Discussion Papers In Economics
And Business

Market Commodity Structure in General Equilibrium Model:
The Commodiﬁcation Technology
Ken Urai
Hiromi Murakami
Weiye Chen
Yijik Oh

Discussion Paper 23-10

September 2023

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANMarket Commodity Structure in General Equilibrium Model: The
Commodiﬁcation Technology
Ken Urai∗
Hiromi Murakami†
Weiye Chen‡
Yijik Oh§

Abstract
In this paper, we formalize the concept of commodiﬁcation technology determining a commodity structure of the market as an extension of the standard Arrow-Debreu general equilibrium framework.
Like the commodity diﬀerentiation approach in Mas-Colell (1975) and the hedonic approach in Rosen(1974), we treat a commodity as a bandle of characteristics. Such quality of each commodity as wellas the market viability problem under asymmetric-information arguments like Dubey et al. (2000)
and Bisin et al. (2011) to determine the market structure will be treated uniformly as an endogenous
equilibrium state in a strict sense of the general equilibrium theory. In our model, a market participant(a consumer or a producer ) is supposed to have a commodiﬁcation technology that enables each agent
to prepare for market commodities based on their endowments and the transformation technology on
basic characteristics, representing the costs for standardization, advertisement, signaling, etc.
Keywords: General Equilibrium Model, Commodity Diﬀerentiation, Market Structure, Asymmetoric
Information, Commodiﬁcation Technology
JEL classiﬁcation: C62; D51; D82; L11; L15∗ Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp† Faculty of Fine Arts, Kyoto City University of Arts, Email: h murakami@kcua.ac.jp‡ Graduate School of Economics, Osaka University, E-mail: chenweiye198966@yahoo.co.jp§ Graduate School of Economics, Osaka University, E-mail: iori032526@gmail.com

1

Introduction
The purpose of this paper is to provide a general equilibrium framework to treat endogenously theformation of market commodity structure. Here, the market commodity structure means both the qualityof commodities (as in the commodity diﬀerentiation under the hedonic approach) and the kinds of commodities emerging in an economy (as in the case with market viability under the asymmetric information
approach). Hence, we provide a uniﬁed general equilibrium viewpoint on both approaches to endogenize
the market structure problem.
For the quality of commodities, there are classical (including general equilibrium) frameworks like Rosen(1974), Lancaster (1966), and Mas-Colell (1975), called the hedonic or product-diﬀerentiation approach.
In such models, commodities embody some properties called characteristics, which are sources of utilities
and implicitly determine the value of commodities. Since discussions in their papers focus on describing
the equilibrium states with commodity diﬀerentiation, they simply set each commodity as an exogenously
given bundle of characteristics. So the framework can be extended to investigate the relationship between
the determination of the quality of commodities and market mechanism. We have similar motivations
as classical monopolystic and oligopolistic product diﬀerentiation frameworks like Chamberlin (1937). In
addition to their implications, our general-equilibrium framework can also make the determination of thequality of commodities tie to social optimality.
On the other hand, problems about what kind of commodities are actually traded have been treated as
the market viability problem. This is one viewpoint of the adverse selection problem discussed by Akerlof(1970) and/or Rothschild and Stiglitz (1976). For the general-equilibrium treatment, the approachesof Dubey et al. (2000) and Bisin and Gottardi (1999) are groundbreaking. Problems such as adverse
selection assume that informational asymmetry is a structure that depends not only on individuals but
also on their selling and/or buying standpoints. Dubey et al. (2000) and Bisin and Gottardi (1999)
realizes such selling-buying informational asymmetry by considering a system of pooled insurance in an
economy. Subsequently to these groundbreaking works, Bisin et al. (2011), Correia-da-Silva (2012), andMeier et al. (2014) treat selling-buying informational asymmetry in a static pure exchange economy with
diﬀerent market frameworks (as contracts for delivery or generalized goods with the description of agents’names). One of the authors also treat this problem (Urai et al. 2017) in a standard static Arrow-Debreu
production economy by considering the concept of commodiﬁcation technology that can treat upper bounds
for individual market trade amounts endogenously through a certain kind of cost structure, which is an
important starting point for this paper’s theme of endogenization of market structure.
What was treated in Bisin et al. (2011) as a general equilibrium problem of future market contracts
under information asymmetry is recast here as a market structure endogenization problem. The key
is commodiﬁcation technology introduced in Urai et al. (2017), where the problem is restricted to theendogenization of delivery upperbounds.
It goes without saying that asymmetric information problems such as signaling and adverse selection
play an important role in determining market structure as a “market viability problem.” However, as
mentioned above, market structure is not only a problem related to them. The “market viablity problem”
caused by information asymmetry and the “market quality problem” for product diﬀerentiation (combining
characteristics such as the hedonic approach) can be treated in an integrated manner by attributing themto the “(production) technology” problem of commodiﬁcation. This is what this paper attempts to do.
In addition, a general and integrated view can be given toward such “technology” in a way that also1

encompasses technology in the normal competitive market. With these various perspectives, the “contract”
model of Bisin et al. (2011) and Urai et al. (2017) are reformulated as our “market commodity” model. The
objective of our model is to address the key issue of “endogenous determination of commodity structure”
in a general equilibrium market.
Here, we give a brief overview of the general equilibrium model of this paper. As in the hedonic
approach, we assume that each commodity is produced from a basic variety of materials, referred to
below as characteristics. The number of all (potentially) possible commodities will be denoted by an
integer λ ≧ 1 and the number of characteristics by ℓ ≧ 1. We assume that exchange is possible only inthe market, through commodities. On the other hand, what gives people utility and serves as an input
for production are the characteristics, i.e., the basic material levels mentioned above. In other words,
consumption and production are considered here as private activities, and the question is how the social
activity (structure) of the market is built upon them. Toward the potential commodities, κ = 1, . . . , λ, foreach seller i (a consumer or a ﬁrm), we suppose that there is a technology to transform characteristics (avector in Rℓ ) into commodities, commodiﬁcation technology Ci . For more detail, technology Ci enables foreach seller i, to transform a list of certain amounts of characteristics, (viκ ∈ Rℓ )λκ=1 , into certain amountsof commodities, (ϕκi ∈ R)λκ=1 , together with certain real deliveries of characteristics, (dκi ∈ Rℓ )ℓκ=1 as the“quality” of commodities, κ = 1, . . . , λ. Hence, the commodiﬁcation technology of i is deﬁned as a set, Ci ,including the triple, ((viκ )λκ=1 , (ϕκi )λκ=1 , (dκi )λκ=1 ) as its element. (See Fig.1 for an image of these.) Notation
ϕκi and dκi for each commodity κ come from the usage of variables for future market contracts in Bisin
et al (2011). Together with the commodiﬁcation technologies, our model also allows the selling-buying
asymmetry by considering a system of pooled insurance. That is, for each commodity κ, each agent i, asa buyer, buys ziκ units of that commodity while expecting the actual delivery of characteristics per unit
from the market to be sκ ∈ Rℓ , which is equal to its average value derived from what all sellers delivered
to the market (see Fig.2). We will treat sκ ∈ Rℓ (i.e., the quality of commodity κ) for each κ = 1, . . . , λ, asa parameter like the price. Then, the determination of qualities and quantities (the market structure) willbe identiﬁed with a rational expectation equilibrium state including parameters s = (s1 , . . . , sλ ) ∈ (Rℓ )λand p ∈ Rλ (see Fig.3).
In summary, this paper provides a general equilibrium model which determines market structure in
the context of the hedonic approach and of the market viability problem. Even though there are many
approaches that are based on imperfect competition frameworks, our model gives a perfect competition
general equilibrium forms on the market structure determination problem. We believe, speciﬁcally, that
this framework can also be a ground for describing an economy with platformization by incorporating aﬁrm-formation argument like Urai et al. (2023).
The rest of the paper is organized as follows. In Section 2, we provide a two-characteristics and onecommodity example to establish a concrete image of our model. In Section 3, The Model, we focus on
agent i’s (i = 1, · · · , m + n) interaction with markets as a seller and a buyer, referring to some assumptions
on the commodiﬁcation technology. We mention producer’s and consumer’s maximization problems andequilibrium in Section 4. In Section 5, the theorem and optimality argument are referred. We provide a
speciﬁc calculation of the example and proof of theorems in Appendix.
We denote by R the set of real numbers and Rn the n-dimensional Euclidean space. On Rn , there arethree types of order relations, ≦, <, and , such that for each x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) in
Rn , we have (x ≦ y) ⇐⇒ (∀i = 1, 2, . . . , n, xi ≦ yi ), (x < y) ⇐⇒ (x ≦ y and x 6= y), and (x nny) ⇐⇒ (∀i = 1, 2, . . . , n, xi < yi ). By R+
and R++, we denote {x ∈ Rn | 0 ≦ x} and {x ∈ Rn | 0  x},2

respectively. For n-dimensional Euclidean space Rn and subset A ⊂ {1, . . . , n}, let RA be the subspace ofRn containing the elements whose k-th coordinate is 0 if k ∈/ A. For each ﬁnite set A, we denote by ♯AAA
the number of elements of A. R+
denote the non-negative orthant, and R++
denote the strictly positiveAA
orthant of R , where we identify R with the ♯A-dimensional Euclidean space.

2

An Example (Two Characteristics and One Commodity)
Consider an economy containing two consumers, i = 1, 2, and two characteristics (Grain and Vegetable)2
k = G, V . We denote the (grain, vegetable) characteristics space by R+. Characteristics are sources of2
utility so that the consumption set Xi is a compact subset of R+
and initially characteristics are endowed

as ω1 = (1, 0), one unit of Grain for agent 1, and ω2 = (0, 1), one unit of vegetable for agent 2. In thisexample we speciﬁcally deﬁne a utility function for each i = 1, 2 asVG Vui (xGi , x i ) = xi xi ,(1)Vwhere xi = (xGi , x i ) ∈ Xi .

Suppose that the characteristics cannot be exchanged directly. Instead, a commodity named LUNCH
formed by these two characteristics will be traded in the market. For simpliﬁcation, in this example, we
assume that there is at most one kind of commodity embodied with characteristics Grain and Vegetabletraded in the market (so λ = 1 and the general equilibrium market-structure discussion is now reduced to
the viability-quality problem for the single commodity market of Lunch).
[As Sellers]: Each consumer i = 1, 2 as a seller has a commodiﬁcation technology, Ci , to produce commodity, LUNCH, from characteristics, Grain and Vegetable. In this example, consumer i’s commodiﬁcation
technology is deﬁned as follows:(ϕi , di , vi ) ∈ Ci ⇔ ϕi = viG + viV and (viG , viV ) ≧ di ≧1 G V(v , v ),
2 i i

(2)

where element (ϕi , di , vi ) ∈ Ci means that for the market of LUNCH, consumer i sells ϕi units of LUNCH
22amount of characteristics while preparing vi ∈ R+
amount of characteristics. (Seeand delivers di ∈ R+

Fig. 1. Note that in this example, all agents as sellers and/or buyers are Consumers. The value of 1/2,
which indicates the lower limit, has no particular signiﬁcance.)

Fig. 1: Interaction with Markets as A Seller

3[As Buyers]: On the other hand, for the commodity, consumers as buyers have a common expectation2s = (sG , sV ) ∈ R+of receiving a certain quantity of each characteristic per unit purchase of commodity
LUNCH through the market. For instance, if zi ∈ R+ is the units of the commodity that consumer ipurchases, consumer i expects to receive zi sG quantities of characteristic Grain and zi sV quantities ofcharacteristic Vegetable. (See Figure 2.)

Fig. 2: Interaction with Markets as A Buyer[Maximization Problems and Equilibria]: Given commodity price p ∈ R++ and common expectations, consumer i’s utility maximization problem can be described as follows:

Max

V
G V
ui (xGi , x i ) = xi xi ,

(3)

G
G
xG
i = s G zi + ω i − v i ,(4)= sV zi + ωiV − viV ,(5)pϕi = pzi ,(6)(ϕκi , dκi , viκ ) ∈ Ciκ .

(7)sub. toxVi

VTwo equations (4) and (5) implies that consumer i decide (i) what to consume xGi , xi , (ii) what to buyzi (as a buyer), and (iii) what to prepare viG , viV (as a seller). The equation (6) is a budget constraint.
Since there is only one market in this economy, budget constraint is ϕi = zi as long as p > 0.
In accordance with the above settings, we now discuss whether equilibrium exists in this economy. A
list ((x∗i , ϕ∗i , d∗i , vi∗ , zi∗ )i=1,2 , p∗ , s∗ ) is an equilibrium if the following conditions are satisﬁed:(i) (utility maximization) given p∗ and s∗ , (x∗i , ϕ∗i , d∗i , vi∗ , zi∗ ) is a solution of consumer i’s utility
maximization problem for each i = 1, 2;∑2∑2(ii) (market clear) the market is clear, i=1 zi∗ = i=1 ϕ∗i ;∑2(iii) (expectation speciﬁcation) as long as i=1 ϕ∗i > 0, the expectation s∗ in an equilibrium expectation corresponds with the average amount of characteristics
(∑
)
∑2
2
∗
∗
i=1 dii=1 di= ∑2= s∗ .
∑2
∗
∗
ϕ
z
i=1 i
i=1 i
4

(8)

For the interpretation of the expectation speciﬁcation condition, see Figure 3.

Fig. 3: Market Equilibrium

Next, we derive equilibria under ŝ∗ = s∗G = s∗V and1≥ ŝ∗ ≥ 0. (Here, we focus on symmetric solutions.)
211(i) When ≥ ŝ∗ ≥ , equilibria are given as
2
(
) 3
( ∗
)
(
)
1
ŝ∗
1
ŝ
1
1
∗
∗
∗
x∗1 =
,
,
ϕ
=
,
d
=
,
0
,
v
=
,
0
, z1∗ =,
1
1
2 2(1 − ŝ∗ )2(1 − ŝ∗ ) 1
1 − ŝ∗2(1 − ŝ∗ )2(1 − ŝ∗ )
(
(
)
)
(
)
ŝ∗
1
ŝ∗
1
1
1
∗
∗
∗
∗
x2 =
,
, d2 = 0,,, ϕ2 =, v2 = 0,, z2∗ =∗∗∗∗
2(1 − ŝ ) 2
2(1 − ŝ )1 − ŝ
2(1 − ŝ )2(1 − ŝ∗ )11≥ ŝ∗ ≥ .
2
3
In section 6, we check that the proﬁle is an equilibrium. If 14 > ŝ > 0, there is no equilibrium due top∗ = 1,

commodiﬁcation technology.
(ii) When ŝ = 0, no characteristics is traded in a market (viG = viV = 0). In this case, althoughx∗1 = (1, 0), x∗2 = (0, 1), ϕ∗1 = ϕ∗2 = z1∗ = z2∗ = 0, d∗1 = v1∗ = d∗2 = v2∗ = (0, 0), p∗ = 1, ŝ∗ = 0 is also an
equilibrium (the market is not viable). Buyer’s expectation might aﬀect the viability of market.
For the discussion of eﬃciency, we discuss the attainability of a consumption state, (x1 , x2 ). A pair
(x1 , x2 ) is said to be attainable if xi ∈ Xi for i = 1, 2 and satisﬁes the following (9).
x1 + x2 ≤ ω1 + ω2 .

(9)

Clearly, the consumption state, (x∗1 , x∗2 ), in the equilibrium proﬁle above is attainable.
A list (xi , ϕi , di , vi , zi )i=1,2 is Pareto optimal if there is no other attainable state (x′1 , x′2 ) such that for
all i = 1, 2, ui (x′i ) ≥ ui (xi ) and at least one i, uj (x′i ) > ui (xi ). If 21 > ŝ > 14 , there is a loss (v11 − d1∗
1 )>0
1
∗
and (v22 − d2∗2 ) > 0. This causes a Pareto ineﬃciency. However, when ŝ = 2 and p = 1, the proﬁlex∗1 = x∗2 = ( 12 , 12 ), ϕ∗1 = ϕ∗2 = z1∗ = z2∗ = 1, d∗1 = (1, 0), d∗2 = (0, 1) establishes the Pareto optimality.

As we have seen above, some equilibria can achieve the Pareto eﬃciency but others are not. Urai et al.
(2017) shows that the ﬁrst welfare theorem can be established under speciﬁc condition, and this is anextension of the research. In contrast to Urai et al. (2017)’s model, di is not uniquely determined byvi . Buyer’s expectation has more signiﬁcant eﬀect on the Pareto eﬃciency. We argue that taking care ofbuyer’s expectation (about s) is a key to achieve the Pareto eﬃciency.

5

3

The Model

3.1

Characteristics and Market Commodity

In this model, we regard commodity as a bundle of characteristics or basic raw materials from which
consumers beneﬁt and producers produce products. 1 In other words, there is a distinction between theprivate use of goods and services and the exchange of “commodities” through social “market” institutions,both for the consumer and for the producer = ﬁrm.2
The number of characteristics are ℓ ≧ 1 indexed by k = 1, . . . ℓ. We denote the set of indices by
L = {1, 2, . . . , ℓ}. From the standpoint of treating a commodity as a bundle of characteristics, it would bereasonable to start with a subset of L as an index of a commodity. Let us consider a family of non-empty
subsets of L, {Lκ }λκ=1 , λ ≧ 1, as the list of all possible commodities. Each commodity κ (= 1, . . . , λ)may include more than one characteristic in Lκ , so is potentially a mixture of ♯Lκ kinds of characteristics.
Prices are identically given to each market commodity, not characteristics. Hence, the price space is
λ-dimensional, and in this sense, we deal with a commodity diﬀerentiation model. 3

3.2

Agents as Sellers and Commodiﬁcation Technologies

There are m > 0 individual consumers indexed by i = 1, . . . , m, and n > 0 ﬁrms indexed by j =m + 1, . . . m + n. We often use a common index i=1,2,...,m+n for consumers and ﬁrms to emphasize thatthey face the market as sellers. Each ﬁrm j has a production technology on characteristics, Yj ⊂ Rℓ . While
only ﬁrms have production technologies, every agents have a commodiﬁcation technology, a set denoted by
∏λ
Lκ
λ
ℓ λ
(ϕi , di , vi ) ∈ Ci ⊂ R+
× κ=1 R+× (R+
) (i = 1, . . . , m + n). For an element, (ϕi , di , vi ), of Ci , where1λ1
λϕi = (ϕi , . . . , ϕi ), di = (di , . . . , di ), and vi = (vi1 , . . . , viλ ), we can provide an interpretation as follows.
LκFor each commodity κ, agent i can sell ϕκi ∈ R+ units of the market commodity κ and deliver dκi ∈ R+ℓ
as inputs. Note that we assign a price to each market
units of characteristics while preparing viκ ∈ R+

commodity but consumers have their preference not over market commodities but over characteristics. 4
We make some assumptions on commodiﬁcation technology Ci as follows. Note that condition (C3)does not necessarily apply to the case ♯Lκ = 1 involving a normal Arrow-Debreu economy, i.e., where one
characteristic can be regarded as one commodity.
(C1)

λCi is a closed and convex set containing (0, 0, 0) ∈ R+
×

∏λ

Lκ
ℓ λ
κ=1 R+ × (R+ )for all i =1, . . . , m + n.
(C2) For any (ϕi , di , vi ) ∈ Ci , viκ ≧ dκi for all i = 1, . . . , m + n and κ = 1, . . . , λ. Moreover, for any(ϕi , di , vi ) ∈ Ci , ϕκi = 0 implies viκ = 0.

1 In this sense, it may be possible to treat all sellers as a kind of producer, but here we will refer speciﬁcally to the agents
who have the transformation technology Yj among the raw materials (characteristics) as producers. Such transformation
technology Yj between raw materials will be distinguished from commodiﬁcation technology Cj in the following.
2 For example, commodiﬁcation in this sense means removing crooked cucumbers, or making them uniform in size. As
such, characteristics and their “market product com-mod-it-ization” are dealt with below.
3 Note that we do not exclude the case where L = L ′ although κ ̸= κ′ .
κκ
4 Among these notation, the pair, (dκ , ϕκ ), for each commodity, κ, can be identiﬁed with a “contract” treated in Bisinjj
et al. (2011) as an element of a compact domain, Φκ × D κ . So our model can be identiﬁed with an extension of theircontract-delivery model.

6(C3) For each i = 1, . . . , m + n, for each κ = 1, . . . , λ and each sequence (ϕνi , dνi , viν )∞ν=1 in Ci ,
∑ℓ
κν
κν
ϕκν
→
∞
(ν
→
∞)
implies
(v
−
d
)
→
∞
(ν
→
∞),
where
the
subscriptk
means
the k-thiik
k=1 ikκνκνκcoordinate, except for the case that ♯Lκ = {k} for a certain k and ϕj = vjk = djk for all i and ν.
Lκ(C4) If (ϕi , di , vi ) ∈ Ci , then for every κ = 1, . . . , λ there exists a compact convex set ∆κ ⊂ R+dκsuch that for all i = 1, . . . , m + n, ϕiκ ∈ ∆κ where ϕκi 6= 0.
iFigures 4 and 5 display a typical form of commodiﬁcation technology.

Fig. 4: d-v

Fig. 5: ϕ-v

The condition (C1) is similar to the standard condition of production technologies. The assumption(C2) is for making the equilibrium ﬁt the reality. Condition (C3) means that except for the market suchthat ♯Lκ = 1 and there are no standardization, selection and/or dressing costs, each agent cannot supply amarket commodity without a positive cost per unit at least for all suﬃciently large amounts. The exception
for market ♯Lκ = 1 is important since by this, our model include the standard market delivery contract
situation for the Arrow-Debreu type model in which unbounded short position supply contracts without
costs are allowed for.5 Condition (C4) is weak, it only says that for one unit of delivery of characteristics
in market κ, the variety of real deliveries can be conﬁned in a bounded set.

3.3

Agents as Buyers and Expectations of Real Receipts

Commodity κ in the market is really a mixture of ♯Lκ kinds of characteristics and agents as buyers are
assumed to have a common expectation of their receipts for each of their trade (demand) contracts before
choosing their actions. For each market commodity κ, let ϕκ ∈ R++ be the expected aggregate amountLκwhich is contracted to be supplied to the market, and let dκ ∈ R+
be the vector of expected aggregate

amounts of real goods and/or services that are actually delivered to the market κ. Then, for each unit ofLκ
willmarket commodity κ that is demanded, agents as buyers expect that the average amount dκ /ϕκ ∈ R+
be delivered. We use valuable sκ ∈ RLκ to represent for such an average, dκ /ϕκ , for each κ = 1, 2, . . . , λ.

By (C4), we can take sκ in ∆κ for an equilibrium.

5 To include the standard unbounded short sales contracts in our model, we have used one of the simplest method that
directly assigns the cost-exception condition in (C3) of commodiﬁcation technologies. It would also be possible to obtainsuch a condition based on the shapes of commodiﬁcation technologies.

7

4

Maximization Problems and Equilibrium

4.1

Producers’ Problems

Producer j = m + 1, . . . , m + n has a production technology Yj ⊂ Rℓ and a commodiﬁcation technology
∏λ
∑λ
Lκ
λ
ℓ
C j ⊂ R+× κ=1 R+× R+
Given a price p ∈ ∆ = {(p1 , . . . , pλ )| p1 ≧ 0, . . . , pλ ≧ 0, κ=1 pκ = 1}
and the expectation of receipts for each market commodity s = (s1 , . . . , sλ ) = ((s1k )k∈L1 , . . . , (sλk )k∈Lλ ) ∈
∏λ
∏λ
Lκ
κ
, producer j chooses a production plan and market transaction plans, (yj , ϕj , dj ,κ=1 R
κ=1 ∆ ⊂
vj , zj ), where zj = (zj1 , . . . , zjλ ), to solve the following proﬁt maximization problem:p · ϕ j − p · zj

max

(10)

subject to
vj = yj + zj1 s1 + zj2 s2 + · · · + zjλ sλ ,(11)yj ∈ Y j ,(12)(ϕj , dj , vj ) ∈ Cj ,(13)

λzj ∈ R +.

(14)

Equation (9) implies that agent j should prepare vj = (vj (1), . . . , vj (ℓ)) through his production yj andpurchase zj1 s1 + · · · + zjλ sλ . Note that zjκ sκ is a point in Rℓ and (9) is a condition in Rℓ . Restrictions (10)and (11) mention that yj is producible and ﬁrm j is able to interact with markets through commodiﬁcation
technology Cj . (12) implies the buying amount should be positive.

4.2

Consumers’ Problems

Consumer i = 1, . . . , m has an initial endowment ωi ∈ Rℓ of characteristics and a consumption set∏λ+λℓ
× κ=1 RLκ ×R+. Given a price system
Xi ⊂ Rℓ with a commodiﬁcation technology (ϕi , di , vi ) ∈ Ci ⊂ R+
p and the expectation of their receipts for each commodity in the market, s = (s1 , . . . , sλ ), consumer iλ(zi = (zi1 , . . . , ziλ )) to solve
chooses a plan with market transaction plans (xi , ϕi , di , vi , zi ) ∈ Xi × Ci × R+

the following maximization problem:max

ui (xi )(15)

subject tovi + xi = ωi + zi1 s1 + zi2 s2 + · · · + ziλ sλ ,m+n∑p · zi = p · ϕ i +θij πj (p, s),

(16)
(17)

j=m+1

xi ∈ Xi ,(18)(ϕi , di , vi ) ∈ Ci ,(19)

λzi ∈ R +.

(20)

where ui is a utility function of i, πj (p, s) is the proﬁt of j under price p and expectation s = (sκ )λκ=1(under the maximization problem (10) – (14) and θij denotes consumer i’s share of the proﬁt of producer∑mj (non-negative real numbers satisfying i=1 θij = 1 for each j). Interpretations of restrictions are almost
same with producers problems. Restriction (15) is the budget constraint in the usual general equilibrium
settings.

84.3

Equilibriumm+nm+nDenote by E := ((Xi , Ci , ωi , ui , (θij )j=m+1)m
i=1 , (Yj , Cj )j=m+1 ) the economy described before.
m+nDeﬁnition 1. (Equilibrium) For an economy E, a state ((xi , ϕi , di , vi , zi )mi=1 , (yj , ϕj , dj , vj , zj )j=m+1 ) ∈) ∏m+n (
)
∏m (
∏λ
λ
λ
κ
i=1 Xi × Ci × R+ ×
j=m+1 Yj × Cj × R+ and (p, s) ∈ ∆ ×
κ=1 ∆ is an equilibrium if it is a solutionof the proﬁt maximization problem (8)–(12), and utility maximization problem (13)–(18), and satisﬁes themarket clearing condition (21) with expectation speciﬁcation (22) for each κ ∈ {1, . . . , λ} and k ∈ Lκ .
The market clearing condition and expectation speciﬁcation are deﬁned as follows.
m+n
∑

∑m+n

κ
i=1 di∑m+nκ
i=1 ϕi

(

ziκ =

i=1

∑m+n

dκi
= ∑i=1m+n κ
i=1 zi

m+n
∑

ϕκi ,

(21)

i=1

)

=s

κ

as long as

m+n
∑
i=1

(
ϕκi

=

m+n
∑

)
ziκ

> 0.

(22)

i=1

Condition (21) is the standard market clearing condition. Condition (22) says that actual mixture ratio of
market κ (the left hand side) is rationally expected by all the agents (the right hand side). Notethat we only
∑m+n κ
∑m+n κ
∑m+n κ ( ∑m+n κ )consider Eq. (22) when i=1 ϕi (that equals i=1 zi ) is positive. Hence, if i=1 ϕi = i=1 zi =0, we have no restriction on the expectation speciﬁcations.
Even though our equilibrium is deﬁned by means of trade quantities (zi )n+m
i=1 and the trading processincludes costs in real (characteristics) goods and/or services that are represented by agents’ commodiﬁcation technologies, it is easy to check (by condition (C3)) that any equilibrium characteristics statem+n((xi )mi=1 , (yj )j=m+1 ) satisﬁes the following condition:
m
∑
i=1

xi ≤

m+n
∑

yj +

j=m+1

m
∑

ωi .

(23)

i=1

m+n
We deﬁne an attainable characteristics state. The pair ((xi )mi=1 , (yj )j=m+1 ) is attainable if conditionsm+nxi ∈ Xi , yj ∈ Yj , and (23) are satisﬁed. We consider an allocation ((xi )mi=1 , (yj )j=m+1 ) without any
restriction and cost of market interaction. Let us deﬁne the attainable characteristics set for each agent

i = 1, . . . , n + m as follows:
def.

m+n
X̃i = {xi ∈ Xi | xi is a component of a pair ((xi )mi=1 , (yj )j=m+1 ) which satisﬁes (23).}
def.

m+n
Ỹj = {yj ∈ Yj | yj is a component of a pair ((xi )mi=1 , (yj )j=m+1 ) which satisﬁes (23).}
m+n
If a state ((xi , ϕi , di , vi , zi )mi=1 , (yj , ϕj , dj , vj , zj , )j=m+1 , p, s) satisﬁes (9), (14), (19), and (20), then eachagents’ real state (xi or yj ) is attainable; xi ∈ X̃i (i = 1, . . . , n) and yj ∈ Ỹj (j = m + 1, . . . , m + n). We
deﬁne the pareto optimality based on the attainable set.
∏m (m+n
Deﬁnition 2. (Pareto Optimality) A state ((xi , ϕi , di , vi , zi )mi=1 , (yj , ϕj , dj , vj , zj , )j=m+1 ) ∈i=1 Xi ×) ∏m+n ()λλ
m
Ci × R+ × j=m+1 Yj × Cj × R+ is pareto optimal if there is no other attainable pair ((x̂i )i=1 , (ŷ)m+n
j=m+1 )
such that for all i = 1, · · · , m, ui (x̂i ) ≥ ui (x) and at least one i, ui (x̂i ) > ui (xi ).
We argue that under speciﬁc conditions, an equilibrium can be pareto optimal next section.

9

5

Equilibrium Existence and Optimality Arguments
We now state a general-equilibrium existence theorem for the economy:

Theorem 1.

m+nm+nEconomy E := ((Xi , Ci , ωi , ui , (θij )j=m+1)m
i=1 , (Yj , Cj )j=m+1 ) has an equilibrium,∗∗ ∗ ∗ ∗ m+n∗ ∗((x∗i , ϕ∗i , d∗i , vi∗ , zi∗ , vi∗ )m
i=1 , (yj , ϕj , dj , vj , zj )j=m+1 , p , s ), if the following conditions are satisﬁed:ℓ(Consumers) Each consumer i = 1, . . . , m has a non-empty closed convex consumption set Xi ⊂ R+

that is bounded from below, a convex preference induced by a strictly monotonic 6 and continuousℓutility function ui : Xi → R+ , and a strictly positive initial endowment ωi ∈ R++.
ℓ 7(Producers) For each j = m + 1, . . . , m + n, Yj is a closed convex set containing −R+.

(Attainable Set) Each agents’ attainable set (X̃i (i = 1, . . . , m) or Ỹj (j = m + 1, . . . , m + n) ) isbounded.
(Commodiﬁcation Technologies) For each agent i = 1, . . . , m + n, commodiﬁcation technology
Ci satisﬁes conditions (C1) – (C4).
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 with transactions in asset markets, and treat producers or
consumers whose actions are restricted not only by their technologies or standard budgets but also by
their buying and selling constraints (9) and (14). Since transaction plans zi and vi are not bounded, andthe expectation s = (sκ )λκ=1 decides the estimation of real receipts, continuity of excess demands withrespect to prices and expectations may not be warranted in some boundary cases such as sκk → 0 for some
κ and k ∈ Lκ together with an agent’s demand like ziκ → ∞. In this paper, we overcome this problem bycondition (C3) for commodiﬁcation technologies, which is a natural condition for them as a cost structure
due to their technological limits. Note that such a discontinuity problem is likely to occur in a market inwhich the adverse selection problem (sκk → 0) exists, but our method do not exclude the existence of theadverse selection behaviors at all. We only exclude the possibility of discontinuity by considering the cost
structure in supplying market commodities.
Condition (C3) is crucial for the existence result. To see this, let us conﬁrm the attainability condition∑m∑m+n
for the real goods and services. Consider each k-th coordinate of i=1 (ωi − xi ) + j=m+1 yj :m∑(ωik − xik ) +

i=1

m+n
∑

yjk

(9),(14)

=

j=m+1

∑m+n
i=1

m+n
∑

1
vik
) + ··· + (

i=1

=
If

(

λ
∑

[m+n
∑

κ=1

i=1

m+n
∑
i=1

κ
vik
−(

m+n
∑

λ
vik
)−(

]

m+n
∑

zi1 )s1k − · · · − (

i=1

m+n
∑

ziλ )sλk

i=1ziκ )sκk .

(24)

i=1

κ
ϕκi = 0 for some κ, we have vik= 0 for all i by condition (C2). If

∑m+n
i=1

ϕκi > 0 for some κ,

the κ-th term of the right hand side of equation (24) is equal by equations (12) and (13) to:
m+n
∑

κ
(vik
− dκik ),

(25)

i=1

≧ x, x′ ̸= x implies ui (x′ ) > ui (x).
a production technology may have a non empty intersection with the positive part of the ﬁrst quadrant.
∗∗ ∗ ∗ ∗ m+n∗ ∗Otherwise, note that there may exist a trivial equilibrium, ((x∗i , ϕ∗i , d∗i , vi∗ , zi∗ , vi∗ )m
i=1 , (yj , ϕj , dj , vj , zj )j=m+1 , p , s ) =6 A function u : Rℓ → R is strictly monotonic if x′
7 Here,m+n((0, 0, 0, 0, 0, 0)mi=1 , (0, 0, 0, 0, 0)j=m+1 , 0, 0).

10κwhich is non-negative by (C2). Hence, the equilibrium state is feasible and the amount vik− dκik for each

i and κ (if it is not equal to zero) necessarily cause a welfare loss as long as the preferences are strictly
monotonic (suboptimality property). 8
Note here that conditions (C2) and (C3) do not exclude the case that viκ − dκi = 0 at least for all viκ in acertain bounded domain, K ⊂ Rl . Furthermore, let us turn our attention to the simple case, where eachcommodiﬁcation technology is represented by a function. That is, on some bounded domain (includingK) of viκ , let dκi and ϕκi be represented by functions, dκi (viκ ) and ϕκi (viκ ), respectively. To be more speciﬁc,κ
suppose dκi be identity function (dκi (viκ ) = viκ ) and ϕκi (viκ ) = Σk∈Lκ vik
in that domain. This still leaves

our model as a generalization of Bisin et al. (2011). This is because the unit of “contract” in their model
can be interpreted as some viκ such that ϕκi (viκ ) = 1, and their model has a delivery upper bound. Wethen obtain the following proposition about the optimality of the equilibrium.
Theorem 2.

(Optimality): Suppose that equilibrium viκ is in the bounded domain mentioned above forκeach i, (i.e. the above simpliﬁcation dκi (viκ ) = viκ and condition of ϕκi = Σk∈Lκ vik
holds for each i inequilibrium). Suppose additionally that at equilibrium, for each characteristic k, there exists a market κsuch that sκk > 0, then the equilibrium state is Pareto-optimal.9
The above suboptimality argument clearly suggested that the avoidance of the condition (C3) (nonessentiality of the adverse selection problem) is a necessary condition for the optimality of equilibrium(non-essentiality of government intervention) in the endogenization problem of market structure. If weconsider this theorem in conjunction with this, it can be said to suggest that the avoidance of the condition(C3) (non-essentiality of the adverse selection problem) is also a suﬃcient condition for the optimality ofequilibrium. More precisely, see the following remark.
Remark (Optimality and Suboptimality) When the presence of adverse selection is essential to ensure
the existence of equilibrium, the condition (C3) is indispensable. Hence as we argued in (25), an equilibrium is sub-optimal. However, consider that (ϕ, d) are functions of v as in theorem 2, the equilibrium belongs to a bounded set, and di = vi is valid. The condition (C3) is not necessary. If (C3) is
not essentially the key to the existence, then the equilibrium is pareto optimal. It implies that (C3),
as a condition of a cost structure between d and v, describes problems of adverse selection. In general, ϕκi and dκi may not be functions of viκ . That is, consider an element (ϕi , di , vi ) ∈ Ci , where
ϕi = (ϕ1 , · · · , ϕλ ) and di = (d1 , · · · , dλ ). There may be a market κ and a pair (ϕ̂κ , dˆκ ) 6= (ϕκ , dκ ) such that
i

i

i

i

i

i

i

i

((ϕ1i , · · · , ϕκ−1
, ϕ̂κi , ϕκ+1
· · · , ϕλi ), (d1i , · · · , dκ−1
, dˆκi , dκ+1
, · · · , dλi ), vi ) is also an element of Ci . In this case,iiii
as we mentioned in section 2, a pareto ineﬃcient equilibrium and pareto eﬃcient equilibrium may exist at
the same time. Thus, the above property may cause the existence of pareto ineﬃcient equilibrium, because
the expectation s is a given paremeter to solve utility maximization and proﬁt maximization problems,

6

Concluding Remarks
1. (An Alternative Notion of Optimality) As we have noted in the end of section 4, our equilibrium

situation with the commodiﬁcation-cost like (C3) is necessarily related to the suboptimality property. It
8 The discussion of optimality is only on the attainable set, not market-attainable. That is, ((x∗ )m , (y ∗ )m+n ) is paretoi i=1
j j=m+1
′ m+n
′
∗
′
∗
optimal if there is no attainable pair (x′i )mi=1 , (yj )j=m+1 such that ∀i = 1, · · · , m, ui (xi ) ≥ ui (xi ) and ∃i, ui (xi ) > ui (xi ).
9 This is because our model do not use the trading upper bounds. For proof, see Theorem 1 of Urai, Yoshimachi, andShiozawa (2017).

11

seems, however, that we may consider two kinds of optimality concepts in this model, (i) the optimalitythat individuals can establish by exchanging their characteristics directly in the economy (without using
the commodiﬁcation technology), as deﬁned in this paper, and (ii) the optimality that individuals canachieve by interacting with markets. While an economy do not generally achieve the Pareto optimality inthe meaning of (i), it may happen that with some conditions, the economy can establish optimality in thesense of (ii). Also, discussion (ii) suggests that an ineﬃciency may happen merely through the expectation
among buyers. The treatment of expectation, therefore, is economically important even if an economy
does not establish the Pareto optimality in the meaning of (i).
2. (Asymmetric Information and Market Viability Problem) The models of Bisin et al.(2011)
and Urai et al. (2017) are generalized applications of the DGS bankruptcy model (Dubey et al. 2005) tothe asymmetric information problem. In these models, asymmetric information is resolved by self-fulﬁlling(average) expectations in the market, and in our model the cost of building such a market structure
corresponds to the (C3)-condition represented by the signaling cost. The problem of endogenization ofmarket structure under various product diﬀerentiation possibilities is here solved as a market viability
problem, i.e., the problem of resolving asymmetric information through self-fulﬁlling expectations.
3. (Incomplete Market Dynamics) It would be worthwhile to extend our result to the dynamic
general equilibrium models with incomplete markets. As is well known that under the incomplete market(rational expectation asset pricing) framework, it is diﬃcult to treat production because of the existenceof new assets based on a future production action (see, e.g., Mas-Colell et al. 1995, Ch. 19). On the other
hand, the need for a model that can include entities creating new markets as an equilibrium action is evengreater today. A, so called, platform ﬁrm like GAFAM can be identiﬁed (from the general equilibrium
viewpoint) with a technology creating new commodities and a new market structure, i.e., based on acommodiﬁcation technology treated here.
4. (Firm Formation Dynamics) The importance of dynamics pointed out in the previous remark,should also be argued with the problem how we could incorporate the creation of new ﬁrms or the endogenized ﬁrm structure. It would be interesting to relate our result with the general equilibrium (together
with the cooperative game) argument including ﬁrm formation like Boehm (1974), Greenburg (1979),
Ichiishi (1997) and some resent researchs of the authors, Urai-Murakami-Chen (2023), Urai et al. (2023)
including Shiozawa, etc. In preparation for such dynamicization, the authors have begun to apply theendogenization problem of market structure treated in this paper to a monetary dynamic economy withan overlapping generation structure (Murakami et al. 2023).
5. (Numbers of Characteristics and Commodities) In this paper, the number of basic characteristics, l, and the possible number of commodities, λ, are taken to be ﬁnite. From the mathematical
viewpoint, it would be natural to consider how our result can be extended to the inﬁnite number of thebasic characteristics (as in Mas-Colell 1975) and/or possible commodities. Finally, to emphasize, ourmodel does not exclude that the same bundle of (the names of) characteristics correspond to two diﬀerent
commodities, so our model includes the treatment of commodity diﬀerentiation like Mas-Colell (1975), in
this respect.

12

Appendix
Appendix A : Example Calculation Detals
This appendix provides the details of the example calculation given in Section 2. The notation forcharacteristics, G and V , used in Section 2 are replaced here by 1 and 2, respectively, to simplify thenotation.
Suppose a pure exchange economy with two consumers indexed by i = 1, 2, and there are two typesof characteristics k = 1, 2 ( ℓ = 2 ) and is only one market (λ = 1). Consumers have preference overcharacteristics and the quality of goods will correspond with a common expectation of consumers s ∈ Rℓin equilibrium. We let ωik ≥ 0 be player i’s initial endowment of characteristics. Initial endowment isgiven as ω1 = (1, 0), ω2 = (0, 1) so that two consumers have incentives to trade through the market. Each
player’s utility function is given as ui (x1i , x2i ) = (x1i )(x2i ), where x1i and x2i are player i’s consumption ofcharacter 1 and 2.

(ϕ1i , d1i , vi1 ) ∈ Ci ⇔ vi1 ≥ 0, vi2 ≥ 0 ϕi = (vi1 + vi2 ) (vi1 , vi2 ) ≥ di ≥1 1 2(v , v ).
2 i i

(26)

In this setting, we consider a situation where consumers have discretion over the delivery of characteristics,
while agents do not have in Urai et al (2017)’s model. In Urai et al (2017), it is veriﬁed that any equilibrium
can achieve pareto eﬃciency. In contrast, our framework can provide the case of ineﬃciency of equilibrium,
because, to a certain extent, people have a control over their delivery. Following the setting, the utility
maximization problem ismax

ui (x1i , x2i ) = (x1i )(x2i ),(27)

sub.tox1i = s1 zi + ωi1 − vi1 ,(28)

x2i = s2 zi + ωi2 − vi2 ,(29)

pϕi = pzi ,(30)(ϕi , di , vi ) ∈ Ci .

(31)

1≥ ŝ := s1 = s2 .
2Deﬁne a Languladian function as follows:

Hereafter, we consider a case ofL(vi1 , vi2 , ν1 , ν2 ) := (ωi1 + (ŝ − 1)vi1 + ŝvi2 )(ωi2 + ŝvi1 + (ŝ − 1)vi2 ) − ν1 vi1 − ν2 vi2 .
By Karush-Kuhn-Tucker condition, we have∂L= (ŝ − 1)(ωi2 + ŝvi1 + (ŝ − 1)vi2 ) + ŝ(ωi1 + (ŝ − 1)vi1 + ŝvi2 ) − ν1 = 0,∂vi1
∂L
= ŝ(ωi2 + ŝvi1 + (ŝ − 1)vi2 ) + (ŝ − 1)(ωi1 + (ŝ − 1)vi1 + ŝvi2 ) − ν2 = 0,∂vi2vi1 = 0 or ν1 = 0,vi2 = 0 or ν2 = 0.

13

(i) vi1 = vi2 = 0. Then x1i = ωi1 , and x2i = ωi2 . Then the utility is ωi1 ωi2 .
(ii) ν1 = 0, ν2 = 0.
(−2ŝ(1 − ŝ)ŝ2 + (1 − ŝ)2

)(

vi1

)

(

−ŝ

1 − ŝ

)(

ωi1

)

=
−2ŝ(1 − ŝ)vi2
1 − ŝ −ŝωi2()−2ŝ(1 − ŝ) ŝ2 + (1 − ŝ)2
Then the determinant of the matrix
is not zero if ŝ 6= 12 .
ŝ2 + (1 − ŝ)2 −2ŝ(1 − ŝ)
When ŝ 6= 12 ,
)
(
)(
)(
)
(
vi1
−ŝ 1 − ŝωi12ŝ(1 − ŝ)ŝ2 + (1 − ŝ)21
= 2.
(ŝ − (1 − ŝ)2 )2vi2ŝ2 + (1 − ŝ)22ŝ(1 − ŝ)
1 − ŝ −ŝ
ωi2
(
)
(
)(
)
x1i − ωi1
1 − ŝ −ŝ
vi1
Since
=−
, and
x2i − ωi2−ŝ 1 − ŝ
vi2
(
)(
)(
)
1 − ŝ −ŝ
2ŝ(1 − ŝ)ŝ2 + (1 − ŝ)2−ŝ 1 − ŝ−1(ŝ2 − (1 − ŝ)2 )2−ŝ 1 − ŝŝ2 + (1 − ŝ)22ŝ(1 − ŝ)
1 − ŝ −ŝ
ŝ2 + (1 − ŝ)21
= 2ŝ − (1 − ŝ)2(

Then, the utility ui = 0.
1
If ŝ = , then (32) yields
2ŝ

1 − ŝ

1 − ŝ

ŝ

(

− 12
1
2

)(

1
2
− 12

−ŝ

1 − ŝ

1 − ŝ

−ŝ

)(

vi1
vi2

)

(
=

)

(
=−

− 21
1
2

1
2
− 12

1 0

(33)

)

0 1

)(

(32)

ωi1

, we have x1i = x2i = 0.

)

ωi2

(34)

There are inﬁnitely many solutions as long as both vi1 and vi2 are non-negative and satisfy vi1 −vi2 = ωi1 −ωi2 .
( 1)2
ωi1 + ωi2
ωi + ωi2
1
2
In this case, xi = xi =. Then ui =.1022(iii) ν1 = 0, vi2 = 0.
The Karush-Kuhn-Tucker condition becomes∂L=(ŝ − 1)(ωi2 + ŝvi1 ) + ŝ(ωi1 + (ŝ − 1)vi1 ) = 0,∂vi1
∂L
= ŝ(ωi2 + ŝvi1 ) + (ŝ − 1)(ωi1 + (ŝ − 1)vi1 ) − ν2 = 0.
∂vi2

(35)
(36)

1 − ŝ 2
ωi > ωi1 , 0 > vi1 . This violates a condition that vi1ŝ
1 − ŝ 2
is non-negative. We consider the case where the initial endowment satisﬁes the condition ωi1 >
ωi .
ŝ
1
2
ŝ ωi + (ŝ − 1) ωi. By (36), ν2 = −(1 − ŝ) ωi1 + ŝ ωi2 + (ŝ2 + (1 − ŝ)2 ) vi1 . Finally, the utility
We have vi1 =
2ŝ
(1
−
ŝ)
(
)(
)
1 1 1 − ŝ 2
ŝ
1
ui =
ωi +
ωi
ωi1 + ωi2 .
22ŝ
2(1 − ŝ)2
By (35), 2ŝ(1 − ŝ)vi1 = ŝ ωi1 − (1 − ŝ) ωi2 . If

10 It will be stated that, in this case, both consumer one and consumer two can achieve Pareto Optimality.

14

(iv) vi1 = 0, ν2 = 0.
The Karush-Kuhn-Tucker condition becomes∂L= (ŝ − 1)(ωi2 + (ŝ − 1)vi2 ) + ŝ(ωi1 + ŝvi2 ) − ν1 = 0,∂vi1
∂L
= ŝ(ωi2 + (ŝ − 1)vi2 ) + (ŝ − 1)(ωi1 + ŝvi2 ) = 0.
∂vi2

(37)
(38)

ŝ
ŝ
ω 2 , 0 > vi2 , which violates that vi2 is non-negative. We suppose that
ω 2 ≥ ωi1 . Then
1 − ŝ i
1 − ŝ i(ŝ − 1) ωi1 + ŝ ωi2(ŝ − 1)ωi1 + ŝωi2vi2 =and ν1 = (ŝ2 + (1 − ŝ)2 )+ ŝωi1 + (1 − ŝ)ωi2 . Then the utility2ŝ (1 − ŝ))()( 2ŝ (1 − ŝ)ŝ
1 − ŝ 1 1 2
1 1
ω +
ω2
ωi + ωi .
ui =2 i
2(1 − ŝ) i2ŝ2
We apply the analysis above for the case where initial endowments are given as ω1 = (1, 0), ω2 = (0, 1). Ifŝ
12
1
2 ≥ ŝ, comsumer 1 and 2’s utility u1 = u2 = 4(1 − ŝ) . Next, we discuss equilibria. When v1 = v2 = 0, then1
v11 = v22 =. In addition, d21 = d12 = 0 by the condition (31). Since the expectation corresponds2(1 − ŝ)the average amount of characteristics,
 1
d1 + d21

 1
( ) 
d1 (1 − ŝ)

ϕ
+
ϕ
1
2

 
ŝ

=
(39)
=



ŝ
1
2
 d +d 2
d2 (1 − ŝ)22
ϕ1 + ϕ2If ωi1 >ŝ
= 2 ŝ v11 = 2 ŝ v22 . By (31), 12 ≥ ŝ ≥ 14 . If ŝ is less than 12 , Pareto eﬃciency
1 − ŝ
cannot be established. The example is an extensive case of Urai et al(2017). According to a proposition

Then, we have d11 = d22 =in Urai et al (2017), Pareto eﬃciency can be achieved under a particular condition. However, we have adiﬀerent conclusion in our model. One reason is that people have a certain discretion over the delivery ofcharacteristics.

Appendix B: Proof of Theorem 1.
The proof arguments on the existence theorem would be reduced to the standard general equilibrium ones
except for the next two points: (i) we have to appropriately select valuables for constructing expectation
parameter sκ for each κ in (22) to deﬁne a ﬁxed-point mapping, and check that for the ﬁxed-point to becalled an equilibrium, such selections are harmless, and (ii) with respect to expectation parameter sκ , theconvergence, sκk → 0 for some k, causes the non-continuity (non-closedness) problem on agents’ constraint
correspondences.
For the ﬁrst problem, we will only use ϕκi in constructing sκ through the equation in (22). For thesecond problem, we can appeal to the ordinary truncation argument, i.e., setting a large cube and taking∑λits limit. The price space is ∆ = {(p1 , . . . , pλ )| p1 ≧ 0, . . . , pλ ≧ 0, κ=1 pκ = 1}. Moreover, by (C4), we
have ∆κ , the set of all real-receipt expectations for market κ, for κ = 1, . . . , λ.
Producers: Each real technology Yj ⊂ Rℓ , j = m + 1, . . . , m + n as well as commodiﬁcaton technology
λl
C j ⊂ R+× Πλκ=1 RLκ × R+
, j = m + 1, . . . , m + n , is assumed to be closed and convex, and to contain 0.

Hence, the set of all solutions to the maximization problem (10)–(14) under price p ∈ ∆ and expectation s =15

(sκ )λκ=1 ∈

∏λ

κ=1 ∆

κ

λ+λℓ+ℓ+λ, ηj (p, s) ⊂ Rℓ ×R+
is closed and convex. Now, take an arbitrarily large numbert > 0 and consider maximization problem (10) subject to (9)–(12) with (yj , ϕj , dj , vj , zj ) ∈ [−t, t]ℓ ×[0, t]λ+λℓ+ℓ+λ . In words, producers problems are restricted to [−t, t]ℓ ×[0, t]λ+λℓ+ℓ+λ . 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 → [−t, t]ℓ ×[0, t]λ+λℓ+ℓ+λ
has a closed graph. Indeed, the constraint correspondence (p, s) 7→ { (yj , ϕj , dj , vj , zj ) | (yj , ϕj , dj , vj , zj ) ∈[−t, t]ℓ × [0, t]λ+λℓ+ℓ+λ satisﬁes (9)–(12) under (p, s) } has a closed graph and is lower semi-continuous,and thus also continuous. Hence, Berge’s maximum theorem (cf.Debreu (1959) , p. 19, Theorem (4)) is
applicable. Here, the continuity of the proﬁt function of this truncated problem, πjt (p, s), is simultaneously
assured by Berge’s theorem.
Consumers: As in the producer case, the set of all solutions to the maximization problem (15), subject∏λto (14)–(18) under p ∈ ∆ and s ∈ κ=1 ∆κ , ξi (p, s), is closed and convex. Let ξit (p, s) denote the set ofsolutions to maximization problem (15) subject to (14)–(18) with (xi , ϕi , di , vi , zi ) ∈ [−t, t]ℓ ×[0, t]λ+λℓ+ℓ+λ ,and with each proﬁt πj (p, s) in Eq. (16) replaced by πjt (p, s), which is the maximized proﬁt of producers
in the truncated maximization problem. We assume that each consumer has a strictly positive initial∏λℓendowment, ωi ∈ R++. Then it is also possible to verify that the correspondence ξit : ∆ × κ=1 ∆κ →[−t, t]ℓ × [0, t]λ+λℓ+ℓ+λ is non-empty closed convex valued and has a closed graph. In particular, for theclosed graph of ξit , check that the constraint correspondence ,(p, s) 7→ {(xi , ϕi , di , vi , zi )| (xi , ϕi , di , vi , zi ) ∈[−t, t]ℓ × [0, t]λ+λℓ+ℓ+λ satisﬁes (14)–(18) under (p, s) where πj in Eq. (9) is replaced by πjt }, has a closed
graph and is lower semi-continuous. Then apply Berge’s maximum theorem again.
Fixed Points and Limit Arguments: Take a number t > 0 suﬃciently large for all the boundedattainable sets, Ỹj (j = m + 1, . . . , m + n) and X̃i (i = 1, . . . , n), to be a subset of the interior of [−t, t]ℓ .
Restrict the individual maximization problems (10)-(14) and (15)-(20) to the set [−t, t]ℓ × [0, t]λ+λℓ+ℓ+λ .
Consider ηjt , ξit , and the product map Φ of these correspondences:
Φ:∆×

λ
∏
κ=1

∆κ 3 (p, s) 7→

m
∏
i=1

ξit (p, s) ×

m+n
∏

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

(40)

j=m+1

The mapping Φ has a closed graph since each of ηjt and ξit is so.
Deﬁne a price-expectation manipulation correspondence as follows:m+nm+nΨ : ([−t, t]ℓ × [0, t]λ+λℓ+ℓ+λ )m+n 3 (ϕi , zi , vi )m+ni=1 7→ Θ((ϕi , zi )i=1 ) × Ξ((vi )i=1 ) ⊂ ∆ ×

λ∏∆κ , (41)κ=1mwhere Θ denotes the price manipulation mapping such that, for each (zi )mi=1 , Θ((zi )i=1 ) assigns a set of∑m+nprices which evaluate the excess-demand with the highest value {p ∈ ∆| ∀q ∈ ∆, q · i=1 (zi − ϕi ) ≦∑m+n
p · i=1 (zi − ϕi )}, and Ξ is the correspondence that assigns the real mixture ratio of the goods for eachmarket. More precisely, we deﬁne the κ-th coordinate of Ξ by∑m+n κ
m+n
i=1 di(42)Ξκ ((vi )i=1 ) = ∑m+n
∈ R Lκ ,
κ
ϕ
i
i=1
∑m+n
κ
as long as i=1 ϕκi 6= 0, and otherwise by Ξκ ((vi )m+ni=1 ) = ∆ . Note that the right hand side of Eq. (42)∑m+nis always an element of ∆κ when i=1 ϕκi 6= 0 by condition (C4) for (ϕκi , dκi )λκ=1 . It is easy to check that
Θ and Ξ are non-empty closed convex valued correspondence with a closed graph. In particular, Ξ has a)∑m+n (∑
closed graph since the right hand side of Eq. (42) is continuous when i=1k∈Lκ ϕik 6= 0.
16

Now, the product of the mappings Φ and Ψ,
λ
λ
(
)m+n
(∏
)
(
)m+n
(∏
)
Φ×Ψ : [−t, t]ℓ ×[0, t]λ+λℓ+ℓ+λ× ∆×∆κ → [−t, t]ℓ ×[0, t]λ+λℓ+ℓ+λ× ∆×∆κ , (43)κ=1

κ=1

is a non-empty closed convex valued correspondence with a closed graph. Then all of the conditions for
(
t
t
t
t
Kakutani’s ﬁxed point theorem are satisﬁed. Φ × Ψ has a ﬁxed point (xti , ϕt , dt , vit , zit )mi=1 , (yj , ϕj , dj , vj ,
)
(
)
(
)
∏
m+n
λ
κ
zjt )nj=1 , pt , st ∈ [−t, t]ℓ × [0, t]λ+λℓ+ℓ+λ× ∆×
κ=1 ∆ . Equation (17) with (10) gives Walras’
Law:(p, s) ∈ ∆ ×λ∏∆κ and z − ϕ ∈ −

κ=1

m
∑

m+n
∑

ξit (p, s) −i=1

ηjt (p, s)=⇒

p · (z − ϕ) = 0.

(44)

j=m+1

Under the standard argument, this means that, by the deﬁnition of Θ, the summation of (ϕti , zit )m+ni=1 must∑m+n t∑m+n ttttsatisfy q · ( i=1 zi − ϕi ) ≦ p · ( i=1 zi − ϕi ) = 0 for all q ∈ ∆, and so for each κ = 1, . . . , λ, the κ-th∑m+n κt∑m+n κtκt
κt κt m+n
coordinates of (ϕti , zit )m+ni=1 , (ϕi , zi )i=1 , must be such that
i=1 (zi − ϕi ) ≦ 0, wherei=1 (zi −tϕκti ) < 0 if and only if the price of κ-th commodity, pκ , equals 0. Moreover each of the price pκ in market κ

is strictly positive since each consumers utility function is strictly monotone. Therefore, it follows that the
(
)
t
t
t
t t m+nt tstate (xti , ϕti , dti , vit , zit )msatisﬁes (21) and (22). We have that this state
i=1 , (yj , ϕj , dj , vj , zj )j=m+1 , p , ssatisﬁes (9), (14), (21) and (22), so the real states xti and yjt are in the bounded attainable sets X̃i and Ỹj(see the end of section 2).
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 (xti , ϕti , dti , vit , zit )m
i=1 ,
)
t t
t t t
(yjt , ϕtj , dtj , vjt , zjt )m+n
,
p
,
s
is
not
an
equilibriumoftheoriginal
economy
E
only
when
(ϕ
,
d
,
v
i i i ) is
j=m+1
λ+λℓ+ℓt t t
a boundary point of [0, t]
for some i = 1, . . . , m + n. Hence, if (ϕi , di , vi ) is bounded for alli = 1, . . . , n + m, then an equilibrium of the original economy E exists.
Suppose, on the contrary, that (ϕti , dti , vit ) can not be rearranged (without changing each agent’s maximization condition) to be an interior point for some i ∈ {1, . . . , n + m} with respect to valuables of a′certain market, κ, for for all t. This implies that ϕκi → ∞ as t → ∞ for some κ′ since the boundednessof {kϕti k| t > 0} means the boundedness of {kdti k| t > 0} and makes {kvit k| t > 0} to be bounded. If this′t
is an exceptional case of (C3), i.e., Lκ′ = {k} and ϕκi = vik
for all t, then sκ is a real number in [0, 1] by(C3) and (22).
Hence, by condition (C2) with equations (8) and (9) (resp., equations (13) and (14)) together with thefree disposal condition for producers (resp., the monotonic preference condition for consumers), they can′realize at least as much as the same proﬁt (resp., the same utility) by decreasing the value of ϕκi as long′as dκi t > 0. Since this is contradictory to the impossibility of such rearrangements, we have to think that′dκi t = 0 for all t, and agent i must obtain good k from other market κ′′ 6= κ′ for all t. For such market κ′′ ,♯Lκ′′ ≧ 2, non-exceptional situation of (C3) would be applicable. It follows that now we can only consider
the non-exceptional case of (C3). Note that we have the following inequality:
ℓ
m
∑
(∑

(ωik − xtik ) +k=1 i=1

m+n
∑
j=m+1

′

yjt

)

≥

ℓ
∑

′

′

κt
(vik
− dκik )

(45)

k=1

for all κ′ .11 However, if ϕκi t → ∞ as t → ∞ for some i and κ′ , then condition (C3) requires that∑m+n κ′ t′
11 See equations (22) and (23). In particular, it is suﬃcient to consider the caseϕi > 0 since ϕtκ
→ ∞. Through
i
i=1
∑m
∑
′
′
m+n
t )+
t ≧ v κ t − dκ t for all i = 1, . . . , n + m and all k = 1, . . . , ℓ.
the above argument, we can get
(ω
−
x
y
ik
ik
ik
ik
i=1
j=m+1 jkHence, (29) follows by taking summation side by side.

17

∑ℓ

κ′ t
κ′ t
t
t
k=1 (vik − dik ) → ∞ as t → ∞. This contradicts to the fact that xi and yj are bounded.

Appendix C: Proof of Theorem 2
We deal with the pareto optimality when dκi (viκ ) = viκ and homogeneity of ϕκi holds for each i inequilibrium and ∀κ = 1, · · · , λ, ∀k = 1, · · · , ℓ, sκk > 0. These conditions are similar to the following
theorem 1. in Urai et al. (2017).12
Theorem 2. (Optimality): Suppose that there exists an equilibrium such that viκ is in the boundedκ
domain K mentioned above for each i, (i.e. the above simpliﬁcation dκi (viκ ) = viκ and ϕκi (viκ ) = Σk∈Lκ vik

holds for each i in equilibrium). Suppose additionally that at equilibrium, for each characteristic k, there
exists a market κ such that sκk > 0, then the equilibrium state is Pareto-optimal.13
m+n(Proof:) Suppose on the contrary that there is an attainable allocation ((x̂i )mi=1 , (ŷ)j=m+1 ) such thatfor all i = 1, · · · , m, ui (x̂i ) ≥ ui (x∗ ) and at least one i′ , ui′ (x̂i′ ) > ui′ (x∗i′ ). Note that we are assumingκ
here that: (i) dκi (viκ ) = viκ , (ii) ϕκi (viκ ) = Σk∈Lκ vik
holds for each i, and (iii) for each characteristic k,there exists a market κ such that sκk > 0. Then, it can be shown that for any attainable allocations((x̂i )m , (ŷ)m+n ), we can take corresponding quadiples (x̂i , ϕ̂i , dˆi , v̂i , ẑi )m+n ∈ (Ci × Rλ ) such that for
i=1

j=m+1

+

i=1

λ
all i = 1, · · · , m, (x̂i , ϕ̂i , dˆi , v̂i , ẑi )mi=1 ∈ (Xi × Ci × R+ ) satisﬁes (16)-(20), and for all j = m + 1, · · · , m + n,(ŷj , ϕ̂j , dˆj , v̂j , ẑj ) ∈ Πm+n (Yj × Cj × Rλ ) satisﬁes (11)-(14). In fact, the implications of assumptions (i),
j=m+1

+

(ii), and (iii) are as follows. All unnecessary characteristics can be sold back in the market where they
were acquired, in the quantities guaranteed in (ii). Through this, any characteristic k can be obtained
independently at price pk = pκ in the market κ where it can be obtained, guaranteed by condition sκk > 0
in (iii). This resale process does not involve any resource loss as guaranteed in (i). Moreover, the pricepk of such characteristic k is unique as long as it is deﬁned from the equilibrium price. This is because
if there can be diﬀerent values, one can arbitrage through the process described above to increase one’s
budget constraint as much as possible. It follows that under the equilibrium price, the budget constraint
for each agent under (16)-(20) or (11)-(14) can be identiﬁed with the budget under price system (pk )ℓk=1
λ
for characteristics space Rℓ . Since (x∗i , ϕ∗i , d∗i , vi∗ , zi∗ )m
i=1 ∈ (Xi × Ci × R+ ) maximizes the utility for alli = 1, · · · , m, we have the following statements.

x̂i i x∗i =⇒ p∗ (ẑi − ϕ̂i ) > Σnj=1 θij πj (p∗ , s∗ )

(46)

∗ ∗
x̂i ≿i x∗i =⇒ p∗ (ẑi − ϕ̂i ) ≥ Σm+n
j=m+1 θij πj (p , s )

(47)

m+n
∗
∗ ∗
The proﬁt maximization assumption of the equilibrium state implies Σ m+n
j=m+1 p (ẑj − ϕ̂j ) ≥ Σj=m+1 p (ziκ− ϕ∗i ). Note that since dκi (viκ ) = viκ ∈ R++L and ϕκi (viκ ) = Σk∈Lκ vik, the sum of the coordinates of
κ

(16)

∗
m
∗
κ
sκ ∈ RL equals to 1 (κ = 1, · · · , λ, Σk∈Lκ sκk = 1). Then Σm
i=1 p (ϕ̂i − ẑi ) = Σi=1 Σk∈Lκ p (v̂ik − ẑk sk ) =κΣk∈Lκ p∗ (ωik − x̂ik )(46),(47)

<

(11)

m+n
m+n
∗
Σj=m+1p∗ (ϕ∗j − zj∗ ) = Σj=m+1
Σk∈Lκ p∗ (v̂ik − ẑi sκk ) = Σm+n
j=m+1 Σk∈Lκ p (−ŷjk ).

m+nThis is a contradiction to the attainability of the allocation ((x̂i )mi=1 , (ŷj )j=m+1 ).

12 We adopt notations in this paper, not same as Urai et al (2017), but it does not essentially matter.
13 This is because our model do not use the trading upperbounds. For proof, see Theorem 1 of Urai et al. (2017).

18

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