma. It is no restriction to assume that the closed limit of (Zn(x,& exists [W. Hildenbrand (1974, B. II)]. Let
lim Z,(x,) = IO. n
Obviously, x0 E I,. Suppose IO # Z(xo). 2A similar result is contained in Mas-Colell(l972).
168
H. Dierker, Equilibriaand core of large economies
Let H denote the hyperplane through 0 which is orthogonal to the diagonal d. Let pr, denote the projection onto H. Because preferences are monotone, we know that pr, : P + H maps 1(x0) and I&J, n = 1, 2, . . .) onto H. Therefore, pr, maps I0 = lirn 1,(x0) onto H as well. As I, # 1(x,), there exist y E I, and z E 1(x,) such that pr, y = pr, z and /y-z1 > 0. (Seefig. 1,) Consider the plane E determined by x,,, y, z. Let (aI, uz) be a Cartesian coordinate system for E n P, where v2 is taken in the direction of A and let t E R denote time. Commodity
2
Fig. 1
Because of monotony, the plane E cuts every indifference surface of g,,, respectively g, transversely. Therefore we define smooth vector-fields fn, respectivelyf, on E n P in the following way : f, assigns to every x E E n P that unit tangent vector of I,(x) n E at x which has positive first coordinate. Similarly, f assigns to every x E E n P that unit tangent vector of I(x) n Eat x which has positive first coordinate. Since (g,) -_)g C’ uniformly on compact sets, also cf,) --f f C1 uniformly on compact sets. Let v”(t) = (q(t), v;(t)) be the integral curve off, starting at time t, in x0 and v(t) = (vi(t), vz(t)) the integral curve off starting at time to in x0. Consider some compact cube K c E n P containing x0, y and z in its interior. Since f,, respectivelyf, are normalized, it follows from monotony that we can find some compact time interval To G R, to E To, such that v”(t), respectively v(t), are outside the compact Kif t $To . We have VGE Iv”(t)-v(t)] + 0 [see Hartman (1964, p. 4)]. This implies &(1,(x,) n K, 1(x,) n K) + 0 for n + co, a contradiction to the choice of y and z. Thus I, = 1(x0). Q.E.D.
H. Dierker, Equilibria and core of large economies
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Remark. The proof does not make use either of the convexity of g,, respectively g, or the strictness of monotony. It uses the following monotony assumption : x i,, x+P, respectively x 4, x+P for all x E P.
It is easy to find examples showing that the C’ topology on G is strictly finer than the topology of closed convergence. References Aumann, R.J., 1964, Markets with a continuum of traders, Econometrica 32,39-50. Bewley, T., 1973, Edgeworth’s conjecture, Econometrica 41,425454. Billingsley, P., 1968, Convergence of probability measures (Wiley, New York). Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38,387-392. Debreu, G., 1972, Smooth preferences, Econometrica 40,603-615. Debreu, G., 1974, Four aspects of the mathematical theory of economic equilibrium, Working Paper IP-211 (Institute of Business and Economic Research, University of California, Berkeley, Calif.). Dierker, E., 1974, Topological methods in Walrasian economics (Springer, Heidelberg). Dierker, H., 1975, Smooth preferences and the regularity of equilibria, to appear in Journal of Mathematical Economics. Fort, M.K., 1949, A unified theory of semi-continuity, Duke Mathematical Journal 16, 237246. Hartman, Ph., 1964, Ordinary differential equations (Wiley, New York). Hausdorff, F., 1957, Set theory (Chelsea Publishing Company, New York). Hicks, N., 1965, Notes on differential geometry (Van Nostrand Reinhold, New York). Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton). Kannai, Y., 1970, Continuity properties of the core of a market, Econometrica 38,791-815. Mas-Colell, A., 1972, Smooth preferences and differentiable demand functions, Working Paper IP-175 (Institute of Business and Economic Research, University of California, Berkeley, Calif.). Mas-Colell, A., 1974, Continuous and smooth consumers: Approximation theorems, Journal of Economic Theory 8,305-336. Parthasarathy, K.R., 1967, Probability measures on metric spaces (Academic Press, New York).