数理解析研究所講究録 1215 巻 2001 年 64-77

64
The Competitive Equilibrium and Pareto Optimum
Allocations in the Economy with Clubs
Shin-Ichi Takekuma
Graduate School of Economics,

Hitotsubashi University

Abstract
Tk competitive equilibrium is defined for an economy with aclub and many identical consumers. In an
example of the economy, the eristence of the competitive equilibium is shown Also, it is proved that any

allocation under the competitive equilibrium in the economy is Pareto optimum

I.

Introduction

Groups of people who share and jointly consume goods are called “clubs”, or consumption
反 oenownels 呻 membership alTangements. G\otimes 出 consumed in clubs are intemediate$\mathrm{g}\infty \mathrm{d}\mathrm{s}$purely private goods and purely public goods. In this paper we shall consider asimple model ofan
economy where there is one club and there are many, but identical consumers. The market ofmembership ofthe dub is analyzed and the competitive equilibri um for the economy is defined In

an example of the economy, the competitive equilibrium is shown to exist. Our definition ofis an extension of the usual competitive equilibrium for economies only
competitive
with private goods. In addition, allocations under the competitive equilibrium are proved to be$\eta \mathrm{u}\mathrm{i}1\mathrm{i}\mathrm{b}\mathrm{r}^{\mathfrak{l}}\mathrm{i}\mathrm{u}\mathrm{m}$Pareto optimum.

Buchanan (1965) presented amodel of economy with clubs, andconsidered Pareto optimality of allocations in the economy. Following his paper, many papers
In his famous paper J.

$\mathrm{M}$have been published (for detail, confer the survey article by T. Sandier and J. T. Tschirhart (1980)).
In most papers such as -K. Ng (1976), 1974, 1978), E. Berglas (1976), and E. Helpman and A. .

$\mathrm{L}\backslash$$\mathrm{Y}$65
Hillman (1977), the optimality of alocations was considered In afew papers, the competitive
equilibrium for economies with clubs is analyzed, for example, by S. Scotchmer and H. Wooders$\mathrm{M}$On the other 1mA acompetitive equilibrium was defined by D. Foley (1967) and D.
Richter (1974) for economies with public goods, which is aspecial case of clubs. However, such
(1987).

$\mathrm{K}$an equilibrium is quite different from the equilibrium in economies with clubs, because clubs are
independent agents and behave for their own purpose.
The definition of competitive equilibrium depends on the behavior of clubs. In this paper we
its members’ utilities. In our model ofeconomy all individual are
assume that the clubare the same and they have initially the same
assumed to be identical in that their utility
we can easiy define anatural concept ofamount of wealth By virture of thiscompetitive equilibrium for the economy. However, in general cases, we exped that many kinds of$\ovalbox{\tt\small REJECT}$$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\alpha \mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$\mathrm{a}\mathrm{s}\mathrm{s}\iota \mathrm{n}\mathrm{n}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{L}$equilibrium concepts might be defined

A General Model

II.

First we present ageneral model of an economy with clubs. There are kinds of commodities,$J$

each of which is shared and consumed in aclub. Thus, there are clubs in the economy and each
, . Clubj is group people who share
one them is indicated by an$n$$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}\dot{\Gamma}^{-1},\cdots$

in consumptioa Also, there are

$n$

$J$

$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{y}/$kinds of other commodities, which are usual private goods andconsumed separately by each single person.
We assume that individuals are “divisibl\"e, and the set of all the persons in the economy isdenoted by$A$, which is aunit interval, i.e., $A=[0,1]$ .

Let us denote the quantity of commodity consumed in club by .
$j$

$j$

fraction$j$people belonging to dub by

is measurable subset$M_{j}\mathrm{o}\mathrm{f}A$,$\theta_{j}$$\theta_{j}=2$, where$0\leqq\theta_{j}\leqq 1$

When the set ofthe members ofdub$X$$\mathrm{f}\mathrm{i}\mathrm{a}\alpha \mathrm{i}\mathrm{o}\mathrm{n}$.

$M_{j}$

The-denotes the number of people$\#\mathrm{y}$.

$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$We assume that people do not care about who are members ofof its members.

Also, let us denote the(A4) where (A4) is the Lebesgue measure of set

total number of indiiduals in the economy is fix4 andparticipating in.

$x_{j}$

Therefore,$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$is described by pair$(x_{j}, \theta_{j})$, but only about the number$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$.

The utility ffinction of person $a\in A$ is denoted by$u=U_{o}((x_{1}, \theta_{1}),\cdots$, $(x_{n}, \theta_{n}),y)$ ,where $y\in K$ denotes the quantities of private goods.

The variable$\theta_{j}$of club$(x_{j}, \theta_{j})$indicates .
$\mathrm{a}$66degree ofcongestion.
To denote the utility of aperson who is not amember of aclub, we assume that people can get

nothing fiom belonging to clubs in which nothing is consumed Namely, when $x_{J}=0$, people in$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$get as the same level ofutility as people out of$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$getThe production set ofthe commodities consumed in clubs and the private goods is denoted by,$Y$

which is asubset ofthe non-negative orthant of a $(J+n)$ -dimensional Euclidean space. When we
write $(x,y)\in Y$, ffiej-th coordinate of vector
denotes an amount of the commodity consumed$x\in R^{J}$in, whereas vector $y\in F$ denotes amounts of private goods.

$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$Ng by measurable$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{c}\dot{\mathrm{u}}\mathrm{o}\mathrm{n}$$m$:$Aarrow R$, let us denote the income

each individual, which arises

fiun production of commodities. The total of incomes is pqud to the valued of commodities
produced in the economy. When production $(x,y)\in Y$ is chosen and the pricetheis
price ofyis $q\in P$ the total value produced
isthe following must hold.
$\mathrm{o}\mathrm{f}x$$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\dot{\mathrm{b}}\mathrm{e}\mathrm{s}$$p\in R^{J},\mathrm{a}\mathrm{n}\mathrm{d}$$px+\emptyset/\mathrm{a}\mathrm{n}\mathrm{d}$$\int_{A}m(a)da$ $=px+\wp$

Finally, we assume that every individual is initially amember of each club where nothing isconsumed in the club, that is,$x_{j}4$

and$\theta_{J}=1$and every individual has the membership ofsells his membership in the market andthe initial income ofindividual is$a$$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$in. Initially, each$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$is specified by $(0, 1)$,. When an individual wants to leave, hebuys it. Ifthe price membership ofclubj is , then$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$$m(a)+ \sum_{J^{1}}^{n}.r_{J}$$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}/$$r_{j}$.

III. A Simple Model
Now, we shall confine ourselves to an economy in which there are two kinds of commodities, say

and commodity 2”. commodity 1is agood shared and consumed in aclub.
The club is ayoup people who share commodity 1in consumption. We ass ume that there is only$” \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{M}^{\cdot}\mathrm{q}$$1$”

one club in the economy. Commodity 2is aprivate good and consumed by each single person.
In what follows, we assume that commodity 2is numeraire and its price is always unity.
We assume that individuals are “divisible” and the set of all the persons in the economy isdenoted by , which is aunit interval, i.e., $A=[0,1]$ . Also, we assume that all individuals are
identical, and their utility ffinctions are the same and their incomes are equal.
Let us denote the quantity of commodity 1consumed in the club by . Also, let us denote the.
$A$

$x$

67
fraction of people belonging to the club by
the club is ameasurable subset$M\mathrm{o}\mathrm{f}A$,$\theta$, wtoeoe $0\leqq\theta\leqq 1$ .

$\theta=2(\lambda 4)$

wlaeoeWhen the set ofthe members of) is the Lebesgue measure of set A#$\mathrm{Z}\{\mathrm{M}$The total number ofindividuals in the economy is fix4 and fiwrion d&notoe the number ofpeople

participating in the club.
We assume that people do not care about who are members ofthe club, but only about the number

ofits members. Therefore, the club is described by pair$(x, \theta)$.

The utility fimction of each person, who becomes amember,$u=U((x,\theta)$ $y)$club

, is denoted by$(x, \theta)$,

where denotes the quantity of commodity 2.
$y$indicates degree of congestion. The following assumption meansthat people prefer aless crowded club.
The variable dot club

Assumion 3.1:
decreasing in 7.

$U$$(x, \theta)$

is acontinuous ffindion and $U((x,\theta),y)$ is increasing in both and , and
$x$

$y$

On the other hand, we denote the utility ofa person who is not amember ofthe club by$u=V(\gamma)$.

Assumption 3.1:

$V(yFU((0,\theta),y)$for ally and 7.

The above assumption implies that people can get nothing ffom belonging to the club in whichnothing is consumed. Namely, when$=0$,people in club$(x, \theta)$

get as the same level of utility as

people out ofthe club get.
In Figure 1, an indifference sufface for the utility$\mathrm{f}\mathrm{i}\mathrm{m}\alpha \mathrm{i}\mathrm{o}\mathrm{n}$satisfying the above assumptions is

illustrated
The production set of commodities 1and 2is denoted by , which is asubset ofthe non-negative$Y$

orthant of a2-dimeniional Euclidean space.
Assumption 3.3:

Set is non-empty, closed, and convex.
$Y$

Next, let us denote by$m$

the income of each individual, which arises ffom production ofcommodities. The total ofincomes is equal to the valued
When production$(x,y)\in \mathrm{Y}$commodities produced in the economy.
is chosen and the price ofcommodity 1is ,the total value ofproduced.
$p$

68

Fig. 1: Indifferent Su&ce for Utility Function $U$

commodities is $px+y$, and the following must hold.
$m= \int_{A}mda=px+y$Namely, the value of produced commodities is diMbutd equally among all the individuals in theeconomy.
Finally, we assume that every individual is initially amember ofthe club and nothing is cons umedin club. Initially, the club is specified by $(0, 1)$ , and every
in the club, that is, $x4$ andindividual has the membership ofclub $(0, 1)$ . When an individual wants to leave the club, he sells
his membership in the market and the club buys it. Ifthe price of membership of club $(0, 1)$ is ,then the initial income ofeach individual is $m+r$.
$\theta=1$$(x, \theta)$

$r$

IV

Competitive Equilibrium

As some individuals leave the club and the club buys some amount of commodity 1club $(0, 1)$changes to club$(x, \theta)$. Let us denote the price of membership of clubadmission fee that individuals have to pay if they join club$(x, \theta)$$(x, \theta)$

by . Price
$q$

$q$

is an

. Since each individual is69.
negligible, and single person does not affect variable Iofclub
The budget consffaint which each individual must satisfy injoining the club, is denoted by$(x, \theta)$$q+y\leqq m+r$,

the amount ofconsumption commodity 2. Thus, each person will continue tojoin thewhereclub if $V(m+r)<U((x,\theta),m+r-q)$ , or leave the club if $V(m+r)>U((x,\theta),m+r-q)$ .
$y\mathrm{i}\mathrm{s}$

Given

$m$

, , andx, let us define$r$$q_{0}$

and$q_{1}$by$q_{0}= \max\{q|V(m+r)\leqq U((x,0),m+r-q)\}$and$q_{1}= \max\{q|V(m\dagger r)\leqq U((x,1),m+r-q)\}$By Assumption 3.1, we have$0$

. On the other hand, when$q>q_{1}$When $q>q_{0}$, nobody will join the club, and therefore. When, everybody willjoin the club, and therefore$q_{0}\geqq q_{1}$$q\leqq q_{1}$.

.

$\theta=1$, some will join the club, but others will not

The fiaction$\theta \mathrm{o}\mathrm{f}$$\theta=$$q_{0}\geqq$

individualsjoining the dub isdetermined by$V(m+r)=U((x,\theta),m+r-q)$ ,and $0\leqq\theta<1$ . We write the above relation as$\theta=\mathrm{X}q,x$, $m$ , ). Thus, the demand for membership$r$ofthe club is defined by1$\theta$$0\leqq q<q_{1}$$=F(q,x, m, r)\equiv\{$ $f(q,x,m, r)$$q_{1}\leqq q<q_{0}$0.

$q_{0}\leqq q$

The demand curve ofF has negative slope with respect to as depicted in Figure 2.
$q$

Now, we assume that the purpose ofthe club is to$\ovalbox{\tt\small REJECT}$

its members’ utilities. In our simple

economy, since individuals are all identical, we can assume that the club chooses , , and$U((x,\theta),m$ \dagger
there is abudget constraint for the club.
so as to$r-q$ ). In

model
$q$

$X$

$\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{i}\dot{\mathrm{u}}04$$\ovalbox{\tt\small REJECT}$Let be the price of commodity 1.
$p$$\sqrt x+r=q\theta$$\theta$Then, the budget constraint for the club is.

The behavior of the club can be interpreted as follows. There is amanager in the club, whose
the utilities of people joining the club. For that purpose, the manager will
job is to$\ovalbox{\tt\small REJECT}$

determine amount of commodity 1consumed in the club, number
price of membership. Thus, given , , and , the club will
$X$

$p$

$q$

respect to , , and
$X$

$q$

$\theta$$m$$\theta$of members ofthe club, and$U((x,\theta),m+r-q)$ with$\ovalbox{\tt\small REJECT}$$r$

under budget constraint $px\dagger r=q\theta$ .

Therefore, the demand for commodity

membership by the club are defined by$G(p, m, r)\equiv\{(x, \theta, q)|px+r=q\theta$ and $U((x,\theta),m+r-q)\geqq U((z,n),m+r-s)$

1and the supply

for all$(\underline{7}, n,s)$with$p^{\underline{7}}+r=sn$

}.

70

price

$\mathrm{o}\mathrm{f}\mathrm{m}\mathrm{e}\mathrm{r}\iota \mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}$Fig. 2: Demand Curve of Membership of the Club

numbe of members$\mathrm{c}$Fig. 3: Behavior ofthe Club

ommo dity 1

71Given , asituation ofthe club is$q$by the club will be adecreasing

increasing function$\mathrm{o}\mathrm{f}q$

Finally, producers

in Figure 3. Usually, the demand for commodity 1
and the suPply
membership by the club will be an$\mathrm{i}\mathrm{u}\ovalbox{\tt\small REJECT}$$\mathrm{f}\mathrm{i}\mathrm{u}\kappa\dot{\mathrm{h}}\mathrm{o}\mathrm{n}$$\mathrm{o}\mathrm{f}p$.

$\mathrm{m}\ovalbox{\tt\small REJECT}$

the value of commodities, and the supply function of commodity 1

and commodity 2is defined by$H(p)\equiv${$|px+y\geqq px’+y$’for all $(x’,y’)\in Y$$(\mathrm{x},\mathrm{y})$}.

In equilibrium, the following must hold:$\theta=F(q,x, m, r)$,$(x, \theta, q)\in\infty$, $m$ , $r)$,$(x,y)\in H(p)$ ,and$m=px+y$.

Thus, the competitive equilibrium for the economy can be described by ($\mathrm{p},$$q,$ $(x, \theta),y,$ $m,$

$r\}$

and

defined as follows:
Definition 4.1: $\{p, q, (x, \theta), y, m, r\}$ is said to be a$\underline{\infty \mathrm{m}\mathrm{o}\mathrm{e}\dot{\mathrm{b}}\dot{\mathrm{h}}\mathrm{v}\mathrm{e}}$equilibrium if the following

conditions are satisfied:$>0$ , then $V(m+r)\leqq U((x,\theta),m+r-q)$ , and if $\theta<1$ , then(1) If$\theta$$V\{m+r)\geqq$$U((x,\theta),m+r-q)$ .

(2)$px\dagger r=q\theta$$\sqrt{}^{\underline{7}}+r\leqq sn$(3)

and $U((x,\theta),m+r-q)\geqq U((z,n),m+r-s)$ for all $(z, n,s)$ will.

$(x,y)\in Y\mathrm{a}\mathrm{n}\mathrm{d}$$m=px+y\geqq\sqrt x’+y$’for all $(x’ y’)\in Y$ .

In the above definition, condition (1) means that each person isconstlaiffi$\ovalbox{\tt\small REJECT} \mathrm{g}$utility under abudget
Condition (3)means that producers ofcommodities are
profits. Conditions (1) and (2) imply that themarket of membership is in equilibrium. Also, conditions (2) and (3) imply that the market ofcommodity 1is in equilibrium. Therefore, by Walras’ law, the market of commodity 2is in$\mathrm{m}\ovalbox{\tt\small REJECT} \mathrm{g}$equilibrium.
The competitive

uilibrium can be defined in more general cases (see Takekuma (1999)).

$\mathrm{e}^{1}\mathrm{q}$condition (1) oflhe above
existing club, or not$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\dot{\mathrm{m}}\dot{\mathfrak{n}}\mathrm{o}\mathrm{l}$each person simply decides whether he (or she) should

In

the

Therefore, our definition of competitive equilibrium is weaker than, ordifferent from that of S. Scotchmer, S. and M. H. Wooders (1987), in which people choose one clubjoin among many potentially existing clubs

72
price ofconu[off.iy–$1$dunuld club,

Fig. 4: Market ofCommodity 1

V. An EXample ofthe Economy
In this section we are going to show an example of the economy in Section I.

Commodity 1,

which is consumed in the club, is interpreted as the fidities ofthe club. Commodity 2, which is a.
private good, is assumed to
The set of all the persons in the economy is denoted by $A=[0,1]$ . Let us denote the size of.
facilities of the club by and the fiaction of people belonging to the club by , where.
Therefore, the club is characterized by apair
All individuals are identical, and their utility functions and the initial holdings of money are the, is assumedof each person, when he (or she) is amember of clubsame. The utilty$\mathrm{k}" \mathrm{m}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{y}"$$k$$0\leqq\theta$ $\leqq 1$$\theta$$(k, \theta)$$\mathrm{f}\mathrm{f}\mathrm{i}^{1}\mathrm{n}$$(k, \theta)$$\mathrm{c}\dot{0}\mathrm{o}\mathrm{n}$to have the following special form.
$u=18\sqrt{k(1-\theta)}+y$ ,

where

$y$

denotes the quantity of money. On ffi other hand, the utility of aperson who is not amember ofthe club is assumed to be$u=y$.

The cost for producing the$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{i}\dot{\mathrm{b}}\mathrm{o}\mathrm{e}$of the club is denoted by acost$\mathrm{f}\mathrm{i}\mathrm{m}\mathrm{c}\dot{\mathrm{h}}\mathrm{o}\mathrm{L}$which has the73following special form.
$c= \frac{1}{3}\emptyset$

Let$p\mathrm{b}\mathrm{e}$.

commodity 1maximize profits,

the price ofcommodityl. Producers$\pi=pk-c=pk^{-}\frac{1}{3}t$.

The condition for profit maximization is$\frac{d\pi}{dk}=p-\frac{2}{3}k=0$

i.e.,

,

$k= \frac{3}{2}$

p.

(5.1)

Therefore, the supply curve of commodity 1is astraight line with apositive slope illustrated in

Figure 4.
Each individual$\mathrm{i}\dot{\mathrm{m}}\dot{\mathrm{b}}\mathrm{a}\mathrm{u}\mathrm{y}$holds the same amount$\overline{y}$The profits obtained in production of commodity 1areof money, and we assume that$\eta \mathrm{u}\mathrm{a}\mathrm{U}\mathrm{y}$economy. Each individual receives the same amount$\pi$$\overline{y}=10$.

distributed to all the individuals in the
profits from producers. In$\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{i}\dot{\mathrm{u}}\mathrm{o}\mathrm{l}$every individual ismember ofthe dub where nothing is consumed Let be the price ofmembership of club $(0, 1)$ . The total income of each indvidual is \dagger $\pi+r$, and the budget$r$$\mathrm{i}\cdot \mathrm{i}\dot{\mathrm{b}}\mathrm{a}\mathrm{u}\mathrm{y}$$\overline{y}$constraint, which each individual must satisfy injoining the club, is denoted by$q+y\leqq\overline{y}$$+$$\pi+r$ ,where is the amount of money and is the price of membership of club
$y$

$q$

$(k, \theta)$. Therefore, eachjoin the club if
person will join the club if $\overline{y}+\pi+r<18\sqrt{k(1-\theta)}$ \dagger $\overline{y}+\pi+r-q$, or will
\dagger $\overline{y}+\pi+r-q$. Hence, theindividuals joining the club$\overline{y}+\pi\dagger r>18\sqrt{k(1-\theta)}$$\mathrm{f}\mathrm{i}\mathrm{a}\alpha \mathrm{i}\mathrm{o}\mathrm{n}\theta \mathrm{o}\mathrm{f}$is determined by, i.e.,$q=18\sqrt{k(1-\theta)}$$\overline{y}+\pi+r=18\sqrt{k(1-\theta)}+\overline{y}+\pi+r-q$

(5.2)

,

ffom which the demand curve ofmembership in Figure 5is derived.
The purpose of the club is to maximize its members’ utility.
$18\sqrt{k(1-\theta)}+\overline{y}+\pi+r^{-}q$$+r=q\theta$.

, is maximized with respect to ,$k$

In club$\theta$

, and

$q$

, members’ utility.

$(k, \theta)$

under budget constraint $pk$

The Lagrangian for the maximization problem is defined by$L=18\sqrt{k(1-\theta)}+\overline{y}+\pi+r^{-}q+a(q\theta-r-\sqrt k)$ ,where 4is Lagrangian multiplier.
$\frac{\partial L}{\partial k}=9\sqrt{\frac{1-\theta}{k}}-ap=0$,

The necessary conditions for maximization are
(5.3)

74
price

$\mathrm{o}\mathrm{f}\mathrm{n}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{h}\dot{\varphi}$–dmond curve

Fig. 5: Market$\frac{\partial L}{\partial\theta}=-9\sqrt{\frac{k}{1-\theta}}+aq=0$$\frac{\partial L}{\Psi}=-1\dagger$.
$\mathrm{o}\mathrm{f}\mathrm{M}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}\psi$

,

(5.4)

a $d.=0$ , and$\frac{\partial L}{\partial a}=q\theta-r^{-}pk=0$

(5.5)

.

(5.6)

From (5.3), (5.5), and (5.6), it follows that$\frac{1-\theta}{k}=_{\mathrm{t}}\frac{p}{q}$and 81$\theta^{2}=m$

.

(5.7)

By (5.6) and (5.7) we have 81 $\theta^{2}(1-2\theta)+pr=0$, which implies that
Thus, we have the stipply curveis determined by and .
membership, which is vertical line in Figure 5.

In equilibrium, by solving six equations ffom (5.1) to (5.6), we have
$r=6$ ,

and$a= \frac{3}{2}$. Furthermore,$\pi^{=}3$$\theta$$\theta=\frac{2}{3}$

$p$

$r$

, $k=3,p=2$ , $q=18$ ,, and the consumption of commodity 2by each member

ofthe club is $\overline{y}+_{X}+r^{-}q=1$ , whereas the consumption ofcommodity 2by each non-member is\dagger $\pi+r=19$ . Thus, acompetitive equilibrium is shown to exist for this example of the$\overline{y}$75Moreover, by (5.7) we have 81 $\theta^{2}(1-\theta)=pk$ which implies that$81 \theta(2-3\theta)\frac{\partial\theta}{\partial p}=k+p\frac{\partial k}{\Phi}$

Therefore, since$\theta=\frac{2}{3}$.

.

equilibrium,$\frac{\partial k}{\Phi}=-\frac{k}{p}<0$

holds in aneighborhood ofthe equilibrium.

Namely, we have the demand curve of commodity 1, which has a$\mathrm{n}\mathrm{e}\dot{\mathrm{g}}\mathrm{v}\mathrm{e}$slope at the$\eta \mathrm{u}\mathrm{i}\mathrm{h}\mathrm{b}\mathrm{i}\mathrm{m}\mathrm{l}$illustrated in Figure 4.

VI.

Pareto Optimum ffocations

To describe an allocation in the economy, we have to specify the amount of commodity 1

consumed in the club, its members, and the$\mathrm{d}\mathrm{i}\mathfrak{W}\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{i}\mathrm{o}\mathrm{n}$of commodity 2among people. Let us

denote the amount of commodity 1consumed in ffie club bymeasurable subset$M\mathrm{o}\mathrm{f}A$$X$

and the set of its members by a

. Then, the club is denoted by ( ,A#).
$x$

To denote the distribution of commodity 2among individuals, we use areal-valued measurable
fimction $f$ on , where ) is the quantity of commodity 2allocated to person $a\in A$ . Thus, an$A$$\mathrm{f}\mathrm{i}\mathrm{a}$allocation in the economy is indicated by these three elements, $\{(x, M),f\}$ . An allocation $\{(x, M)$,$]$in the economy is said to be$\underline{\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}}$, the utility ofmember \^a Mis $U((x, \lambda(M)),f(a))$ , whereas the utility

In allocation$\{(x,\emptyset,J\}$

non-member$a\in A\backslash M\mathrm{i}\mathrm{s}V((a))$.

Definition 6.1: Afeasible allocation {feasible allocation

if $(x, \int_{A}fda)\in Y$.

$\{(z,N),\mathrm{g}\}$$(x$, A4, $f$} is said to be Pareto$\mathrm{o}\mathrm{D}\dot{\mathfrak{g}}\mathrm{m}\iota \mathrm{m}$if there is no othersuch that(1)$U((x, \lambda(M)),f(a))\leqq U((z, \lambda(N)),\mathrm{g}(a))$(2)$U((x, \lambda(M)),f(a))\leqq V(g(a))$(3)$\nabla(Ka))\leqq U((z, \lambda(N)),\mathrm{g}(a))$(4)$V(Ka))\leqq V(g(a))$

for all

for all \^a,$\mathrm{M}\mathrm{O}\mathrm{N}$for all $a\in M\cap(A\psi$ ,for all $a\in(A\backslash w\cap N$,$a\in A\backslash (M\cup N)$,

and strict inequalities hold for some a&A (with positive measure).
Now we can prove the basic theorem of welfare economics for economies with clubs

76
Theorem 6.1: Any allocation in the competitive equilibrium is Pareto optimum.

Proof Let $\{p, q, (x, \theta),y, m,r\}$ be competitive equilibrium. Define aset Mand a$M=[0, \theta]$

By (2) and (3)and $f(a)=\{$$\mathrm{o}\mathrm{f}\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\cdot \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$4.1$$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}/\mathrm{b}\mathrm{y}$for $a\in M$
for$m+r-q$$m+r$$a\in A\backslash M$,$\int_{A}fda=\theta(m+r^{-}q)+(1-d)$$(m+r)=m+r-dq=px+y+r-dq=y$,and therefore, $(x, \int_{A}fda)\in Y$. Namely, allocation { $(x,M),fl$ is feasible.
Now, suppose that alocation {( , A#), 7} were not Pareto optimum. Then, by Definition 6.1,
$x$there is afeasible allocationsuch that$\{(z,N),\mathrm{g}\}$for all $a\in M\cap N$,$U((x, \theta),m+r-q)\leqq U((z, \lambda(N)),\mathrm{g}(a))$(6.1)$U((x, \theta),m+r-q)\leqq V(\mathrm{g}(a))$

for all $a\in M\cap(A\backslash N)$,

(6.2)

$V(m+r)\leqq U((z, \lambda(N)),\mathrm{g}(a))$

for all $a\in(AW)\cap N$,

(6.3)

,

(6.4)

$V(m+r)\leqq V(\mathrm{g}(a))$

for all$a\in A\backslash (M\cup N)$

and skid inequalities hold for some a&A with positive measure.
By (1) Definition 4.1, $U((x, \theta),m+r-q)\leqq V(m+r)$ holds in (6.3). TlrlefoIe, from (6.1)
and (6.3), it follows thatfor all $a\in N$,$U((x, \theta),m+r-q)\leqq U((z, \lambda(N)),\mathrm{g}(a))$

which implies, by (2)$\mathrm{o}\mathrm{f}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{i}\dot{\mathrm{u}}\mathrm{o}\mathrm{n}4.1$, thatfor all $a\in N$.

$\mu\dagger r\geqq(m+r^{-}\mathrm{g}(a))A(N)$By (1)(6.3)

Definition 4.1, $V(m+r)\leqq U((x, \theta),m+r-q)$ holds in (6.2). Therefore, ffom (6.2), it

follows that

$V(m+r)\leqq V\ovalbox{\tt\small REJECT} a))$$m+r\leqq \mathrm{g}(a)$forfor all, which implies, by Assumption 2.2, that$a\in M\cap(A\backslash N)$$\mathrm{a}11a\in M\cap(A\mathrm{W})$

.

(6.6)

Moreover, (6.4) and Assumption 2.2 imply that$m+r\leqq \mathrm{g}(a)$forall$a\in A\backslash (M\cup N)$

Since strict inequalities hold for some \^a$m<pz+ \int_{A}\mathrm{g}da$

.

$A$

(6.7)

in (6.5), or (6.6), or (6.7), we have, by integration,,

which contradicts (3) ofOefinition 4.1.

$\blacksquare$

77
References
Berglas, E.(1976), On the theory of clubs, American Economic Review 66, pp. 116-121.
An economic theory clubs, Economica 32, pp. 1-14.
Buchanan, J.
Foley, D.(1967), Resource allocation and the public sector, Yale Economic Essays 7, pp.43-98.
Helpman, E. and A. Hillman (1977), Two remarks on optimal club size, Economica 44, pp.293-96.
$\mathrm{M}(1965)$

Ng,-K.(1973), The economic theory of clubs: Pareto optimality conditions, Economica 40,$\mathrm{Y}$pp.291-298.
.(1974), The economic theory of clubs: Optimal,Economica 41, pp.308-321.

$\mathrm{b}\mathrm{x}/\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{y}$. (1978), Optimal club size: Areply, Economica 45, pp.407A10.

Richter, D. K.(1974), The core of apublic goods economy, International Economic Review 15,

pp.131-142.

Smuelsok P. A.(1954), The pure theory of public expenditure, Review ofEconomics and$\theta atistics$

36, pp.387-89.
Sandier, T. and J. T. Tschiihart (1980), The economic theory of clubs: An evaluative survey, Journal

Economic Literature 18, pp. 1481-1521.
Scotchmer, S. and M H. Wooders (1987), Competitive equilibrium and the core in club economies

with anonymous crowding, Journal ofPubtic Economics 34, PP. 159-173.
Takekuma, S. (1999), Pareto optimum allocations in the economy with clubs, Hitotsubashi Journal
Economics 40, pp.29A0.