Proﬁt Share Equilibria for Economies withPublic Goods�� ����������2019 � 8 � 7 ������� ���������� �����������Hiromi MurakamiCore Limit Theorem for Economies with Production

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ContentsIntroduction
Model
Dividend Lindahl Equilibrium
Public Production CoreTheorem 2: Optimality and Core Property
Theorem 1: Core Limit Theorem
Conclusion
References

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Introduction (1)����������� Lindahl �������������������������������
���������������������Lindahl (1919) ���� Samuelson (1954) ����������������������������������������������
����������������������������������������������������������
����

Lindahl ������������������������������������������������������
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Introduction (2)Lindahl ����������Lindahl �����������Arrow-Debre ��������
��������������������Lindahl-Foley �������������������������������
��� satiation ������������������������������������������
• Foley (1970) · · · ��������������������
������������ cone ����⇒ ���������
������Free disposability ��������Milleron 1972��
• Mas-Colell and Silvestre (1989) · · · ������������
����������������������� 1 �����
��������������������
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Introduction (3)Dividend Lindahl Equilibrium ������
��� Dividend Lindahl Equilibrium � Mas-Colell and Silvestre(1989) � Lindahl-Foley ����������� cost-share ��������� proﬁt-share ��������������∗ �������satiation ��∗ �����������������������∗ ���������������������������������������������������������������������
��������������������������

Hiromi MurakamiCore Limit Theorem for Economies with Production

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R is the set of real numbers.
K ∪ L is the ﬁnite set of commodities.
· · · K ̸= ∅ is the index set of public goods.
· · · L ̸= ∅ is the set of private goods (K ∩ L = ∅).
I is the index set of agents (=consumers).
J is the index set of private ﬁrms.
For ﬁnite set A, denote by ♯A the number of elements of A.
Write RK instead of R♯K to represent ♯K-dimensional vector
space.

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Public GoodsModel (1)As in Foley (1970), a vector of public and private goods iswritten as (x1 , . . . , x♯K ; z1 , . . . , z♯L ) = (x; z) ∈ RK∪L .
Each agent i ∈ I is written by (≿i , ωi ), where ≿i is preference
relation on closed convex consumption set Xi ⊂ RK∪L
L is the initial endowment.
bounded from below, and ωi ∈ R++
The preferences are assumed to satisfy reﬂexivity, transitivity,completeness, continuity, and strict convexity.
For each agent, preference can be satiated, and locally nonsatiated at every point except for the maximal satiation point.

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(K個人1

個人2

個人#I・非競合性
・非排除性
純粋公共財;L)Model (2)An economy E is a ﬁnite lists of agents, (≿i , ωi )i∈I , private ﬁrm
technologies, (Yj )j∈J , and a public ﬁrm technology Y0 .
(1-1) For each j ∈ J, Yj is a closed convex subset of RK∪Ls.t. Yj has no positive coordinate for public goods.
(1-2) Technology Y0 is a closed convex subset of RK∪L s.t.
0 ∈ Y0 and Y0 has no positive coordinate for private goods.
(1-3) For each Yj , public goods are harmless in the sensethat if (x; z) ∈ Yj , then (x′ ; z) ∈ Yj for all x′ ≧ x.
N.B. Compare (1-3) with the condition in Foley (1970), i.e.,(B.5) No public good is necessary as a production input.
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私企業 j

公企業 0（j=1,2,･･･,#J）（公共財の生産技術）

LK0K0LModel (3)The set of all technically possible production plans in theeconomy, Y , is deﬁned as the following set:Y = {(x; y)|y = y0 +j ∈ J }.

∑j∈J yj , (x; y0 ) ∈ Y0 , (x; yj ) ∈ Yj for all

For private ﬁrms (Yj )j∈J , the list of sets of the technology
owners is represented by (Bj )j∈J .
Y and Y0 are owned by I, the set of all members of economy.
The pure exchange case is Yj = {0} and Bj = ∅ for all j ∈ J.
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私企業 j（j=1,2,･･･,#J）

LControlRights0K価値財（R. Musgrave）は、人間社会のあるべき姿 − それが国家によって異なるなら個人の信条への侵害? −という目的に向けられた財として、それを普遍的かつ十全に行き渡らせる役割を国家が担うという意味で、目指された公共財であるといえる。

公企業 0（公共財の生産技術）
公共財 は全員に（非排除性）、等量消費される（非競合性）。

Model (4)A consumption allocaion for a list of agents I is a sequenceof elements of consumption sets, ((xi ; zi ) ∈ Xi ⊂ RK∪L )i∈I .

For economy E = ((≿i , ωi )i∈I , Y, (Bj )j∈J ), if (x; zi )i∈I satisﬁes∑i∈I zi =∑i∈I ωi + y for some (x; y) ∈ Y ,we say that (x; zi )i∈I is feasible under (x; y) ∈ Y .

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(K;K0LL)

Dividend Lindahl Equilibrium
Dividend Lindahl equilibrium for economy E is the list ofprice vector p∗ = (p∗K ; p∗L ) ∈ RK∪L , private price vectors
for consumers (piI ∗ ∈ RK )i∈I and producers (pjJ ∗ ∈ RK )j∈J∑∑∗i∗such that pK = i∈I pI + j∈J pjJ ∗ ,I ,non-negative dividends d∗ = (d∗i )i∈I ∈ R+feasible consumption allocation (x∗ ; zi∗ )i∈I under∑∑(L)
∗∗
(L)
∗∗
∗,
Ywithy=y+y(x ; y ) ∈ Y = Y0 +j0j∈J jj∈Jwhere (x∗ ; y0∗ ) ∈ Y0 and (x∗ ; yj∗ ) ∈ Yj for each j ∈ J,
which satisﬁes the following three conditions:(i) p∗L · yj∗ − pjJ ∗ · x∗ ≧ p∗L · yj − pjJ ∗ · x for all (x; yj ) ∈ Yj and j ∈ J,(ii) p∗ · (x∗ ; y0∗ ) ≧ p∗ · (x; y0 ) for all (x; y0 ) ∈ Y0 ,(iii) for each i ∈ I, (x∗ ; zi∗ ) is the ≿i -greatest element in the set,{(x; zi ) ∈ RK∪L | piI ∗ · x + p∗L · zi ≦ p∗L · ωi + d∗i }.
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Lemma 1

Lemma 1 A dividend Lindahl equilibrium satisﬁes∑∑∗∗ · y ∗ − pj ∗ · x∗ ) + p∗ · (x∗ ; y ∗ ),d=(p0ji∈I ij∈J L
Jif the equilibrium allocation has no satiation point. In general,by the feasibility condition and the budget constraint (iii), we∑∑∗have i∈I di ≧ j∈J (p∗L · yj∗ − pjJ ∗ · x∗ ) + p∗ · (x∗ ; y0∗ ),Proﬁt �������������������������������������������������������Walras Law �������Hiromi MurakamiCore Limit Theorem for Economies with Production

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Dividend Lindahl Equilibrium (Profit Share)･･･Profit は Closed Production Groupで分けている。

Dividend Lindahl Equilibrium (2)Dividend Lindahl Equilibrium is a generalized concept ofLindahl-Foley Equilibrium of Mas-Colell and Silvestre (1989)
in the following sense:
There are multiple private goods and public goods.
Preferences are possibly satiated.
There are multiple private ﬁrms. Consumers and private
frims pay the cost of public goods.
Public ﬁrm maximizes its proﬁts. Our equilibrium concept
focuses on the proﬁt-shares of public and private ﬁrms.

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Closed Production Group (1)Given feasible (x∗ ; zi∗ )i∈I under (x∗ ; y0∗ ) ∈ Y0 and (x∗ ; yj∗ ) ∈ Yj
for each j ∈ J in E, a group of members of I, C ⊂ I, is called aclosed production group for (x∗ ; zi∗ )i∈I under (x∗ ; y ∗ ) with∑∗∗y = y0 + j∈J yj∗ if there exists J(C) = {j ∈ J | Bj ⊂ C} s.t.
∑∗ =zi∈C i∑∗+ω+y0i∈C i∑∗,yj∈J(C) j

where (x∗ ; y0∗ ) ∈ Y0 and (x∗ ; yj∗ ) ∈ Yj for all j ∈ J(C).

�For closed production group C, their status quo allocationsof public and private goods can be supplied through public∑
goods technology Y0 , private ﬁrms’ technologies j∈J(C) Yj
owned merely by members of C, and resources owned by C.
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Closed Production Group (2)The total members group I of E is always a closed productiongroup. It is quite natural to consider that such a group has full∑control rights to change their production j∈J(C) yj∗ to another.
From the deﬁnition of CPG and the budget constraint (iii) ofdividend Lindahl equilibrium, we have the next lemma.
∑Lemma 2 For each closed production group C ⊂ I, i∈C d∗i ≧∑∑∑j∗ ∗j∗ ∗∗∗i∗∗ ; y ∗ ).
(p·y−p·x)+(p+p;p)·(x0j∈J(C) L ji∈C Ij∈J(C) JLJ· · · All proﬁts are completely distributed in each CPG.

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Admissible Production Decision Groups

Suppose that E = ((≿i , ωi )i∈I , Y, (Bj )j∈J ) is associated with thestructure of production decision groups for private goods.
Admissible production decision groups C(E, (x; zi )i∈I , y) =(Ch )h∈H for each (x; zi )i∈I feasible under (x; y) with y = y0 +∑j∈J yj , (x; y0 ) ∈ Y0 , (x; yj ) ∈ Yj for each j ∈ J, of E as follows:∅ and I are elements of {Ch | h ∈ H},

for every h ∈ H, there is a private technology, YCh ⊂ RK∪L ,∑if Ch is CPG, YCh = {(x; y)|y = j∈J(Ch ) yj , (x; yj ) ∈ Yj }.

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Public Production Core
Given economy E = ((≿i , ωi )i∈I , Y, (Bj )j∈J ), (x; zi )i∈I feasible∑
under (x; y) ∈ Y with y = y0 + j∈J yj , is a public production
core allocation with C(E, (x; zi )i∈I , y) = (Ch )h∈H , if there are nocoalition S with partition (S1 , S2 ), S = S1 ∪ S2 , where S1 ⊂ I \ Cand S2 ⊂ C for a certain C = Ch , h ∈ H, and no allocation(x′ ; zi′ )i∈S∪C for S ∪ C with (x′ ; y ′ ) ∈ Y0 +(L) YCh , where′ , satisfying the following conditions:y ′ = y0′ + yCh(f-1)∑′ = y′ + y′ +z0i∈S∪C iCh∑i∈(S∪C) ωi ,(g) (x′ ; zi′ ) ≻i (x̄; z̄i ) for all i ∈ S, and(x′ , zi′ ) ≿i (x̄; z̄i ) for all i ∈ C \ S.

Hiromi Murakami

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Optimality and Core

[Equilibrium ⇒ Core]

Theorem 2* If feasible allocation (x∗ ; zi∗ )i∈I under (x∗ ; y ∗ )∈ Y is
dividend Lindahl equilibrium for E = ((≿i , ωi )i∈I , Y, (Bj )j∈J )under p∗ ̸= 0 with C(E, (x∗ ; zi∗ )i∈I , y ∗ ) = (Ch∗ )h∈H ∗ , where Ch is
CPG for (x∗ ; zi∗ )i∈I under (x∗ ; y ∗ ) for each h ∈ H ∗, then (x∗ ; z ∗ )nis a public production core allocation of E n for every n ≧ 1.
(1) Pareto optimality can be proved as the case with C = I.
(2) If the amount of public goods is ﬁxed, for each n ≧ 1, coreproperty is proved when p∗L · zi∗ ≧ p∗L · ωi for all i ∈ S1 .

(3) If the amount of public goods is not ﬁxed, even for n = 1,
core property is difﬁcult to prove. · · · � The result of Mas-Colelland Silvestre (1989) is hard to extend.
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Counter Example for Case (2) of Theorem 2HPublicpx∗

Y0Private

0Public

Agent A(pAK , pL )

x∗
ωA0z′

PrivateHiromi Murakami

Agent A �������
∗ < p∗ · ω .
· · · p∗L · zA
A
L
Dividend Lindahl ���
S1 = {A}, C = {A, B ��� I ������� }���������Public

Agent Bx∗

(pBK , pL )
ωB0z′

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Core Limit Theorem[Core ⇒ Equilibrium]

Theorem 1 A feasible allocation (x∗ ; zi∗ )i∈I under (x∗ ; y ∗ ) ∈ Y∑∗∗
with y = y0 + j∈J yj∗ for E = ((≿i , ωi )i∈I , Y, (Bj )j∈J ), with
C(E, (x∗ ; zi∗ )i∈I , y ∗ )j∈J ) = (Ch∗ )h∈H , where {Ch∗ | h ∈ H} ⊃ {I},is a dividend Lindahl equilibrium if its n-fold replica allocation
belongs to the public production core of E n for every n ≧ 1.
· · · Public Production Core ����������������
������Dividend Lindahl Equilibrium����������
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Conclusion

Dividend Lindahl Equilibrium �����������������������������������������
�����������������������
Dividend Lindahl Equilibrium �������������
�������������Public Production Core � Limit����������
Dividend Lindahl Equilibrium ����������������������������������������
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Dividend Lindahl Equilibrium ���∑Lemma 2 For each closed production group C ⊂ I, i∈C d∗i ≧∑∑∑j∗ ∗j∗ ∗∗∗i∗∗ ; y ∗ ).
(p·y−p·x)+(p+p;p)·(x0jj∈J(C) Li∈C Ij∈J(C) JLJ· · · Satiation �����������������Proﬁt �Closed Production Group C ���������������������������������������������
��������Walras Law �����Hiromi MurakamiCore Limit Theorem for Economies with Production

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例： 医療費による財政赤字と私企業

国が治療費を定める ⇒高額医療を実施 ⇒ 税で補填される公共財としての医療を提供している国家という観点から、公企業は赤字の内部留保として定着する形で均衡となっているReferences
Debreu and Scarf (1963): A limit theorem on the core of an economy,
International Economic Review.
Foley (1970): Lindahl’s Solution and the Core of an Economy withPublic Goods, Econometrica.
Mas-Colell and Silvestre (1989): Cost Share Equilibria: A Lindahlian
Approach, Journal of Economic Theory.
Milleron (1992): Theory of Value with Public Goods: A Survey Article,
Journal of Economic Theory.
Murakami and Urai (2017): Replica core limit theorem for economies
with satiation, Economic Theory Bulletin.
Murakami and Urai (2019): An axiomatic characterization of the pricemoney message mechanism for economies with satiation, Journal of
Mathematical Economics.
Samuelson (1954): The Pure Theory of Public Expenditure, The
Review of Economics and Statistics.
Yamamoto (2019): ���������������������, ����������������.
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