Discussion Papers In Economics
And Business

General Equilibrium Model with Information Asymmetry andCommodity-Information Technologies

Ken Urai, Akihiko Yoshimachi and Kohei Shiozawa

Discussion Paper 14-02

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGeneral Equilibrium Model with Information Asymmetry andCommodity-Information Technologies

Ken Urai, Akihiko Yoshimachi and Kohei Shiozawa

Discussion Paper 14-02
January 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGeneral Equilibrium Model with Information Asymmetry andCommodity-Information Technologies
Ken Urai∗

Akihiko Yoshimachi†

Kohei Shiozawa‡

Abstract
In this paper, we investigate a new concept of a market’s commodity-information structure (a partitionof the set of real goods that are treated as one commodity for market exchanges) and technologies relating to it, commodity-information technologies. Using this concept, we can always aﬃrmatively answer
the market viability problem, concerning the existence of general equilibrium even when information
asymmetry among agents such as adverse selection prevails in the economy. Some Pareto-optimality
problems and policy implications are also discussed.
Keywords: General Equilibrium Model, Asymmetric Information, Adverse Selection, Market Viability Problem, Commodity-information Structure
JEL classification: C62; D51; D82∗ Graduate

School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
urai@econ.osaka-u.ac.jp† Department of Commerce, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan‡ Graduate School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
nge013sk@student.econ.osaka-u.ac.jp

E-mail:E-mail:1

Introduction

Adverse selection problems (e.g., Akerlof (1970), Rothschild and Stiglitz (1976),) have traditionally beentreated through static partial equilibrium arguments. The usual argument is that the price mechanism ina market with asymmetric information is not suﬃcient to assure the existence of an equilibrium if agents
rationally expect their average receipts. From a general equilibrium theoretic viewpoint, however, partial
equilibrium arguments on market unraveling or the adverse selection problem ignore several important
total-market closed-income-circulation features of an economy. Even in adverse-selection cases, the ownersof high-quality commodities must sell them elsewhere, so their opportunity costs in giving up trading highquality goods must be treated endogenously.
Problems such as adverse selection assume that information asymmetry is a structure that depends not
only on individuals but also on their selling and/or buying standpoints. This premise influences individuals
to change their actions in a way that cannot be described under the traditional general equilibrium
structure. In a recent paper (Urai et al. 2013), the new concept of the commodity-information structureof a market (a partition of the set of real goods that are treated as one commodity for market exchanges)
is introduced, on which can be based the ordinary static general equilibrium arguments and settings
for analyzing asymmetric information problems such as adverse selection.1 We showed the existence ofequilibria and examples of the non-existence of equilibria, discussed some equilibrium-optimality problems,and emphasized that the existence of an exogenous upper bound for market-trade amounts is essential forthe existence and optimality of the general equilibrium.
The purpose of this paper is to construct a model that can treat such a market-trade upper boundendogenously, and to provide one of the most general answers to the market-viability problem by investigating conditions under which market-disappearance never happens, even with information asymmetry.
The key concept used here is commodity-information technology. In this paper, we suppose that eachcommodity-information structure has an associated class of technologies that enable each agent (a possible seller) to supply in certain amounts of their outputs as a unit of market commodity, and that sucha process necessarily uses a positive amount of real goods or services. We emphasize that one should
not view such a technology as an ad hoc or special device for an asymmetric information economy. Evenin the standard setting of general equilibrium theory, it would be interesting (from both an historicaland realistic perspective) to recognize the market as a special place where the available commodity iscompletely diﬀerent from the goods and services with which we are familiar. This paper gives a radicalaﬃrmative answer to the market mechanism even in such cases, that is, whether a situation exists where
all buyers obtain exactly what they expect, (given rational expectations that reflect the aggregate real features of society), and where the ordinary equilibrium conditions for all agents and the demand-supply are
1 We believe that the approaches of Dubey et al. (2000) and Bisin and Gottardi (1999) are groundbreaking. Our model

might (technically) be classified as a complete static version of their futures-market or probabilistic-dynamics models.

1

satisfied.
If we view the market as a mechanism that is administrated by the government, some parts of thecommodity-information technologies may be treated as policy variables. If we consider the market as anautonomous system, commodity-information technology may also be considered one of the fundamental
factors behind the market commodity-information structure itself. For general equilibrium arguments, thecommodity-information technology gives a normative and descriptive unified viewpoint of both symmetricand asymmetric information economies. From this perspective, it is worth noting that market unravelingunder asymmetric information should not be seen as the disappearance or collapse of competitive markets,
but as a discussion of complementary changes and replacements among several individual markets basedon the strong viability of the market mechanism.

2

The Model

Denote the set of real numbers by R, n-dimensional Euclidean space by Rn , the non-negative orthantnof Rn , {x = (xk )nk=1 ∈ Rn | xk ≧ 0, k = 1, . . . , n}, by R+, and the strictly positive orthant of Rn ,n{x = (xk )nk=1 ∈ Rn | xk > 0, k = 1, . . . , n}, by R++. For each finite set A, we denote by ♯A the number of
elements of A.
Market Structure and Commodity Information Structure: There are m types of consumers
and n types of producers indexed by i = 1, . . . , m and j = m + 1, . . . , m + n. In this model, ℓ types ofreal goods and services are indexed by k = 1, . . . , ℓ and λ types of market commodities are indexed byκ = 1, . . . , λ. Let L be the set of real commodity indices {1, 2, . . . , ℓ}, partitioned as L = L1 ∪ L2 ∪ · · · ∪ Lλwhere Lκ ∩ Lκ′ = ∅ for all κ ̸= κ′ . This describes a market that cannot distinguish between realcommodities k ∈ L and k ′ ∈ L if and only if k and k ′ belong to the same Lκ . We assume that Lκ ̸= ∅for all κ = 1, · · · , λ and call the partition {L1 , . . . , Lλ } the market commodity-information structure. (Inour model, agents as sellers may have finer individual information structures than the market, but asbuyers, they must follow the common market commodity-information. Here, we can find the structure of
the seller-buyer information asymmetry.) For each κ = 1, . . . , λ, we denote by RLκ the subspace of Rℓcontaining the elements whose k-th coordinate is 0 if k ∈/ Lκ . Since we assume that agents can distinguish
all ℓ kinds of real commodity, we treat both their consumption behavior and production behavior as points
in Rℓ and the market demand and market supply as points in Rλ .
Agents as Buyers and Expectations of Real Receipts: In our setting, commodity κ in themarket is really a mixture of ♯Lκ kinds of real commodity. We assume that agents as buyers have an
expectation of their real receipts for each of their trade (demand) contracts before choosing their actions.
The expectation for each agent is assumed to be an identity function of the following given aggregate
parameter. For each market commodity κ, let sκ > 0 be the aggregate amount contracted to be supplied

2

Lκ
ℓ
⊂ R+
be the aggregate amount that is actually supplied. Then, forto the market, and let ŝκ ∈ R+
Lκeach unit of commodity κ that is demanded, an average amount ŝκ /sκ ∈ R+
is delivered, and thatamount will be expected by all members of the economy. (In this sense, we are considering a rational
expectation equilibrium.) To simplify the model, we assume that ∥ŝκ /sκ ∥ = 1 and let sκ = ŝκ /sκ foreach κ = 1, 2, . . . , λ.2
Agents as Sellers and Commodity-information Technologies: Given a market commodityinformation structure {L1 , . . . , Lλ }, each agent must sell their real goods as λ types of market commodity. Therefore, they should standardize real goods to sell in the market depending on the commodityinformation structure. We describe such situations by assuming that each agent has a certain kindℓℓof technology for standardizing real goods. For example, the function F : R+
→ R+
defined as
F (v1 , . . . , vℓ ) = (v1 , · · · , vℓ ) may be a candidate for such technologies. An agent having this F canstandardize real goods with no cost. However, in the real-world market economy, no one can standardize
and supply their goods with no cost. Hence, we put some natural restrictions on these standardizing
technologies.
ℓℓ
Each agent i = 1, . . . , n + m has a function Fi : R+
→ R+
which satisfies the following conditions:(C1) Fi is a continuous function.
(C2) Fik is concave function for all k = 1, . . . , ℓ, where Fik is k-th coordinate of Fi .
(C3) Fik is monotone for all k = 1, . . . , ℓ.
ℓ(C4) Fi (v) ≤ v for all v ∈ R+.

∑νℓ(C5) For each κ such that ♯Lκ ≧ 2 and for each sequence {v ν }∞
ν=1 ⊂ R , if ∥k∈Lκ Fik (v )∥ → ∞∑as ν → ∞ then there exists some Lκ′ such that | k∈Lκ ∪Lκ′ (vkν − Fik (v ν ))| → ∞ as ν → ∞.
We call Fi for i = 1, . . . , n + m agent i’s commodity-information technology. As we will formalize later,∑∑
each agent supplies their goods and services to the market as a vector ( k∈L1 Fik (v), . . . , k∈Lλ Fik (v))λ
in R+
. Therefore, condition (C5) says that each agent cannot sell their real goods in the market with
commodity-information structure {L1 , . . . , Lλ } with no cost. Note that (C5) requires such standardizing
costs to exist only if Lκ is not a singleton set. If some Lκ is a singleton, then there is no diﬀerence
between a real good and a market commodity in the corresponding market. Therefore, in such cases, theargument about standardizing costs reduces to the argument about usual production costs.3 As noted,
condition (C5) allows such arguments.
∑ℓ2ℓNote that ∥x∥ =|xi | for x = (x1 , . . . , xℓ ) ∈ R . For our results (the existence of equilibrium), the boundednessi=1
κ
of ŝ /sκ is suﬃcient.
3 This treatment of a single good market never harms existence of the equilibrium since we can make trade quantities
bounded in such markets without loss of generality. We formally show this in the existence proof.

3

Producers’ Problems: Producer j = m+1, . . . , m+n has production technology Yj ⊂ Rℓ such that
∑λYj is closed, Yj is convex, and 0 ∈ Yj . Given a price p ∈ ∆ = {(p1 , . . . , pλ )| p1 ≧ 0, . . . , pλ ≧ 0, κ=1 pκ =LλL1
, . . . , sλ ∈ R+,
1} and the expectation of their receipts for each commodity through market s1 ∈ R++−1
λ∥s ∥ = 1, . . . , ∥s ∥ = 1, producer j chooses a production plan yj = yj −yj ∈ Yj with sales and purchasingℓplans zj ∈ Rλ and vj+ ∈ R+
, so that (yj , zj , vj+ ) solves the maximization problem below: 4
max

p · zj+ − p · zj−(1)subject tozj+ ≦ (∑∑

Fjk (vj+ ))

(2)

k∈L1
k∈Lλ
− 2
− λ
− 1
+
−
+
s ,
vj + yj = yj + zj1 s + zj2 s + · · · + zjλ(3)yj = yj+ − yj− ∈ Yj ,

(4)

±
±
ℓ
λ
vj+ ∈ R+, zj± = (zj1
, . . . , zjλ, zj = zj+ − zj− .
) ∈ R+(5)

Fjk (vj+ ), . . . ,ℓof realConsumers’ Problems: Consumer i = 1, . . . , m has an initial endowment ωi ∈ R++ℓcommodities and a consumption set Xi ⊂ R . We assume that Xi is a non-empty closed convex subsetℓbounded from below such that Xi ⊃ R+
for each i. Given a price p ∈ ∆ and the expectation of their
∏λ
Lκ
ℓ λ
) , where ∥sκ ∥ = 1 for⊂ (R+receipts for each commodity in the market, s = (s1 , . . . , sλ ) ∈ κ=1 R+
κ = 1, . . . , λ, consumer i chooses consumption plan xi with market transaction plans zi and vi+ , so that
(xi , zi , vi+ ) solves the following maximization problem:max

ui (xi )(6)subject tozi+ ≦ (∑Fik (vi+ ), . . . ,

k∈L1

∑

Fik (vi+ ))

(7)

k∈Lλ

−
− 1
− 2
− λ
vi+ + x+i = xi + ωi + zi1 s + zi2 s + · · · + ziλ s ,n∑p · zi− = p · zi+ +θij πj (p, s),

(8)
(9)

j=1

xi ∈ Xi ,

(10)

±
±
ℓ
λ
vi+ ∈ R+, zi± = (zi1
, . . . , ziℓ) ∈ R+, zi = zi+ − zi− ,(11)where ui is a continuous concave utility function of i, πj (p, s) is the profit of j under price p and expectations = (sκ )λκ=1 (under the maximization problem (1) – (5)) and θij denotes consumer i’s share of the profit∑mof producer j (a non-negative real number satisfying i=1 θij = 1 for each j).
4 The variables x+ and x− always represent x+ = sup {x, 0} and x− = sup {−x, 0}. We sometimes use the notation x+

and/or x− without referring to x to emphasize the non-negativity constraint.

4m+n
Equilibrium: Denote by E = ((Xi , ωi , ui , Fi , (θij )nj=1 )mi=1 , (Yj , Fj )j=m+1 ) the economy we have de∏m (+ m
λscribed above. An equilibrium for economy E is ((xi , zi , vi )i=1 , (yj , zj , vj+ )m+n
j=m+1 ) ∈i=1 Xi × R ×) ∏m+n (
)
∏λ
Lκ
λ
ℓ λ
ℓ
λ
ℓ
R+ × j=m+1 Yj × R × R+ and (p, s) ∈ ∆ × κ=1 R+ ⊂ R+ × (R+ ) satisfying (1) – (11) and themarket clearing condition (12) with expectation specification (13) for each κ ∈ {1, . . . , λ} and k ∈ Lκ :m+n∑ziκ = 0,

(12)

i=1

∑m+n

+
i=1 Fik (vi )= sκk∑m+n +ziκ
i=1

as long as

m+n
∑

+
ziκ
> 0.

(13)

i=1

We use the notation zi = (zi1 , . . . , ziλ ) and sκk for the k-th coordinate of sκ . Note that we only consider∑m+n +∑m+n +
Eq. (13) when
i=1 ziκ > 0. Hence, if
i=1 ziκ = 0, we have no restriction on the expectation
specifications.

3

The Existence Theorem

We now state a general-equilibrium existence theorem for economies with asymmetric information:
m+n
m
∗ ∗
Theorem 1. Economy E = ((Xi , ωi , ui , Fi , (θij )m+n
j=m+1 )i=1 , (Yj , Fj )j=m+1 ) has an equilibrium, ((xi , zi ,∗ ∗ +∗ m+n∗ ∗vi+∗ )mi=1 , (yj , zj , vj )j=m+1 , p , s ), if the following conditions are satisfied:(Consumers) Each consumer i = 1, . . . , m has a non-empty closed convex consumption set Xi ⊃ℓR+that is bounded from below with a convex preference induced by a strictly monotone (i.e., x′ ≧ x,x′ ̸= x implies ui (x′ ) > ui (x)) and continuous utility function ui : Xi → R+ and a strictly positiveℓinitial endowment ωi ∈ R++.
(Producers) For each j = m + 1, . . . , m + n, Yj is a closed convex set containing 0.
∑m
∑m+n
∑m
ℓ
(Attainable Set) The attainable set for real state
j=m+1 Yj +i=1 ωi ) ⊂ R is
i=1 Xi ∩ (bounded.
ℓℓ, i = 1, . . . , m+n satisfies→ R+(Commodity-information Technologies) Each function Fi : R+
conditions (C1) – (C5).

There are some diﬃculties with the existence proof. In our setting, we must treat demand and supply(zi and vi+ ) as being distinguished from consumption and production (xi and yj ), as in the case oftransactions in asset markets, and treat producers or consumers whose actions are restricted not only by5their technologies or standard budgets but also by their buying and selling constraints. Since transactionplans zi and vi+ are not bounded, and the expectation s = (sκ )λκ=1 decides the estimation of real receipts,
continuity of demands with respect to prices and expectations may not be warranted in some boundary
cases. However, there is no appropriate reason for limiting such quantities of transactions. In this paper,
we overcome such problems by (C5), which is a natural condition for commodity-information technologies.
Condition (C5) is essential for the result of the theorem. For example, if we take Fi as Fi (v) = (v1 , . . . , vℓ )for each agent i = 1, . . . , m + n, then we can construct non-existence examples.5 Indeed, this form ofFi (v) satisfies conditions (C1)–(C4) but not (C5).

4

Proof of Theorem

Producers: We denote by ∆κ the set of all real-receipt expectations, so that ∆κ = {sκ ∈ RLκ | ∥sκ ∥ = 1}
for κ = 1, . . . , λ. As stated in the previous section, each real technology Yj ⊂ Rℓ , j = m + 1, . . . , m + n, isassumed to be closed and convex, and to contain 0, with Fj satisfying continuity (C1) and concavity (C2).
Hence, we can check that the set of all solutions to the maximization problem (1)–(5) under price p ∈ ∆∏λand expectation s = (sκ )λκ=1 ∈ κ=1 ∆κ , ηj (p, s) ⊂ Rℓ is closed and convex. Now, assume an arbitrarily
large number t > 0 and consider maximization problem (1) subject to (2)–(5) with (yj , vj+ , zj+ , zj− ) ∈[−t, t]ℓ × [0, t]ℓ+2λ . The maximization problem is restricted to [−t, t]ℓ × [0, t]ℓ+2λ . We denote by ηjt (p, s)
the set of solutions to the restricted maximization problem. The non-emptiness, closedness, and convexity∏λof ηjt (p, s) are clear. We can also prove that the correspondence ηjt : ∆ × κ=1 ∆κ → R2ℓ+2λ has a closed
graph. Indeed, the constraint correspondence (p, s) 7→ { (yj , vj+ , zj+ , zj− ) | (yj , vj+ , zj+ , zj− ) ∈ [−t, t]ℓ ×[0, t]ℓ+2λ satisfies (2)–(5) under (p, s) } has a closed graph and is lower semi-continuous, and thus also
continuous. Note that the continuity (C1) and monotonicity (C3) of Fj are used to check this. Hence,
Berge’s maximum theorem (cf. Debreu (1959), p. 19, Theorem (4)) is applicable. In this case, it may
simultaneously be confirmed that the profit function of this truncated problem, πjt (p, s), is continuous.
Consumers: As in the producer case, the set of all solutions to the maximization problem (6),∏λsubject to (7)–(11) under p ∈ ∆ and s ∈ κ=1 ∆κ , ξi (p, s), is closed and convex. Denote by ξit (p, s) the
set of solution trades to maximization problem (6) subject to (7)–(11) with (xi , vi+ , zi+ , zi− ) ∈ [−t, t]ℓ ×[0, t]ℓ+2λ , and with each profit πj (p, s) in Eq. (9) replaced by πjt (p, s), which is the maximized profit
of producers in the truncated maximization problem. We assume that each consumer has a strictlyℓpositive initial endowment, ωi ∈ R++. Then it is also possible to verify that the correspondence ξit :∏λ∆ × κ=1 ∆κ → R2ℓ+2λ is non-empty closed convex valued and has a closed graph. The non-emptiness,∏λclosedness, and convexity of ξit (p, s) are easy to confirm. For the closed graph of ξit : ∆ × κ=1 →5 See Urai, Yoshimachi, and Shiozawa (2013).

6

R2ℓ+2λ , check that the constraint correspondence (p, s) 7→ { (xi , vi+ , zi+ , zi− ) | (xi , vi+ , zi+ , zi− ) ∈ [−t, t]ℓ ×[0, t]ℓ+2λ satisfies (7)–(11) under (p, s) }, where πj in (9) is replaced by πjt , has a closed graph, and islower semi-continuous. Then apply Berge’s maximum theorem again.
Fixed Points and Limit Arguments: Take a number t > 0 suﬃciently large for the boundedattainable set to be a subset of the interior of [−t, t]ℓ , and restrict the individual maximization problems(1)-(5) and (6)-(11) to the set [−t, t]ℓ × [0, t]ℓ+2λ . We have defined correspondences ηjt and ξit to be sets ofsolutions (xi , vi+ , zi+ , zi− ) and (yj , vj+ , zj+ , zj− ) for each maximization problem under (p, s). Consider the
product map Φ of these correspondences:
Φ:∆×

λ
∏

∆κ ∋ (p, s) 7→

κ=1

m
∏

m+n
∏

ξit (p, s) ×

i=1

(
)m+n
ηjt (p, s) ⊂ [−t, t]ℓ × [0, t]ℓ+2λ
.

(14)

j=m+1

The mapping Φ has a closed graph. Define a price-expectation manipulation correspondence as follows:
m+n
+ m+n
Ψ : ([0, t]ℓ+2λ )m+n ∋ (vi+ , zi+ , zi− )m+ni=1 7→ Θ((zi )i=1 ) × Ξ((vi )i=1 ) ⊂ ∆ ×

λ
∏

∆κ ,

(15)

κ=1
m
where Θ denotes the price manipulation mapping such that, for each (zi )mi=1 , Θ((zi )i=1 ) assigns a set of
∑m+n
∑m+n
prices {p ∈ ∆| ∀q ∈ ∆, q · i=1 −zi ≦ p · i=1 −zi }, and Ξ is the correspondence that assigns the real
mixture ratio of the goods for each market. More precisely, we define the κ-th coordinate of Ξ by∑m+n+i=1 prLκ (Fi (vi ))
+ m+n
(16)
Ξκ ((vi )i=1 ) = ∑m+n
(∑
),
+
i=1
k∈Lκ Fik (vi ))∑m+n (∑+m+nκ
as long as i=1k∈Lκ Fik (vi ) ̸= 0, and otherwise by Ξκ ((vi , wi )i=1 ) = ∆ . (We use the notationLκℓprLκ for the projection onto subspace R of R for each κ = 1, . . . , λ.) Note that the right hand side of)∑m+n (∑+
Eq. (16) is always an element of ∆κ when i=1k∈Lκ Fik (vi ) ̸= 0. It is routine to check that Θ isa non-empty closed convex valued correspondence with a closed graph and that the correspondence Ξ isnon-empty closed convex valued. It is also easy to check that Ξ has a closed graph since the right hand)∑m+n (∑+
side of Eq. (16) is continuous when i=1k∈Lκ Fik (vi ) ̸= 0. Now, the product of the mappings Φand Ψ,

Φ×Ψ:∆×

λ
(∏

λ
) (
)m+n
(∏
) (
)m+n
∆κ × [−t, t]ℓ × [0, t]ℓ+2λ→ ∆×∆κ × [−t, t]ℓ × [0, t]ℓ+2λ
,

κ=1

(17)

κ=1

is a non-empty closed convex valued correspondence with a closed graph. By Kakutani’s fixed point
(
)
( ∏λ
)
t t t m+nκtheorem, Φ × Ψ has a fixed point pt , st , (xti , vit , zit )m× ([−t, t]ℓ ×
i=1 , (yj , vj , zj )j=m+1 ∈ ∆ ×
κ=1 ∆[0, t]ℓ+2λ )m+n . Equation (9) with (1) gives Walras’ Law:∀(p, s) ∈ ∆ ×

λ
∏
κ=1

∆κ , ∀z ∈ −m∑ξit (p, s) −

i=1

m+n
∑
j=m+1

7

ηjt (p, s), p · z = 0.

(18)

Under the standard argument, this means that, by the definition of Θ, the summation of (zit )m+ni=1 must∑m+n t∑m+n ttsatisfy q · (− i=1 zi ) ≦ p · (− i=1 zi ) = 0 for all q ∈ ∆, and so for each κ = 1, . . . , λ, the κ-th∑m+n t∑m+n tt m+ncoordinates of (zit )m+ni=1 , (ziκ )i=1 , must be such that −i=1 ziκ < 0 if and only
i=1 ziκ ≦ 0, where −t
if the price of κ-th commodity, pκ , equals 0.
Considering that each individual maximizes his profit (or utility), condition (2) (or (7)) is satisfied()
t t t m+nwith equality. Therefore, it follows that the state pt , st , (xti , vit , zit )mi=1 , (yj , vj , zj )j=m+1 satisfies (12) and)( t t t t t m(13). We have that the state p , s , (xi , vi , zi )i=1 , (yjt , vjt , zjt )m+n
j=m+1 satisfies (2) with equality, (3), (7)t m+nwith equality, (8), (12) and (13), and hence it follows that the real state ((xti )mi=1 , (yj )j=m+1 ) satisfies∑m t ∑m+n∑m
x =yjt + i=1 ωi , or that xti and yjt are in the attainable set which is bounded. We call
( ti=1t i t t j=m+1)
t m
p , s , (xi , vi , zi )i=1 , (yjt , vjt , zjt )m+n
j=m+1 a t-equilibrium state.
Since we take t > 0 suﬃciently large for the bounded attainable set to be a subset of the interior of([−t, t]ℓ , all xti or yjt are interior points of [−t, t]ℓ . Therefore, the t-equilibrium state pt , st , (xti , vit , zit )mi=1 ,)t t t m+nt tℓ(yj , vj , zj )j=m+1 is not an equilibrium of the original economy E only when (vi , zi ) ∈ [0, t] × [−t, t]λis a boundary point of [0, t]ℓ × [−t, t]λ for some i = 1, . . . , m + n. Hence, if (vit , zit ) is bounded for alli = 1, . . . , n + m, then an equilibrium of the original economy E exists.
Suppose that (vit , zit ) is not bounded for some i. If Lκ = {k} then sκk = 1 and restriction (3) requires+−that vjk− zjκ= yjk . (The same argument is relevant for consumers.) Therefore, we can also supposettthat (vik , zk ) is bounded for such singleton markets Lκ without loss of generality. This implies that∑∥ k∈Lκ Fik (vit+ )∥ → ∞ as t → ∞ for some κ ∈ {1, . . . , λ} such that ♯Lκ ≧ 2. Note first that
m
∑
(

∑

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

j=m+1 k∈Lκ ∪Lκ′

)
t
yik
≧

∑

t+
t+
t+
vik
− (ziκ
+ ziκ
′)

(19)

k∈Lκ ∪Lκ′

holds for all i = 1, . . . , m + n and all κ, κ′ = 1, . . . , λ by considering conditions (2), (3), (7), (8), (12),∑∑t+− k∈Lκ ∪Lκ′ Fik (vit+ ), sinceand (C4). Moreover, the right hand side of Eq. (19) equals k∈Lκ ∪Lκ′ vik∑
condition (2) (or (7)) holds with equality. However, if ∥ k∈Lκ Fik (vit+ )∥ → ∞ as t → ∞ for someκ ∈ {1, . . . , λ} such that ♯Lκ ≧ 2, then condition (C5) requires that there exists some Lκ′ such that
∑
∑
t+
t+
k∈Lκ ∪Lκ′ vik −k∈Lκ ∪Lκ′ Fik (vi ) → ∞ as t → ∞. This implies that the right-hand side and hence
the left-hand side of Eq. (19) tend to ∞ as t → ∞, contradicting the fact that xti and yjt are bounded.

5

Concluding Remarks

1. In this paper, we concentrate our attention on the case where the average amount of real goods and/or
services is delivered to each buyer. To give an explanation of this averaging process for each commodity,
we use probabilistic arguments including the law of large numbers for cases with daily perishable goods
for consumers during a month, and input commodities for big buyers such as companies.

8Such arguments are plausible when the cost structure (C5) for commodity-information technology is
not essential for small market transaction amounts.
2. Cost condition (C5) may not necessarily mean the existence of total welfare loss under the use of+commodity-information technologies. Indeed, we may assume Fik (vi+ ) = vik
for each i = 1, . . . , m + n and+++| is not too big
k = 1, . . . , ℓ, where Fik (vi ) represents the k-th coordinate of value Fi (vi ), as long as |vik
(e.g., it is in the real attainable set). In this sense, the existence of the equilibrium result in our previous
paper (Urai et al. 2013) may be identified as a special case of the theorem in this paper by defining Fik ,++i = 1, . . . , m + n and k = 1, . . . , ℓ, as Fik (vi+ ) = vik
if vik≦ b and Fik (vi+ ) = b otherwise, for an arbitrarytransaction upper bound b.
3. As we state in the introduction, some parts of the commodity-information structure and technologies may be considered to be administrated by the government. By regarding commodity-information
structures and/or technologies as policy variables, we can implement some comparative statics analysis.
4. Under the equilibrium in this paper, given the structure of commodity-information technology,
the transaction upper bound is treated endogenously so that we succeed in describing the situationthat “bad money drives out good” as a natural equilibrium situation of the model. It would also be
possible, however, to ask what defines the commodity-information technology. Specifically, while we havetreated the market commodity information structure {L1 , . . . , Lλ } as given, it would be desirable for theconcepts of market equilibrium, market information structure, and commodity-information technologies tobe mutually related and to be treated simultaneously as an equilibrium under a more general framework.

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