A General Equilibrium Model with Firm Formation and Social Coalitional
Equilibrium Structure
Ken Urai (Osaka University) · Kohei Shiozawa (Osaka University PD)
March 30, 2017
1 Abstract

4 Equilibrium with Firm Formation

A General equilibrium model with the firm formation andthe determination of share holdings rates is provided througha generalization of the coalition production equilibrium (CPE)framework in Böhm (1973) and the social coalitional equilibrium (SCE) concept in Ichiishi (1981).

A list ((xi , (zit )λt=1 )ni=1 , θ, p, T ) is an equilibrium state if thefollowing three conditions are satisfied.

2 Specific Features
1. The share holdings rates are treated as given (like prices)
for all economic members although each coalition is allowed
to change the rates as SCE deviation strategy so that the formation of firms and their share holdings rates are determined
as a generalized sense of the social coalitional equilibrium.
2. Our equilibrium concept generalizes the notions ofBöhm’s CPE and Ichiishi’s SCE in that (i) technologies ofcoalitions are assumed to depend on their investment levels,and (ii) multiple coalition structures having diﬀerent investment purposes (such as diﬀerent industries having independent coalition-deviation opportunities) are simultaneously determined.
3. Our model also gives an extension of the standard
Arrow-Debreu private ownership economy (Debreu, 1959)
and an answer to the firm formation problem including thedetermination of share holdings rates.

3 Agents, Coalitions and Technologies
There are n agents, ℓ types of commodities, and λ typeseof investment purposes. Let N = {1, . . . , n} be the set ofagents. For each investment purpose t = 1, . . . , λ, a coalition
structure T t (a partition of N ) is determined as a generalized
sense of the SCE equilibrium.
Profit function for S ⊂ N under∑ investment
π t,S (p, (zit )i∈S ) ≡ max{p · y|y ∈ Y t,S ( i∈S zit )}.
Income for i of S under share-holdings rate θIit,S (p, (zit )i∈S ; θt,S ) ≡ θit,S π t,S (p, (zit )i∈S ).

t,S(zit )i∈S :≡ (θit,S )i∈S :∏nℓ λA list (xi , (zit )λt=1 )n(Xi × (R+) ) of agents’ coni=1 ∈i=1
sumptions and investments is said to be feasible under coalition structure T = (T 1 , . . . , T λ ), price p and list of share holdings rates θ ≡ ((θt,S )S∈T t )λt=1 if it satisfies the following twoconditions.

(Investment Feasibility): for each i ∈ S, S ∈ T t and∑λt = 1, . . . , λ, ωi − t=1 zit ∈ Xi .
(Budget Feasibility): for each i ∈ N , p · xi ≦ p · (ωi −∑λ ∑z t ) + t=1 S:i∈S∈T t Iit,S (p, (zit )i∈S ; θt,S ).
t=1 i∑λ(Feasibility Condition) (xi , (zit )λt=1 )ni=1 is feasible underT, p and θ.

(Market-Clearing Condition)∑λ ∑ψ t,S (p, (zit )i∈S ).
t=1S∈T t∑ni=1xi ∈∑ni=1

ωi +(Market-Stability Condition) There are no investment
type k ∈ {1, . . . , λ}, coalition D ⊂ N , list of consumptions(x̄i )i∈D , list of investments (z̄ik )i∈D , and share-holdings rateθ̄k,D satisfying the following three conditions:(i) ui (x̄i ) > ui (xi ) for all i ∈ D,∑(ii) ωi − t̸=k zit − z̄ik ∈ Xi for all i ∈ D,(iii) p·x̄i ≦ p·ωi +∑Iik,D (p, (z̄ik )i∈D ; θ̄∑t̸=kk,DS:i∈S∈T t

Iit,S (p, (zit )i∈S ; θt,S )+) for all i ∈ D.

5 Theorems
Lemma 1: Let ((xi , (zit )λt=1 )ni=1 , θ, p, T ) be an equilibrium state. Then, for any i ∈ N , xi maximizes∑
ui subjecλzt ) +t to the constraint {x′ ∈ Xi |p · x′ ≦ p · (ωi −t=1 i∑λ ∑t,St,SI (p, (zi )i∈S ; θ )}.
t=1S:i∈S∈T t i
The lemma says ∑that if we redefine each agent’s∑initial enλdowments as ωi − t=1 zit and recognize Y t,S ( i∈S zit ) as
the technology of each firm S, our equilibrium state is an equilibrium of the standard Arrow-Debreu Private Ownership
Economy.
Lemma 2: Our equilibrium state can be identified with ageneralized Social Coalitional Equilibrium.
For such an identification, we consider an additional agent,a price and shareholdings rate manipulator.
Theorem: A Generalized social coalitional equilibrium exists under a certain balancedness condition.
We investigate how the balancedness condition is satisfied
or violated in our natural economic settings and assumptions.

ReferencesBöhm, V. (1973), ” Firms and Market Equilibria in a Private Ownership Economy ”, Zeitschrift für Nationalökonomie
33, pp. 87–102.
Debreu, G. (1959), Theory of Value, Yale University Press,
New Haven, CT.
Ichiishi, T. (1981), “A social coalitional equilibrium lemma,” Econometrica 49 (2), pp. 369–377.
(JSME Methodology Subcommittee - Spring 2017 Meeting)