Title

Social structure, capital accumulation, anddistribution of wealth

Author(s)Chen, Weiye

Citation

大阪大学経済学. 2023, 73(2・3), p. 123-134

Version Type VoR
URL

https://doi.org/10.18910/93336

rights
Note

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

Vol. 73 Nos. 2・3

OSAKA ECONOMIC PAPERS

December 2023

Social structure, capital accumulation, and distribution of
wealth＊Weiye Chen †Abstract
This paper explores a two-period Allow-Debreu production economy where agents cooperatively
form firms (coalitions) by jointly determining investments and shareholding rates. The firm
structure can be perceived as a social structure based on cooperative decision-making among
agents. We introduce the concept of social coalitional general equilibrium, which is regarded asa typical social coalitional equilibrium initially presented by Ichiishi (1981), and characterize
all equilibrium states. Our findings demonstrate that an equilibrium social structure is organizedwith dependence on initial capital accumulation, aiming to generate maximum societal wealth.
Furthermore, this paper shows that in equilibrium, society tolerates maldistribution, which isendogenously determined by market mechanisms.
JEL Classification: C71, D51, D63
Keywords: Coalition Production Economy, General Equilibrium, Social Structure, Firm
Formation, Capital Accumulation, Maldistribution.

1 Introduction
Social structure refers to structurally related groups determined by the interrelationships andinteractions of agents within society. Naturally, various groups (coalitions) exist for different purposes
and benefits, and agents are allowed to belong to groups serving different purposes. Within thetheoretical framework of economics, specifically the general equilibrium concept, the structural grouporganized through these interactions among agents is commonly referred to as a firm. Consequently,social structure can be interpreted as industrial structure, and the process of forming these structuralgroups can be characterized as a problem of firm formation. Therefore, it is reasonable to address
the issue of social structure formation within the general equilibrium framework incorporating firm＊†

I am sincerely grateful to my supervisor, Ken Urai, for his invaluable support and guidance throughout my academic
research endeavors. His substantial contributions to the preliminary research of this study have played a pivotal role inshaping its trajectory and outcomes. I would also like to express my appreciation to Hiromi Murakami, for her helpful
and motivating suggestions and comments, which have inspired me to improve the paper to a greater extent.
Visiting Researcher, Graduate School of Economics, Osaka University, E-mail: chenweiye198966@yahoo.co.jp

－ 124 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

formation.
A firm-formation problem is regarded as a fundamental core solution in the coalition games
developed by Aumann and Peleg (1960), Aumann (1961), Scarf (1967), and Scarf (1971).
Subsequently, Ichiishi (1981) extended the concept of a coalition game and introduced a generalizedcooperative concept known as social coalitional equilibrium. This concept encompasses economic
settings (abstract market equilibrium) and cooperative game-theoretic arguments, ultimately aiming
to identify a social structure. Simultaneously, the coalition game framework was extended to address
core issues within production economies, including general equilibrium economies with coalition
production (Boehm, 1974), with labor-managed market (Greenberg 1979; Ichiishi, 1977), andwith decentralization issue (Ichiishi, 1981). However, despite the use of these abstract conceptual
frameworks to broadly characterize economies (societies), there is still no comprehensive formulationthat captures the relationships among exogenous parameters in society, social (firm) structure, andagents’ decision-making within organizations. More specifically, the formation of social structure is
intricately intertwined with both capital accumulation and wealth distribution. Elaborating on theseinterrelationships could offer a precise theoretical interpretation of the form of social structure.
Inspired by the above context, we introduce a two-period Arrow-Debreu general equilibrium
economy incorporating firm formation. In this economy, there are two (groups of) agents who livein two periods. Each period features a single type of good that can be either consumed or invested
in. Each agent forms a firm (coalition) in cooperation with the other agent, jointly determining theirinvestment levels and shareholding rates. Consequently, a firm (social) structure is considered a
representative partition of agents. Specifically, a firm formed by two agents is called a cooperativetype firm, whereas a firm organized by a single agent is called a non-cooperative-type firm.
Accordingly, a structure comprising cooperative-type firms is labeled a cooperative social structure,
while one consisting of non-cooperative-type firms is termed a non-cooperative social structure. Eachfirm produces goods of period

based on the investments received during period

so as to maximize

its profits. In this paper, the productivity of each type of firm varies, with non-cooperative firms
exhibiting higher productivity at lower levels of investment, while cooperative firms prove to be more
profitable in the opposite condition.1
This paper introduces the concept of social coalition general equilibrium within the establishedframework and provides a comprehensive characterization of all equilibrium states. The findings
demonstrate that the existence of equilibrium is contingent upon initial endowments (capital
accumulation). Specifically, a low level of initial endowments in society results in a non-cooperativesocial structure, whereas a high level of initial endowments leads to the formation of a cooperativesocial structure. This phenomenon arises because an equilibrium social structure is comprised onlyof firms that are capable of achieving maximum net profit.2 This implies that only a social structure
capable of generating maximum societal wealth will be established. Unfortunately, variations in
1

2

This observation highlights the fact that the production technologies of the two types of firms adhere to the singlecrossing property.
Net profit here refers to the difference between firm profit and investment cost.

December 2023

Social structure, capital accumulation, and distribution of wealth－ 125 －

productivity among different types of firms may internally determine an uneven distribution of firm
profits in the case where the cooperative social structure prevails. However, the absence of a social
structure that internally alleviates such inequality in wealth distribution among agents compels
society to accept such maldistribution in equilibrium. As a result, this paper is anticipated to aid informulating the relationship between social structure, capital accumulation, and wealth distribution
within a dynamic general equilibrium framework.
Section 2 establishes a two-period Arrow-Debreu production economy that incorporates the firm
formation problem. In section 3, we present the primary result, which pertains to the characterizationof social coalitional general equilibrium. We thoroughly examine the properties exhibited byequilibrium states and delve into a comprehensive discussion on this issue. The proofs for all of theresults are provided in Appendix, ensuring a rigorous and thorough treatment of the subject matter.
as the set of real numbers andWe use, we denote by

the number of elements ofas the set of-dimensional vector space. For finite set. The simplex, a subset of, is defined.

by the set
2 ModelConsider a dynamic Arrow-Debreu production economy with two periods. In each period,

there exists a single type of good that can be either consumed by agents or invested to produce, eachgoods for the subsequent period. There are two (groups of) agents, denoted by,

endowed with initial endowments of
In period.

, agents form firms (coalitions) to engage in cooperative investments. These firms,

established in period , exclusively use the investments made by their respective members to produce
goods for period . Furthermore, we assume that each agent can only belong to a single firm. Thus,a firm structure, denoted by, represents a partition of. To simplify our discussion, we use thefollowing notation to represent the admissible firm structures:

For each, there exists a technology correspondence denoted bycoalition
firm.

represents the total amount of the investments made by the agents inHere,, wheredenotes the investment made by consumer

in period. For a, we define a production technology as.3 Similarly, for a coalition, theproduction technology is defined as. For each firm

, the agents incollectively determine a shareholding rate

for its members. Additionally, we assume that3

Due to the symmetry ofa priceand. It is clear, the optimal inputs and outputs are inherently equal. Given, the aggregate total social optimal inputs and outputs satisfy, andNaturally, a generalization of the production technology will not disrupt our discussion.

holds for every.

－ 126 －OSAKA ECONOMIC PAPERS

that

is more efficient thanVol. 73 Nos. 2・3

when

but is more.

inefficient whenbe the price of goods, and given the investmentsLet,

the profit maximization program for firm,

, we can define, as follows:(1)It can be easily verified that for each, the profit of firm

is maximized only when equality, the profit of firmholds for all of the inequalities. Therefore, given the investments
,

, is

,

.
selects consumption

Each agent

over the two periods, taking into account his/her budget constraint. The consumer’s utility function, representing their preferences, is denoted as. Consequently, consumer ’s maximization problem can be formulated as follows:(2)

Based on the above definition, we establish the following definition of the social coalitional generalequilibrium in a two-period Arrow-Debreu production economy.
Definition 1. For a two-period Arrow-Debreu production economy, a social coalitional generalsatisfying theequilibrium state is represented as a list
following conditions:(Utility Maximization)

is a solution of (2) under,andfor every.
(Market Feasibility)
(3)
(4)
where

is a solution of (1) under(Market Stability) There exists no coalition, and shareholding rate
the following condition:such that.
, consumptions, investment
for all, satisfying

December 2023

Social structure, capital accumulation, and distribution of wealth－ 127 －

for all(5).

Utility Maximization and Market Feasibility are standard conditions that do not necessitate further
can improve the utility ofelaboration. The Market Stability condition states that no coalition
its members by dismantling the existing firm structure. The inequality in this condition can bewhen considering a deviationinterpreted as the budget constraints faced by the members of coalition

andfrom the current structure. These members are able to freely utilize their investments, thereby abandoning their existing relationships within

determine their shareholding rate.

3 Results
In this section, we present the main results of our paper.
Theorem 2. For a two-period Arrow-Debreu production economy described in Section 2,, a list(1) whenis a social coalitional,

equilibrium state that satisfies the following conditions:
every

,

fulfills (23),,.

, a list(2) whenis a social coalitional general,equilibrium state that satisfies the following conditions:,for every
for all(3) iffulfills (22), and for

fulfills (31),,fulfills (30), andfulfills (33) whereand.

, no social coalitional general equilibrium exists.

Proof: See Appendix.
The main results indicate that the equilibrium, particularly the equilibrium firm (social) structure, isdetermined by the initial endowments (capital accumulation). Specifically, although the formationof firms in equilibrium is influenced by the agents' investment behaviors
governed by exogenous parameterson the value of

,

, and

, it is ultimately. It is not surprising that equilibrium is conditional, since the convexity assumption of the aggregate total production possibility set,
, is not imposed in this economy.4 Furthermore, it is noteworthy

that the production technology functions of the two types of firms intersect at a single point, satisfying
the condition of the single-crossing property. Consequently, a cooperative structure,,is established when the initial endowment level is high, while a non-cooperative structure,, is implemented in the opposite case.
4

The convexity assumption of the aggregate total production possibility set is deduced from the Balancedness Condition
proposed by Boehm (1974) and Greenberg (1979), which is a necessary condition for the existence of firm-formation
equilibrium in a general context.

－ 128 －OSAKA ECONOMIC PAPERS

Corollary 1. Let

be the solution ofVol. 73 Nos. 2・3

for the firm

, then.
This result shows that only a firm structure with high productivity, which maximizes social net
profit, is formed thus demonstrating that agents' investment behavior not only yields individual gains
but also contributes to the overall improvement of societal wealth. Furthermore, this result is derived
exclusively from the Market Stability condition, underscoring the role of market mechanisms inshaping the formation of the firm (social) structure and enhancing social wealth.
Corollary 2. Let a list

be a social coalitional generalequilibrium state, and let the social coalitional social equilibrium state
be generated through Lemma 2. Then,This result suggests that when the (cooperative) firm structure

may not be equal to .
prevails, the presence of income

inequality in the distribution of firm profits may arise endogenously. However, from the perspectiveof each consumer, the firm structure that allows for income inequality is advantageous. In essence,a firm (social) structure formed solely through market mechanisms will tolerate a certain degree ofmaldistribution.
4 Conclusion
This paper developed a two-period Arrow-Debreu production framework that incorporates firm
formation. Agents jointly determine their investments and shareholding rates to form firms. The
coalition-deviation within a firm structure allows us to model the social cooperative structure
as a standard firm structure. Following this framework, the concept of social coalitional generalequilibrium is introduced, and the equilibrium states are characterized. Consequently, our findings
demonstrate that the equilibrium social structure, which leads to higher social wealth, is determinedby the initial endowments, although this may lead to income inequality. Future research focuses on
extending the analysis to dynamic frameworks with infinite periods. As discussed in this study, theformation of the social structure in each period is solely influenced by the initial capital accumulated
from the previous period. The introduction of dynamics into the model is expected to provide insights
into the relationship between capital accumulation and changes in the social structure.
REFERENCES
Aumann, R. J. (1961). The core of a cooperative game without side payments. Transactions of theAmerican Mathematical Society, 98(3):539–552.
Aumann, R. J. and Peleg, B. (1960). Von neumann-morgenstern solutions to cooperative games
without side payments. Bulletin of the American Mathematical Society, 66(3):173–179.
Boehm, V. (1974). The core of an economy with production. The Review of Economic Studies,
41(3):429–436.

December 2023

Social structure, capital accumulation, and distribution of wealth－ 129 －

Greenberg, J. (1979). Existence and optimality of equilibrium in labour-managed economies. The
Review of Economic Studies, 46:419–433.
Ichiishi, T. (1977). Coalition structure in a labor-managed market economy. Econometrica, 45(2):369–
377.
Ichiishi, T. (1981). A social coalitional equilibrium lemma. Econometrica, 49(2):369–377.
Scarf, H. E. (1967). The core of an n person game. Econometrica: Journal of the Econometric Society,pages 50–69.
Scarf, H. E. (1971). On the existence of a coopertive solution for a general class of n-person games.
Journal of Economic Theory, 3(2):169–181.
Appendix
Lemma 1. Let the list

be a social coalitional generalequilibrium state, then the following conditions are satisfied:(i) for any, it holds that

where

;

(ii) for any

,

,

,

, and for any,, the inequality
(6)is satisfied.
Proof: (i) Clearly, according to the definition of the maximization program, inequality
must hold. Thus, it suffices to
is false. Suppose,demonstrate thatsuch thatas a contradiction, that there exists
holds. Let

and let. We now define a new shareholding rate

be the solution to

as
(7)

for all. Then, for every

other words, for every, we have. In, the condition
(8)holds. This condition states that for every
investment triple

such that, there exists an alternative consumption and, whereand(9)This fact follows from the strict concavity of. Consequently, this constitutes a deviation,

contradicting the condition of Market Stability. The proof of (i) is thus completed.

－ 130 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3(ii) From (i), we havefor any, for anytrue if both

and
and

and

. Therefore, it is sufficient to verify that for a partition,such that inequality (6) holds. It is evident that the result isare equilibrium firm structures. Thus, we consider the case where only one. Suppose that there exists

type of structure is an equilibrium structure. Let

such that
(10)That is, there must exist a coalition

such that. For agent , there exists alternative consumption and investment triple, wheresuch that

(11)Note that. Hence, we obtain a contradiction. The case where

can be proved using

the same logic. Thus, the proof of (ii) is completed.
Lemma 2. Let the list

be a social coalitional generalequilibrium state, then an alternative social coalitional general equilibrium state
exists in the same economy. In this new equilibrium state, for every, thefollowing conditions are satisfied:
(12)
and

.

Proof: According to the newly definedand, we can observe that
. This demonstrates that the budget

underconstraint remains unchanged for every
maximizes agent ’s utility underandFurthermore, the condition

and. Hence,, satisfying the Market Stability condition.

implies(13)The final equality holds because
also holds under the state

for all. Thus, the Market Feasibility condition. Hence,is a social coalitional general equilibrium state. This completes the proof.
According to Lemma 1, for a given coalition
the following condition:and , the optimal consumptions undersatisfyDecember 2023

Social structure, capital accumulation, and distribution of wealth－ 131 －

(14)Solving this program, we obtain, and hence,. Since, , and, we have. Therefore, given, agent ’s utility maximization program is given by (15)

Solving the above program, the optimal consumption satisfies the following conditions:
(16)
(17)
Furthermore, since, the optimal consumption can be rewritten as follows:
(18)
(19)

Substituting the optimal consumption and optimal investment into the conditions in Market Feasibility,
we have
(20)
and

(21)

. Substituting

Solving equation (21), we find
. Substituting

we obtain
have. Since

into equation (20),

into, we can express the price under

, we
as
(22)

Given a pricethat satisfies condition (22), the optimal consumptions underare given by(23)Similarly, for a given coalition

and, we can calculate the optimal investments

－ 132 －OSAKA ECONOMIC PAPERS

through the firmVol. 73 Nos. 2・3’s maximum program(24)Solving this program, we have

, and hence,. According to Lemma 2, it is sufficient to use an arbitrary pairconverted according to the logic of Lemma 2. Therefore, given, , and,agent ’s utility maximization program is given by (25)

Solving the above program, the optimal consumption satisfies the following conditions:
(26)
(27)
Substituting the optimal consumption and optimal investment into the Market Feasibility conditions,
we have


(28)

and



. Calculating the equation (29), we find

It is important to note that. Substituting

into equation (28), we obtain. Substituting

have. Since

(29)

and

into

, we can express the price under

, we
as
(30)

Given a pricethat satisfies this condition (30), the optimal consumptions undersatisfy

the following condition:
(31)December 2023

Social structure, capital accumulation, and distribution of wealth－ 133 －

Now, we examine the Market Stability condition for a given firm structure

,

,

.

, we only need to determine under what conditions agents do not deviate to

Since
from

. Let

. Given a price, and that for coalition
for every

have,, the optimal investment for coalitionis. By the definition of

, we

is

, which implies. Similarly,.
Lemma 2 shows that every equilibrium investment and shareholding rate pairis equivalent to the pair

, wheresatisfies condition (12) and. Therefore, it is sufficient to verify the Market Stability condition using the pair
. Furthermore, as previously shown, the maximum profit of coalitionisindependent of the investment level. Thus, according to the Market Stability condition, the condition
tounder which agents do not deviate from
. Sinceis,is given by. That, we have
(32)It is evident that
, wherealso implies. Moreover, the equation (28) demonstrates that the Market Feasibility. Therefore, suchcondition is independent of the value ofholds for every

will notLemma 2, agents with an investment and shareholding rate pairdeviate if there exists

for an agent. Recallingsuch that
(33)Finally, since, we can observe thatand thatforfor. This implies that theaggregate total production technologies satisfy the single-crossing property. Moreover, since theoptimal investment level depends only on the price and endowment level, following the singlecrossing property, there must exist a price
and

under

for which the optimal investment levels

undersatisfy the following condition:
(34)Furthermore, since the price is the same, the marginal productivity of the optimal investment levelmust also be the same for each coalition structure:Combining the equations (34) and (35), we find that(35)

－ 134 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

(36)This condition suggests that the firm structure
and thatis an equilibrium structure if

is an equilibrium structure if

Moreover, since the optimal investment levels.

andare determined solely by the

initial endowments, we can express the conditions as follows:
(37)
and
(38)
Since
if

, it follows that, and. Hence, the firm structure

is an equilibrium structure whenis an equilibrium structure.

In summary, the social coalitional general equilibrium state
can be formulated as follows:(1) if, then the following conditions are holding:(22), and for every(2) ifsatisfies (23),condition (33), where,satisfies condition,, then the following conditions are holding:(30), and for every(3) if,,satisfies (31),and, no social coalitional general equilibrium exists.

.
,satisfies condition

and.

satisfy