Discussion Papers In Economics
And Business

General Equilibrium Model for an Asymmetric Information
Economy without Delivery Upper Bounds

Ken Urai, Akihiko Yoshimachi and Kohei Shiozawa

Discussion Paper 13-27-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGeneral Equilibrium Model for an Asymmetric Information
Economy without Delivery Upper Bounds

Ken Urai, Akihiko Yoshimachi and Kohei Shiozawa

Discussion Paper 13-27-Rev.2
March 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGeneral Equilibrium Model for an Asymmetric Information
Economy without Delivery Upper Bounds
Ken Urai∗
Akihiko Yoshimachi†
Kohei Shiozawa‡

Abstract
In this paper, we introduce production into the standard general equilibrium model with asymmetric information, which was proposed by Dubey et al. (Cowles Foundation Discussion Paper
2000; Econometrica 2005). In such an economy, there is no rational explanation for producers’ delivery upper bounds while the endowments naturally limit consumers’ deliveries. However, we showthat the typical equilibrium allocation of the asymmetric information economy necessarily and substantially depends on such exogenous upper bounds (Example 1 and Theorem 1). In other words,
an equilibrium existence theorem without such upper bounds, even if such exists, will typically failto treat the asymmetric information problem, e.g., the adverse selection problem. Hence, to treat
the equilibrium existence problem under the informational asymmetry appropriately, we have toextend the standard model so that the delivery upper bounds need not to be specified explicitly.
For this purpose, we propose a quite natural and realistic assumption with respect to the technological condition related to the market delivery, i.e., the existence of some small standardization,commoditization, and/or transaction costs of market deliveries is shown to be suﬃcient (Theorem
3).
Keywords: general equilibrium model, asymmetric information, adverse selection, market viability
problem
JEL Classification Numbers: C62; D51; D82∗ Graduate

School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560-0043, Japan; E-mail:urai@econ.osaka-u.ac.jp† Doshisha University‡ Graduate School of Economics, Osaka University; E-mail: nge013sk@student.econ.osaka-u.ac.jp

1 Introduction
Informational asymmetry problems have been traditionally treated using static partial equilibrium
arguments (e.g., Akerlof 1970, Rothschild and Stiglitz 1976). The approaches of Dubey et al. (2005) are
groundbreaking general equilibrium treatments for asymmetric information problems.1 In the context
of asset markets, that work discussed how a certain system of pooled insurance may solve equilibriumexistence problems in a default economy. In their model, as a buyer, every agent obtains an average
receipt (including defaults) in each asset market. As sellers, they can choose not to deliver their fullobligations. If we consider a rational expectations equilibrium, the default problem does not harm theexistence of a market equilibrium, i.e., the market viability problem is solved aﬃrmatively under seller–buyer informational asymmetry.
Several researchers have investigated this seller–buyer informational asymmetry using a general competitive equilibrium model. Bisin and Gottardi (1999) considered a similar problem to Dubey et al.
(2005) in a probabilistic and dynamic setting. Bisin et al. (2011), Correia-da-Silva (2012), and Meier etal. (2014) most recently considered a static setting.2 Bisin et al. (2011) considered a situation where
agents only have finer information (than the market) when they are sellers. Correia-da-Silva (2012) andMeier et al. (2014) investigated a situation where agents also have finer information as buyers and theinformation varies among agents, although their models do not have the standard general equilibrium
setting. In this paper, we adopt the general equilibrium approach of Dubey et al. (2000, 2005) and Bisin
et al. (2011).
Bisin et al. (2011) considered a model of an exchange economy where agents (consumers) can makea limited amount of delivery contracts, related to their endowments.3 However, if we consider a production economy, there is no natural counterpart for these delivery upper bounds for producers andthis fact causes some problematic situation for general equilibrium modeling of the economy. In thispaper, we show that an equilibrium may not exist if producers have no limit on delivery contracts, even

1 See also Dubey et al. (2000) and their earlier draft, Dubey et al. (1989).
2 The authors investigated a static general competitive equilibrium model independently from these recent works (Uraiand Yoshimachi 2005).
3 All of the previously mentioned works also used such upper bounds for delivery contracts.

2

if consumers do (Example 1); that an equilibrium exists if each agent has an exogenously given upperbound for delivery contracts (Theorem 2); and that, unfortunately, the equilibria typically depend on
the exogenous upper bounds if the asymmetric information does actually have an eﬀect (Theorem 1).4
Theorem 1 implies that we cannot treat asymmetric information problems successfully without using
such delivery upper bounds. Therefore, to treat the market viability problem that determines an active market structure under the asymmetric information, appropriately, we must consider the existence
problem without explicitly specified upper bounds for the delivery. Hence, to solve this problem, we
extended the model by introducing some technological conditions that represent realistic costs related
to delivery contracts such as standardization, commoditization, and/or transaction costs. We show thatthe equilibrium existence problem under asymmetric information is solved aﬃrmatively in this extended
model (Theorem 3) and that all the conditions for the market viability problem are endogenized.5
The paper is organized as follows. In Section 2, we establish the basic model and present some
substantial results for this basic setting. In Section 3, we extend the model and state the existence
theorem. All proofs are consigned to Appendix.

2 Basic Model and Results
2.1 Basic Model
The basic model and its setup are counterparts of Dubey et al. (2005) and Bisin et al. (2011), which
we modified for the standard Arrow–Debreu production economy. In the following, we introduce a classof possible markets, agents’ optimization problems, and an equilibrium.6

4 Even in exchange economies, as long as consumers can resale their goods arbitrarily, the same problems arise.

Inparticular, we can construct equilibrium non-existence examples of such economies (see footnote 12). Note that oursolution to the non-existence problem, explicit modeling of costs that are related to delivery contracts, is relevant
also in such pure exchange cases.
5 We note that the determination problem of an active market structure is simplified here among a given partition of
the set of real commodities L = {1, . . . , ℓ}. Extension of this point to more general subclass of L, which may include
sets with non empty intersection, is straightforward and will be discussed in our another paper (Urai et al. 2016).
6 Let R denote the set of real numbers, let Rn denote n-dimensional Euclidean space, let Rn be the non-negative+nnnorthant of Rn , {x = (xk )n
k=1 ∈ R | xk ≧ 0, k = 1, . . . , n}, and let R++ be the strictly positive orthant of R ,n| xn , we let x ≧ y :⇔ x − y ∈ Rn and{x = (xk )n
∈
R
>
0,
k
=
1,
.
.
.
,
n}.
For
any
vectors
x,
y
∈
R
k
+
k=1
n . For any vector x ∈ Rn , we use notations x+ := max{x, 0} and x− := max{−x, 0}. Notex ≫ y :⇔ x − y ∈ R++
n . For each finite set A, let ♯A denote the number ofthat for all x ∈ Rn , we have x = x+ − x− and x+ , x− ∈ R+
elements in A.

3

Class of possible markets:

There are ℓ types of real commodities (goods and services) indexedby k = 1, . . . , ℓ. Let L := {1, 2, . . . , ℓ} denote the set of real commodity indices and M:= {L1 , . . . , Lλ }∪λdenote a class of non-empty subset of L such that L = κ=1 Lκ . The class M is interpreted here as
the class of all possible markets and, by using this, we describe a situation that in market Lκ ∈ M realcommodities k, k ′ ∈ L are not distinguished as long as k, k ′ ∈ Lκ . In our model, a market structure will
be determined as the set of markets in M with non-zero amount of trade in an equilibrium. We also
assume that each commodity is traded in exactly one market, i.e., Lκ ∩ Lκ′ = ∅ for all κ, κ′ ∈ {1, . . . , λ}with κ ̸= κ′ . Although we can treat a more general class of possible markets, here we adapt thesimplest partitioned case to concentrate our attention on the problem concerning the exogenous delivery
upper bounds. Moreover, in most standard economic settings such as adverse selection, we only need
the partitioned possible markets framework as argued in Bisin et al. (2011). We also note that thispartitioned possible markets framework contains the standard Arrow–Debreu economy as a special casewith the finest partition: {{1}, . . . , {ℓ}}.
Since there are only λ types of markets indexed by κ = 1, . . . , λ and the economy evaluates λ typesof marketed contracts, or standardized commodities, the price space is a subset of the λ-dimensional∑λλEuclidean space Rλ , instead of the ℓ-dimensional one. Let ∆ := {(p1 , . . . , pλ ) ∈ R+
| κ=1 pκ = 1}
be the price space. For each market Lκ ∈ M, all types of real commodities k ∈ Lκ are evaluated by
identical price pκ , given a price system p = (p1 , . . . , pλ ) ∈ ∆.
Agents:There are m types of consumers and n types of producers in this economy and they areindexed by i = 1, . . . , m and j = 1, . . . , n, respectively. In our model, we assume that each agent (eachconsumer and each producer) can buy and sell standardized commodities in all the markets. We also
assume that, for all market Lκ ∈ M and all types of real commodities that belong to Lκ , every agents canidentify each of these commodity types as long as it is possessed by themselves, and use each of them as anunit of delivery to the market Lκ . However, they can only obtain a mixture of such real commodities when
they buy a standardized commodity from the market Lκ . Hence, we treat their behaviors (consumptions
and productions) and their transactions (demands and supplies) as diﬀerent variables. More precisely,
we describe consumption plans xi (i = 1, . . . , m) and production plans yj (j = 1, . . . , n) as points of Rℓ
λand demand plans zi− (i = 1, . . . , m + n) and supply plans zi+ (i = 1, . . . , m + n) as points of R+
.

4

+
+
Agents as sellers: For all agents i = 1, . . . , m+n, we assume that the supply plan zi+ = (zi1
, . . . , ziλ)must satisfy++(zi1, . . . , ziλ
)=

( ∑
k∈L1

vik , . . . ,

∑

)
vik ,

k∈Lλ

ℓ
where (vi1 , . . . , vik , . . . , viℓ ) ∈ R+
is a bundle of real commodities that agent i plans to deliver to themarkets. This equation describes the above mentioned situation that agents can identify each of the realcommodity types in a market Lκ ∈ M and use each of them as an unit of delivery to the same market
Lκ , although the market Lκ does not distinguish them and hence recognizes the delivery from agent i as∑+
the sum ziκ= k∈Lκ vik .
Agents as buyers and expectations of real receipts: As we described above, in this setting, amarketed contract κ is actually a mixture of ♯Lκ kinds of real commodities. Therefore, we assume thatagents have a certain kind of expectation for their real receipts. So we introduce an additional exogenous
parameter, sκ , which represents the ratio of the ♯Lκ types of real commodities in the total quantity ofcontracts supplied to market κ. Formally, for each κ = 1, . . . , λ, we let RLκ denote the subspace of Rℓconstructed by elements with k-th coordinates equal to 0 if k ∈/ Lκ ; that is, RLκ := {x = (xk )ℓk=1 ∈Rℓ | xk = 0 if k ∈/ Lκ }, and we take expectation s = (s1 , . . . , sλ ) as sκ ∈ ∆κ := {x = (x1 , . . . , xℓ ) ∈Lκ ∑ℓR+| k=1 xk = 1 } for all κ = 1, . . . , λ. This allows us to parametrize the uncertainty about real
receipts in the same way that we treat the prices of goods in general equilibrium modelings.

Next, we describe agents’ optimization problems. In this model, agents’ optimization problems have
two kinds of macro parameters: price p = (p1 , . . . , pλ ) and expectations of real receipts s = (s1 , . . . , sλ ).
Note that given these two parameters, which are determined in equilibrium, agents choose micro variables:
consumption or production plan xi /yj , demand plan zi− , and actual delivery of real commodities vi (whichconstitutes supply plan zi+ ).
ℓConsumers’ problems: Consumer i = 1, . . . , m has initial endowment ωi ∈ R++of real commodi-

ties, consumption set Xi ⊂ Rℓ , and utility function ui : Xi → R. Given price p ∈ ∆ and the expectation∏λof their receipts for each real commodity through the market, s = (s1 , . . . , sλ ) ∈ κ=1 ∆κ , consumer ichooses consumption plan xi with market transaction plans (vi , zi ), where zi = zi+ − zi− , to solve thefollowing maximization problem:

5

max

ui (xi )(1)

sub. toℓ(xi , vi , zi ) ∈ Xi × R+
× Rλ

(2)

v i ≦ bi ,

(3)

zi+ =

( ∑

vik , . . . ,

k∈L1

∑

)
vik ,

(4)

k∈Lλ

− 1
− λ
xi + vi = ωi + zi1
s + · · · + ziλs ,n∑p · zi− = p · zi+ +θij πj (p, s),

(5)
(6)

j=1

where πj (p, s) is the profit of producer j under price p and expectation s (in the maximization problems
described below), θij denotes i’s share of the profit of producer j (a non-negative real number satisfying∑mℓi=1 θij = 1 for each j), and bi ∈ R++ is an arbitrary taken upper bound for the consumer’s delivery.
For example, it is natural to take bi = ωi for all i = 1, . . . , m.
The following are the interpretations of the constraints. Formula (2) indicates the domains of thevariables; consumption plan xi must be taken from consumption set Xi . Eq. (3) expresses that noneof the consumer i can supply arbitrarily large amounts of real commodities. Eq. (4) means that supply
plan zi+ consists of the delivery of real commodity vi . Eq. (5) denotes that consumption xi and actual− 1− λdelivery vi must be covered by the consumer’s own endowments ωi and purchase zi1
s + · · · + ziλs . Note− κℓℓ. Finally, Eq. (6) is the budget constraint underfor κ = 1, . . . , λ because sκ ∈ ∆κ ⊂ R+
that ziκs ∈ R+p ∈ ∆.
In the above, we can chose ωi as a natural candidate of upper bound bi in the constraint (3). Thisis a natural assumption when we only consider the consumers in an economy, i.e., when we consider apure exchange economy, and such an assumption was also used in all of the early works: Dubey et al.
(2005), Bisin et al. (2011), Correia-da-Silva (2012), and Meier et al. (2014). For a production economy,
however, since producers do not have endowments, they have no such natural upper bounds. However,
Example 1 below shows that without upper bounds for the producers, an equilibrium may fail to exist
even if consumers’ delivery amounts are bounded, as in Formula (3) with bi = ωi . For this reason, in thebasic model, we also introduce the upper bound condition (Eq. (9)) for each producer using given upperℓbound parameter bj ∈ R++
although it does not have a natural interpretation.

6

Producers’ problems:

Producer j = 1, . . . , n has production technology Yj ⊂ Rℓ . Given two

exogenous parameters, price p and the expectations of their real receipts, s = (s1 , . . . , sλ ), producer jchooses production plan yj with market transaction plans (vj , zj ) to solve the following maximization
problem:

max

p·z

(7)

sub. toℓ(yj , vj , zj ) ∈ Yj × R+
× Rλ ,(8)v j ≦ bj ,

(9)

( ∑

∑

)
vjk ,

(10)

− 1
− λ
vj = yj + zj1
s + · · · + zjλ
s .

(11)

zj+ =

vjk , . . . ,

k∈L1

k∈Lλ

Constraints (8)–(11) can be interpreted in the same way as in the consumers’ problems. In Formulaℓas an exogenously given upper bound, as previously discussed.
(9), we use bj ∈ R++

Finally, we define the equilibrium.
n
Let E = ((Xi , ωi , ui , bi , (θij )nj=1 )mi=1 , (Yj , bj )j=1 , M) denote the above economy. An) ∏n (∏m (nℓ
λequilibrium for economy E is a pair ((xi , vi , zi )m× j=1 Yj ×
i=1 , (yj , vj , zj )j=1 ) ∈i=1 Xi × R+ × R
)
∏λ
ℓ
R+
× Rλ and (p, s) ∈ ∆ × κ=1 ∆κ , which satisfies Eqs. (1)–(11) and the market clearing condition (12)Equilibrium:with expectation specification (13) for each κ ∈ {1, . . . , λ}:m+n∑ziκ = 0,

(12)

i=1

∑m+n

i=1 prLκ (vi )= sκ∑m+n +ziκ
i=1

as long as

m+n
∑

+
ziκ
> 0,

(13)

i=1

where prLκ denotes the projection onto subspace RLκ of Rℓ for each κ = 1, . . . , λ, i.e., for each x ∈ Rℓ ,/ Lκ and xk if k ∈ Lκ . Note that we only consider Eq. (13) when
the k-th coordinate of prLκ (x) is 0 if k ∈∑m+n +∑m+n +
i=1 ziκ > 0. Hence, if
i=1 ziκ = 0 we have no restrictions on the expectation specifications.

72.2 Results for the Basic Model
As previously mentioned, the upper bound condition (9) is problematic because it is diﬃcult to interpret
and/or to justify it from a natural economic point of view. However, as shown in the next theorem andthe non-existence example, the condition is not only necessary but also essential to analyze asymmetric
information problems. More precisely, an equilibrium allocation of the model typically depends on theupper bound parameters, bi , as long as we want to analyze a situation where the asymmetric informational
problem actually matters. In other words, we can prove that if an equilibrium for this standard setting
does not depend on the upper bound parameters, the equilibrium allocation is typically Pareto-optimal.7
To state this fact formally, we introduce some standard definitions.
n
We say that an allocation ((xi )mi=1 , (yj )j=1 ), where xi ∈ Xi (i = 1, . . . , m) and yj ∈ Yj (j = 1, . . . , n),
∑m
∑n
∑m
n
is feasible if it satisfies i=1 xi = j=1 yj + i=1 ωi . Feasible allocation ((xi )mi=1 , (yj )j=1 ) is Pareto′ n′optimal if there is no other feasible allocation ((x′i )mi=1 , (yj )j=1 ) such that xi ≿i xi for all i = 1, . . . , mand x′i ≻i xi for some i = 1, . . . , m, where ≿i and ≻i are respectively agent i’s preference and strict
preference relations.
Theorem 1. Suppose that, for each i = 1, . . . , m, consumer i’s preference ≿i is monotone and convex.8∗ ∗ ∗ n∗ ∗∗If equilibrium state ((x∗i , vi∗ , zi∗ )m
i=1 , (yj , vj , zj )j=1 , p , s ) satisfies vi ≪ bi for all i = 1, · · · , m + n andLκ∗ nfor all κ = 1, · · · , λ, then equilibrium allocation ((x∗i )msκ∗ ∈ R++i=1 , (yj )j=1 ) is Pareto-optimal.

Hence, in economies with asymmetric information, where Pareto-optimality typically does not hold,there must exist some agent i that delivers real commodity k with its limit amount bik or real commodity
k ′ that is vanishing from the market, i.e., sκk′ = 0. In this sense, to treat the asymmetric information
problem such as the adverse selection problem in the standard setting, we cannot avoid the arbitrarinesscaused by exogenous upper bound parameters bi .9
7 Suppose that the asymmetric information is based on the diﬀerent quality of commodities in a certain standardized

market. Assuming that each agent’s expectation for a standardized commodity typically satisfies the condition thatfor each delivery of the standardized commodity there is a non-zero possibility to obtain all real commodities thatbelong to this market, then an equilibrium, which does not depend on the delivery upper bound constraints, is always
Pareto-optimal.
8 Preference relation ≿ is monotone if x ≫ y implies x ≻ y. Preference relation ≿ is convex if x ≻ y implies
tx + (1 − t)y ≻ y for all t ∈ (0, 1]. (For the definition of a preference relation’s convexity, see also Debreu (1959).)
9 Note that here, the situation is diﬀerent from the one concerning the short selling upper bound in the incomplete

8Furthermore, as we noted above, we cannot even ensure the equilibrium existence without the upperbound condition (9). In the following, we describe a non-existence example. The setting is the same as
for the existence theorem (Theorem 2 below), but without the upper bound condition for the producer(Condition (9)). Hence, without the upper bounds for the producers, an equilibrium may not exist even
if the consumer’s delivery amount is bounded, as in Formula (3) with bi = ωi .
Example 1: Non-existence of equilibrium without an upper boundSuppose we have two real goods, one consumer, and one producer in an economy. Additionally, suppose
there is only one market, and hence the two goods are traded at an identical price: p ∈ R+ . These agents
expect that they will receive s ∈ [0, 1] units of good 1 and (1 − s) units of good 2 when they contract tobuy one unit from the market. The producer has technology Y = {(y1 , −y2 ) ∈ R2 : y1 ≦ 2y2 , y2 ≧ 0},
and there is no upper bound on its delivery. (Hence this producer’s problem is specified by Eqs. (7), (8),2, utility function u(x1 , x2 ) = 2x1 + x2 , and(10), and (11)). The consumer has consumption set X = R+

endowment (1, 1), and his actual delivery amounts must be bounded by his endowments. (This problem
is specified by Eqs. (1)–(6) with bi = ωi = (1, 1). ) In this economy, the consumer prefers real good 1 togood 2, and real good 2 is the raw material for the producer. Let (vp1 , vp2 , zp± ) and (vc1 , vc2 , zc± ) denotethe producer’s and consumer’s transaction plans.
First, note that price p = 0 never constitutes an equilibrium, because the consumer can make infinite
purchases; i.e., the consumer problem has no maximum. Next, note that for any price p > 0, theproducer’s problem is reduced to max{py2 |(1 − s)zp− ≧ y2 ≧ 0 and zp− ≧ 0}, and the producer’s actual
deliveries are characterized by (vp1 , vp2 ) = (2y2 + szp− , −y2 + (1 − s)zp− ).10
If p > 0 and s ̸= 1, then the producer can get his raw material (real good 2) from the market and makemarket model (Hart 1975). In the incomplete market model, it is just desirable from an economic modeling pointof view to remove the short selling upper bound, whereas in the model of an asymmetric information economy, theremoval of the upper bound implies the removal of the asymmetric information problem itself, as shown by Theorem
1.
10 To see that the original problem described by Eqs. (7), (8), (10), and (11) can be reduced to the stated problem withtwo variables (y2 and zp− ), we eliminate other variables as follows. First, we can eliminate vp1 and vp2 by substituting
vp1 = y1 + szp− and vp2 = −y2 + (1 − s)zp− into all the relevant restrictions (i.e., substituting Eq. (11) into Eqs. (8) and(10)). We can also eliminate zp+ by substituting Eq. (10) into the profit function of Eq. (7). Finally, if p > 0, then y1 =2y2 must hold in an optimum, so we can also eliminate y1 by substituting it into all the relevant restrictions and the
profit function. Then the reduced problem is actually max{py2 |(1 − s)zp− ≧ y2 ≧ 0 and zp− ≧ 0}, and this problem is
equivalent to the original problem. The actual deliveries are characterized by (vp1 , vp2 ) = (2y2 +szp− , −y2 +(1−s)zp− )by substituting the equation y1 = 2y2 to eliminate y1 .

9

twice as much product (real good 1) as raw material. Moreover, the prices of these two goods are thesame, p > 0, and there is no upper bound for his delivery, so the producer can make an infinite amount of
profit. Indeed, if s ̸= 1, then the reduced problem has no maximum because y2 can be made arbitrarilyy2large by taking zp− such that zp− ≧ 1−s. Hence, an equilibrium only exists when p > 0 and s = 1.

If p > 0 and s = 1, then the producer expects that he cannot get raw material for production.
Hence, no-production is his optimal production plan. The reduced problem has maximum solutions(y2 = 0, zp− = arbitrary) with maximum profit π = 0, and the actual deliveries are (vp1 , vp2 ) = (zp− , 0).11
However, the consumer wants to exchange endowed good 2 with more preferable good 1 through themarket, because he only expects to get good 1 when he purchases it from the market, s = 1. So in the
optimum, vc2 > 0 must hold. Hence, there is a positive delivery of commodity 2, while the agents expect
no delivery for good 2, (1 − s) = 0. Therefore, Eq. (13) never holds. This means that if p > 0 and s = 1,
there is never an equilibrium. Consequently, no equilibrium exists in this economy.12□

Now we state a general equilibrium existence theorem for production economies with asymmetric
information. We can ensure that an equilibrium exists as long as all agents have upper bounds for theirdeliveries.
n∗ ∗ ∗ mTheorem 2. Economy E = ((Xi , ωi , ui , bi , (θij )nj=1 )mi=1 , (Yj , bj )j=1 , M) has an equilibrium, ((xi , vi , zi )i=1 ,(yj∗ , vj∗ , zj∗ )nj=1 , p∗ , s∗ ), if the following conditions are satisfied:ℓ(Consumers) Each consumer i = 1, . . . , m has (i) a closed convex consumption set, Xi ⊃ R+,

bounded from below; (ii) a convex preference induced by a strictly monotone and continuous utility
function ui : Xi → R; and (iii) endowment ωi ∈ intXi .13(Producers) For each j = 1, . . . , n, Yj ⊂ Rℓ is a closed convex set that contains 0.
(Attainable Set) The attainable sets for each agents are bounded.14
11 Here we observe that the producer’s constraint correspondence does not satisfy upper semicontinuity at s = 1.
12Note that even in exchange economies, the same problem arises if consumers can resale their goods arbitrarily.
Suppose two goods, one market, two consumers, and no producer in an economy. The two goods are traded at
identical price p ∈ R+ . Suppose that consumers can resale their goods arbitrarily, i.e., consumers’ problems are
2 , u (x , x ) = 2x + x ,specified by Eqs. (1), (2), (4), (5) and (6). Assume that consumer 1’s attributes are X1 = R+
1 1
2
1
2
2 , u (x , x ) = x + 2x and ω = (1, 1). It isand ω1 = (1, 2), and that consumer 2’s attributes are X2 = R+
2 1
2
1
2
2
straightforward to see that no equilibrium exists in this economy because they always can get a more preferable
consumption bundle through the unbounded transactions.
13 Preference relation ≿ is strictly monotone if x′ ≧ x, x′ ̸= x implies x′ ≻ x.
∑m∑n∑m
14 Allocation ((x )m , (y )n ) is attainable if
i i=1
j j=1
i=1 xi ≦j=1 yj +i=1 ωi . The attainable sets for agents, i =10Although this theorem ensures the equilibrium existence, as shown in Theorem 1, the equilibrium
typically depends on exogenous upper bounds bi , and their removal induces the removal of the asymmetric
information problem itself. Moreover, as shown in Example 1, we cannot even technically remove theupper bound condition because an equilibrium may not exist. In the next section, we endogenize thisupper bound condition by introducing natural cost for the market deliveries and solve this problem.

3 Model without Exogenous Upper Bounds and Equilibrium Existence
Given the class of possible markets M= {L1 , . . . , Lλ }, each agent must sell his real goods as λ typesof market contracts. Therefore, he should standardize or commoditize his real goods to sell them in themarket depending on the class of possible markets M. It is quite natural to assume that this procedure
will cost each agent some loss of goods or services that may be delivered to the market, because it will
cost each agent at least some labor to standardize, package, and/or deliver real commodities to themarket, which has the class of possible markets. We describe such standardization, commoditization,
and/or transaction costs by assuming that each agent has a certain kind of technology for standardizing
his real goods.
ℓℓAssume that each agent i = 1, . . . , n+m (each of consumers and producers) has function Fi : R+
→ R+

that satisfies the following conditions:(C1) Fi is a continuous function;(C2) for all k = 1, . . . , ℓ, Fik is a concave function, where Fik is a k-th coordinate of Fi ;∑ℓ(C3) for each κ = 1, . . . , λ and v ∈ R+
, k∈Lκ Fik (v) = 0 is equivalent to vLκ := (vk )k∈Lκ = 0;∑ℓ′′
for each κ = 1, . . . , λ and v, v ′ ∈ R+
, vLκ ≦ vL
and vLκ ̸= vL
impliesk∈Lκ Fik (v) <κκ∑′k∈Lκ Fik (v );ℓ(C4) Fi (v) ≦ v for all v ∈ R+
; and(C5) for each κ such that Lκ = {k} for some k, Fik (v) = vk ;ℓfor each κ such that ♯Lκ ≧ 2, and for each sequence {v ν }∞
ν=1 ⊂ R+ , if∑k∈Lκ Fik (v

ν

) → ∞

1, . . . , n + m, are defined as X̃i := {xi ∈ Xi |xi constitutes an attainable allocation with some xi′ ∈ Xi′ (∀i′ ̸=i), yj ∈ Yj (∀j = 1, . . . , m)}, and Ỹj := {yj ∈ Yj |yj constitutes an attainable allocation with some xi ∈ Xi (∀i =1, . . . , n), yj ′ ∈ Yj ′ (∀j ′ ̸= j)}.

11

(ν → ∞), then there exists some Lκ′ such that∑ννk∈Lκ ∪Lκ′ (vk − Fik (v )) → ∞ (ν → ∞).

As we will formalize later, each agent supplies his real goods and services to the market with the class of∑∑
λpossible markets M= {L1 , . . . , Lλ } as a λ-tuple ( k∈L1 Fik (v), . . . , k∈Lλ Fik (v)) ∈ R+
(see Eqs. (16)and (20) below). We call Fi agent i’s standardizing technology. Condition (C5) means that no agent cansell his real goods in the market with the class of possible markets M ={L1 , . . . , Lλ } at no cost, and thisis the crucial assumption as we see in the following example.

Example 1 revisited: Non-existence of equilibrium without the cost condition (C5)
Recall that there are two goods, one market, one consumer, and one producer in Example 1. Note thatin Example 1, only the consumer has an upper bound for delivery contract. Consider a model without
the upper bound condition for each agents by removing consumer’s upper bound in Example 1. In such amodel, each agent’s “standardizing technology” can be formalized as (Fi1 (v1 , v2 ), Fi2 (v1 , v2 )) := (v1 , v2 )and this formalization means that agents can standardize their goods at no cost.15 Formally, these2
2
2
satisfy all the conditions but (C5). Indeed, for each sequence {v ν }∞→ R+
functions Fi : R+
ν=1 ⊂ R+ ,∑∑we have k∈{1,2} (vkν − Fik (v ν )) = k∈{1,2} (vkν − vkν ) = 0 for all ν = 1, 2, . . ., and hence condition (C5)must be violated. (Note that there is only one market L1 := {1, 2} in this economy.) Moreover, by asimilar argument in Example 1, it is straightforward to see that there is no equilibrium in this model,i.e., we cannot ensure the existence of an equilibrium without condition (C5).

□Note that the cost imposed by condition (C5) of standardizing technologies Fi (or, the transaction cost)for consumers and producers (i = 1, . . . , m + n) and the transformation cost represented by production
technologies Yj for producers (j = 1, . . . , n) are two independent concepts, and in general we specify
these two independently. In particular, in Example 1, while the transformation cost represented byproduction set Y = {(y1 , −y2 ) ∈ R2 : y1 ≦ 2y2 , y2 ≧ 0} is positive, the transaction cost represented by∑∑
Fi is zero since k∈{1,2} (vkν − Fik (v ν )) = k∈{1,2} (vkν − vkν ) = 0 and because of the latter fact we haveno equilibrium. In the following, we show that by introducing the transaction costs expressed by (C5),
we can ensure the equilibrium existence.16
15 Note that with this notation and the agents’ attributes given in Example 1, the model specified here is identical witha model specified by Eqs. (14)–(22) that define our extended model below.
16 Also, note that (C5) requires such standardizing costs only if L is not a singleton set because if some L is a singleton,κκ12Now we propose the following modifications to the model. The formalization is almost the same as thebasic model in Eqs. (1)–(11), except that we remove the upper bound conditions (3) and (9) and replace∑∑(4) and (10) by “zi+ = ( k∈L1 Fik (vi ), . . . , k∈Lλ Fik (vi ))”.
Agents’ Problems: The producers’ problems are defined as:max

p · zj+ − p · zj−(14)sub. toℓ(yj , vj , zj ) ∈ Yj × R+
× Rλ ,

zj+ =

( ∑

∑

(15)

)
Fjk (vj ) ,

(16)

− 1
− λ
vj = yj + zj1
s + · · · + zjλ
s .

(17)

Fjk (vj ), . . . ,

k∈L1

k∈Lλ

Similarly, the consumers’ problems are defined as:max

ui (xi )(18)

sub. toℓ(xi , vi , zi ) ∈ Xi × R+
× Rλ ,zi+ =( ∑Fik (vi ), . . . ,

k∈L1

∑

)
Fik (vi ) ,

(19)
(20)

k∈Lλ

− 1
− λ
xi + vi = ωi + zi1
s + · · · + ziλs ,n∑θij πj (p, s).
p · zi− = p · zi+ +

(21)
(22)

j=1

n
Equilibrium: Let E = ((Xi , ωi , ui , Fi , (θij )nj=1 )mi=1 , (Yj , Fj )j=1 , M) denote the above economy. Annequilibrium for economy E is a pair ((xi , vi , zi )mi=1 , (yj , vj , zj )j=1 ) and (p, s), satisfying (14)–(22) and themarket clearing condition (23) with an expectation specification (24) for each κ ∈ {1, . . . , λ} and k ∈ Lκ :m+n∑ziκ = 0,

(23)

i=1

∑m+n

i=1 Fik (vi )= sκk as long as
∑
m+n +
z
iκ
i=1

m+n
∑

+
ziκ
> 0.

(24)

i=1

We state the existence theorem for this modified economy.
then there is no diﬀerence between a real good and a standardized commodity in the corresponding market, and hence,in such cases, the argument about standardizing costs can be negligible. Condition (C5) expresses this point using
the case ♯Lκ = 1, i.e., Lκ = {k} for some k.

13

n
∗ ∗
Theorem 3. Economy E = ((Xi , ωi , ui , Fi , (θij )nj=1 )mi=1 , (Yj , Fj )j=1 , M) has an equilibrium, ((xi , zi ,∗ ∗ ∗ n∗ ∗vi∗ )mi=1 , (yj , zj , vj )j=1 , p , s ), if the following conditions are satisfied.

(Consumers) Each consumer i = 1, . . . , m has a non-empty closed convex consumption setℓ
Xi ⊃ R+
that is bounded from below with a convex preference induced by a strictly monotoneand continuous utility function ui : Xi → R+ , and initial endowment ωi ∈ intXi .
(Producers) For each j = 1, . . . , n, Yj ⊂ Rℓ is a closed convex set containing 0.
(Attainable Set) The attainable sets for all agents (X̃i (i = 1, . . . , m) or Ỹj (j = 1, . . . , n) ) arebounded.17ℓℓ(Standardizing Technologies) For each agent i = 1, . . . , m + n Fi : R+
→ R+
satisfies condi-

tions (C1) – (C5).
ℓℓ(i = 1, . . . , n + m), we→ R+
As shown by Theorem 3, with standardizing technologies Fi : R+

can remove the exogenous upper bound conditions (3) and (9). Agents choose to deliver vi so thatℓℓnever harms their own total payoﬀ. Note that→ R+
the standardizing cost represented by Fi : R+

the assumptions on production structure are quite standard as a general equilibrium economy of theArrow–Debreu type. Therefore, it would be possible to interpret our existence result as follows: we candispense with concerns over the production structure or the delivery upper bounds as long as we impose
restrictions on the structure of standardization, commoditization, and/or transaction costs.

Appendix
Proof of Theorem 1nSuppose, on the contrary, that there is an allocation ((xi )mi=1 , (yj )j=1 ) which Pareto-dominates∗ nℓ
λ((x∗i )mi=1 , (yj )j=1 ). Take (vi , zi ) ∈ R+ × R (i = 1, · · · , m + n) to satisfy conditions (4), (5), (10), and(11) with allocation xi or yi .18 Note, here, that (3) and/or (9) are not necessarily satisfied. Since wenassume that vi∗ ≪ bi for all i = 1, · · · , m + n, we can construct state ((x̄i , v̄i , z̄i )mi=1 , (ȳj , v̄j , z̄j )j=1 ) such

17 For the definitions of X̃ and Ỹ , see footnote 14 of Theorem 2.
ij
18 Since we assume sκ∗ ∈ RLκ for all κ = 1, · · · , λ, we can take z − and v to satisfy condition (5) (or (11)) for eachi++ii = 1, · · · , m + n. Then, take zi+ to satisfy (4) (or (10)).

14nthat ((x̄i )mi=1 , (ȳj )j=1 ) is feasible and Pareto-dominates the equilibrium allocation and satisfies conditions(3), (4), (5), (9), (10), and (11). (For example, if we denote (x̄i , v̄i , z̄i ) = (1 − t)(x∗i , vi∗ , zi∗ ) + t(xi , vi , zi )and let t → 0, then v̄i ≪ bi , in particular condition (3) (or (9)), holds for suﬃciently small t. By taking
a suﬃciently small and identical t for all agents i = 1, · · · , m + n we can construct a feasible staten((x̄i )mi=1 , (ȳj )j=1 ) that satisfies these conditions. Pareto-dominance is particularly preserved since eachconsumer’s preference is convex.)
Note that the following two conditions hold:x̄i ≻i x∗i =⇒ p∗ · z̄i− > p∗ · z̄i+ +x̄i ≿i x∗i =⇒ p∗ · z̄i− ≧ p∗ · z̄i+ +

n
∑
j=1
n
∑

θij πj (p∗ , s∗ )(25)

θij πj (p∗ , s∗ )

(26)

j=1

(where πj (p∗ , s∗ ) is the maximized profit of producer j = 1, · · · , n under (p∗ , s∗ )) since, if not, we have aLκcontradiction to the fact that x∗i is an utility maximizing consumption plan under (p∗ , s∗ ), sκ∗ ∈ R++
fornall κ, and each consumer’s preference is monotone. Hence, the Pareto-dominance of ((x̄i )mi=1 , (ȳj )j=1 ) with∑m(25), (26), and the profit maximization assumption of the equilibrium state implies that i=1 p∗ · z̄i >∑n∗19
j=1 p · z̄j holds. We can rewrite this inequality as follows:
m {∑
∑
i=1

p∗1 (ωik − x̄ik ) + · · · +

k∈L1

∑

n {∑
} ∑
}
∑
p∗1 ȳjk + · · · +p∗λ ȳjk .
p∗λ (ωik − x̄ik ) >j=1 k∈L1

k∈Lλ

(27)

k∈Lλ

n
However, (27) contradicts the fact that ((x̄i )mi=1 , (ȳj )j=1 ) is feasible. Indeed, if we define “extended price”p̄ = (p̄1 , . . . , p̄ℓ ) ∈ Rℓ as p̄k = p∗κ if k ∈ Lκ (k = 1, . . . , ℓ, κ = 1, . . . , λ) and evaluate feasibility condition∑m∑ni=1 (x̄i − ωi ) =j=1 ȳj by extended price p̄, then (27) must hold with equality.

Proof of Existence Theorems
Here, we make some notes regarding the proof of Theorem 2. Theorem 2 corresponds to the caseℓFi (vi ) = vi for all vi ∈ {v ∈ R+|v ≦ bi } in the extended standardizing technology setting of Section 3.
ℓThis Fi satisfies (C1)–(C4) on the compact domain {v ∈ R+|v ≦ bi }. Although it does not satisfy (C5),this assumption is not needed if the domain of Fi is compact on Rℓ . (Footnote 22 clarifies this point.)

19 From conditions (4), (5), (10), and (11).

15Now, we proceed to the proof of Theorem 3. There are some diﬃculties related to the agents’ problems(14)–(22). In particular, we cannot ensure that the constraint correspondences satisfy upper and lower
semicontinuity and have convex values. More specifically, first, the diﬃculty related to the upper semicontinuity is due to non-bounded deliveries, and was observed in Example 1. Next, equality constrains(17), (21), and (22) cause some diﬃculties in showing the lower semi-continuity with respect to the vector
valued parameters s = (s1 , . . . , sλ ). Finally, equality constraints (16) and (20) create problems whenensuring the convexity of the values of the constraint correspondences, because Fi (i = 1, . . . , n + m) are
generally not linear.
To avoid these diﬃculties, we first modify the original problems in two ways; we truncate the variables
and relax the equality constraints (in Producers’ Problems and Consumers’ Problems). Then,
we obtain a kind of an “equilibrium” for the modified problem as a fixed point of a correspondence(in Fixed Point Argument). Finally, we show that the fixed point is actually an equilibrium of ouroriginal model in the limit of truncation argument (in Relationship to the Original Problem and
Limit Argument).
ℓ
Producers’ Problems: For any t > 1, define subset Ωt ⊂ Rℓ × R+
× Rλ asΩt := [−t, t]ℓ × [0, t]ℓ × [−t, t]λ .

(28)

Now, take an arbitrarily large number t > 1. We consider a modified version of the original problem(14)–(17). Specifically, we replace the variable constraint (15) by the following truncated version:ℓ(yj , vj , zj ) ∈ (Yj × R+
× R λ ) ∩ Ωt .

Moreover, we relax the constraints (16) and (17) by replacing equalities “ = ” with inequalities “ ≦ ”. We
refer those modified constraints as (15’), (16’), and (17’), respectively. Recall that Yj ⊂ Rℓ is closed andconvex, and contains 0, and Fj satisfies (C1) – (C5) for each j = 1, . . . , n.
We denote by ηjt (p, s) the set of solutions to the modified maximization problem; (14) subject to (15’),(16’), and (17’) under (p, s). It is clear that ηjt (p, s) is non-empty, closed, and convex. (In particular,
convexity is assured since we relaxed equality constraint (16) by inequality constraint (16’) and Fik is aconcave function for each k = 1, . . . , ℓ from condition (C2).) Also, we can prove that the correspondence

16ηjt : ∆ ×

∏λ

κ=1 ∆

κ

→ Ωt has a closed graph. Indeed, we can see that the constraint correspondence(p, s) 7→ { (yj , vj , zj ) ∈ Ωt | (yj , vj , zj ) satisfies (15’), (16’), and (17’) under (p, s) }has a closed graph and that its value and range are compact. (The compactness of the value is assured
since we truncated the variables and hence the diﬃculty on upper semi-continuity which we pointed out
in Example 1 can be avoided. Note that the continuity of Fj , condition (C1), is used here.) Therefore,
the constraint correspondence is upper semi-continuous. Moreover, it is clear that the constraint correspondence is lower semi-continuous because Fj satisfies (C3) and constraints are inequality, and hence
the standard argument is applicable. Thus, the constraint correspondence is continuous. Therefore,
Berge’s maximum theorem is applicable (cf. Debreu (1959, p. 19)). In this case, it is simultaneously
assured that the profit function of this truncated problem, πjt (p, s), is continuous.
Consumers’ Problems: As in the producer case, we consider the modified version of the original
problem (18)–(22). First, for each t > 1, define number t1 (t) > 1 ast1 (t) := t · max{1 +n∑πjt (p, s)|(p, s) ∈ ∆ ×

∆κ }.

(29)

κ=1

j=1

Note that such number t1 (t) exists since ∆ ×

λ
∏

∏λ

κ=1 ∆

κ

is compact and each πjt (p, s) is continuous. Also,

note that, since 0 ∈ Yj ∩ [−t, t]ℓ implies πjt (p, s) ≧ 0 for all j = 1, . . . , n, we havet1 (t) = t · max{1 +n∑πjt (p, s)|(p, s) ∈ ∆ ×

λ∏∆κ } ≧ t · 1 = t

κ=1

j=1

for all t > 1. Hence, we can take an arbitrarily large t1 (t) > 1 by taking suﬃciently large t > 1.
Now, consider the modified version of the original problem. Namely, variables are truncated asℓ(xi , vi , zi ) ∈ (Xi × R+
× Rλ ) ∩ Ωt1 (t) where Ωt1 (t) is defined as in (28), and all the equalities “ = ”

in constrains (20)–(22) are replaced with inequalities “ ≦ ”. Moreover, we replace each profit πj (p, s) inconstraint (22) by πjt (p, s), which is the maximized profit of each producer in the modified maximization
problem we argued above. We refer those modified constraints as (19’), (20’), (21’), and (22’), respectively. Denote by ξit (p, s) the set of all solutions to the modified maximization problem; (18) subject to17(19’)–(22’) under (p, s).20 The correspondence ξit : ∆ ×

∏λ

κ=1 ∆

κ

→ Ωt1 (t) is non-empty closed convex

valued and has a closed graph. The argument is almost the same as the producer case and hence weomit.
∏λFixed Point Argument: We have defined solution correspondences ηjt : ∆ × κ=1 ∆κ → Ωt and∏λξit : ∆ × κ=1 ∆κ → Ωt1 (t) for agents’ modified problems for any t > 1. Now, consider the product map
Φ of these correspondences:

Φ:∆×

λ
∏
κ=1

∆κ ∋ (p, s) 7→m∏ξit (p, s) ×

i=1

n
∏

ηjt (p, s) ⊂ (Ωt1 (t) )m × (Ωt )n .

(30)

j=1

The mapping Φ has a closed graph. Then, define a price-expectation manipulation correspondence Ψ asfollows:m+nm+nΨ : ([0, t1 (t)]ℓ ×[−t1 (t), t1 (t)]λ )m ×([0, t]ℓ ×[−t, t]λ )n ∋ (vi , zi )m+ni=1 7→ Θ((zi )i=1 )×Ξ((vi )i=1 ) ⊂ ∆×

λ
∏

∆κ ,

κ=1

(31)
where Θ is the price manipulation mapping and Ξ is the correspondence that assigns the real mixture
ratio of the goods for each market. More precisely, we defineΘ((zi )m+ni=1 ) := {p ∈ ∆ | ∀q ∈ ∆, q ·

m+n
∑
i=1

zi ≧ p ·

m+n
∑

zi }

i=1

for each (zi )n+m
i=1 , and the κ-th coordinate of Ξ by∑m+ni=1 prL (Fi (vi ))Ξκ ((vi )m+n):=,
∑m+n ∑ κi=1
i=1 (k∈Lκ Fik (vi ) )as long as(32)∑m+n ∑
m+n
m+n
κ
i=1 (
k∈Lκ Fik (vi ) ) ̸= 0, and otherwise by Ξκ ((vi )i=1 ) := ∆ for each (vi )i=1 . (We use

the notation prLκ for the projection onto subspace RLκ of Rℓ for each κ = 1, . . . , λ.) Note that the right)∑m+n (∑
hand side of Eq. (32) is always an element of ∆κ when i=1k∈Lκ Fik (vi ) ̸= 0. It is routine to checkthat Θ and Ξ are non-empty closed convex valued correspondence with a closed graph. In particular, Ξ)∑m+n (∑
has a closed graph since the right hand side of Eq. (32) is continuous when i=1k∈Lκ Fik (vi ) ̸= 0.

20 We denote the solution set by ξ t (p, s) instead of ξ t1 (t) (p, s) for notational simplicity.
ii18

Now, the product of the mappings Φ and Ψ,Φ×Ψ:∆×

λ
(∏

λ
)
(∏
)
∆κ × (Ωt1 (t) )m × (Ωt )n → ∆ ×∆κ × (Ωt1 (t) )m × (Ωt )n ,

κ=1

(33)

κ=1

is a non-empty closed convex valued correspondence with a closed graph. By Kakutani’s fixed pointt t t ntheorem, Φ × Ψ has a fixed point ( pt , st , (xti , vit , zit )mi=1 , (yj , vj , zj )j=1 ) for any t > 1.

Relationship to the Original Problem and Limit Argument:

In this part, we show thatthe fixed point we obtained above is an equilibrium point of our original model for some numbert > 1. Take number t > 1 suﬃciently large so that the bounded attainable sets, X̃i ⊂ Rℓ andỸj ⊂ Rℓ , to be subsets of the interiors of [−t1 (t), t1 (t)]ℓ and [−t, t]ℓ , respectively. Take a fixed pointt t t n( pt , st , (xti , vit , zit )mi=1 , (yj , vj , zj )j=1 ) we obtained above.

First, we show that (22’) holds with equality, i.e., for all i = 1, . . . , m, we havept · zit− = pt · zit+ +n∑θij πjt (pt , st ).

(34)

j=1

Indeed, if we assume pt · zit− < pt · zit+ +

∑n

t t t
j=1 θij πj (p , s ) for some i = 1, . . . , m, we have a contradiction

to the consumer’s utility maximization. More precisely, we have three cases; zit− ̸= (t1 (t), . . . , t1 (t)),
zit− = (t1 (t), . . . , t1 (t)) and zit+ ̸= (0, . . . , 0), and, zit− = (t1 (t), . . . , t1 (t)) and zit+ = (0, . . . , 0). For
the first case, since consumer’s utility function is strictly monotone, we can make a preferable choice(x̂i , vit , ẑi ) by purchasing more in some market κ and consumes more. For the second case, also, we
can make a preferable choice (x̂i , v̂i , ẑi ) by selling less in some market κ by using condition (C3) of thestandardizing technology Fi and consumes more. For the third case, we havept · (t1 (t), . . . , t1 (t)) = pt · zit− < pt · zit+ +n∑θij πjt (pt , st ) =j=1

and this implies t1 (t) <n∑θij πjt (pt , st )

j=1

∑n

t t t
t
t
j=1 πj (p , s ) since p1 + · · · + pλ = 1 and θij ≦ 1 for all j = 1, . . . , n. However,this inequality contradicts to the definition (29) of t1 (t) and the assumption that t > 1.
Therefore, from condition (34) and equalities πjt (pt , st ) = pt · zjt for all j = 1, . . . , n, it is clear that we∑n+mhave Walras’ Law; pt · ( i=1 zit ) = 0. Then, by the definition of Θ, we also have
m+n
∑

q·(

i=1

zit ) ≧ pt · (m+n∑zit ) = 0i=1

19

for all q ∈ ∆.

Hnece, for each κ = 1, . . . , λ, the κ-th coordinates of (
m+n
∑

∑m+n
i=1

zit ) must satisfy that

t
ziκ
≧0

(35)

t
ziκ
> 0 if and only if ptκ = 0.

(36)

i=1

and

m+n
∑
i=1

t n
Here, we note that the real state ((xti )mi=1 , (yj )j=1 ) of the fixed point satisfies
m
∑

xti ≦

i=1

n
∑

yjt +

j=1

m
∑

ωi ,

(37)

i=1

i.e., xti and yjt are in the bounded attainable sets X̃i and Ỹj , respectively. Indeed, for each κ = 1, . . . , λ∑m+nsuch that i=1 zκt+ > 0 and each k ∈ Lκ , we have the following inequalities:m∑(ωik − xtik ) +

n
∑
j=1

i=1

t
yjk

(17′ ),(21′ ) m+n
∑

t
vik
−

≧

m+n
∑

i=1

i=1

m+n
∑

m+n
∑

zκt− stκk∑m+n
Fik (vit )zκt− ∑m+n i=1
∑
t
′
i=1
k′ ∈Lκ Fik (vi )
i=1
i=1
∑
m+n
m+n
m+n
(35)
∑
∑
Fik (vit )
t
≧
vik
−
zκt+ ∑m+n i=1
∑
t
′
i=1
k′ ∈Lκ Fik (vi )
i=1
i=1
∑m+n
m+n
t
(16′ ),(20′ ) m+n∑∑t+ti=1 Fik (vi )
zκ ∑
vik −
≧
m+n t+
i=1 zκ
i=1
i=1
(32)

=

=

m+n
∑

t
vik
−

t
vik
−

i=1

m+n
∑

Fik (vit ).

i=1

Hence, from condition (C4), we have (37).21 Also, note that the attainability condition (37) implies thatthe relaxed constraint (21’) holds with equality, i.e., for all i = 1, . . . , m, we have
t− t1
t− tλ
xti + vit = ωi + zi1
s + · · · + ziλs(38)

because consumers’ utility functions are strictly monotone and xti ̸= (t1 (t), . . . , t1 (t)) since we took
number t1 (t) > 1 suﬃciently large so that the bounded attainable set X̃i ⊂ Rℓ to be a subset of theinterior of [−t1 (t), t1 (t)]ℓ .
Furthermore, we have pt ≫ 0. Indeed, if we assume that there is some market κ such that ptκ = 0
21 The case

∑m+n
i=1

t+
= 0 is obvious since (C3) implies
zκ

∑m+n
i=1

20

t = 0 for all k ∈ L and (35) implies
vik
κ

∑m+n
i=1

t−
= 0.
zκ

then, we have a contradiction. More precisely, we have three cases; for some consumer i = 1, . . . , m wet−t+have ziκ
< t1 (t), for some consumer i = 1, . . . , m we have ziκ> 0, and for all consumers i = 1, . . . , mt−t+
we have ziκ
= t1 (t) and ziκ
= 0. For the former two cases, we can make preferable choices for consumer

i in similar ways to the above arguments that we used to show equality (34). For the third case, because∑t+ttk∈Lκ Fi (vi ) = ziκ = 0 implies vik = 0 for all k ∈ Lκ by (C3) and we have equality (38), we also havethat for all k ∈ Lκ ,

m
∑
i=1

xtik =

m
∑

t− tκ
(ziκ
sk + ωik ) = mt1 (t)stκ
k +

i=1

m
∑

ωik .

i=1

However, because stκ ∈ ∆κ and we can take t1 (t) > 1 arbitrarily large, so it contradicts to the fact that,for all i = 1, . . . , m, xti belongs to attainable set X̃i , which is bounded in Rℓ .
∑m+n tTherefore, because condition (36) and pt ≫ 0 imply
i=1 zi = 0, we have the market clearing
condition. Moreover, pt ≫ 0 implies that inequality constraints (16’), (17’), and (20’) in the modified
problems actually hold with equalities since if not, a contradiction to the utility or profit maximization
clearly follows. Hence, these equalities, together with (34) and (38), mean that all of our modified
inequality constraints actually hold with equalities for suﬃciently large t > 1.
()
t t t nThus far, we see that the state pt , st , (xti , vit , zit )mi=1 , (yj , vj , zj )j=1 satisfies (16), (17), (20), (21),()
t t t n(22), (23), and (24). We call pt , st , (xti , vit , zit )mi=1 , (yj , vj , zj )j=1 a t-equilibrium state. Since we taket > 1 suﬃciently large so that the bounded attainable sets to be subsets of the interior of [−t, t]ℓ or[−t1 (t), t1 (t)]ℓ , all xti or yjt are interior points of [−t, t]ℓ or [−t1 (t), t1 (t)]ℓ . Therefore, the t-equilibrium()
t t t nstate pt , st , (xti , vit , zit )mi=1 , (yj , vj , zj )j=1 is not an equilibrium of the original economy E only when(vit , zit ) is a boundary point of [0, t]ℓ × [−t, t]λ or [0, t1 (t)]ℓ × [−t1 (t), t1 (t)]λ for some i = 1, . . . , m + n.22
Suppose that, for some i = 1, . . . , m+n, (vit , zit ) is a boundary point of [0, t]ℓ ×[−t, t]λ for all t > 1. If somet+tmarket κ consists of a single real commodity k, Lκ = {k}, then restriction (20) requires that vik
= ziκ
by
t−
t
(C5), and skκ = 1 and restriction (21) require that xtik + vik= ωik + ziκ
. (The same argument is relevanttfor producers.) Therefore, we can also suppose that (vik, zκt ) is bounded for such singleton markets Lκ
t−
t+ t−
t
without loss of generality, since we can decrease the amounts of (vik, ziκ
) = (ziκ, ziκ ) without any

change of utility value (and/or profit value) of i or any harm in conditions we get so far. This implies
22

If the domain of Fi is compact for all i = 1, . . . , m + n then (vi , zi ) (i = 1, . . . , n + m) are in a fixed bounded area by
the continuity of Fi (i = 1, . . . , n + m). Hence, in such cases, the proof is completed here. As we mentioned before,
the proof of Theorem 2 corresponds to this case.

21

that ∥∑

tk∈Lκ Fik (vi )∥ → ∞ as t → ∞ for some κ ∈ {1, . . . , λ} such that ♯Lκ ≧ 2. Note first that, for alli = 1, . . . , m + n and all κ, κ′ = 1, . . . , λ, we have
m
∑
(

∑

i=1 k∈Lκ ∪Lκ′n
) ∑((ωik − xtik ) +∑j=1 k∈Lκ ∪Lκ′

)
t
≧
yik

∑

t+
t+
t
vik
− (ziκ
+ ziκ
′)

(39)

k∈Lκ ∪Lκ′by considering conditions (16), (17), (20), (21), (23), (24), and (C4). Moreover, the right hand∑∑ttside of Eq. (39) isk∈Lκ ∪Lκ′ vik −k∈Lκ ∪Lκ′ Fik (vi ), from conditions (16) and (20). However, if∑∥ k∈Lκ Fik (vit )∥ → ∞ as t → ∞ for some κ ∈ {1, . . . , λ} such that ♯Lκ ≧ 2, then condition (C5)∑∑ttrequires that there exists some Lκ′ such thatk∈Lκ ∪Lκ′ Fik (vi ) → ∞ as t → ∞.
k∈Lκ ∪Lκ′ vik −This implies that the right-hand side and hence the left-hand side of Eq. (39) tends to ∞ as t → ∞,contradicting the fact that each of xti and yjt is in a bounded area in Rℓ . Thus, we complete the proofof our main theorem.

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