Title

Fixed points and social equilibrium existence
without convexity conditions

Author(s)Urai, Ken; Yoshimachi, Akihiko

Citation

大阪大学経済学. 2023, 73(2・3), p. 135-147

Version Type VoR
URL

https://doi.org/10.18910/93337

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

Vol. 73 Nos. 2・3

OSAKA ECONOMIC PAPERS

December 2023

Fixed points and social equilibrium existence without
convexity conditions＊Ken Urai † and Akihiko Yoshimachi‡

Abstract
The purpose of this paper is to show the existence of equilibrium for economic models with
non-convex constraint correspondences. Our proof is based on a new fixed point theorem (ageneralization of Kakutani-Fan-Glicksberg) depending merely on conditions for local directionsof correspondences, so that we may obtain a general condition that may not depend on any global
continuity and convexity on values of mappings for the existence of economic equilibria. We also
apply the result for the existence of competitive equilibrium in economies with default and/or
bankruptcy.
JEL Classification: C62; D51; D52; D82; G33
Keywords: Fixed point theorem, Abstract economy, Non-convexity, Competitive equilibrium,
Default, Bankruptcy.

1 INTRODUCTION
In this paper, we prove the existence of equilibrium for an abstract economy with non-convex
constraint correspondences. Moreover, we apply the theorem to the existence of general equilibrium
problem in economies with default and/or bankruptcy.
The method we have used here is based on a new fixed point theorem depending merely on theconcept of local directions of correspondences (Section 2, Theorem 1), so that we may obtain ageneral condition which does not depend on any global continuity and/or convexity on values of＊†‡This paper was originally written by the late Akihiko Yoshimachi and myself, Ken Urai, and submitted to the Journalof Mathematical Economics in 2004. However, about four years later, the manuscript was returned because we could
not find a reviewer. Fortunately, through the kindness of Professor Hiromi Murakami, I was given the opportunity topresent this paper in the Osaka Economic Papers, a publication commemorating my 60th birthday. I firmly believe thatthe late Professor Yoshimachi would have shared in our delight. It is with great respect for his memory that I hope thispaper to be a valuable addition to this precious collection.
Professor, Graduate School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN.
(Internet e-mail address: urai@econ.osaka-u.ac.jp)
The earliest draft of this manuscript was written on March 14, 2002 and revised on March 25, 2004. He was a lecturerof Faculty of Commerce, Doshisha University until January 2013.

－ 136 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

mappings for the existence of social equilibria (Section 2, Theorem 2). We further generalize the result(Section 2, Corollary 2.1), and apply it to the general equilibrium existence problem for the default
economy (Section 3, Theorem 3) as a generalized model for economies with asymmetric information(see, e.g., Dubey-Geanakoplos-Shubik (2000)), which is one of the most important framework under
the recent development in the general equilibrium theory.
Under the settings of social equilibrium, our result may be stated as follows: If we select appropriately
for each player

and a strategy profile, a non-empty subsetof (possibly) non-, the condition for existence of equilibria may be obtained merely as aconvex constraint set

and the better set

local condition on directions to the set

fromfor eachwhich isweaker than to assume any convexity and/or global continuity. Hence, in the competitive equilibriumwith the budget constraint without default

framework (Section 3, Theorem 3), if we identify

with the constraint allowing for default, the equilibrium existence condition may directly

andbe characterized as consumer ’s tastes for keeping promises (moral actions and plans) under
exogenously specified default penalties.
It seems somewhat incredible that we have shown the existence of equilibrium without using any
convexity condition for budgets. As we shall see in Section 2, mathematically, it is not the convexity
but the direction of correspondence that is important for the existence of fixed points, so that, theconvexity of the budget is superfluous as long as we use conditions for the direction of better sets andconstraint sets.
2 EQUILIBRIUM FOR AN ABSTRACT ECONOMY
In this section, we show the existence of an equilibrium for an abstract economy having non-convex
constraints. Subsection 2.1 is to develop a fixed point theorem based on the concept of local directionsof correspondence. Theorem 2 in subsection 2.2 provides a general result on economic equilibrium
existence in which the choice space for the abstract economy is taken to be a subset of locally convex
Hausdorff topological vector spacehaving topological dual

. Constraint correspondences aretaken to be (possibly) non-convex. Preferences of agents are also taken to be (possibly) non-ordered
and non-convex. They are not even supposed to be continuous in the ordinary sense. What we haveassumed here is merely conditions on local directions of correspondences.1
2.1 A Fixed Point Theorem
be a locally convex Hausdorff topological vector space having topological dualLet

say that a correspondenceneighborhoodofthat a correspondenceof1

has a fixed direction
such that for all

near

and

,

has a continuous direction nearand a continuous function. We

if there exists a.We also say

if there exists a neighborhood

such thatfor all

The mathematical results in this paper are extensions of theorems in Urai (2000) and Urai-Hayashi (2000). See also
these papers for other game theoretic applications and market equilibrium existence results.

December 2023

Fixed points and social equilibrium existence without convexity conditions

and, where－ 137 －

is given a topology of compact convergence. Moreover, we saythat a correspondence

has a compact valued upper semi-continuous direction near

if there exists a neighborhoodofcorrespondence

such that, where the topology on
It is clear that if

and a non-empty compact valued upper semi-continuous

has a fixed direction near

has a continuous direction nearfor all,andis a topology of compact convergence.
, then, thenhas a continuous direction near

, and if

has a compact valued upper semi-continuous direction near

. The first result of this paper is the following fixed point theorem which, as we shall see below, may
be considered as a generalization of Kakutani-Fan-Glicksberg’s fixed point theorem.
Theorem 1. Let

be a non-empty compact convex subset of a locally convex Hausdorff topological

vector spacehaving topological dual,

Suppose thathas a compact valued upper semi-continuous direction near

. Then,. Let

be a non-empty valued correspondence.

does not have a fixed point. Then, since

pointsand their open neighborhoods, and for each,is compact, we haveinsuch that, there is a non-empty compact valued upper semisatisf ying that

continuous cor respondence
. Let

,

,

, be a partition of unity subordinated. For eachas.

is non-empty and compact, and the correspondence

is upper semi-to, define a set

for eachcontinuous. Let, and for all

such that, we havethat, we havefor all

for all. Then, for all

for all, sofor all

for all . That is,. Hence,is a non-empty
and for anconvex valued correspondence having no fixed point. On the other hand, for each
arbitrary element, there is an open neighborhood. (Indeed, since
is compact, we have neighborhoodof, we have
of

and

of

in

satisfying,

for all. Sincesuch that for all

andby the definition of the topology of compact convergence. Moreover, sincesemi-continuous, we may obtain a neighbourhood
Hence,

such thathas a fixed point.

Proof. Suppose that

The setfor eachofsuch that,is upper,has a fixed point by Browder’s fixed point theorem (Browder (1968)), a contradiction.

.)▪

Note that the above theorem is a generalization of well known Kakutani-Fan-Glicksberg’s fixed
point theorem2 since if
2

is a non-empty compact convex valued upper semi-continuousLet
be a non-empty compact subset of a locally convex Hausdorff topological vector space. Every non-empty
has a fixed point. (c.f.
closed convex valued upper semi-continuous correspondence
Glicksberg (1952).)

－ 138 －OSAKA ECONOMIC PAPERS

correspondence,satisfies all conditions in Theorem 1. Indeed, if, there is aifVol. 73 Nos. 2・3

such thatfor allmeans that

Then, the upper semi-continuity ofis compact and convex andby the separation theorem.

has a fixed direction

near .

In view of economics, the better-set correspondence for a continuously differentiable quasi-concave
utility function may be considered as a typical mapping having a continuous direction near every
point. It is also easy to check that a better-set correspondence which may be represented (at least
locally) by quasi-concave utility functions have a compact valued upper semi-continuous direction
near every point. Since the budget correspondence is ordinarily assumed to be non-empty compact
convex valued upper semi-continuous, the generalized fixed point theorem enables us to treat these
two mappings (the better-set correspondence and the budget constraint correspondence) in a unified
manner so that we may characterize the existence of economic equilibrium merely through conditions
on local directions for these two mappings (see Theorem 2).
2.2 Existence of Equilibrium for an Abstract Economy, whereLet us define an abstract economy as a list,
is the choice set ofof agents,for each,is the constraint

correspondence of , andandFor eachis the finite index set

is the preference correspondence of .

, we say thatis a maximal point for

under

3

. An equilibrium of the abstract economy

andsuch that for eachunder

point forfor all,iff
is a point

and, i.e., a maximal.

The second result of this paper is the following equilibrium existence theorem. Assume that for
,

each
dual

is a subset of locally convex Hausdorff topological vector space. We say that a correspondence

direction near

has a compact valued upper semi-continuous

if there exist a neighborhoodofand a non-empty compact
such thatvalued upper semi-continuous correspondence,

having topologicalfor all4, and. The next theorem characterizes the existence of equilibrium

merely through conditions on local directions of correspondences,’s and’s. The condition forincludes all cases such that preferences are locally represented by continuous utility functions.
The condition forincludes all cases with continuous closed convex valued upper semi-continuous

constraint correspondences.
Theorem 2. Suppose that abstract economy
For each,

topological vector space
3

4

satisfies the following conditions:is a non-empty compact convex subset of locally convex Hausdorff
having a topological dual.

In the following, given a vector, we denote by
the -th coordinate of as long as there is noambiguity.
The definition in the previous section may be identified with the special case such that
and.

December 2023

Fixed points and social equilibrium existence without convexity conditions

For each,

is a (possibly empty valued) correspondence having anear every

compact valued upper semi-continuous direction
For each

,

Then,

.

is a non-empty valued correspondence having a compact
near every

valued upper semi-continuous direction
For each－ 139 －

such that, the set.
.5

is open in

has an equilibrium.

The theorem says that as long as the feasible set and the better set at a certain action level havecontinuous directions, (i) the convexity of these sets and the global continuity for these mappings are
not necessary, and (ii) merely the continuous local direction property of each mapping completely
characterize the existence of equilibrium.
We provide a proof of the theorem under more general settings in the next corollary. In corollaryas2.1, if we take, condition (B5) is nothing but a tautology, hence, we have

Theorem 2. The corollary is used in the next section, where the existence of the default leads to more.

serious non-convexity for

Corollary 2.1. Suppose that abstract economy
For each,

is a non-empty compact convex subset of locally convex Hausdorff
having a topological dual

topological vector space
For eachsatisfies the following conditions:,is a (possibly empty valued) correspondence having anear every

compact valued upper semi-continuous direction
For each.

,.

is a correspondence such that there is a non-empty valued

correspondence,

for allsemi-continuous direction

near every

such that

For each, the set

For eachandan open neighborhood, having a compact valued upper.
is open insuch thatofsuch that for all

and.
, there exists,

implies.
Then,

has an equilibrium.

Proof. In this proof, we denote by
set of direction
5

atfor eachthe setand, the. Assume that there is no equilibrium for

Under (A2) and (A3), condition (A4) may be dropped if “for each,is a subset of theclosure of the interior of ” and “for each, the value,, is not equal toand does notdepend on the -th component, , of .” In view of economics, these settings are quite natural. Of
course, (A4) is automatically satisfied if
is lower semi-continuous and has an open graph.

－ 140 －OSAKA ECONOMIC PAPERS. Then, for eachVol. 73 Nos. 2・3, there is at least one

satisfying one and only

one of the following three conditions:
.

(a)
(b)

is an interior point of.

(c)is a boundary point of.

In the following, we fix such an agent(satisfying one and only one of the conditions, (a),. For each(b), and (c)) for eachand case (a)–(c) of, define, locally, on a certain openof , a correspondenceneighborhoodasfor case (a),for case (b), andfor case (c),wherefor case (a) is the local direction for(use (B4)),

impliessuch that

open set

on a certain open set

for case (b) is the local direction for

impliesandon a certain open neighborhood(c) is the local direction forso thatmay chosesuch that, andfor allfor case (a). Moreover,so thatfor all(c). Note also that for each,is convex valued and. Sincebe such a sub-covering, and letin defining

respectively forbeand (c). Let

for cases (b), be a partition of unity subordinated to,. For eachand

. Moreover, for eachfor case (a) and besatisfying thatand , denote byand, (i.e.,the set ofasif. Note that since for each), the mapping

is clearly convex valued. Moreover,onand, respectively, and define a correspondence, whereifforbe names of agents fixedand, let correspondence

indices

for cases (b) andis compact, there is a finite sub-covering of the covering

. Let

of

for caseof . Under (B2) and (B4), we

under (B3), we may choseeach

on a certain

is non-empty valued since every
for eachandis non-empty valued
such thatis clearly non-empty when all elements ofor (c). We may also check the set to be non-empty when the elements ofunder condition (B5). Note thatsuch thathas no fixed point. On the other hand, the mappingand since(Indeed,, there is at least one

, and

.
are type (b)are type (a) or (c),’s of type (a) and type (b) never intersect.) Furthermore,

a compact valued upper semi-continuous direction near eachfor all

since

has

implies that
. Hence,has a fixedDecember 2023

Fixed points and social equilibrium existence without convexity conditions－ 141 －▪

point by Theorem 1, so we have a contradiction.
3 APPLICATION FOR THE DEFAULT ECONOMY

In the previous section, we have shown the theorem on the existence of equilibria for an abstract
economy under non-convex constraint correspondence. We apply in this section the theorem to provethe existence of equilibria allowing for default and/or bankruptcy.
Under general equilibrium settings, problems with asymmetric information (including default and/or bankruptcy as well as adverse selection and moral hazard) are closely related to non-convex agents’budget constraints since we may settle the problem into the situation that the constraint (the price) asa buyer is different from the constraint from the view point of a seller. Especially, the non-convexity
problem in economies with default and/or bankruptcy is classical and has been treated in many
literature (see, e.g., Green (1974), Grandmont (1982), Eichberger (1989)).
Recent several authors (see, e.g., Zame (1993), Dubey-Geanakoplos-Shubik (2000)) have avoided
the problem by considering a completely anonymous asset market in which the average default rate (foreach asset in the market) is perfectly expected and each agent is allowed simultaneously to buy andsell (go long and short in) the same asset. It is interesting that they have changed the problem about
moral hazard into the choice problem under the taste for morals of each agent. Such an approach,however, (1) fails to describe an accidental or unintended default and/or bankruptcy, (2) implicitlychanges the concept of standard economic equilibrium where for each commodity, demand and supply
are treated in a total net amount, and, especially, (3) they have used the concept of the “complete
anonymity” as a “necessary” condition for the existence of equilibria. Of course, it is more desirable
to show an existence of equilibrium directlywith non-convex budget constraints with or without thecomplete anonymity and perfect foresights.
To overcome the non-convexity problem on budget constraints in economies with default or
bankruptcy, many authors use the smoothing method under continuum of traders or the methods ofStarr (1969). An exceptional approach (in temporary general equilibrium model) with money and thecentral bank under a special bankruptcy rule is Eichberger (1989).6 Also, Green (1974) proved theexistence of an approximate equilibrium, following the Starr method, by introducing a bankruptcy
rule and disutility from the extent of bankruptcy. In the following, we treat the problem by using theconcept we have used in sections 1 and 2, the direction of correspondences.
3.1 The Default Economy
Now let us consider a two-period economy, periodof agents indexed by. In periods

commodities. We also assume that there are(present) andandand purchased in the asset market in each period. Asset7

there are, respectively,

types of assets indexed by
7

6(future). There are

,

types

types of realthat is sold, specifies bundles of goods to

A further development of the model in Eichberger (1989) is treated in Yoshimachi (1999).
We suppose that the number of states in period 2 is one for the sake of notational simplicity.

－ 142 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

be delivered in period . We assume that the payoff of each asset is specified to a certain commodity.
Hence, in each period, two types of markets, the spot market for
asset market for

types of real commodities and thetypes of assets, exist.

Prices for commodities in the spot market and the asset market in periodandare denoted by, respectively. The set of prices in period

is defined by

be the set ofprices.
Agent

has an initial endowment,, of real commodities, and a. We assume that

consumption set,

is a closed convex subset bounded from below

for each . The preference of agent

such thatby a continuous concave utility function

is represented at each
assume that at each

which may possibly be non-ordered, the value of utility function

with respect toat, i.e.,. In order to determine his plan, agent

commodities and the default rate of each asset in period. We also
is continuous
forecast the prices of. We assume that agent

has continuous

expectation functions for prices,
(1)
(2)
and for delivery rates,
8

For a list of other person’s consumption plans,
,

,

(3)

,

, agent, andunder his budget constraints, where

his utility function

choosesso thatmaximizeandindicate thequantity of assets purchased and sold, respectively. Notice that for each agent , each asset
limit on the quantity for sales,,,. (An elementin the following.) In period , agent

has a. We now define the budget set of agent ,ofwill be denoted byshould satisfy
(4)In the same way, we may suppose an expected budget constraint at period

8

for

to be

In the domains and ranges of these functions, the first
denotes the set of prices in period , the second
denotes the set of prices in period , anddenotes the set of delivery rates. Of course, the case with rational
expectations is treated as the special (identity function) case of these expectation functions. For the concept of delivery
rates, see Zame (1993) and Dubey-Geanakoplos-Shubik (2000).

December 2023

Fixed points and social equilibrium existence without convexity conditions－ 143 －



where

,

, and the(5)operation allows for the

possibility of default. Also, we have to allow for the bound on sales for asset

for agent

and anatural condition on the quantity of assets purchased and sold, respectively.


(6)
(7)

Thus, the budget set of agentis

As in Dubey, Geanakoplos, and Shubik (2000), we define for each agent

a constant,

(8)
, the

real default penalty on . We assume that the default penalties are assessed directly in terms of utility
and are proportional to the size of the default. Hence, underof agent
of

for

at

,

, the utility levelis as follows: (9)denotes anwhere constraints (4)–(7) are not taken into consideration and the term

appropriately defined indirect utility for assets in the second period. In the following, we neglect every
terms onandin equation (9) for the sake of simplicity. When condition (4) is satisfied, thesecond entry in the last term in (9) (i.e.,,) represents the size of the default of
under . We denote the amount by.
the economy stated in the above. An

Denote byequilibrium for economy
such that for

associated with the action and planis,over ’s budget under, and market clearing conditions together with a specification of expectation functions such that

(10)
(11)

－ 144 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3(for the short run equilibrium), or
(12)
(13)
are identity functions,

(14)(for the long run equilibrium with perfect foresights). As in Eichberger (1989) and Dubey,
Geanakoplos, and Shubik (2000), consumers or agents having budget constraints like the above
have a common problem that the budget set may not be convex as long as we suppose thenatural setting (4).
3.2 Existence of Equilibrium
sufficiently small, we define an abstract economy

For eachbe the set of all vectors inLetthe setdenote byfor the economy

as follows.

whose all coordinates are greater than or equal to. If we restrict the set of prices for economy, andon the compact set, then the set of all consumption and asset holding levels that are not associated. Under thewith default is also confined in a certain compact cube,
default penalty, we may take such awhich maximizes

inso large that for eachmaximizesas, and let the choice setof abstract.

For each agent
correspondence at(8) on.9 Forin, leteconomy, every action, define the preference correspondence
under

on

as the better set, and define the constraint correspondenceas.

For -th agent, an auctioneer, define a utility (payoff) function,

, as


(15)



(16)

for the short run equilibrium, and as

for the long run equilibrium with perfect foresights. For agent , define the choice set as(), and the constraint correspondence as
Let the set of players for

9

be

,

.
. Then, the following two additional structures

As stated before, the last two terms(the amount of assets purchased and sold in the second period) together
will be neglected in the following for the sake of simplicity.
with their all entries inDecember 2023

Fixed points and social equilibrium existence without convexity conditions－ 145 －

(the minimum wealth structure and the minimum delivery rate structure) enable us to show that for a, an equilibrium forsufficiently small

is an equilibrium for the default economy.

Assumption (Minimum Wealth) : There is a continuous wealth transfer mechanism (a function) which assures for each agent

and, a wealth level which is.

strictly greater than the minimum level under

Assumption (Minimum Delivery Rate) : There is a continuous insurance mechanism (a function
financed by the transfer of wealth) which assures for each asset, a strictly positive minimum delivery rate underand.

It is true that in considering these two mechanisms, we have to rewrite the budget for (equations (4)and (5)) and reconsider all related assertions. We shall omit, however, the process since (i) it is a routine), andtask (e.g., use a continuous transfer function on endowments
10(ii) they are not indispensable structures.

We now state the main theorem in this section. Let
for eachbe the set of elements of. It is clear that, the set of moral actions and plans,,

is convex. It is also easy to verify that for each
respect to

,

,

,

,

is continuous with, and , (see equation (9) for the definition of

better set correspondence defined bysuch that

on

into

,) and at each

and , theis convex valued. Hence, each, is a convex valued correspondence having continuous local utility representation,

so that has a compact valued upper semi-continuous direction at every point.
Theorem 3. Assume the minimum wealth condition and the minimum delivery rate condition.
has a long run (perfect foresight)

Then, the economy
equilibrium if the following condition is satisfied.
(Local Directions for Morals) At each

,

and

, and

such that for allofimpliessuch that, there is an open neighbourhood,.

Adding to the above condition, if we suppose for eachand, that the range ofis a subset ofstrictly positive reals, then a short run equilibrium exists.

10

It is always possible to replace these two mechanisms with more individualistic conditions on endowments, weakerboundary conditions, and standard limit arguments. The mechanism on default penalties, however, is rather essential inthis paper. The ordinary cubic truncation method fails to assure demands under truncation to be true maximal elements
under the non-convexity of constraints.

－ 146 －OSAKA ECONOMIC PAPERS

Proof. The compactness ofVol. 73 Nos. 2・3

assures that there are positive numbers

such that all agents have an ability to pay (without default) greater thanrates are greater thanandand all delivery. For the case with short run equilibrium, we may suppose without loss ofgenerality that the range ofis a subset of. Since utility functions are strictly positive and, and since feasible utility levels are bounded, the existence offor anall equilibrium prices, if such exist, should be inandmeans thatsufficiently small. Moreover, since

has an upper bound, and since the existence of an upper boundthe aggregate supply in economy

for the aggregate demand means the existence of an upper bound for the utility levels (without
default) for all , we may take such anso small that under any boundary price, for the short run case) and a delivery rate(

, the summation of excess demands
is positive. (Indeed, byof all commodities (in period 1, resp., for the short run case) undertakingarbitrarily small, sinceincludes all non default actions and plans, the utility level underboundary prices can be taken arbitrarily large. Hence, the total excess demand under such boundary
andprices cannot be bounded.) That is, the price system(and, resp., for the short run) appreciates all boundary excess demands positively. This fact together
with the Walras’ law in all periods (in period 1, resp., for the short run) implies that the maximal point
for player

in abstract economy

have chosen the boundbe a solution inforfor such anis the market clearing price. Moreover, since weso that for each , every solution to the utility maximization in, we have shown that an equilibrium foris an equilibrium for.

for economy

Next, we have to show that abstract economy

tosatisfies allconditions in Corollary 2.1. Condition (B1) is clearly satisfied. For consumers, the minimum
andwealth condition assures the continuity ofvalue oftogether with the non-emptiness for the. By the convexity and closedness for the value of, (B3) is automatically satisfied.

Moreover, as stated before, the better set correspondence under

has a compact valued uppersemi-continuous direction near every point, so that (B2) is also satisfied. Furthermore, the continuity
with respect tooftogether with the continuity of

,

,

,

, and

means thathas an open graph. This,, assures (B4). It is clear that the condition on Local Directionsfor Morals assures condition (B5). For the auctioneer, the upper semi-continuity and closed convex
non-empty valuedness of price transformation correspondence satisfies condition (B2). Moreover, by
for all , (B3), (B4), and (B5) are automatically satisfied. Hence,

considering▪

satisfies all conditions in Theorem 2.

A typical example for preferences assuring the condition of local directions in Theorem 3 is thefamily of CES utility functions on the positive orthant consumption sets. For any utility function,
however, we may expect that an appropriate level of utility punishment always assures the condition
since at eachsuch that, the better set at

eventually include a point infor allsufficiently large.
(Graduate School of Economics, Osaka University)

December 2023

Fixed points and social equilibrium existence without convexity conditions－ 147 －

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