Continuity of equilibria in exchange economies

Journal

of Mathematical

Economics

22 (1993) 27-34.

North-Holland

Continuity of equilibria in exchange economies* Zihao Liang Chinese Academy of Sciences, Beijing, China Submitted

April 1990, accepted

January

1992

The purpose of this paper is to extend the theory of regular economies to the space of exchange economies with continuous (Co) preferences. It is shown that economies with equilibria that are contained in a finite number of arbitrarily small disjoint balls form an open dense set in the space of Co economies.

1. Introduction Debreu (1970) pioneered the study of regular economies and proved that for economies described by initial endowments, the set of non-regular economies forms a negligible set in the sense that it is a closed set of Lebesgue measure zero. In the past two decades, the study of regular economies has been an active area. Mas-Cole11 (1985) summarises the theoretical developments since 1970. But all such models have been in a differentiable framework. In the space of Co economies, there is no generic subset such as open dense set in which economies possess only finitely many equilibria. Since the c’ (r-2 1) topology is much liner than the Co topology, economies which are close together in the Co topology are probably far away from each other in the C’ (t-2 1) topology. Therefore, even though two regular economies are very close in the Co topology, we cannot expect that they have the same number of equilibria. Furthermore, in a continuous framework, if indifference curves are kinked, we can find a whole range of prices that are equilibrium prices; hence near some regular economy there are economies with infinite Correspondence to: Mr. Zihao Liang, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, China. *This paper contains some results from my M.Sc. thesis submitted to the Institute of Systems Science of the Chinese Academy of Sciences in 1987. I want to thank Professor Wang Yu-Yun, my supervisor, for guidance and encouragement. I also thank Professor W. Shafer and anonymous referees for encouragement and helpful suggestions on previous versions. Of course they are exempt from responsibility, and remaining errors and shortcomings are mine. 03044068/93/$06.00

0

1993-Elsevier

Science Publishers

B.V. All rights reserved

28

Z. Liang, Continuity

of equilibria

in exchange

economies

equilibria. However, if we change our attention from the number of equilibria to the distribution of equilibria and replace finiteness of equilibria with being contained in finite disjoint small balls, we can show that the set of economies with equilibria which are contained in a finite number of sufficiently small disjoint balls is an open dense set in the space of Co economies. In this paper we deal with pure exchange economies with continuous (Co) preferences. In section 2, we introduce some notation and the economic model. In section 3, it is shown that economies with equilibria that are contained in a finite number of arbitrarily small disjoint balls form an open set in the space of Co economies. It is also proved that the equilibrium price correspondence, which assigns economy E to the set of equilibria of E, is lower semi-continuous at every regular economy and upper semi-continuous in the whole space of economies.

2. The economic model The economies we consider are exchange economies. In an exchange economy, there are I commodities and m consumers. Each consumer is described by his preferences and initial endowment. We assume that the consumption set is iw$+ = {x=(x1,. . . , xr) E 5X’1Xi>O for all i}, the price set is S’;: = {x E rW$+ ( 1(x1(=I!= 1x: = l}, and the initial endowment is WE@+ +. The following notation is the same as in Mas-Cole11 (1985): 2 denotes preferences on lP+ + ; is Co strictly monotone and strictly convex and satisfies the boundary condition};’ P,‘, = { 2 E P,“,( 2 is a Cz preference relation}; Pz= and P& are endowed with the compact-open topology [see Mas-Cole11 (1985)]. P,“,= { 2 ( 2

Let &‘=( Pi, x F@++)” (r=O, 2) denote the space of Cr (r =O, 2) economies. They are endowed with the product topology. For consumer (2, w) E Pfc x l@++, his demand function is d( t, w, .) : Sy: -+ @++. Here we restate a result from Mas-Cole11 (1985) in the following form. Theorem 1.

[Mas-Colell(198.5,

d:P,O,xR:+

p. 74 and p. 90)]

(i) The demand function

xS:-:+R:+,

is continuous: and ‘A preference relation {y E W’++ 1ykx) XEK.3

t on R’+ + is said to satisfy is closed relative to R’.

the boundary

condition

if, for every

29

Z. Liang, Continuity of equilibria in exchange economies

(ii) PfC is dense in P,“,.

Since preferences satisfy the boundary condition, we know that for each consumer (&w), ((d(>,w,p)jl++cc as p+pO~aS~:=S’+I’\S~:. Furthermore, we have the following proposition. For any k>O, there is a compact set Proposition 1. Let (to,w,)EP,OcxW++. M cS$-: and a neighborhood U x V of (kO, w,), such that for each p# M, ((d(t,w,p)lll>k, where ((~j(~=rnax{Ix~l: i=l,...,I}.

This proposition will be proved in the appendix. 3. Continuity of equilibria w,,,)}EP, and d(>i,Wi, .):Sy:+R’++ Let E={(2:lr~1),~..,(2,, ith consumer’s demand function. The map

be the

<(E,P)= f Cd(ki,wt,P)-wtl, i=l

is the aggregate excess demand function. Let ~:PxS’;j-+R’-’ be obtained from t by deleting the last component. By Walras’ Law, t(p) = 0 if and only if p(p) = 0. Therefore we only need to study the properties of r. e is called the quasi-excess demand function. Economy E E p is called a regular economy if 0 E Iw’-’ is a regular value of 5”((E,.):S:_:+Rw’-‘. Let R(c?) denote the set of regular economies in t?. Lemma 1.

R(t?)

Proof.

is a dense set in p.

Let U EC?’ that on P’, V= U n R(C) is dense in nonempty. Therefore

be an open set. Since the topology on 6 is finer than p is an open set in 6. V n R(p) is nonempty because fl. Hence U n R(c)= U n (p n R(p))= Vn R(p) is 0 R(p) is dense in Pp.

The equilibrium correspondence P : p -+S$-: is defined by P(E) = {p E ST: 1p is an equilibrium of E). Proposition 2. semi-continuous.

J.Math-B

The

equilibrium

correspondence

P: t;“-S$-j

is

upper

30

2. Liang, Continuity

of equilibria in exchange economies

Proof Let E, E 8” and Us Syj be an open set such that P{E,} c U and its closure rf ES’;: is compact. Then b =mir~,,,~+-~,, ((~@,,p)~] >O. Take an E such that 0 -CE< b/2. (i) By Proposition 1 there is a compact set M ES::, k>O and a neighborhood V(E,) of E, in pa, such that for each E E I/(&,) and p $ M, IIQE, p)(J > k. (ii) By Theorem l(i), r: &e x Sy: +R’- ’ is continuous; hence for each p’ E s:_: there are neighborhoods U(E,) of E, and V(p’) of p’ in Syj such that for each (414~ U(-%) x VP’), ~~~(E,P)-~(E,,P)~~~~~~“((E,P)-~(E,P’)~~+I) ~((E,P’)--(E,,P’))I<(E/~)+(E/~)=E. S’mce E, x { rf LJ M} is compact, there are a finite number of neighborhoods Ui(E,) x V,(p:) for i= 1,. . . , k, which cover E,x{UuM}. Let U(EO)=nLzl Ui(E,). Then for (E,~)EU(&) x {M\U), \\Z(E,p)-F(E,,p)llI~~(E~,P)~I-E>O; hence 5”E~)f@ (iii) Let W(E,)= I/(&,) n {n:= 1 Ui(E,)}. By (i) and (ii) f(E,p)#O for each EE W(E,) and p# U. This implies P(E)E U. Hence the equilibrium correspondence P is upper semi-continuous. •l Definition.

equilibria at E.

An economy E is called an essential economy if E has finite and the equilibrium correspondence P : ~5”-+Sy: is continuous

Proposition 3. (i) The equilibrium correspondence P: t”--+Sy: semi-continuous at each regular economy E E R(p). (ii) Each regular economy is an essential economy. (iii) The set of essential economies is a dense set in c?‘.

In order to prove this proposition,

is lower

we need the following lemma.

Lemma 2. Let B,(O)=(x~[W”Illxll~~}, I/c[w” be an open set such that the closure P and B,(O) are homeomorphic, and f : P-+B,(O) is a continuous map is a homeomorphism. Let g: P+Iw” be a continuous such that flav : db3f?,(O) x )I1
This lemma will be proved in the appendix. Proof of Proposition 3. (i) Let E0 be a regular economy, P(E,)= {pl,. . . ,pks} the equilibrium set of EO, and U EST: an open set such that U n P(E,) #8. We assume P( E,) n U = (pl,. . . .pk}, k 5 k’. Since E, is a regular economy, for i = 1,. . . , k’, there exist neighborhoods E(pi) E U of pi and B,(O) = {X E 5P1 1 llxll ~a} for a sufficiently small number a>O, such that F(,P,: ~i-B,(O) is a homeomorphism, where F is the closure of r/;.

Z. Liang, Continuity of equilibria in exchange economies

31

Let V= U:= 1 vi. Since each preference relation P,“, satisfies the boundary condition, there exists E’>O such that

Let s=min {a’,a}. By Theorem l(i), we can find a neighborhood N(E,) c &0 of E, such that, for each EEW%), (/5”(&p)-E@,,p)(l
is a set {xES~:

III~--x~ll

for some T
Definition. E, E & is a E-essential economy if there is an open neighborhood N(E,) of E, and finite disjoint s-balls Vy,. . . , V? such that, for each EEN(E& the equilibrium set of E is contained in Uy= 1 VP, and each such s-ball I/p contains at least one equilibrium of E.

For an s-essential economy, its equilibria are contained in a finite number of disjoint balls whose radii are less than E. Loosely speaking, the equilibria of an s-essential economy are ‘locally deterministic’. Proposition 4. IfE0 is an essential economy, then, for given E~0, there is a neighborhood N(E,) such that each E E N(E,) is an E-essential economy. Proof. Let P(EJ={p,,... disjoint s-balls such that dence P is continuous at that for each EEN(EJ, equilibrium of E. This economy. 0 Proposition 5. Proof.

,pk} be the equilibrium set of E,, and VI,. . . , 4 be P(E,) E Uf= I &. Since the equilibrium corresponE,, there is a neighborhood N(E,) of E0 in p such P(E)zUf=, I$, and each v contains at least an implies that every E E N(E,) is an E-essential

The set of &-essential economies is an open dense set in p.

Combine Proposition

4 and Proposition

3(iii).

Appendix Proof of Proposition I.

Let V C W+ + be a neighborhood

of w,, such that the

32

2. Liang, Continuity of equilibria in exchange economies

closure B C R’++ is compact. Let ae = (a,. . . , a) be a lower bound of V, i.e., aelx for all XEV, where e=(l,...,l)E@++. For ksa, the proposition is obvious. Now, we assume k>a. First we show that for consumer (k,,,ae), there is a compact set such that for each p $ Sy:\M, he can afford some consumption point on the indifference surface I&, = {x E RI++ 1x N o 2ke}, which means that there is an x E I$_ such that p. x 5 p. (ae). Let C be the smallest cone which includes the set (&--ae), i.e., C={l(x-ae))xE12ke, AER++}. Since k0 satisfies the boundary condition, C is a closed convex cone. Let C* = {p E [w’ ) p. y 20 for all y E C}. C* is a closed convex cone because it is the polar of (-C), a closed convex cone. Since 2 is strictly monotone, {xE@++jx>2ke)sC’. Hence C*r@++. Let M=C*nS’-‘. Then McSy: and M is compact in Sy:. Let p E ST:\ M, i.e., p 6 C*. Then there is an x E Iike such that (i)

M GS~:

p’(x-ae)sO. This implies that consumer (k,,, ae) can afford some consumption point x E I;,,. (ii) Since k0 satisfy the boundary condition, cone C is determined only by a bounded subset of the indifference surface &, i.e., there is some d> 0 and D={x~l!j~~(llx\I~d) such that C={I(x-ae)(xED}. Let B= C n D cI&,. Then B is a compact subset. By (i), each consumer (2, ae) can afford some consumption point in B. (iii) For each 2 E Pfc, let ug denote the normalized utility function for 2, i.e., u*(Ae) = 1.. for 1.E R, +. Let 0
N(B,~,t,)=(tEP,q.lIlu*,(x)-u*,,(x)l(
of to.

Therefore, for each 2 EN(B, E,>,},

each consumer i.e., xtke. Since for pcS’,I:\M consumption point in B, we know d(t, w,p)Zke.

(>,ue)

x E B,

can afford some

Z. Liang, Continuity of equilibria in exchange economies

Because 2 is strictly monotone,

Ild(2, w,P)(/I >k.

33

0

In order to prove Lemma 2, we need the following properties. A.1. Let B” be a unit ball in R”, and f: B”-+B” be continuous such that f(x)=x for euery xESn_l. Then there exists some x E B” such that f(x) = 0. Proof. Assume that for every XE B”, f(x)#O. Let g(x)=f(x)/fIl(x)ll. Then the map g: B”-+S n-1 is continuous and satisfies g(x) =x for all x~S’-r, i.e., g would be a retraction on S”- ‘. It is well known that there does not exist such a map [see Hirsh (1976, p. 72)]. q A.2. Let B” be a unit ball in IX”, and f: B”+B” be continuous such that f Is”-I : s”-’ -S-l is a homeomorphism. Then there exists some x0 E B” such that f (x0) = 0. Proof.

Define F: B”-tB” by F(x) =

0

for

x=0,

11x//f -‘(xlj(xl()

for

x+0.

It is easy to verify that F: B”+B” is a homeomorphism. Define f ‘: B”+B” by f’(x) = F(f (x)). It is easy to verify that the map f’ is continuous and f’(x)=x for x~S”-i. By A.l, there exists X~E B” such that f ‘(x,) = F(f(x,)) =O. From the definition of F, F(y) =0 if and only if y =O. Hence f(xo)=O.

0

Proof of Lemma 2.

For the sake of simplicity, we assume B,(O) = v= B”=

{xq((xIjSq. Let B,={x~R”~~lx~~~2}.

G(x) =

Define the map G:B,dB,

by for

g(x) (2-((~(l)g(~III~II)+Cl-~~-((~((~lC~f~~/((~l(~l

[(XII5 1

for 1S((x((S

It is easy to verify that G: B2+B, is continuous and that GI,,,:dB,+aB, is a homeomorphism. By A.2, there exists some X~E B, such that G(x,,) =O. From the definition of the map G, we know that for every XE B,\B”, G(x) #O, i.e., x,, E B”. Therefore, g(x,,) = G(x,) = 0. 0 References Allen, B., 1981, Utility perturbations Economics 8, 277-307.

and

the equilibrium

price

set, Journal

of Mathematical

34

2. Liang, Continuity

of equilibria in exchange economies

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