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\begin{document}

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\title{Clubs and the Market: \\
Continuum 
Economies\thanks{We thank Robert Anderson for
tutelage, Kenneth Arrow and Joe Ostroy for helpful arguments, and
seminar audiences at CalTech, the Conference on City Formation at 
Washington University in St. Louis, the Midwest Mathematical Economics 
Meetings (Fall 1996), the Public Choice Society Meetings, the Santa Fe
Institute,  the Stanford Institute for Theoretical Economics (Summer
1996), UC Berkeley, UC Santa Cruz and UCLA, especially Mike Akemann,
Larry Blume,
Sara Castellanos, John Conley, David Cooper, Mike Ryall and Federico Weinschelbaum,
for  comments.  We thank the UCLA and UC Berkeley Academic Senate
Committees on Research, the National Science Foundation and the
Fullbright Foundation for financial support and the UCLA and UC
Berkeley Departments of Economics and the Institute of Economics of
the University of Copenhagen for gracious hospitality (to Grodal,  to
Zame, and to Scotchmer and Zame, respectively)  during preparation of
this paper.}} 

\author{\vspace{-.18in}  Bryan Ellickson
 \\ {\parbox{3.5in}{\begin{center} \small Department of Economics\\
 University of California, Los Angeles \end{center}}} \vspace{-.1in}
 \and
 \vspace{-.18in} Birgit Grodal \\ {\parbox{3.5in}{\begin{center}
 \small Institute of Economics\\ University of Copenhagen
 \end{center}}} \vspace{-.1in}  
 \and  
 \vspace{-.18in} Suzanne Scotchmer \\ {\parbox{3.5in}{\begin{center}
 \small Department of Economics and\\ 
Graduate School of Public Policy \\
University of California,
 Berkeley \end{center}}} \vspace{-.1in}  \and
 \vspace{-.18in} William R. Zame \\ {\parbox{3.5in}{\begin{center}
 \small Department of Economics\\ University of California, Los
 Angeles \end{center}}}  }     

\date{April 10, 1997}  
 
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\begin{abstract}
This paper defines a general equilibrium model with exchange and
club formation.  Agents trade multiple private goods widely in the
market, can belong to several clubs, and care about the characteristics
of the other members of their clubs.  The space of agents is 
a continuum, but clubs are finite.  It is shown that
(i) competitive equilibria exist, and 
(ii) the core coincides with the set of equilibrium states.
The central subtlety is in modeling club memberships 
and expressing the notion that membership choices are consistent
across the population.
\end{abstract}

\bigskip

\noindent JEL Classifications:  D2, D5, H4 \\
{\em Keywords:}  Clubs, Continuum Models, Public Goods
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\section{Introduction}

Consumption is typically a social activity.  The company we keep affects 
our demand for private goods, and our consumption of private goods
affects  the company we seek.  General equilibrium theory in the
tradition of Arrow and Debreu focuses on  the anonymous interactions
of consumers with the market, largely ignoring the  social aspect of
consumption.  Club theory in the tradition of  Buchanan, on the other
hand, focuses  on the social activity of consumption, largely ignoring
the anonymous  interactions of individuals with the market.  The
principal purpose of  this paper, and also of our (1997) companion
paper, is to integrate club theory and general
equilibrium  theory, constructing a framework which incorporates
widespread trading of  private goods in competitive markets and
individual consumption in small  groups chosen voluntarily in
equilibrium.  This paper treats continuum economies and the companion
paper treats large finite economies.

Cornes and Sandler (1986, p. 159) define a club as `` \ldots a voluntary
group deriving mutual benefit from sharing  \ldots production costs, the
members' characteristics, or a good characterized by excludable
benefits.''  Following Tiebout (1956), one tradition in  the literature 
focuses on clubs as political jurisdictions,
assumes that each agent can belong to at most one jurisdiction and
takes a partition into jurisdictions as part of the
basic description of a feasible state of the economy.\footnote{See for
example Berglas (1976), Berglas and Pines (1981), Bewley (1981),
Brueckner (1994), Conley and Wooders (1994), Gilles and Scotchmer
(1997), Greenberg and Weber (1986), Greenberg and Shitovitz (1988)
Ellickson (1973, 1979), Konishi (1996), Scotchmer (1985a, 1985b, 1994,
1996), Scotchmer and Wooders (1987a, 1987b),  Wooders (1978, 1989).} 
A different tradition, following an idea of Buchanan (1965),  focuses on
small clubs: a marriage, a gym, an academic department, a golfing
foursome, or the clientele of a restaurant.  When clubs are not to be
thought of as political jurisdictions, we see no reason why each agent
should belong to a single club nor why the club structure should be (or
induce) a partition.  In keeping with this view, we build here a framework
in which each agent may belong to several clubs (partitions are a
special case).  

Our work builds on a long tradition in the club literature, beginning with
Buchanan (1965), that seeks to demonstrate that club activities can be
interpreted as competitive, for example, Gilles and Scotchmer  
(1997), who studied replica economies.  
In keeping with the tradition in general
equilibrium theory that perfect competition is best demonstrated in the
continuum, we  build a model in which the space of agents is
a continuum, but we restrict clubs to be finite.  Thus, as suggested 
by Buchanan (1965), clubs are ``small'' compared to society.
In the continuum framework, the ``integer problem'' and other
non-convexities disappear.  As a result, we establish the
existence of equilibrium and verify a fundamental test of perfect
competition, the coincidence of the core with the set of equilibrium
states.  Central to our work is that we view clubs and club memberships
as primitives on equal footing with more  conventional primitives of
general equilibrium theory.  This view leads to a fuller integration 
of club theory into general equilibrium theory, and 
to a more general interpretation of clubs.
Other papers in the same spirit include Makowski (1978) and Cole and
Prescott (1994).

We describe  a (type of) club as a pair consisting of a description of
the external characteristics of its members and a specified activity;
thus we follow Ellickson (1979) and Mas-Colell (1981) as viewing the
activity of a club  as a {\em public project\/} rather than as provision of
some level of a public good.  A club membership is an opening in a club
available to agents with specified characteristics.   Agents choose both 
private goods and  club memberships, and private goods and club
memberships are treated and priced in  parallel fashion.  

Despite the parallel treatment of  club memberships and private
goods, there are important differences from exchange economies.
First, club memberships are indivisible.  Cole and Prescott (1994)
deal with this indivisibility by viewing the objects of
choice as lotteries on private goods and club memberships.  
It seems to us that the
indivisibility of club memberships is central to understanding clubs, and we
prefer to address it directly.  

Second, club membership choices must be consistent
across the population.  If
a third of the population are women married to men, for example,
then a third of the population must be men married to
women.  Consistency must hold simultaneously for all types of
clubs, and allow for the possibility that every individual may belong to several clubs.

Finally, there is an important difference in the pricing of
private goods and of club memberships:  private good prices must be
positive, but club membership prices may be positive, negative or zero.

Our proofs follow lines that are typical of
general equilibrium theory, but there
are many subtleties.  The central subtlety
is in accommodating the club consistency condition, which has no analog 
in exchange economies.

Our proof of equivalence of the core  with the set of equilibrium
states follows an outline parallel to
Schmeidler's (1969) proof of Aumann's (1964) core equivalence
theorem:  Begin with a core state, construct individual net preferred
sets in the space of private goods and club memberships, and then an
aggregate net preferred set.   Use the Lyapunov convexity theorem to
show that the aggregate net preferred set is convex and  the core
property of the given allocation to show that the aggregate net
preferred set is disjoint from an appropriate cone.  Obtain equilibrium
prices  by separating this aggregate net preferred set from an
appropriate cone.  Because we work in the space of private goods and club memberships, however, our argument differs from Schmeidler's, and we must work much harder to be
certain that the private good prices we construct are not zero. 
And we must restrict the space of club memberships to accommodate the 
matching property.\footnote{In contrast to the proof outlined above, decentralization is usually
accomplished in the club literature by first constructing prices for
private goods and then defining prices for club memberships in terms
of willingness to pay.  Because we allow agents to belong to more than one
club, the sequential construction does not work.}

Our proof that equilibrium exists also follows a familiar outline:  Construct an excess demand correspondence in
the space of private goods and club memberships.  Use the Lyapunov
convexity theorem to show that this correspondence is convex valued.
 Apply Kakutani's fixed point theorem to find a zero.  Because we work in the space of private goods and club memberships, however, and club
membership prices may be positive, negative or zero, there is
no natural price domain on which to work.  We must therefore work in
perturbations of the original economy which have the property that
equilibrium prices are known {\em a priori\/} to lie in some compact set,
construct equilibria for these perturbed economies, show that
equilibrium prices for the perturbed economies can be chosen to
be bounded, and take limits as we relax the perturbations.

Following this Introduction, Section 2 provides some  motivating
examples. The formal model is described in Section 3.  Section 4
 establishes the first welfare theorem and Section 5 shows that
the second welfare theorem may fail.  This is not surprising
in finite economies, but our examples show that it may fail even
in atomless economies.  Section 6 establishes the
equivalence of the core and the set of equilibrium states and Section 7
establishes the existence of equilibrium.  The text outlines 
the main proofs; details are collected in \hbox{Section 8.} 

\vfill \newpage


\section{Examples}

In this section we present four examples illustrating various aspects
of competitive equilibrium in a club economy with a continuum
of agents.  The first example, a version of a familiar crowding story, 
illustrates the nature of competitive equilibrium in a setting where the 
composition of club memberships does not matter.

\begin{example}  {\bf  Crowding} \\
Consider an economy with a continuum of consumers uniformly distributed on
$[0,10]$ and a single private good.  The endowment of consumer $k$ is
$e_k = k$.  In addition to the private good, consumers have the option of 
using a swimming pool which they can enjoy alone or in a club.  All 
consumers have the same preferences: a consumer who consumes $x$ units of 
the private good derives utility $u(x;0)= x$ if using no pool and
$u(x;n)= 4x/n$ if she belongs to a swimming pool club with $n$ members.
(We assume a consumer can belong to at most one such club.) Building a 
swimming pool requires an input of $6$ units of the private good.

Although swimming pool clubs could in principle be arbitrarily large, in 
equilibrium there will be no clubs of size greater than 4.  Since consumers 
care only about the number and not other characteristics of fellow pool 
club members, all consumers belonging to the same club share equally in its 
cost. Normalizing the price of the private good to one, the price of a 
membership is $q_n = 6/n$ for $n = 2$, 3 or 4. (Consistent with this 
formula, a swimming pool costs 6, but we prefer to treat singleton 
``clubs'' separately.) The normalization also implies that consumer $k$ has 
wealth $k$.  Choosing no pool yields utility $k$ and enjoying a pool by 
herself yields utility $4(k-6)$. After paying her share of the cost, sharing a 
pool in a club with $n$ members yields utility
$$ u(k-q_n;n) = \cases{2(k-3)          & if $n=2$;\cr
                       \frac{4}{3}(k-2)& if $n=3$;\cr
                       k-\frac{3}{2}   & if $n=4$.\cr}
$$

Solving for the equilibrium choices of individuals is easy: the wealthiest 
consumers, with wealth $k \in (9,10]$, have a pool of their own; consumers 
with wealth $k \in (6,9]$ share a pool with one other person; and the 
poorest consumers, with wealth $k\in [0,6]$, consume the private good but 
do not enjoy the use of a pool. Clubs of size greater than two do not form 
in equilibrium.  $\clubsuit$
\end{example}

The second example, motivated by the commentary by Arrow (1972) on Becker
(1957), illustrates the importance of allowing for membership prices which 
discriminate among types of membership (i.e., on  external 
characteristics).

\begin{example}  {\bf Segregation } \\
Consider an economy with a continuum of consumers uniformly distributed on 
$[0,1]$: consumers in $[0,.3)$ are blue, consumers in $[.3,1]$ are green. 
There is a single private good. All consumers have endowment $e_a = 2$.  In 
addition to the private good, duplex apartments are available.  The utility 
of a consumer depends on his {\sl external characteristic} (blue or 
green), on consumption of the private good, on whether or not housing is 
consumed, and on the external characteristic of the consumer with whom the 
housing is shared. (We assume that no consumer desires more than one unit 
of housing.) A blue or green consumer who consumes no housing and $x$ 
units of the private good derives utility
$$ u_B(x;0) =  u_G(x;0) = x  $$
Using the obvious notation for the external characteristics of the 
occupants of a duplex, a consumer who lives in a duplex and consumes $x$ 
units of the private good derives utility
$$ u_B(x; BB) = 4x \qquad\hbox{and}\qquad u_B(x; BG) = 6x  $$
if blue and
$$ u_G(x; GG) = 6x \qquad\hbox{and}\qquad u_G(x; BG) = 4x  $$
if green.  Note that a blue (respectively, green) consumer cannot consume 
housing in a duplex with two green (respectively, blue) consumers because 
there would be no space for her.

Assuming that a duplex can be produced using two units of the private good 
and that race-discriminatory pricing is possible, the prices blue or green 
consumers pay for segregated and integrated duplexes
must satisfy:
 \begin{eqnarray*}
2q_B(BB)&=& 2 \\
2q_G(GG)&=& 2 \\
q_B(BG) + q_G(BG) &=& 2
\end{eqnarray*}
(Again the notation should be self explanatory.) At these prices, a blue 
or a green consumer can obtain utility $2$ by choosing no housing. 
 Alternatively, a blue consumer can obtain utility 4 by choosing a 
segregated duplex at price 1 or utility $6(2 - q_B(BG))$  by choosing an 
integrated duplex at price $q_B(BG)$.  Green consumers can obtain utility 6 
by choosing a segregated duplex for price 1 or utility
$4(2 - q_G(BG))$ by choosing an integrated duplex for price $q_G(BG)$.
In order that integrated housing be chosen at equilibrium it is necessary 
that $6(2 - q_B(BG)) \geq 4$ and $4(2 - q_G(BG)) \geq 6$ or, equivalently, 
$q_B(BG)) \leq 4/3$ and $q_G(BG) \leq 1/2$ whence
$q_B(BG) + q_G(BG) \leq \frac{11}{6}$.  However, we already know that 
$q_B(BG) + q_G(BG) = 2$, so no integrated housing will be chosen at 
equilibrium.  Equilibrium prices for segregated housing are $q_B(BB) = 
 q_G(GG) = 1$ while equilibrium prices for integrated housing are 
indeterminate, constrained only by the requirements
$$
q_B(BG) \geq \frac{4}{3} \ , \ q_G(BG) \geq \frac{1}{2}
 \ , \ q_B(BG) + q_G(BG) = 2
$$
At equilibrium, all consumers choose segregated housing.

Suppose the government offers a subsidy  $s > 0$ for integrated housing, 
reducing its price to $2 - s$.  Equilibrium prices must then satisfy
\begin{eqnarray*}
2q_B(BB)&=& 2 \\
2q_G(GG)&=& 2 \\
q_B(BG) + q_G(BG) &=& 2-s
\end{eqnarray*}
 In order that integrated housing be chosen at equilibrium it remains 
necessary that $q_B(BG)) \leq 4/3$ and $q_G(BG) \leq 1/2$. Because $q_B(BG) 
+ q_G(BG) = 2 - s$, integration is possible only when
$s \geq 1/6$.

Suppose that $1/6 \leq s < 1 $.  The number of green consumers choosing 
integregated housing must equal the number of blue consumers choosing 
integregated housing. Because there are more green consumers than blue 
consumers, some  green consumers {\em must\/} choose segregated housing. 
 Because all green consumers must enjoy the same equilibrium utility, it 
follows that $q_G(BG) = 1/2$ and hence $q_B(BG) = 3/2 - s > 1/2$.  In 
equilibrium all blue consumers and 3/7 of the green consumers will choose 
integrated housing; the remaining green consumers will choose segregated 
housing.\footnote{If $s = 1/6$ equilibrium choices are indeterminate, 
constrained only by the requirement that the same number of blue consumers 
and green consumers choose integrated housing.}

Thus the government subsidy achieves integration --- {\em but only if 
housing prices are discriminatory\/}:  $q_B(BG) > 1/2 = q_G(BG)$.  In
order to achieve integration with {\em non-discriminatory prices\/}, the 
government must raise the subsidy to $s = 1$.\footnote{We leave it to the 
reader to examine the welfare consequences of various methods by which the 
government could tax individuals to provide the necessary subsidy.} 
 $\clubsuit$
\end{example}

Our third example illustrates some of the subtleties inherent in allowing
for membership in several clubs.

\begin{example}  {\bf Monogamy, Polygamy and Group Marriage }\label{ex:manyclubs} \\
Consider an economy comprised of a continuum of consumers uniformly 
distributed on $[0,11]$; consumers in $[0,6)$ are male, consumers in 
$[6,11]$ are female.  There is a single consumption good; endowments are
$$
e_a = \cases{a & if $0 \leq a < 6$ \cr
            1 & if $ 6 \leq a \leq 11$ \cr}
$$
(Thus, male endowments are uniformly distributed between 0 and 6; female 
endowments are identically 1.)  In addition to consuming the private good, 
individuals may enter into several kinds of marriage:  exclusive monogamy 
(1 male and 1 female, symbolized $m_e$); non-exclusive monogamy (1 male and 
1 female, symbolized $m_{ne}$); and a group marriage (1 male and 2 females, 
symbolized $m_g$).  Males have the option of belonging to one or two 
non-exclusive marriages; females can be in only one.  (We could incorporate
these restrictions into consumption sets or into preferences.)  
A consumer consuming 
$x$ units of the private good derives utility according to sex and marital 
status shown in the following table:
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
    & single & $m_e$ & $m_{ne}$ & $2m_{ne}$ & $m_g$ \\ \hline
M   & $x$ & $3x/2$ & $3x/2$ & $3x$ & $15x/4$
 \\  \hline
F   & $x$ & $9x$ & $8x$ & & $36x/5$  \\  \hline
\end{tabular}
\end{center}
Note that the choice of two non-exclusive marriages and the choice of a 
group marriage are distinct and males prefer the latter (holding 
consumption fixed).

We assume that all marriages are costless activities.  Hence,
if $q_M(m_e)$, $ q_M(m_{ne})$, $q_M(m_g)$ are the prices paid by males
to enter an exclusive, non-exclusive  or group
marriage, respectively, and $q_F(m_e)$, $q_F(m_{ne})$, $q_F(m_g)$
are the corresponding prices paid by females, it follows that:
\begin{eqnarray}
q_M(m_e) + q_F(m_e) &=& 0 \\
q_M(m_{ne}) + q_F(m_{ne}) &=& 0 \\
q_M(m_g) + 2q_F(m_g) &=& 0
\end{eqnarray}

To determine the equilibrium, note first that, because both males and 
females find any form of marriage preferable to being single, having both 
unmarried males and unmarried females would contradict the Pareto 
optimality of an equilibrium.  Because males outnumber females, some
males must necessarily be single and hence all females must be married
at equilibrium.  Write $a, b, c, d$ for the fraction of males choosing 1
exclusive marriage, 1 non-exclusive marriage, 2 non-exclusive marriages and 
1 group marriage, respectively.  Keeping in mind that males who choose two 
non-exclusive marriages or one group marriage are involved with two 
females, it follows that
\begin{equation}
a + b + 2c + 2d = 5
\end{equation}
Because non-exclusive marriage is less desirable than exclusive marriage 
for females, non-exclusive marriage will be more expensive at equilibrium. 
Because males find one non-exclusive or one exclusive marriage to be 
perfect substitutes, no men will choose one non-exclusive marriage. 
Therefore,
\begin{equation}
b = 0
\end{equation}

It is evident that wealthier males choose more desirable marriage 
arrangements.  Thus, males in $[6-d,6]$ choose 1 group marriage, males in 
$[6-c-d, 6-d)$ choose 2 non-exclusive marriages, males in
$[6-a-c-d, 6-c-d)$ choose 1 exclusive marriage, and the remaining males 
choose no marriage at all.  Because male $6-d$ must be exactly indifferent 
between choosing 1 group marriage or 2 non-exclusive marriages, it follows 
that:
\begin{equation}
3(6-d - 2q_M(m_{ne})) = \frac{15}{4}(6-d -q_M(m_g))
\end{equation}
Similarly, the indifference of male $6-c-d$ between choosing 2 
non-exclusive marriages or 1 exclusive marriage implies
\begin{equation}
\frac{3}{2}(6-c-d - q_M(m_e)) = 3(6-c-d - 2q_M(m_{ne}))
\end{equation}
and the indifference of male $6-a-c-d$ between choosing 1 exclusive 
marriage or remaining single implies
\begin{equation}
6-a-c-d = \frac{3}{2}(6-a-c-d - q_M(m_e))
\end{equation}

Because all females are identical, their equilibrium utility is independent 
of marital state. Consequently,
\begin{equation}
9(1 - q_F(m_e)) = 8(1 - q_F(m_{ne})) = \frac{36}{5}(1 - q_F(m_g))
\end{equation}

Solving equations (1)--(9) yields
$$ a = 1 \qquad b = 0 \qquad c = 1 \qquad d = 1 $$
$$ q_M(m_e) = 1 \qquad q_M(m_{ne}) = \frac{5}{4} \qquad q_M(m_g) = 3 $$
and
$$ q_F(m_e) = -1 \qquad  q_F(m_{ne}) = - \frac{5}{4} \qquad
 q_F(m_g) = -\frac{3}{2}
$$

Thus, at equilibrium, the wealthiest $\frac{1}{6}$ of males enter into 
group marriage, the next wealthiest $\frac{1}{6}$ enter into 2 
non-exclusive marriages, the next wealthiest $\frac{1}{6}$ enter into 1 
exclusive marriage, and the poorest $\frac{1}{2}$ of males remain single.
$\clubsuit$
\end{example}

To this point, all of our examples have involved a single private good, so
that there is no trade between clubs.  Our final example shows that the
interaction between the demand  for club memberships and the demand for 
private goods can have profound and unexpected consequences when there are 
multiple private goods.

\begin{example} {\bf Marriage and the Market} \\
Consider an economy with a continuum of consumers uniformly distributed on 
$[0,1]$; consumers in $[0,\beta)$ are male, consumers in $[\beta,1]$ are 
female.  There are 2 private goods and each consumers has endowment  $e_a = 
(10,10)$. For consumers remaining single,
$$ u_M(x_1,x_2;0) = x_1 \qquad\hbox{and}\qquad
u_F(x_1,x_2;0) = x_2
$$
while for those who marry
$$ u_M(x_1,x_2; m) = u_F(x_1,x_2; m) = \frac{5}{2}\sqrt{x_1 x_2} $$

Write $q_M, q_F$ for the marriage prices paid by males and females, 
respectively.
(Because sex is the only  characteristic that matters to others, we assume that all males pay the same price and all females pay the same
price.)  Because marriage is costless, marriage prices $q_M + q_F = 0$ 
(i.e., one sex bribes the other to be married).

To solve for equilibrium, it is convenient to work backwards from a 
hypothesized distribution of marriages and single individuals.  To give the 
reader a flavor of the solution, consider a hypothetical distribution in 
which all males are married and some females remain unmarried; of course 
this requires $\beta < 1/2$.

Normalizing so that private good prices sum to 1, all individuals have 
wealth 10. Unmarried females spend all their wealth on good 2. Married 
females spend $q_F$ and married males $q_M$ to enter a marriage, each 
spending half of their remaining income on each of the private goods.
Consequently, the
market clearing conditions for $x_1$ and $x_2$ become:
\begin{eqnarray*}
\beta \left[ \frac{10 - q_M}{2p_1} + \frac{10 - q_F}{2p_1} \right]
   &=& 10 \\
(1- 2\beta)\frac{10}{p_2} +
  \beta \left[ \frac{10 - q_M}{2p_2} + \frac{10 - q_F}{2p_2} \right]
    &=& 10
\end{eqnarray*}
Solving yields $p_1 = \beta$ and $\ p_2 = 1 - \beta$.

To solve for marriage prices, keep in mind that unmarried females and 
married females must obtain the same equilibrium utilities and that males,
all married, must obtain at least as much utility in the marriage as
they would if they were single.  These considerations lead to:
$$
\frac{10}{1 - \beta} =
   \frac{5}{2}\sqrt{\left(\frac{10 - q_F}{2\beta}\right)
            \left( \frac{10 - q_F}{2(1-\beta)}\right) }
$$
and
$$\frac{10}{\beta} \leq
   \frac{5}{2}\sqrt{\left(\frac{10 - q_M}{2\beta}\right)
      \left( \frac{10 - q_M}{2(1-\beta)} \right)}
$$
Solving  yields
$$q_F = 10 - 8 \sqrt{\frac{\beta}{1-\beta}} $$
and
$$ q_M \leq 10 - 8 \sqrt{\frac{1 - \beta}{\beta}} $$
Because $q_M + q_F = 0$, this entails $\beta \geq 1/5$ and hence
$1/5 \leq \beta < 1/2$.

Proceeding in similar fashion for other possible distributions of marriages 
and single individuals, we can work out the equilibrium correspondence as 
$\beta$ varies from 0 to 1.  As in the case above, private good prices vary 
linearly with $\beta$:
$$ p_1 = \beta \qquad\hbox{and}\qquad \ p_2 = 1-\beta $$
The proportion of married males is somewhat more complex:
$$ m =  \cases{0         & if $0 \le \beta < 1/2$;\cr
              [0,\beta]  & if $\beta = 1/5$;\cr
              \beta      & if $1/5 < \beta \le 1/2$;\cr
              1-\beta    & if $1/2 < \beta < 4/5$;\cr
              [0,1-\beta]& if $\beta = 4/5$;\cr
               0         & if $4/5 < \beta \le 1$.\cr}
$$
(If $\beta=1/5$ or $4/5$, the proportion of married males is indeterminate.)
Note that, when the males and females are too far out of balance, marriage 
is priced out of existence!  The price $q_F$ females pay to be in a 
marriage varies with $\beta$ as follows:
$$ q_F =
\cases{10 - 8\sqrt{{\beta\over 1-\beta}} & if $0 \le \beta < 1/2$;\cr
            [-2,2]                      & if $\beta = 1/2$;\cr
      -10 + 8\sqrt{{\beta\over 1-\beta}}& if $1/2 \le \beta \le 1$.\cr}
$$
As the proportion of males increases from 0 to 1/2, the price females pay 
for marriage decreases toward 2, and becomes indeterminate in the interval $[-2,2]$ at
$\beta = 1/2$.  Once $\beta$ exceeds 1/2, the position of males and females 
is reversed: females receive the subsidy, and it increases as the 
proportion of males increase.
$\clubsuit$
\end{example}

\vfill
\newpage

\section{Club Economies}    

In this section we describe  a club economy and define Pareto optimality, the core and equilibrium for such economies.

\subsection{Private Goods} We assume throughout that there are $N
\geq 1$ private goods, each perfectly divisible and publicly traded; the
space of private goods is therefore $\R^N$.  For $x, x' \in \R^N$, we write $x \geq
x'$ to mean $x_i \geq x'_i$ for each $i$, $x > x'$ to mean that $x \geq x'$ but $x
\not= x'$, and $x >> x'$ to mean that $x_i > x'_i$ for each $i$. We write $|x| =
\sum_{n = 1}^N |x_n|$.


\subsection{Clubs} 

We will describe a {\em type of club\/} by the number and characteristics of its members and the activity in which the club is engaged.  

Formally, we let $\Omega$ be a finite set\footnote{We 
could relax the constraint that $\Omega$ be finite, and allow  $\Omega$ to be a
compact metric space.  However, this would 
increase the complexity of the model and of the arguments.}  
of {\em external characteristics\/} (of potential members of a club). 
An element $\omega \in \Omega$  is (or encodes) a complete description of the characteristics of an individual that are relevant for the {\em other\/}
members of a club.
For further discussion of the interpretation of external characteristics, see Section \ref{subsection:discussion}.

A {\em profile\/} is a function  $\pi :\Omega \rightarrow \Z_+ = \{0, 1,
\ldots \}$  describing the members of a club.  For
$\omega \in \Omega$, $\pi(\omega)$ represents the number of members
of the club having external characteristic  $\omega$.  For $\pi$ a profile,
write $|\pi| = \sum_{\omega \in \Omega} \pi(\omega)$ for the total
number of members.  We write $0$ for the zero  profile (representing the
empty club). 

The {\em activities\/} available to a profile of agents belong to a finite
set $\Act$.  We interpret the elements $\act \in \Act$ as public projects
in the sense of Ellickson (1979) and Mas-Colell (1980), rather than as
public goods in the sense of Samuelson. 

A {\em club type\/} is a pair $c = (\pi, \act)$ consisting of a profile and an activity.   We 
take as given a finite set of possible club types
$\Clubs = \{ (\pi, \act)\}$.  We find it convenient to treat singleton clubs separately, 
so we assume that $|\pi| \geq 2$ for all $(\pi, \act) \in
\Clubs$.\footnote{Since activities are not traded, the choice of
activities of singleton clubs can be incorporated into preferences.} 
Formation of the club $(\pi, \act)$  requires a total input of private
goods equal to  $\inp(\pi, \act) \in \R^N_+$.\footnote{More generally, we
could assume that each project could be produced from any input vector
from some specified set and incorporate the choice of production
technology into our notion of feasibility.}  

A {\em club membership\/} is an opening in a particular type of club for
an agent of a particular external characteristic; i.e., a triple $m =
(\omega, \pi,\act)$ such that $(\pi, \act) \in \Clubs$ and $\pi(\omega)
\geq 1$.   (An agent can belong to a club only if the description of that
club type includes one or more members of his/her external
characteristics.)  Write $\mem$ for the set of club memberships.  

Each agent may choose to belong to many clubs or to none.  A {\em list\/} is a function  $\ell : \mem \rightarrow \{0, 1, \ldots \}$;  $\ell(\omega, \pi,\act)$ specifies the number of  memberships of type $(\omega, \pi,\act)$ chosen by an
agent.  Write:
$$ 
\Lists =  \left\{ \ell: \ell \hbox{ is a list } \right\} 
$$
for the set of lists. We frequently find it convenient to view $\Lists$ (which is a set of functions from $\mem$ to $\{0, 1, \ldots \}$) as a subset of $\Mem$ (which is the set of functions from $\mem$ to $\R$).
 For $m \in \mem$ we write  $\delta_m$ for the list defined by
$$
\delta_m(m') = \cases{ 1 \mbox{ if } m = m' \cr
                        0 \mbox{ otherwise } }
$$
That is, $\delta_m$ is the list specifying 1 membership of type $m$ and no others.

\subsection{Agents}

  The set of agents is a nonatomic finite measure space  $(A,{\cal F},
\lambda)$; that is, $A$ is a set, $\cal F$ is a $\sigma$-algebra 
 of subsets of $A$ and $\lambda$ 
 is a non-atomic measure on $\cal F$ with $\lambda(A)<\infty$.\footnote{There would be 
 little loss of generality in assuming that $A = [0,1]$, ${\cal F}$ is the 
 $\sigma$-algebra of Lebesgue measurable sets, and $\lambda$ is Lebesgue 
 measure.}
 
A complete description of an agent $a \in A$ consists
of  his/her external characteristics,  choice set, endowment of private
goods and utility function.\footnote{We use utility functions rather than
preferences as a matter of convenience;  under the
assumptions made here, the two specifications are equivalent.}   An external
characteristic is an element $\omega_a \in \Omega$.  The choice set $X_a$ for an agent $a \in A$ 
specifies which bundles of private goods and which choices of club 
memberships are feasible, so $X_a \subset \R^N \times \Lists$.  For 
simplicity, we assume that the only restriction on private good consumption 
is that it be non-negative, so that $X_a = \R^N_+ \times 
\Lists(a)$ for some subset $\Lists(a) \subset \Lists$.\footnote{Thus 
we incorporate into consumption sets various kinds of restrictions on club 
memberships.  For instance, we may forbid membership in 2 marriages.  More 
general specifications of consumption sets would be easily accommodated
at the  cost of complicating
some definitions (of a linked state and  of an irreducible economy) and proofs (of
the coincidence of weak  and strong Pareto optimality and of quasi-equilibrium
and equilibrium).} 
We assume that $\ell(\omega,\pi,\act) = 0$ for every $(\omega,\pi,\act) \in \mem$ for which $\omega \not= \omega_a$; that is, no individual may choose membership in 
any club type containing no members of his/her external characteristic.  
We also assume throughout that there is an exogenously given 
upper bound $M$ on the number of memberships an individual 
may choose, so $|\ell| \leq M$ for each $\ell \in \Lists(a)$.
The utility function for agent
$a$ is defined over private goods consumptions and club memberships
and is thus a mapping $u_a : X_a \rightarrow \R$.  

We assume throughout that utility functions are strictly monotone in
private goods; i.e., $u_a(x,\ell) > u_a(x', \ell)$ for $a \in A, x, x' \in \R^N_+, x > x'$. 
However, we make no assumption
 that utility is monotone in the level of any
activity;  indeed, in our framework it is meaningless to talk about the
level of an activity.  
 The ranking of activities may be different for different 
individuals, and an individual's ranking of activities may depend on his/her consumption of private goods.   
We take the view that an agent's preferences for private goods and for club memberships are interdependent and 
cannot be disentangled (except for monotonicity in private goods).%
\footnote{See Diamantaras and Gilles (1996), Gilles and Scotchmer (1997) and
Diamantaras, Gilles and Scotchmer (1996) for further discussion on this point.}  Activities are not traded.  

\subsection{Club Economies}

A {\em club economy\/} $\cal E$ is a  mapping  
$a \mapsto (\omega_a,X_a,e_a, u_a)$    for which:
 \begin{itemize}
\item the external characteristic mapping  
$  a \mapsto \omega_a $
is a measurable function
\item the consumption set correspondence $a \mapsto X_a  $ 
is a measurable correspondence
\item the endowment mapping $a \mapsto e_a$  is an integrable function
\item the utility mapping $(a, x, \ell) \mapsto u_a(x, \ell)  $ 
is a (jointly) measurable function (of all its arguments)\footnote{It can be shown
that this measurability requirement is equivalent to the usual requirement on
measurability of preferences.} \end{itemize}
As above, we assume
that utility functions are continuous and strictly monotone in private
goods.

We assume that the {\em aggregate endowment\/}  
$$ \bar{e} = \int_A e_a \, d\lambda(a)
$$ 
is strictly positive, so all
private goods are represented in the aggregate.


\subsection{States} 

A {\em state\/} of a club economy is a measurable
mapping   $$   f = (x,\mu) : A \rightarrow \R^N \times \R^\mem   $$     A
state describes choices for each individual agent, ignoring feasibility at
the level of the individual and at the level of society.   {\em Individual
feasibility\/} means that  $(x_a,\mu_a) \in X_a$.   {\em Social
feasibility\/} entails market clearing for private goods and consistent
matching of agents.  

We define consistency as a property of choice functions
$\mu : B \rightarrow \Lists$, and show that it is equivalent to a 
property of aggregate membership vectors $\int_B \mu_a d \lambda(a)$.
For each integer $j$ and each membership $(\omega, \pi, \act)$, let 
$$ 
E^j_\mu(\omega, \pi, \act) = \{ a \in B : \mu_a(\omega, \pi, \act) =j \}
$$
be the set of all agents who choose $j$ memberships $(\omega, \pi, \act)$.  
If we interpret $\lambda(E)$ as the proportion of agents who 
belong to a set $E \subset A$, then $\lambda(E^j_\mu(\omega, \pi, \act))$ is the proportion of agents who choose $j$ memberships of type $(\omega, \pi, \act)$ and 
$j \lambda(E^j_\mu(\omega, \pi, \act))$ is the proportion of memberships of type $(\omega, \pi, \act)$ chosen by these agents.  Hence the sum
$$
\sum_{j=1}^\infty j \lambda(E^j_\mu(\omega, \pi, \act))
$$
is the proportion of memberships of type $(\omega, \pi, \act)$ chosen by agents in $A$.  We therefore say that a function $\mu : B \rightarrow \Lists$
is {\em consistent for $B$}  if  
$$
\frac{\sum_{j = 1}^\infty j\lambda(E_\mu(\omega, \pi,\act))}
{\sum_{j = 1}^\infty j \lambda(E_\mu(\omega',\pi,\act))}
  =  \frac{\pi(\omega)}{\pi(\omega')}
$$ 
for each 
$(\pi, \act) \in \Clubs$ and each $\omega, \omega' \in \Omega$.  Equivalently, $\mu$ is consistent for $B$ if for each $(\pi, \act)$ there is a real number $\alpha(\pi, \act)$ such that
$$
{\sum_{j = 1}^\infty j\lambda(E_\mu(\omega, \pi,\act))}
  = \alpha(\pi, \act) \pi(\omega)
$$
for each $\omega \in \Omega$.  
Thus, consistency means that the distribution of club membership choices in the population is the same as in the club itself.\footnote{Consider Example \ref{ex:manyclubs} for instance.  Keeping in mind that some males may choose one non-exclusive marriage, and some may choose two, consistency entails that the number of non-exclusive marriages chosen by males is the same as the number of non-exclusive marriages chosen by females.} 
   
 
We say that a club membership vector $\bar{\mu} \in \Mem$ is {\em
consistent\/} if for every club type $(\pi, \act)$ there
is a real number $\alpha(\pi,\act)$ such that  
$$
\bar{\mu}(\omega, \pi,\act) = \alpha(\pi,\act) \, \pi(\omega) 
$$ 
for every $\omega \in \Omega$.  
Write 
$$  
\Cons = \{ \bar{\mu} \in \Mem : \bar{\mu} \hbox{ is consistent } \}  
$$
Note that $\Cons$ is a subspace of $\Mem$.  Because  agents will choose lists of memberships that are nonnegative, the 
feasible states will have membership vectors in the positive part of
$\Cons$.

The following lemma, whose simple proof is left to the reader, 
states the relationship between these two notions.

\begin{lemma}  Let $\cal E$ be a non-atomic club economy, let $B \subset A$ be a measurable set,  and let  $\mu : 
B \rightarrow \Lists$ be an integrable function.  Then the function
$\mu$ is consistent for $B$ if and only if the membership vector
$\int_B \mu_a \, d\lambda(a)$ is consistent. \end{lemma}

We say that the state $f = (x,\mu)$ is {\em feasible for\/} the measurable subset
$B \subset A$ if it satisfies the following requirements: 
\begin{itemize} 
\item[(i)] {\bf Individual Feasibility } 
$$
(x_a,\mu_a) \in X_a \mbox{ for each } a \in A  
$$ 
\item[(ii)]  
 {\bf Material Balance \footnote{Material balance means
that the social consumption of private goods (within $B$) plus the
quantity of private goods used as inputs to club activities (by
members of $B$) is equal to the social endowment of private goods
(within  $B$).}}
$$
\int_B
x_a \, d\lambda(a)  \ + \ 
     \int_B \sum_{(\omega,\pi,\act) \in \mem} 
          \frac{1}{|\pi|}\inp(\pi,\act)\, \mu_a(\omega,\pi,\act) \, d\lambda(a)
      = \int_B e_a \, d\lambda(a)  
$$
\vfill
\pagebreak
   \item[(iii)]  {\bf Consistency } 
\begin{quote}
$\mu$ is consistent for $B$
\end{quote}
\end{itemize}   
We say the state $f$ is {\em feasible\/} if it is feasible for the
set $A$ itself.

Our description of feasible states of the economy is different from the
description of feasible states in most of club theory, where the analog
of consistency is expressed by a requirement that clubs
form a partitition of the set of agents.  Our description allows for the
possibility that agents belong to many clubs, that different agents
belong to different numbers of clubs, and that clubs have overlapping
memberships.  For instance, agents may  be married, have employment
in a firm, belong to a gym, attend movies and concerts, take meals in
a restaurant, and so forth.   In the special case that agents can
belong to only one club ($M$=1), the
consistency condition reduces to the assertion that clubs form a partition
that is ``measure consistent'' in the sense of Hammond, Kaneko and Wooders (1989).

 We do not keep track of which person
belongs to which club, nor do we need to do so:  every function $\mu : A \rightarrow \lists$ that satisfies the consistency condition corresponds to a consistent assignment of individuals to 
clubs (and vice versa).  Of course, a given $\mu$ may correspond to 
many  consistent assignments, but we do not need to distinguish them, because 
we assume that individuals care only about the external characteristics of their 
consumption partners, not about their identities. (See Section 
\ref{subsection:discussion}.) 


\subsection{Pareto Optimality and the Core}

As in the exchange setting, we distinguish
two notions of Pareto optimality and the core; the stronger notion allows
blocking if some agents (in the relevant group) are made better off and
none are made worse off,  the weaker notion requires that all agents be
made better off.  For exchange economies, strict monotonicity of
preferences guarantees that the two notions coincide.   Because choices of club memberships are
indivisible, however, the notions may be
distinct, even if preferences are strictly monotone in private goods.  In
this subsection we define two notions of Pareto optimality and the core
and give a natural condition that guarantees that they coincide.

Let $f$ be a feasible state.  We say that
$f$ is {\em weakly Pareto optimal\/} if there is no feasible state $g$
such that $u_a(g(a)) > u_a(f(a))$ for almost all $a \in A$;  $f$ is {\em
strongly Pareto optimal\/} if there is no feasible state $h$ such that
$u_a(h(a)) \geq u_a(f(a))$ for almost all $a \in A$ and $u_{a'}(h(a)) >
u_{a'}(f(a))$ for all $a$ in some subset $A' \subset A$ having positive
measure.  Note that strong Pareto optimality is  a more restrictive 
notion than weak Pareto optimality.  Similarly,  $f$ is in the {\em weak
core\/} if there is no  subset $B \subset A$ of positive measure and  state
$g$ that is feasible for $B$ such that  $u_b(g(b)) > u_b (f(b))$ for
almost every every $b \in B$; $f$ is  in the {\em strong
core\/} if there is no subset $B \subset A$ of positive
measure and  state $h$ that is feasible for $B$ such that $u_b(h(b))
\geq u_b (f(b))$ for every $b \in B$ and $u_{b'}(h(b')) > u_{b'} (f(b'))$ for all $b'$ in some subset $B' \subset B$ having positive measure. 
The strong core is a subset of  the weak core.

In general,  weakly Pareto optimal allocations may not be strongly
Pareto optimal, and the
weak core may be a proper superset of the strong core.  The following
assumption, adapted from Gilles and Scotchmer (1997), guarantees that
weak and strong Pareto optimality coincide and that the weak and strong
cores coincide.

We say that {\em endowments are desirable\/} if for every agent $a$ and
every list $\ell \in \Lists(a)$, $ u_a(e_a,0) > u_a(0, \ell) $.  
That is, each agent would prefer to remain single and consume his
endowment rather than to belong to any feasible set of clubs and
consume no private goods.  Desirability of endowments is weaker than
the assumption  Mas-Colell (1980) refers to as {\em essentiality\/} of
private goods, which in our framework would be:    
$$
u_a(0,\ell) = \min_{(x^*,\ell^*) \in X_a} u_a(x^*,\ell^*) 
$$ 
for every $\ell \in \lists(a)$.  

\begin{proposition}    If endowments are desirable, then weak and
strong Pareto optimality coincide and the weak and strong core
coincide.
\end{proposition}   
\begin{proof}   Let $f$ be a feasible state not in the
strong core.  By definition, there exists a subset $B \subset A$ of
positive measure and a state $g$ that is feasible for $B$ such that
$u_b(g(b)) \geq u_b( f(b))$ for every $b \in B$ and $u_b(g(b)) >
u_b(f(b)$ for all $b$ in some subset $B' \subset B$ having positive
measure.  Because endowments are desirable, in the state $g$ all
members of $B'$ must be consuming strictly positive amounts of
private goods.  Making
use of continuity and strict monotonicity of preferences, we can find 
a small transfer of private goods from members of $B \setminus B'$
that leads to a state $g'$ which is feasible for $B$ and which all members
of $B$ strictly prefer to $f$.  That is, $f$ is not in the weak core.  

The same argument with $B = A$ establishes coincidence of weak and strong Pareto optimality.  \end{proof}  


When endowments are desirable, we omit the modifiers and refer unambiguously
to Pareto optimality and the core.

 \subsection{Equilibrium}

Our notion of equilibrium involves the pricing of private goods and of
club memberships.  Private goods prices $p$ lie in $\R^N$;  prices for
club memberships $q$ lie in $\Mem$, so the vector of all prices lies in
$\R^N_+ \times \Mem$.  Because we assume that preferences are monotone in 
private goods, we will require  that private goods prices be non-negative.  However, 
prices for club memberships may be positive, negative or zero; prices for 
club memberships include  transfers between agents in a given club ---  some 
agents may subsidize others.  For $ (x,\bar{\mu}) \in \R^N \times \Mem$ a 
vector of private goods and club memberships and $(p,q)\in \R^N \times \Mem$ 
a vector of  prices, write  
$$ (p,q) \cdot (\bar{x}, \bar{\mu}) = 
  p \cdot \bar{x} + q \cdot \bar{\mu} $$ 
for the cost of  $(\bar{x}, \bar{\mu})$.

A {\em club equilibrium\/} consists of a feasible state $f = (x,\mu)$,
private good prices $p \in \R^N_+ \setminus \{0\}$ and club membership
prices $q \in \Mem$, satisfying the conditions:  
\begin{itemize} 
\item[(1)] {\bf Budget Feasibility for Individuals } 
\begin{quote}
For
almost all $a \in A$:    
     $$ p \cdot x_a + q \cdot \mu_a \leq p \cdot e_a   $$  
\end{quote}
\item[(2)] {\bf Optimization }  
\begin{quote} For almost all $a \in A$:
$$ 
(x'_a,\mu'_a) \in X_a \mbox{ and }
 u_a(x'_a,\mu'_a) > u_a(x_a,\mu_a)    
\Rightarrow  p \cdot x'_a + q \cdot \mu'_a > p \cdot e_a    $$ 
\end{quote} 
 \item[(3)] {\bf Budget Balance for Clubs }  \begin{quote}  For each club type
$(\pi,\act) \in \Clubs$: 
$$  \sum_{\omega \in \Omega} \pi(\omega)q
(\omega,\pi,\act) = p \cdot \inp(\pi, \act)  $$   
\end{quote} 
\end{itemize}  Thus, at an equilibrium, individuals 
optimize subject to their budget constraint and the total cost of memberships in a given club is just enough to pay for the inputs to the given activity.  

A {\em club quasi-equilibrium\/} differs from a club equilibrium only in the optimization condition (2) above is replaced by the weaker
quasi-optimization condition:
\begin{itemize}  
\item[(2${}'$)]  {\bf Quasi-Optimization } \begin{quote}  
For almost all $a \in A$:
$$ (x'_a,\mu'_a) \in X_a \mbox{ and }
 u_a(x'_a,\mu'_a) > u_a(x_a,\mu_a)    
 \Rightarrow  p \cdot x'_a + q \cdot \mu'_a \geq p \cdot e_a    $$\end{quote}
\end{itemize}  
That is, nothing that is feasible and strictly preferred can cost strictly
less than agent $a$'s wealth.   An equilibrium is necessarily a
quasi-equilibrium.  


\subsection{Equilibrium and Quasi-Equilibrium}\label{subsection:qe}

In the exchange case, the possibility of a quasi-equilibrium that is not an
equilibrium is frequently viewed as a mere technical problem; the
combination of strictly monotone preferences and strictly positive
aggregate endowments is enough to assure that this problem does not
occur.  However,  the indivisibilities and activities in our setting make
the issue more subtle.  The following
example illustrates the problems that may arise when private goods are used 
as inputs to club activities;  see Gilles and Scotchmer (1997) 
for an example illustrating the problems that may arise  when endowments 
are not desirable .

\begin{example}\label{example:qe} Consider an economy with two
private goods, a single external characteristic $\omega$ and a single
club $c = (2,\gamma)$ consisting of two people, requiring inputs $\inp (c)
= (2,0)$.  We assume agents are constrained to choose at most one club membership.  All agents are identical, with
endowments $e_a = (1,1)$ and utility functions: 
\begin{eqnarray*} 
u_a(x,0) & = & 1-e^{-x_1 -  x_2}   \\
u_a(x,\delta_{(\omega,c)})    & = & \sqrt{x_1} + \sqrt{x_2}  
\end{eqnarray*}
Reminder:  $\delta_{(\omega,c)}$ is the list specifying choice of the unique
membership $(\omega,c)$.   Because endowments are desirable, the weak and
strong cores coincide.  Indeed, there is a unique state $f$ in the core:  all agents
belong to clubs, consume none of good 1 and 1 unit of good 2, and the
entire supply of  good 1 is used to provide the input to the club
activity.  However, the state $f$ cannot be supported as an equilibrium,
because the marginal rate of substitution of good 1 for good 2 is
infinite, so the equilibrium price ratio would have to be infinite also. 
On the other hand, $f$ can be supported as a quasi-equilibrium: 
quasi-equilibrium prices are $p = (1,0), \ q(\omega,c) = 1$.  (This is {\em
not\/} an equilibrium, because good 2 is free and every agent desires
more of it.)   $\clubsuit$  \end{example}


In the familiar exchange setting, a quasi-equilibrium may fail to be an equilibrium
if some agents are in the ``minimum expenditure situation;'' that is, when
quasi-equilibrium consumptions require expenditures exactly equal to
wealth and slightly smaller expenditures are not possible.  As the
example above illustrates, it is easier for this minimum expenditure situation to arise in
club economies, because private goods are used as inputs to club
activities and club choices are indivisible.  Various assumptions would enable us to avoid the minimum expenditure setting and guarantee that a quasi-equilibrium is an equilibrium; we take a route parallel to the exchange setting.

Let ${\cal E}$ be a club economy and let $f = (x, \mu)$ be a feasible state. Write $\delta_j$ for the consumption bundle consisting of one unit of
good $j$ and nothing else.  Say that $f$ is {\em club linked\/} if whenever  
$$ I \cup J = \{ 1, \ldots , N \}$$ 
is a partition  of the set of private goods and  $x_{ai} = 0$ for all
$i \in I$ and almost all $a \in A$, then for  almost all $a \in A$ there exist
$r \in \R_+, \,   j \in J $ such that    
$$
             u_a(e_a + r\delta_j, 0) > u_a(x_a, \mu_a)   
$$  
That is, if  (as in Example \ref{example:qe})
the {\em entire\/} social endowment of the private goods in $I$ is used
in  the production of club activities, then for almost all agents $a$, there
is some good $j \notin I$ and some sufficiently large level of
consumption of good $j$ such that agent $a$ would prefer consuming his
endowment together with this large level of good $j$, and belong to no
clubs, rather than  consume the bundle $x_a$ in the club memberships
$\mu_a$.  Say that $\cal E$ is {\em club irreducible\/} if every feasible
allocation is club linked.\footnote{We use the terms ``club linked'' and ``club irreducible'' because these notions play the same role for us that linked allocations and irreducibility play in the exchange setting; see Mas-Colell (1985) for instance.}

\begin{proposition}\label{prop:qe} 
 Let $\cal E$ be a club economy for which  endowments are desirable.  If $(f, p, q)$
is a club quasi-equilibrium and $f$ is  club linked, then $p >> 0$ and 
$(f, p,q)$ is an equilibrium.  
 \end{proposition}
\begin{proof} We show first that all private good prices are strictly
positive.  If not, let $I$ be the set of indices for which $p_i > 0$, and let $J
\not= \emptyset$ be the complementary set of indices.  Fix $i \in I$.  If
$x_{ai} \not= 0$ for some set of consumers having positive measure, then
some of these consumers could sell a small amount of their consumption
of $x_i$ and buy an unlimited quantity of $x_j$ (for any $j \in J$) and be
strictly better off with a lower expenditure; this would contradict the
quasi-equilibrium conditions.  We conclude that, for each $i \in I$, $x_{ai}
= 0$ for almost all $a \in A$.  Club linkedness guarantees that all consumers
would prefer to consume their endowments plus a large quantity of
some commodity $x_j$ rather than their quasi-equilibrium consumption. 
Since aggregate endowments of private goods are strictly positive, the
endowments of some consumers have a strictly positive value and those
consumers would (by continuity of preferences) prefer to consume a
very large fraction of  their endowment plus a large quantity of
commodity $x_j$, rather than their quasi-equilibrium consumption. 
Again, this would contradict the quasi-equilibrium conditions, so we
conclude that all private good prices are strictly positive.

If $(f, p, q)$ is not an equilibrium, then there is an agent $a$ who is
quasi-optimizing, but not optimizing.  Hence there is a choice $(x', \mu') \in X_a$
which is strictly preferred to agent $a$'s quasi-equilibrium choice and
costs no more than his endowment.  Desirability of endowments entails
that $x'  \not= 0$, so $p \cdot x'  >  0$.  Continuity of preferences
entails that there is a bundle $x''$ such that $p \cdot x'' < p \cdot x'$,   
$(x'', \mu') \in X_a$ and $(x'', \mu')$ is  strictly preferred to agent $a$'s quasi-equilibrium choice
 --- but costs strictly less than his endowment.  This is a contradiction, so the proof is complete.  \end{proof} 


\subsection{Pure Transfers}

 Our formulation of equilibrium requires that the sum of  membership
prices in each club type be exactly sufficient to pay for the inputs required
for production of the club activity.  An equivalent notion makes clear the
role of membership prices as taxes and subsidies (and will prove to be
more convenient in proofs).

\noindent Say that $q \in \Mem$ is a {\em pure transfer\/} if
$q\in \Trans$, defined as:
\[ \Trans = \{ q \in \Mem : q \cdot \mu = 0 \hbox{ for each }\mu \in \Cons\} 
\]
\noindent Thus for each club type $(\pi,\act)$ and $q\in \Trans$,
\[ 
\sum_{\omega \in \Omega} \pi(\omega) q(\omega,\pi,\act) = 0 
\] 

A {\em pure transfer equilibrium\/} is a triple $(f, p, q)$ where $f$ is a
feasible state, \mbox{$p \in \R^N_+ \setminus \{0\}$} is a vector of private
good prices and $q \in \Mem$ is a vector of membership prices satisfying
the conditions:  
\begin{itemize} 
\item[(1)] {\bf Budget Feasibility }  \begin{quote} For
almost all $a \in A$,    
$$ p \cdot x_a + 
    q \cdot \mu_a + 
      \sum_{(\omega,\pi,\act)} 
         p \cdot \frac{1}{|\pi|}\,  \inp (\pi,\act) \mu_a(\omega,\pi,\act) \ 
 \leq \ p \cdot e_a   
$$ \end{quote}
   \item[(2)]  {\bf Optimization } \begin{quote}  For almost all $a \in A$,
if $(x'_a,\mu'_a) \in X_a$ and   $$
u_a(x'_a,\mu'_a) > u_a(x_a,\mu_a) $$   then  
$$  p \cdot x'_a + q \cdot \mu'_a + 
\sum_{(\omega,\pi,\act)} 
p \cdot \frac{1}{|\pi|}\,  \inp (\pi,\act)  \mu'_a(\omega,\pi,\act)  \ > \ 
p \cdot e_a    
$$   \end{quote}
\item[(3)]  {\bf Pure Transfers }   $$ q \in \Trans $$  \end{itemize}  We
define a pure
transfer quasi-equilibrium in the obvious way. 

The following lemma tells us that equilibrium (respectively quasi-equi\-lib\-rium) and  pure transfer equilibrium (respectively quasi-equi\-lib\-rium) are equivalent notions; we leave the simple proof to the reader.

\begin{lemma}  Let ${\cal E}$ be a club economy.  For $q^* \in \Trans$ define 
$q \in \Mem$ by
$$
q^*(\omega, \pi, \gamma) = q(\omega, \pi, \gamma) +                                                                p \cdot \frac{1}{|\pi|} \inp(\pi, \gamma)
$$
Then: $(f,p,q)$ is a pure transfer equilibrium (respectively, pure transfer
quasi-equilibrium) if and only if $(f,p,q^*)$ is an  equilibrium (respectively,
quasi-equilibrium).  \end{lemma}

\subsection{Discussion}\label{subsection:discussion}

In our model, agents care about their own consumption and about the
external characteristics of others in their clubs.  The characteristics we 
have in mind should be observable to others in the club, which is why we 
call them external.  Such characteristics might include sex,
intelligence, appearance, even tastes and endowments, to the extent that such
characteristics can be observed by others.\footnote{But keep in mind that we
assume in this paper that the set of external characteristics is finite.}  On
the other hand, we exclude private characteristics which are known only to the
individual.  Because we assume that memberships are priced according to
external characteristics, our construction can be viewed as  a compromise between
the non-discriminatory pricing of competitive  equilibrium and the personalized
prices of Lindahl.  To capture the essence  of club theory, we regard as
essential a certain degree of anonymity, but  we also think it important to
recognize that clubs offer different types of  membership.\footnote{Much of the club literature indexes both the
external characteristics and the tastes and endowments by a single ``type;''
see Berglas (1976), Gilles and Scotchmer (1997) for instance.  
Our use of  external characteristics is closer in spirit to  Conley and Wooders
(1994), Engl and Scotchmer (1996) and Scotchmer (1996), where prices are
understood as ``externality prices.''  However, these latter papers treat only
finite TU economies with a single private good, restrict agents to belong to at
most one club, and do not discuss existence.}

One restriction in this model, which would be particularly desirable to eliminate in future
work is that external
characteristics are  ascriptive, not acquired.  Intelligence and
endowments (if observable) are possible external characteristics, skill and
consumption are not.  

Of course we could formulate a model in which preferences for club memberships 
depend  on various characteristics of club partners, but insist that prices
be  independent of those characteristics.  In that case, however, and in contrast to the results proved here, core allocations might not be 
decentralizable by prices, and equilibria could fail to exist.  (A similar comment applies to the possibility of preferences that depend  on the {\em consumptions\/} of club partners.)

\vfill
\newpage
%%%%%%%%%%

\section{The First Welfare Theorem and the Core}

In our club context, as in the exchange case, we easily
obtain the first welfare theorem.

\begin{theorem}\label{thm:1stwelfare}
Every equilibrium state of a club economy belongs to
the weak core and, in particular, is weakly Pareto
optimal.  If endowments are desirable, every equilibrium state
belongs to the strong core and, in particular, is strongly Pareto
optimal.
 \end{theorem}  
\begin{proof}
Let $\cal E$ be a club economy and let $f = (x, \mu)$  be an
equilibrium state, supported by the prices $p \in \R^N_+
\setminus \{0\},q \in \Mem$.  If $f$ is not in the weak core, there is a
subset $B \subset A$ of positive measure and a state $g = (y, \nu)$
that is feasible for $B$ and
 preferred to $f$ by every member of $B$.  Feasibility of $g$ for
the coalition $B$ entails the
material balance condition:
$$ \int_B y_a \, d\lambda(a)  \ + \ 
     \int_B \sum_{(\omega,\pi,\act) \in \mem} 
          \frac{1}{|\pi|}\inp(\pi,\act)\, \nu_a(\omega,\pi,\act) \,
d\lambda(a)
      = \int_B e_a \, d\lambda(a)  
$$
and the budget balance condition for each club type $(\pi, \act)$:
$$  
\sum_{\omega \in \Omega} \pi(\omega)q
(\omega,\pi,\act) = p \cdot \inp(\pi, \act)  
$$
 Combining these with the consistency condition, we conclude that
$$
\int_B (p,q) \cdot (y_a, \nu_a) \, d\lambda(a) \leq 
   \int_B p \cdot e_a \, d\lambda(a)
$$
Hence there is a set $B' \subset B$ having positive measure for
which
$$
(p,q) \cdot (y_b, \nu_b) \leq p \cdot e_b
$$ 
for every $b \in B'$.  Since $g$ is unanimously preferred
to $f$ by members of $B$, this contradicts the
equilibrium nature of $f$.  We conclude that $f$ is in the
weak core, as desired.

That $f$ is weakly Pareto optimal follows immediately
by taking $B = A$ in the argument above.

If endowments are desirable, the weak and strong cores coincide
and weak and strong Pareto optimality coincide, so the proof is
complete.  \end{proof}

\vfill
\newpage
%%%%%%%%%


\section{Failure of the Second Welfare Theorem} 

For exchange economies, the second welfare theorem  asserts
that every Pareto optimal allocation can be realized  as an
equilibrium allocation after a suitable redistribution of
endowments.  For finite economies, the second welfare theorem depends on
convexity of preferences;  because the indivisibility of
club memberships introduces an essential
non-convexity in our context, it should come as no surprise
that the second welfare theorem may fail for finite club economies. 
More surprising is that the second welfare theorem may
fail even for {\em non-atomic\/} club economies.  The following
simple examples (see also  Example
\ref{example:bounded}) illustrate what may go wrong.

\begin{example}\label{example:2ndwelfare1} We consider an
economy with a single consumption good.  Agents have one of two
external characteristics,  $\Omega=\{M,F\}$ (males and females); a
single club $c$, a monogamous marriage,
requiring no inputs, is possible, and agents are constrained to choose at most
one club membership.  In this case $\mem = \{(M,c), (F,c)\}$.  
Agents in
the interval $[0,1/2)$ are male, agents in the interval
$[1/2,1]$ are female.     Males love marriage and females hate it:   
$$ \begin{array}{rclll}   u_a(x,0) & =& x
& \mbox{ for all }a\in [0,1] & \\  
u_a(x,\delta_{(M,c)}) & = & 2x & \mbox{ for all
}a\in [0,1/2) &\mbox{ (males)} \\ 
u_a(x,\delta_{(F,c)}) &=& 1 - e^{-x} & \mbox{
for all }a\in [1/2,1] & \mbox{ (females)}  \end{array} 
$$ 
Endowments are  $ e_a = 1$ for all $a$.  Define $x$ by
$$
x_a = \cases{    
\frac{1}{\sqrt{2a}} \quad \mbox{ for all }a\in
[0,1/2)\mbox{ (males)}  \cr 
                 1 \quad \quad \mbox{ for all }a\in [1/2,1]\mbox{
(females)}   } $$
and set $f = (x,0)$.  It is easily checked that $f$ is a
Pareto optimal feasible state, but cannot be
supported as an equilibrium following any redistribution of
endowments:  Whatever the marriage price, some
males will be rich enough to desire and afford marriage; those
males will not be optimizing.  $\clubsuit$
 \end{example}

\bigskip

\begin{example}\label{example:2ndwelfare2} The economy is as
described in Example \ref{example:2ndwelfare1}.  Consider the
feasible state in which there is no exchange of the consumption
good, but all agents are married.  That is, $g = ({\bf 1},\nu)$, where
$\nu$ is defined by: 
$$
\nu_a = \cases{    
\delta_{(M,c)} \quad \mbox{ for all }a\in [0,1/2)\mbox{ (males)}  \cr 
                \delta_{(F,c)} \ \quad \mbox{ for all }a\in [1/2,1]\mbox{
(females)}   } $$
Again, it is easily checked that $g$ is Pareto optimal, but cannot
arise as an equilibrium state, no matter what the endowments:  No
matter what the prices, females --- who hate marriage --- cannot
be optimizing when they are married.  $\clubsuit$
 \end{example}

\bigskip

The role of unbounded consumption in the failure of the 2nd
welfare theorem seen in Example \ref{example:2ndwelfare1} foreshadows
the role unbounded endowments will play in the failure of core equivalence in
Example \ref{example:bounded}.   The failure of the 2nd welfare theorem
seen in Example \ref{example:2ndwelfare2} reflects the fundamental
asymmetry between initial states (in which agents choose no clubs) and other
feasible states (in which agents may choose various clubs).\footnote{This
problem might be ``solved'' by allowing for endowments of clubs, but it is not
clear what endowments of clubs should mean.  Here we follows tradition in the club literature and assume endowments consist of private goods only.}

\vfill
\newpage

%%%%%%%%%%%%%%%
\section{Core/Equilibrium Equivalence}
 
In this section  we establish that non-atomic club economies pass a familiar
test of perfect competition:  coincidence of the core with the set of
equilibrium states.

 \begin{theorem}{\label{thm:equivalence}}
 Let $\cal E$ be a non-atomic club economy in which endowments are desirable
and uniformly bounded above.  Then every core state can be supported as a
quasi-equilibrium and every core state that is club linked can be supported as an equilibrium.  In particular, if $\cal E$ is club irreducible, then the core coincides with the set of equilibrium states.
 \end{theorem}
 
In the proof (which we defer to Section \ref{sect:proofs}), we find it convenient to construct a pure transfer
equilibrium.  The  argument parallels Schmeidler's (1969) proof of Aumann's core
equivalence theorem for exchange economies: 
 \begin{itemize} 
\item[{\bf 1}]
Construct a preferred net trade correspondence and aggregate net preferred
set.  
\item[{\bf 2}] Apply the Lyapunov convexity theorem to show that the
aggregate net preferred set is convex.   
\item[{\bf 3}]  Use the core property
to show that the aggregate net preferred set  is disjoint from a cone that represents feasible net trades (for all coalitions). 
\item[{\bf 4}]  Construct a quasi-equilibrium price as a
price that separates the net aggregate preferred set from this cone.  Use linkedness to conclude that the quasi-equilibrium is an equilibrium.  
\end{itemize}

 The argument contains two surprises.  The first is that we require
endowments to be bounded; no such assumption is required in the familiar 
exchange case.  This is not merely an artifact of the proof, however:  if 
endowments are unbounded, the core may not coincide with the set 
of equilibrium states and equilibrium may not exist.  The following variant of  
Examples 5.1 and 5.2 illustrates the problem.

\begin{example}\label{example:bounded} We consider an economy with a
single consumption good.  Agents have one of two external characteristics, 
$\Omega=\{M,F\}$ (males and females);  a single club $c$, a monogamous
marriage requiring no inputs, is possible, and we we assume agents are constrained
to choose at most one membership.  
Agents in the interval $[0,1/2)$ are male, agents in
the interval $[1/2,1]$ are female.     Males love marriage and females hate it:  
$$ \begin{array}{rcll}  u_a(x,0) & =& x & \mbox{ for all }a\in [0,1] \\  u_a(x,\delta_{(M,c)})
& = & 2x & \mbox{ for all }a\in [0,1/2)\mbox{ (males)} \\ u_a(x,\delta_{(F,c)}) &=& 1 -
e^{-x} & \mbox{ for all }a\in [1/2,1]\mbox{ (females)}  \end{array} $$
Endowments are  $ e_a = \frac{1}{\sqrt{a}} $.  It is easily checked that the
initial state is the unique element of the core but cannot be supported as an
equilibrium:  there is no upper bound on the amount men would pay to enter a
marriage (because  males are willing to give up half their endowment to enter
a marriage, and male endowments are unbounded), but no female is willing to
enter a marriage at any price.   $\clubsuit$  \end{example}

The other surprise in the proof is that it will not be quite good enough to find prices $(p,q)$ that separate the aggregate net preferred set from the cone representing feasible net trades; we must also be sure that $p \not=0$.  
To achieve this we will separate the aggregate net preferred set from a  cone that is larger than the cone representing feasible net trades.  To show that the aggregate net preferred set is disjoint 
from this cone, we will need to show that if $g = (y,\nu)$ is a state, 
$B \subset A$ is a coalition, and $\nu$ nearly satisfies the consistency condition with respect to $B$, then there is a large subset $B' \subset B$ such that $\nu$ {\em exactly\/} satisfies the consistency condition with respect to $B'$.  

 We  formalize this idea in the following Lemma, but we must 
first introduce  a little notation.  For $L \subset \Mem$ write $\co(L)$ 
for its convex hull.  We have assumed that individuals are constrained to 
choose lists with no more than $M$ memberships; write  
$$\lists_M = \{
\ell \in \lists: |\ell| \leq M \} $$ (Recall that $M$ is the given upper bound on the
number of memberships that may be chosen by any individual.)  Set
$$
{\cal D} = 
   \{ L \subset \lists_M, \co(L) \cap \Cons = \emptyset \}
$$
and 
$$
D = \inf \,  \Bigl\{\dist (\co (L), \Cons) : 
           L \in {\cal D} \Bigr\}  
$$

\begin{lemma}\label{lemma:adjust}   Let $B \subset A$ be a 
measurable set of positive measure and let $\nu : B \rightarrow \lists_M$
be a measurable function.  Then there is a measurable subset $B' \subset B$
such that $$ \int_{B'} \nu_b \, d\lambda(b) \ \in \ \cons $$ and
\begin{equation}\label{eqn:D} \lambda(B') \geq \lambda(B) - 
  \frac{1}{D}
   \dist (\int_{B} \nu_b \, d\lambda(b) \ ,\cons) \end{equation}
 \end{lemma}   

\vfill
\newpage
%%%%%%%%%%

\section{Existence of Equilibrium}

In this Section we establish the existence of equilibrium for non-atomic
economies. 

\begin{theorem}\label{thm:existence}
Let $\cal E$ be a non-atomic club economy.  If endowments are desirable
and uniformly bounded above,  then a quasi-equilibrium exists.  If in
addition ${\cal E}$ is club irreducible, then an equilibrium exists.
\end{theorem}

The basic idea of the argument will be familiar:  construct an excess
demand correspondence, use a fixed point theorem to find a zero, and
show that this zero is an equilibrium price.  However, the club structure
gives rise to many subtleties: 
\begin{itemize} 
\item The balance condition
for private goods translates to the requirement that the excess demand
for private goods be 0; the balance condition for club memberships
translates to the requirement that the
demand for club memberships be in $\Cons$, a more subtle condition to
verify.  
\item  In equilibrium, prices for private goods must be 
positive, but prices for club membership prices may be positive, negative or
0.  Hence the relevant space of all prices is not a proper cone, and the
usual forms of the excess demand lemma will not apply.  \item Private
good prices can be normalized to sum to 1; club membership prices admit
no obvious normalization or bound.  Hence it is not clear how to
construct a  domain for prices on which to apply a fixed point
theorem. 
\item We assume that all private goods are present in the
aggregate, but we do not assume that all external characteristics are
present in the aggregate, and some clubs may not be chosen at equilibrium.  In effect, therefore, we must construct
reservation prices for unavailable club memberships. \end{itemize} 

One reason that membership prices may be unbounded is that they may
be indeterminate.  The following variant on Example
\ref{example:2ndwelfare1} illustrates the point.



\begin{example} We consider an economy with a
single consumption good.  Agents have one of two external characteristics, 
$\Omega=\{M,F\}$ (males and females);  a single club $c$, a monogamous
marriage requiring no inputs, is possible.  Agents are constrained
to choose at most one membership.  The agent space is $A = [0,1]$; agents in $[0,1/2]$ are
male, agents in $[1/2,1]$ are female.  Endowments are $e_a = 1$ and
utility functions for all agents are:  
$$\begin{array}{llll}   
u_a(x,0) &=& x & \hbox{ for  all } a\\    
u_a(x,\delta_{(M, c)}) &=& 1 -e^{-x} &  \hbox{ for } a \in [0,1/2) \\ 
u_a(x,\delta_{(F,c)}) &=& 1 - e^{-x}  &   \hbox{ for } a \in [1/2,1]
\end{array} $$   
In this example both males and females hate marriage.  The core
consists of the single autarkic state, and is supported as an equilibrium by 
{\em any\/} prices $p, q$ such that $p > 0$ and $q(M,c) + q(F,c) = 0$ 
(because agents will never choose to marry, no matter what the
subsidy).  $\clubsuit$ \end{example}

With multiple club memberships the problem is more subtle, as the
following example illustrates.

\begin{example}  We consider an economy with a
single consumption good.  Agents have one of two external characteristics, 
$\Omega=\{M,F\}$ (males and females);   two
club types $c_1, c_2$, each consisting of one male and one female and
requiring no inputs, are possible.  Agents are allowed to choose at most 2
memberships.  The agent space is $A = [0,1]$; agents in $[0,1/2]$ 
are male, agents in $[1/2,1]$ are female.  Endowments 
are $e_a = 1$.  Utility functions for males $a \in [0,1/2)$ are:  
$$ \begin{array}{lllll}  u_a(x,0) &=&  x   \\   u_a(x,\delta_{(M, c_1)})
&=&u_a(x,\delta_{(M, c_2)}) &=& 1 - e^{-x}  \\ u_a(x,2\delta_{(M,
c_1)}) &=&u_a(x,2\delta_{(M, c_2)}) &=& 1 - e^{-x}  \\
u_a(x, \delta_{(M, c_1)} + \delta_{(M, c_2)}) & =& 2x \\
\end{array}
 $$ 
 (Note that $\delta_{(M, c_1)} + \delta_{(M, c_2)}$ is the list representing choice of one membership of type $(M,c_1)$ and one membership of type $(M, c_2)$.)  Utility functions for females $a \in [1/2, 1]$ are:
$$ \begin{array}{lllll}  u_a(x,0) &=&  x   \\   u_a(x,\delta_{(F, c_1)})
&=&u_a(x,\delta_{(F, c_2)}) &=& 1 - e^{-x}  \\ u_a(x,2\delta_{(F,
c_1)}) &=&u_a(x,2\delta_{(F, c_2)}) &=& 1 - e^{-x}  \\
 u_a(x, \delta_{(F, c_1)} + \delta_{(F, c_2)}) & =& 2x \\
 \end{array}
 $$  Thus, both males and females hate belonging to a single club or two
clubs of the same type, but love belonging to two clubs of different
types.  The core consists of a single point:  all agents choose one club of
each type and consume their endowments.  This state is supported as an
equilibrium by {\em any\/} private good prices and club membership
prices such that $p > 0$ and  
\begin{eqnarray*} q(M,c_1) + q(F, c_1) &=& 0  \\ q(M,c_2) + q(F,
c_2) &=& 0   \\ q(M,c_1) + q(M, c_2) &=& 0 \\
q(F,c_1) + q(F, c_2) &=& 0
    \end{eqnarray*}
(Because agents will never choose to belong to only one club or to two
clubs of the same type, no matter how big the subsidy.  $\clubsuit$\end{example}

As these examples suggest,  some of the indeterminacy would disappear if we regarded lists as 
the primary objects and priced them directly.  However, doing so would lead to a 
 less appealing notion of equilibrium, in which the price of a list
might not be the sum of the prices of its component memberships.  
We shall therefore  work  directly with membership prices, keeping list prices in the
background.

Finding upper and lower bounds  for list prices is much simpler than finding a bounded domain for membership prices:  If we normalize private good
prices to sum to 1 and assume that individual endowments are uniformly bounded, then individual incomes also will be uniformly 
bounded.  Hence if $q \in \Mem$ is a vector of membership 
prices, $\ell \in \Lists$ is a list, and $q \cdot \ell$ exceeds the bound on individual incomes, then 
no agent will be able to afford the list $\ell$, and the demand for $\ell$ will be 0.  Thus the upper bound on individual incomes provides an upper
bound for list prices.  To construct a lower bound for  list prices (keeping in mind that list prices might be negative), we show that, if some individuals are paying large negative list 
prices then others are paying large positive list prices, which is impossible.  The construction we use is formalized in Lemma \ref{lemma:lowerbound}.

 As does the proof of core/equilibrium equivalence (Theorem \ref{thm:equivalence}), the argument for  Theorem \ref{thm:existence} proceeds by constructing a pure transfer 
quasi-equilibrium.  To deal with the difficulties identified earlier,  
we work with perturbations of the true economy, and then take limits as we 
make the perturbation disappear.  The argument  is divided into 8 steps:  
\begin{itemize}  \item[\bf 1 ] For each $k$, construct a  perturbed
economy ${\cal E}^k$ by adjoining to $A$ a few agents of each external
characteristic, with  utility functions  unbounded in private good
consumption.    
 \item[\bf 2 ] For the perturbed economy ${\cal E}^k$,  identify a
compact set of  prices in which an equilibrium  price will be found.
  \item[\bf 3 ] Construct an excess demand correspondence. 
 \item[\bf 4 ]  Find a fixed point of the correspondence that maximizes
the value of excess demand. \item[\bf 5 ] Show that, at this fixed point,
excess demand for private goods is equal to 0 and  demand for club
memberships is an element  in $\Cons$.
 \item[\bf 6 ] Construct an equilibrium for  ${\cal E}^k$.  \item[\bf 7 ]
Show that the equilibrium state can be supported by prices satisfying a 
uniform bound independent of $k$. \item[\bf 8 ] Take limits of these
uniformly bounded equilibrium prices as $k \rightarrow \infty$  and apply
Fatou's lemma to construct an equilibrium  for $\cal E$.
 \end{itemize}

\vfill
\newpage

%%%%%%%%%%%%%%%

\section{Proofs}\label{sect:proofs}

Here we collect proofs of most of the results in the text. 
We first show that if $\nu$ is ``almost'' consistent for $B$ then it
is exactly consistent for a large subset of $B$.

\noindent {\bf Proof of Lemma \ref{lemma:adjust} }  If $\Lists_M \subset \Cons$ then $\dist (\int_{B} \nu_b \, d\lambda(b) \ ,\cons) = 0$, $\cal D = \emptyset$ and $D = \infty$, so we may take $B' = B$.  Assume therefore that $\Lists_M \not\subset \Cons$. 

For each $\ell \in \lists_M$, write  $$ B_\ell = \nu^{-1}(\ell) = \{ b \in B :
\nu_b = \ell \} $$ and let  $L = \{\ell \in \lists_M : \lambda(B_\ell) > 0
\}$.  Note that $ \sum_{\ell \in L} \lambda(B_\ell) = \lambda(B) $ and
that 
$$  \int_B \nu_b \, d\lambda(b) = 
\sum_{\ell \in L} \lambda(B_\ell) \, \ell 
   = \frac{1}{\lambda(B)} \, 
      \sum_{\ell \in L} \frac{\lambda(B_\ell)}{\lambda(B)} \, \ell
$$
 In particular $$ \int_B \nu_b \, d\lambda(b) \in
\frac{1}{\lambda(B)} \co(L) $$ If $\co(L) \cap \Cons = \emptyset$ then
the right hand side of (\ref{eqn:D}) is
non-positive, so we may take $B_\ell = \emptyset$ for each $\ell$.  We
therefore assume  $\co(L) \cap \Cons \not= \emptyset$.


Consider the linear programming problem:
 \begin{eqnarray*}    &\hbox{ maximize} \quad & \sum_{\ell \in L} \beta_\ell
\\   & \hbox{ subject to } \quad &  0 \leq \beta_\ell \leq \lambda(B_\ell) 
\\  & &\sum_{\ell \in L} \beta_\ell \ell \in \Cons   \end{eqnarray*}   
The feasible set for this problem is non-empty (it contains the
origin), so this problem has a solution; let $\{\beta_\ell : \ell \in L \}$ be
any such solution.

For each $\ell$, write $\alpha_\ell = \lambda(B_\ell) - \beta_\ell 
\geq 0$.  Write $L' = \{ \ell : \alpha_\ell > 0 \}$.  If $L' =
\emptyset$ we are done, so assume not.  Write $\alpha = \min\{\alpha_\ell : \ell \in L' \}$.  If $\co(L') \cap \Cons \not=
\emptyset$ there are non-negative real numbers $\epsilon_\ell$ summing to
1 with  $\sum_{L'} \epsilon_\ell \ell \in  \Cons$.    Set $\beta^*_\ell =
\beta_\ell + \epsilon_\ell  \alpha_\ell$ for $\ell \in L'$ and $\beta^*_\ell =
\beta_\ell$ for $\ell \notin L'$.  Then $\{\beta^*_\ell : \ell \in L \}$
satisfies the constraints in the  linear programming problem and yields a
larger value of the objective, contradicting the choice of $\{\beta_\ell \}$ as
the solution.  We conclude that $\co (L') \cap \Cons = \emptyset$.  

For each $\ell \in \lists_M$, non-atomicity of $\lambda$
guarantees that we can choose $B'_\ell \subset B_\ell$ such that
$\lambda(B'_\ell) = \beta_\ell$.  Set 
$$ 
B' = \bigcup_{\ell\in L} B'_\ell 
$$
By construction, 
$$ 
\int_{B'} \nu_b \, d\lambda(b) = 
  \sum_{\ell\in L} \lambda(B'_\ell) \ell  \ \in \ \cons 
$$
 We need only estimate $\lambda(B')$.  To this end, note first that, because
$\Cons$ is a linear subspace,    $$   \dist(x - y, \Cons) = \dist(x,\Cons)  $$  
 and   $$  \dist(rx, \Cons) = r \dist(x, \Cons)  $$  for every $x \in \Mem, y
\in \Cons, r \in \R_+$.  Hence    
\begin{eqnarray*} 
 \dist(\sum_{\ell\in L} \lambda(B_\ell) \ell, \Cons) 
   & = & 
       \dist ((\sum_{\ell\in L} \lambda(B_\ell) \ell - \sum_{\ell\in L} \beta_\ell
\ell) \, , \, \Cons) \\
&=&  \dist(\sum_{\ell\in L'} (\lambda(B_\ell) -\beta_\ell) \ell , \, \Cons)
\\
    &=&  \dist(\sum_{\ell\in L'} \alpha_\ell \ell , \, \Cons)
\\
   &=&  \dist\left(\sum_{\ell\in L'} \alpha_\ell \sum_{\ell\in L'} \left[\frac{\alpha_\ell}{\sum_{\ell\in L'}
\alpha_\ell}\right] \ell \, , \, \Cons \right) \\
    &=& \sum_{\ell\in L'} \alpha_\ell \ 
  \dist\left( \left[ \sum_{\ell\in L'} \frac{\alpha_\ell}{\sum_{\ell\in L'} \alpha_\ell}\right] \ell \, , \,
\Cons \right) \\
     & \geq & \sum_{\ell\in L'} \alpha_\ell \  \dist ( \co
(L'),\Cons)  \\
& =& \sum_{\ell\in L'} (\lambda(B_\ell) - \beta_\ell) \  \dist ( \co
(L'),\Cons)  \\
    & \geq & D \sum_{\ell\in L'} (\lambda(B_\ell) - \beta_\ell) \\
     & = & D \sum_{\ell\in L} (\lambda(B_\ell) - \beta_\ell) \\
& = & D [\lambda(B) - \lambda(B')]
 \end{eqnarray*} Rearranging terms yields the desired inequality (\ref{eqn:D}). \qed
 
With this lemma in hand, we turn to the proof of core/equilibrium equivalence.


 \noindent {\bf Proof of Theorem \ref{thm:equivalence} }  Let $f = (x, \mu)$ be a
core state.  We show that $f$ can be supported as  a pure transfer
quasi-equilibrium.

\noindent {\bf Step 1 }  For each agent $a$, consider the preferred set    $$
\Phi(a) = \{(x, \ell) \in X_a:  u_a(x,\ell) > u_a (x_a,\mu_a)\}
 $$   For each club  $(\pi,\act) \in \Clubs$, consider the bundle $\frac{1}{|\pi|}
\inp(\pi,\act)  $;  this is what each agent would be required to contribute to
the club $(\pi, \act)$ if inputs were imputed equally to all members.  For $\ell
\in  \lists(\omega_a)$, define 
$$
 \tau(\ell) = \sum_{(\omega, \pi, \gamma) \in \mem}
                          \ell(\omega, \pi, \gamma)\frac{1}{|\pi|} \inp(\pi,\act)  $$ This is
the total an individual would be required to contribute to all clubs if $\ell$ is
the chosen list of memberships and inputs were imputed equally to all
members.

Define the net preferred set for agent $a$ as:    $$ \psi(a) = \{(x,\ell)\in \R^N
\times \Mem : 
     (x + e_a - \tau(\ell) , \ell) \in \Phi(a) \}   $$    and set   $$  \Psi(a) = \psi(a)
\cup \{0\}  $$ 
 It is easily checked that $\Psi$ is a measurable correspondence.  Define the
aggregate net preferred set to be  the integral of the correspondence $\Psi$:
$$  Z = \int_A \Psi(a) \, d\lambda(a)  $$  (We refer the reader to Hildenbrand
(1974) for  discussion of the integral of a correspondence.)

\medskip

\noindent {\bf Step 2 } In view of the Lyapunov convexity theorem, $Z$ is a
non-empty convex subset of $\R^N \times \Mem$.  (See Hildenbrand (1974).)  

\medskip

\noindent {\bf Step 3 }  Write ${\bf 1} = (1, \ldots , 1) \in \R^N_+$.  By
assumption, endowments are uniformly bounded; say $e_a \leq W{\bf 1}$ for
each $a \in A$.  Set   $$ C = \{ (\bar{x}, \bar{\mu}) \in \R^N \times \Mem : 
     \bar{x} \leq 0 \, , \, \bar{\mu} \in \cons \}   $$
 $C$ is a  convex cone in $\R^N \times \Mem$.  The core property of $f$ implies
that \mbox{$Z \cap C = \emptyset$} and hence that $Z$ can be separated from
$C$ by a price pair $(p,q)$.  Unfortunately, it might happen that the separating
price has $p = 0$.  (See Example \ref{example:bounded}.)  In order to guarantee
that $p \not= 0$, we separate $Z$ from a ``fatter'' cone.    

Define  $$ C^* = \{ (\bar{x}, \bar{\mu}) \in \R^L \times \Mem : 
 \bar{x} < - \frac{W}{D}\dist( \bar{\mu}, \Cons)\mbox{\bf 1} \}  $$ 
We claim that $Z \cap C^* = \emptyset$.  

To see this, suppose not; we construct a blocking coalition.  Choose 
 $z^* = (x^*, \mu^*) \in Z \cap C^*$.  By definition, there is a measurable
selection    $  a \mapsto (y_a,\nu_a)  $
 from the correspondence $\Psi$ such that    
$$
 z^* = \int_A (y_a,\nu_a) \, d\lambda(a)    $$
 Let $B = \{ a \in A : (y_a,\nu_a) \in \psi(a) \}$;  this is the set of agents for
whom $(y_a,\nu_a)$ is in their net preferred set.  Note that  $\lambda(B) > 0$
and     
$$
 z^* = \int_{B} (y_a,\nu_a) \, d\lambda(a)   $$
so \begin{equation}\label{eqn:x*}
x^* = \int_{B} y_a \, d\lambda(a)  \ ,  \  
  \mu^* = \int_{B} \nu_a \, d\lambda(a)
\end{equation}
We now apply Lemma \ref{lemma:adjust} to choose $B' \subset B$ such that
\begin{equation}\label{eqn:consistency} \int_{B'} \nu_a \, d\lambda(a)  \ \in \
\cons \end{equation} and 
\begin{equation}\label{B'} \lambda(B') \geq
\lambda(B) - 
  \frac{1}{D}
   \dist (\int_{B} \nu_b \, d\lambda(b) \ ,\cons) \end{equation} 


We assert that $B'$ is a blocking coalition.  To see this, note first that, because endowments are bounded above by $W {\bf 1}$, net preferred sets are bounded below by $- W {\bf 1}$.  Hence 
$$
\int_B y_a d\lambda(a) \geq - \lambda(B) W {\bf 1}
$$
Because $z^* = (x^*, \mu^*) \in C^*$, equation (\ref{eqn:x*}) entails that
$$
\int_B y_a d\lambda(a) < 
   - \frac{W}{D} \dist(\int_B \nu_a d\lambda(a) \, , \Cons)
$$
Hence
$$
\dist(\int_B \nu_a d\lambda(a) \, , \Cons) < \lambda(B) D
$$
so $\lambda(B') > 0$.  
Define a state $g$  by  $$ g(a) = \left(y_a + e_a - \tau(\nu_a), \, \nu_a \right)    $$
 To see that the state $g$ is feasible for $B'$ note first that equation
(\ref{eqn:x*}) and the definition of $C^*$ entail that 
\begin{equation}\label{z^*}  
x^* = \int_{B} y_a \, d\lambda
 \leq - \, \frac{W}{D} \dist(\mu^*, \Cons) {\bf 1}  
 = - \, \frac{W}{D} \dist(\int_B \nu_a d\lambda(a) \, , \Cons) {\bf 1}
 \end{equation}  
Additivity of integration entails that  \begin{equation}\label{int}   \int_{B} y_a \,
d\lambda =  \int_{B'} y_a \, d\lambda   +  \int_{B \setminus B'} y_a \, d\lambda 
\end{equation}   Our bound on endowments and the definition of individual excess
demand entails that
 \begin{equation}\label{end}
 \int_{B \setminus B'} y_a \, d\lambda \geq - \, \lambda(B \setminus B') W
\mbox{\bf 1}  \end{equation}  Combining equations (\ref{B'}), (\ref{z^*}),
(\ref{int}) and (\ref{end}) yields  $$
 \int_{B'} y_a \, d\lambda(a) \leq 0  $$ and hence that  $$  \int_{B'} [y_a +
\tau(\ell_a)] \, d\lambda(a) \leq \int_{B'} e_a \, d\lambda  $$  which is the
material balance condition.  Since equation (\ref{eqn:consistency}) entails 
consistency for $B'$,  we conclude that the state 
$g$ is feasible for $B'$.  By construction, $g$ is  preferred to $f = (x, \mu)$ by
every member of $B'$, so this contradicts the assumption that $f$ is a core state. 
We conclude that $Z \cap C^* = \emptyset$, as asserted.

\noindent {\bf Step 4 }  We now use the separation theorem to find prices
$(p,q) \in \R^N_+ \times \Mem$, $(p,q) \not= (0,0)$ such that  
\begin{eqnarray*}  (p,q) \cdot (\bar{x},\bar{\mu}) \leq 0 \quad & & \hbox{ for
each }   (\bar{x},\bar{\mu})  \in C^* \\  (p,q) \cdot z \geq 0 \quad & & \hbox{ for
each } z \in Z   \end{eqnarray*}  Because $C^*$  contains the cone $- \R^N_+
\times \{0\}$, it follows that that $p \geq 0$. Because $C^*$
 contains the subspace $\{0\} \times \Cons$, it follows that $q$ vanishes on
$\Cons$ and hence that $q \in \Trans$.  To see that $p \not= 0$, supppose to the contrary that $p = 0$.  By construction, 
$(p,q^*) \not= (0,0)$ so $q \not=0$.  Hence there is a $\bar{\mu} \in \Mem$ such that $q \cdot \bar{\mu} > 0$.  For $\ep > 0$ sufficiently small, $(-{\bf 1},\ep\bar{\mu}) \in C^*$, so that
$(p,q) \cdot (-{\bf 1},\ep\bar{\mu}) \leq 0$.  However    
\begin{eqnarray*}
 (p,q) \cdot (-{\bf 1},\ep\bar{\mu}) &=& 
     (0,q) \cdot (-{\bf 1},\ep\bar{\mu}) \\
      &=&  \ep q \cdot \bar{\mu} 
\end{eqnarray*}
which, by our choice of $\bar{\mu}$, is positive.  This is a contradiction, so we conclude that $p \not=0$, as desired.


We claim that $(f,p,q)$ is a pure transfer  quasi-equilibrium.  Feasibility of $f$
is guaranteed by assumption; we need to check budget feasibility and
quasi-optimization.  To this end, let $E_1 \subset A$ be the set of agents for
whom $f(a) = (x_a, \mu_a)$ is not in their budget set; that is, $a \in E_1$ if and
only if expenditure strictly exceeds income:  \begin{eqnarray*} {\rm
expenditure}(a) & = & p \cdot [ x_a + \tau(\mu_a)] + 
   q \cdot \mu_a \\ & > &
          p \cdot e_a = {\rm income}(a) \end{eqnarray*}  Write $E_2$ for the 
 set of agents for whom income strictly exceeds 
expenditure.   Measurability of the endowment mapping $e$ implies that $E_1,
E_2$ are measurable sets.  Feasibility of $f$ implies that the integral (over
$A$) of expenditure must equal the integral of income.  Hence, if
$\lambda(E_1) > 0$ it must also be the case that  $\lambda(E_2) > 0$.  Strict
monotonicity of preferences in private goods means that, for each $a \in A$
and each $\ep > 0$,  the choice vector   $(x_a + \ep \bar{e}, \mu_a)$ is strictly
preferred to $f(a) = (x_a, \mu_a)$.  Hence if $a \in E_2$ then there is an $\ep_a
> 0$ such that $(x_a + \ep_a \bar{e}, \mu_a)$ costs strictly less than $e_a$ and
is strictly preferred to $f(a)$; we may choose $\ep_a$ to be a measurable
function of $a$.  Define $g : A \rightarrow \R^N_+ \times \Mem$ by  $$ g(a)=
\left\{\begin{array}{ll}
                       (x_a + \ep_a \bar{e} - e_a, \mu_a) & \hbox{ if } a \in E_2 \cr
                                                                           (0,0)  &\hbox{ otherwise} 
 \end{array} \right. $$  By construction, $g$ is a measurable selection from the
correspondence $\Psi$, so $\int_A g(a) \, d\lambda(a) \in Z$.  However, our
construction guarantees that  $$ (p,q) \cdot \int_A g(a) \, d\lambda(a) = 
  \int_A (p,q) \cdot g(a) \, d\lambda(a) < 0 $$  which contradicts the fact that
$(p,q)$ separates $Z$ from $C^*$.  We conclude that $\lambda(E_1) = 0$; that
is, almost all agents are choosing in their budget set.

To check the quasi-optimization conditions, write $E_3$ for the set of agents
who are not quasi-optimizing in their budget set; suppose that $\lambda(E_3)
> 0$.  Note that  $a \in E_3$ if and only if there is a choice vector $(y_a, \nu_a)$
which is strictly preferred to $(x_a, \mu_a)$ and costs strictly less than $a$'s
endowment; we may choose these choice vectors so that the mapping
 $a \mapsto (y_a, \nu_a)$ is measurable.  Define $h:A \rightarrow \R^N_+
\times \Mem$ by  $$ h(a) = \left\{ \begin{array}{ll}
                          (y_a - e_a, \nu_a) & \hbox{ if } a \in E_3 \cr
                                           (0,0)       & \hbox{ otherwise }  \end{array} \right.
 $$  By construction, $h$ is a measurable selection from $\Psi$ so $\int_A h(a) \,
d\lambda(a) \in Z$.  However, our construction guarantees that $$ (p,q) \cdot
\int_A g(a) \, d\lambda(a) = 
  \int_A (p,q) \cdot g(a) \, d\lambda(a) < 0 $$  which contradicts the fact that
$(p,q)$ separates $Z$ from $C^*$.  We conclude that $\lambda(E_3) = 0$; that
is, almost all agents are quasi-optimizing in their budget set.  
  
It follows that $(f,p,q)$ is a pure transfer quasi-equilibrium.  Setting 
 $$ q^*_m = q_m + \frac{1}{|\pi|} \, p \cdot \inp(\pi,\act) $$  for each $m \in
\mem$ yields a quasi-equilibrium $f,p,q^*$.  If $f$ is club linked, it follows  from
Proposition \ref{prop:qe} that $(f, p, q^*)$ is an equilibrium.  

Finally, if $\cal E$ is club irreducible, then every feasible state is club linked and
hence every core state can be supported as an equilibrium.  By Theorem
\ref{thm:1stwelfare}, every equilibrium state belongs to the core.  Hence the
core coincides with the set of equilibrium states.  \qed

\bigskip

We now turn to the task of establishing existence of equilibrium.
We begin by finding upper and lower bounds  for list prices.

Write $\lists_M = \{ \ell \in \lists : |\ell| \leq M \}$.  By analogy with
a notion from cooperative game theory, we say that a set $L \subset
\Lists_M$ is {\em strictly balanced\/} if there are strictly positive real
numbers $\{\epsilon_L(\ell) : \ell \in L \}$ (which we call {\em balancing
weights\/}) such that     
$$  
\sum_{\ell \in L} \epsilon_L(\ell) \ell \ \in \ \Cons   
$$

\begin{lemma}\label{lemma:lowerbound} There is a constant $R^*$ with
the following property:  
\begin{quote}
If $L \subset \Lists_M$ is a strictly balanced
collection and $q \in \Trans$ is a pure transfer then   $$  \max_{\ell \in
L} q \cdot \ell  \ \geq 
    \ -R^* \, \min_{\ell \in L} q \cdot \ell  $$ 
\end{quote}
 \end{lemma} 

 \begin{proof} For each strictly balanced collection $L$,
choose strictly positive balancing weights $\{\epsilon_L(\ell) : \ell \in L \}$
 and set  
$$
R(L) = 
\frac{\max \{\epsilon_L(\ell) : \ell \in L \} }{\min \{\epsilon_L(\ell) : \ell \in L\}
} $$  
Define  
$$ 
R^* \, =  \, \max_L \Bigl(R(L)(|L| -1)\Bigr) 
$$  
where the maximum
extends over the finite set of strictly balanced collections $L$.

To see that $R^*$ has the desired property, let $q \in\Trans$ be a pure
transfer  and observe that \begin{equation}\label{eqn:qdot} \sum_{\ell
\in L} \epsilon_L(\ell) (q \cdot \ell) = q \cdot \sum_{\ell \in L}
    \epsilon_L(\ell) \ell  = 0  \end{equation}  Choose a list $\ell^* \in L$
such that   $$q \cdot \ell^* = \max_{\ell \in L} q \cdot \ell$$
Rearranging terms in equation (\ref{eqn:qdot}) and carefully keeping 
track of signs, we find:    \begin{eqnarray*}   q \cdot \ell^* &=&  - \,
\frac{1}{\epsilon_L(\ell^*)}\sum_{\ell \in L, \ell \not= \ell^*}
\epsilon_L(\ell) q \cdot \ell \\  &=&  - \sum_{\ell \in L, \ell \not= \ell^*} 
\frac{\epsilon_L(\ell)}{\epsilon_L(\ell^*)} q \cdot \ell \\  &\geq&  - \, (|L| -
1)  \min_{\ell \in L} \Bigl( R(L)(q \cdot \ell) \Bigr) \\  &\geq&  - \,
\Bigl(\min_{\ell \in L} q \cdot \ell \Bigr) R^*   \end{eqnarray*}  which is
the desired inequality.  \end{proof}

With this lemma in hand we establish the existence of equilibrium.

\noindent {\bf Proof of Theorem \ref{thm:existence} }   By assumption,
aggregate endowment $\bar{e}$ is strictly positive and individual
endowments are uniformly bounded above; say that  $\bar{e} \geq w{\bf
1} >> 0$ and that $e_a \leq W_0{\bf 1}$ for all $a \in A$.  Write  $W = \max \{
W_0,1\}$.  We  assume without loss that $\lambda(A) = 1$.

\noindent {\bf Step 1 }  Fix an integer $k > 0$.  Choose a family $\{A^k_\omega
: \omega \in \Omega \}$ of pairwise disjoint intervals in $\R$, each of length
$1/k$.  Write  $$ A^* = \bigcup_{\omega \in \Omega} A^k_\omega $$

We define the agent space $(A^k, {\cal F}^k, \lambda^k)$ for the perturbed
economy    ${\cal E}^k$ by setting $ A^k = A \cup A^*$,  defining ${\cal F}^k$ to
be the $\sigma$-algebra generated by $\cal F$  and the Lebesgue measurable
subsets of $A^*$, and defining $\lambda^k$ to  be $\lambda$ on $A$ and
Lebesgue measure on $A^*$.   Note that $\lambda^k(A^k) =  1 +
\frac{|\Omega|}{k}$.  External characteristics, consumption sets, endowments 
and utility functions of agents in $A$ are just as in the original club  economy
$\cal E$.  For agents $a \in A^k_\omega$, we define: 
$$ \begin{array}{rcl}
\omega_a &=& \omega \\ X_a &=& \R^N_+ \times \{\ell \in \Lists_M : 
    \ell(\omega',\pi,\act) = 0 \hbox{ if } \omega' \not= \omega \} \\ e_a &=&
W{\bf 1}   \\ u_a(x,\ell) &=& |x| \end{array} $$ 

\noindent {\bf Step 2 } The demand functions of these added agents is such
that, for commodity prices near the boundary of the simplex and for
membership prices that are large in absolute value, their commodity excess
demand will be very large.  This will lead to aggregate excess demands that
are impossibly large.   As a consequence, we can write down compact price
sets that contain an equilibrium price for ${\cal E}^k$.  

To define these sets, set
$$ M^* = \max \{ |\pi| : (\pi,\act) \in \Clubs \} $$ 
Choose a real number $\ep > 0$ so small that      
$$
\left[1 - (N-1)\ep \right] 
    \left[ \frac{W}{kN\ep} - W(1 + \frac{|\Omega|}{k}) \right]  
   - \ep (N-1)W(1 + \frac{|\Omega|}{k}) > 0
 $$ 
 Having chosen $\ep$, choose a real number $R > 0$ so big that  
$$ \left[
\frac{R}{kN M^*} - W(1 + \frac{|\Omega|}{k}) \right] \left[1 - (N-1)\ep \right] 
    - \ep (N-1) W(1 + \frac{|\Omega|}{k}) > 0 
$$
  Of course $\ep, R$ depend on $k$.
 Define a  price simplex for private goods and a bounded price set for club
memberships:
 \begin{eqnarray*}  \Delta_\ep &=& \{ p \in \R^N_+ : p_n \geq \ep \hbox{ for
each }  \ n \}  \\ Q_R &=& \{q \in \Trans :  |q_m| \leq R \hbox{ for all } \ m \in
\mem \}  \end{eqnarray*}

\noindent {\bf Step 3 } We define an excess demand correspondence. As in
the proof of Theorem \ref{thm:equivalence}, define $$
 \tau(\ell) = \sum_{(\omega, \pi, \gamma) \in \mem}
                          \ell(\omega, \pi, \gamma)\frac{1}{|\pi|} \inp(\pi,\act)  $$ Let $p
\in \Delta_\ep, q \in Q_R$.  For each agent $a \in A$, write   $$ B(a,p,q) = \{
(x,\ell) \in X_a : 
                     p \cdot x + q \cdot \ell  + p \cdot \tau(\ell)
                         \leq p \cdot e_a \} $$   As in the proof of Theorem
\ref{thm:equivalence}, this is agent $a$'s budget set, assuming that he is
required to pay his share of the inputs to club activities.   Let  
$$  
d(a,p,q) = \argmax  \{ u_a(x,\ell) : (x,\ell) \in  B(a,p,q) \}
$$
 be the set of
utility optimal choices in agent $a$'s budget set; that is, $d(a,p,q)$ is agent
$a$'s demand set.  Define agent $a$'s excess demand set to be
 $$  \zeta(a,p,q) = d(a,p,q) - (e_a, 0)  $$   It is easily checked that excess
demand sets are uniformly bounded (because endowments are bounded,
private good prices are bounded away from 0 and club membership prices
are bounded above and below).  Moreover the correspondence  $  (a,p,q)
\rightarrow \zeta(a,p,q)  $   is measurable and, for each fixed $a$,  is upper
hemi-continuous (in $p,q$).   Define the aggregate excess demand
correspondence    $$  Z : \Delta_\ep \times Q_R \rightarrow \R^N_+ \times
\Mem  $$
 to be the integral of the individual excess demand correspondences:   $$  
Z(p,q) = \int_A \zeta(a,p,q) \, d\lambda(a) 
 $$  As the integral of an upper hemi-continuous correspondence with
respect to a non-atomic measure, $Z$ is upper hemi-continuous, with
compact, convex, non-empty values.

\noindent {\bf Step 4 } We find a fixed point of the excess demand
correspondence,  in a slightly roundabout way.  Note first that individual
income comes from selling private good endowments and receiving
subsidies for club memberships; because private good endowments are
bounded by $W$, private good prices are bounded below by $\ep$ and sum to
1, and club membership prices lie in the interval $[-R, + R]$, this means that
individual demand for private goods is bounded above by $\frac{1}{\ep} (W  +
RM)$.  Hence individual (and aggregate) excess demands for private goods lie
in the compact set   $$ X  = \{x \in \R^N : - W \leq x_n \leq \frac{1}{\ep} (W  +
RM) \hbox{ for each } n \}   $$  By assumption, agents can choose at most $M$
memberships, so individual and aggregate demands for club memberships lie
in the set   
$$ C = \{ \bar{\mu} \in \Mem_+ : 
    \sum_{m \in \mem} \bar{\mu}(m) \leq M \} $$
 
Define a correspondence   
 $$ 
 \Phi : \Delta_\ep \times Q_{R} \times X \times C \rightarrow 
    \Delta_\ep \times Q_{R} \times X \times C  $$    by  
 $$  \Phi(p,q,x,\bar{\mu}) = 
   \Bigl[\argmax\{(p^*,q^*) \cdot (x,\bar{\mu}):(p^*,q^*) \in \Delta_\ep
\times Q_{R} \} \Bigr] \times Z(p,q) $$   It is easily checked that $\Phi$ is
upper hemi-continuous with compact convex values.  Hence Kakutani's fixed
point theorem guarantees that $\Phi$ has a fixed point.  Thus there is a price
pair  $(p^k, q^k) \in \Delta_\ep \times Q_{R}$ and a consumption/club
membership pair  $(z^k, \bar{\mu}^k) \in Z(p^k, q^k)$ such that  
 $$  (p^k, q^k) \cdot (z^k, \bar{\mu}^k) =  \max \Bigl\{ (p^*,q^*) \cdot
(z,\bar{\mu}) \, : \, (p^*,q^*) \in \Delta_\ep \times Q_{R},
    (z,\bar{\mu}) \in Z(p^k, q^k) \Bigr\}  $$

Walras's law implies that  $$ (p^k,q^k) \cdot (z^k, \bar{\mu}^k) = 0 $$

\bigskip

\noindent {\bf Step 5 }  We show that $z^k = 0$ and $\bar{\mu}^k \in \cons$. 
The argument is in several parts.

\noindent {\bf Step 5.1 } We show first that $ q^k \cdot \bar{\mu}^k = 0 $.  Suppose that this is not so.  We
 obtain a contradiction by looking at excess demands (at prices $p^k, q^k$)
of agents in $A^k \setminus A$.  Maximality and the definition of $\Phi$ entail
that  $q^k \cdot \bar{\mu}^k > 0$ (because $0 \cdot \bar{\mu}^k = 0$). 
Maximality entails that $q^k \in \bdy Q_R$ so that $|q^k_{m}| = R$ for some 
$m \in \mem$.  The  budget balance condition for clubs means that if some
price has large magnitude and is positive then some other price must have
large magnitude and be negative.  Thus there is a membership $m^*$ such
that $q^k_{m^*} \leq - R/M^*$.  The agents in $A^k_{\omega^*}$ (whom we
have adjoined to the original set of agents, and whose external
characteristic is $\omega^*$), could obtain a subsidy of $R/M^*$ by choosing
the membership $m^*$ (and no other).  Because  agents in $A^k_{\omega^*}$
don't care at all about club memberships and find all private goods to be
perfect substitutes, they will choose to consume only the least expensive
private good and to choose all club memberships whose prices are negative
and no club memberships whose prices are positive.  It follows that their
excess demand for the least expensive private good --- which we may as
well suppose is good 1 --- is at least    
$$  
\zeta_1(b,p^k,q^k) \geq  \frac{R}{N  M^*}  
 $$   
Keeping in mind that $\lambda(A^k_{\omega^*}) =1/k$ and that the
excess demand of each agent is bounded below by $-W {\bf 1}$, it follows
that the aggregate excess demand for good 1 and for other private goods
satisfy:  
\begin{eqnarray*}
 z^k_1 & \geq & \frac{1}{k}\frac{R}{N M^*}  - W(1 + \frac{|\Omega|}{k}) \\ 
{}\\
z^k_n & \geq & - W(1 + \frac{|\Omega|}{k}) 
\end{eqnarray*}  
Define $p \in \Delta_\ep $  by: 
$$ p_n =
\left\{ \begin{array}{lrl}
                      1 - (N-1)\ep &\hbox{ if }& n = 1 \\
                       \ep& \hbox{if }& n > 1               
                      \end{array} \right. 
$$  
Calculation shows that  $$ p \cdot z^k \geq 
     \left[1 - (N-1)\ep \right]
          \left[ \frac{R}{kN M^*} - W(1 + \frac{|\Omega|}{k}) \right]         
    - \ep (N-1) W(1 + \frac{|\Omega|}{k}) 
 $$  
 Our choices of $R, \ep$ guarantee that this is strictly positive, so that 
$$
(p,0) \cdot (z^k, \bar{\mu}^k) > 0 = (p^k, q^k)  \cdot (z^k, \bar{\mu}^k) $$
which contradicts maximality.  We conclude that  $q^k \cdot \bar{\mu}^k = 0$,
as desired.

\noindent {\bf Step 5.2  }   We show next that $\bar{\mu}^k \in {\Cons}$.  If
not,we could find a pure transfer $q^* \in \Trans$ such that $q^* \cdot
\bar{\mu}^k > 0$ and hence could find a $q^{**} \in Q_R$ such that  $q^{**}
\cdot \bar{\mu}^k > 0$, contradicting maximality.
 
\noindent {\bf Step 5.3 }  We claim that $p^k_n > \ep$ for each $n$.   Suppose
not; we once again  obtain a contradiction by considering the excess demand
of agents in $A^* = A^k \setminus A$.   Every agent in  $A^*$ finds all
commodities to be perfect substitutes, and therefore demands only the
least expensive commodities.  Because  agents in $A^*$ have  endowment
$W{\bf 1}$ and hence wealth $W$,   there is at least one commodity, say
commodity 1, for which the excess demand of each agent in $A^*$ is at least  
$$ 
\zeta_1(a,p^k,q^k) \geq \frac{W}{N \ep} 
$$
 Integrating over all agents and
keeping in mind that individual excess demands
are bounded below by $-W{\bf 1}$ and  that $\lambda^k(A^K_\omega) = 1/k$,
we conclude that  
\begin{eqnarray*} 
z^k_1 & \geq & \frac{1}{k} \frac{W}{N \ep}  - W(1 + \frac{|\Omega|}{k})  \\  
{}\\
z^k_n & \geq & - W(1 + \frac{|\Omega|}{k})
 \end{eqnarray*} 
 Define $p \in
\Delta_\ep$ by  $$ p_n = \left\{ \begin{array}{lrl}
                      1 - (N-1)\ep &\hbox{ if }& n = 1 \\
                       \ep& \hbox{if }& n > 1               
                      \end{array} \right. $$  Calculation gives  
$$ p \cdot z^k \geq  
 \left[1 - (N-1)\ep \right] 
    \left[ \frac{W}{kN\ep} - W(1 + \frac{|\Omega|}{k}) \right]  
   - \ep (N-1)W(1 + \frac{|\Omega|}{k})
 $$  
Our choice of $\ep$ guarantees that this is strictly positive and hence that 
 $$
(p,0) \cdot (z^k, \bar{\mu}^k) > 0 = (p^k,q^k) \cdot (z^k, \bar{\mu}^k) 
$$ 
which again contradicts maximality.We conclude that $p^k_n > \ep$ for each
$n$.
 



\noindent {\bf Step 5.4 }  We show that $z^k = 0$.  If $z^k \not= 0$ there are
indices $i,j$ such that $z^k_i < 0$ and $z^k_j > 0$.  Since $(p^k, q^k) \cdot (z^k,
\bar{\mu}^k) = 0$ and  $q^k \cdot  \bar{\mu}^k = 0$ it follows that $p^k \cdot
z^k= 0$.  Since $p^k_i > \ep$, we can construct a price $\hat{p} \in \Delta_\ep$
by setting  $$ \hat{p}_n = \left\{  \begin{array}{ll}
           p^k_i - \frac{1}{2} (p^k_i - \ep) &\hbox{ if } n = i \\ 
           p^k_j + \frac{1}{2}(p^k_i - \ep) &  \hbox{ if }  n = j \\
              p^k_n   & \hbox{ otherwise }
               \end{array}  \right. $$ Since $p^k \cdot z^k = 0$, it follows that
$\hat{p} \cdot z^k > 0$, a contradiction.  We conclude that $z^k = 0$.  

\noindent {\bf Step 6 }  By definition, there is a selection $g(a) =
(y_a,\mu_a)$ from the individual excess demand sets which integrates to
$(z^k, \bar{\mu}^k)$.  Set  
$$ 
y^*_a = y_a + e_a - \tau(\mu_a) 
$$ 
Setting $f^k =
(y^*, \mu)$ yields  a state of the economy ${\cal E}^k$.  Since we have just
shown that commodity excess demand $z^k = 0$ and that $\bar{\mu}^k \in
\Cons$, we conclude that   $(f^k, p^k, q^k)$ constitutes a pure transfer
quasi-equilibrium for ${\cal E}^k$.  Since  ${\cal E}^k$ is club irreducible, $(f^k,
p^k, q^k)$ in fact constitutes a pure transfer equilibrium for ${\cal E}^k $.


\noindent {\bf Step 7 }  Our price normalization entails that private good
prices $p^k$ are bounded by 1; our construction entails that club membership
prices $q^k$ are bounded by $R$, but $R$ depends on $k$.   We now replace the
sequence of membership prices $q^k$ by
 membership prices $\bar{q}^k$ which lead to the same demands and are
bounded independently of $k$.  



Passing to a subsequence if necessary, we may assume that for each $\ell \in
\Lists_M$ the sequence $(q^k \cdot \ell)$ converges to a limit $G_\ell$, which
may be finite or infinite.  Define:   
$$ \begin{array}{lcl}
 L & = & \{ \ell \in \Lists_M : q^k \cdot \ell \rightarrow G_\ell \in \R \} \\ L_+
&=&   \{ \ell \in \Lists_M : q^k \cdot \ell \rightarrow + \, \infty \} \\ L_- &=&  
\{ \ell \in \Lists_M : q^k \cdot \ell \rightarrow - \, \infty \}   \end{array}  $$ 
Choose  $\bar{G} \in \R$ so large that $|q^k \cdot \ell| \leq\bar{G}$ for each $k$,
each $\ell \in L$.


Define a linear transformation $T : \Trans \rightarrow \R^L$ by $T(q)_\ell = q
\cdot \ell$.  Write $\ran T = T(\Trans) \subset \R^L$ for the range of $T$ and 
$\ker T = T^{-1}(0) \subset \Trans$ for the kernel (null space) of $T$.  The
fundamental theorem of linear algebra implies that we can choose a subspace $H
\subset \Trans$ so that $H \cap \ker T = \{0\}$ and $H + \ker T = \Trans$.  Write
$\TH$ for the restriction of $T$ to $H$.  Note that $\TH : H \rightarrow \ran T$ is a
one-to-one and onto linear transformation, so it has an inverse $S : \ran T
\rightarrow H$.  Because $S$ is a linear transformation, it is continuous, so there
is a constant $K$ such that $|S(x)| \leq K |x|$ for each $x \in \ran T$.  

Let $R^*$ be the constant constructed in Lemma \ref{lemma:lowerbound}.
Choose $k_0$ so large that $k\ge k_0$ implies
\begin{eqnarray*}
q^{k} \cdot \ell \ & > & \ +2K\bar{G} + W  \ \qquad \hbox{ if } \ell \in L_+
\cr
q^{k} \cdot \ell \ & < & \ -2K\bar{G} - R^* W  \quad \hbox{ if } \ell \in L_- 
\end{eqnarray*}
 
Write $ST$ for the composition of $S$ with $T$.  For each $k \geq k_0$ set 
$$\bar{q}^k = ST(q^k) - ST(q^{k_0}) + q^{k_0} \in  \Trans$$
Because $S, \TH$ are inverses the composition $TS$ is
the identity, so  
$$T(\bar{q}^k) = TST(q^k) - 
     TST(q^{k_0}) + T(q^{k_0}) = T(q^k)$$ 

We claim that for $k > k_0$, the triple $(f^k, p^k, \bar{q}^k)$ constitutes a pure
transfer equilibrium.  To see this, we first consider the prices of lists.  For $\ell
\in L$, $\bar{q}^k \cdot \ell = q^k \cdot \ell$ because $T(\bar{q}^k) = T(q^k)$. 
For $\ell \in L_+$, $\bar{q}^k \cdot \ell > W$ because $|ST(q^k)| \leq K\bar{G}$,
 $|ST(q^{k_0})| \leq K\bar{G}$ and $q^{k_0} \cdot \ell > W + 2K\bar{G}$.  For 
$\ell \in L_+$, $\bar{q}^k \cdot \ell < - R^*W$ because $|ST(q^k)| \leq K\bar{G}$,
 $|ST(q^{k_0})| \leq K\bar{G}$ and $q^{k_0} \cdot \ell < - R^*W - 2K\bar{G}$.

To check the equilibrium conditions, keep in mind that individual demands
for private goods and club memberships depend only on the prices of private
goods and of lists, not directly on the prices of memberships.  Because
endowments are bounded above by $W$ and private goods prices sum to 1, 
individual wealth is also bounded above by $W$.  Hence no list whose price
exceeds $W$ is ever demanded; in particular, no list in $L_+$ is demanded at
prices $p^k, q^k$ or at prices $p^k, \bar{q}^k$.  Moreover, because the 
set of lists demanded at
an equilibrium is strictly balanced, it follows from Lemma
\ref{lemma:lowerbound} that no list in $L_-$ is demanded at prices $p^k, q^k$.  By
construction,  prices for lists in $L_-$ are higher with respect to $\bar{q}^k$ than
with respect to $q^k$, so no lists in $L_-$ are demanded at prices $p^k,
\bar{q}^k$:  if no one is willing to buy a list when a large subsidy is provided, no
one will be willing to buy it when the subsidy is reduced.  Since prices for lists in
$L$ are the same with respect to $\bar{q}^k$ as with respect to $q^k$, it follows
that demands are the same with respect to $p^k, \bar{q}^k$ as they are with
respect to $p^k, q^k$.  (In words:  When we replace membership prices $q^k$ with 
membership prices $\bar{q}^k$ we lower the prices of some unaffordable lists, but we keep them so high that they remain unaffordable.  We also lower the
subsidies of some lists, but lists that are not demanded when subsidies are
large will not be demanded when subsidies are smaller.  Hence we do not change
demands.)  It follows that $(f^k, p^k, \bar{q}^k)$ is a pure transfer equilibrium
for ${\cal E}^k$.  By construction, $|\bar{q}^k \cdot \ell| \leq 2K\bar{G} + |q^{k_0}
\cdot \ell|$ for $k, \ell$;  because  singleton memberships are themselves lists, it
follows that $(\bar{q}^k)$ is a bounded sequence in $\Trans$.

\noindent {\bf Step 8 }  In view of this construction, we have bounded
sequences $(p^k)$ of private goods prices, $(\bar{q}^k)$ of membership
prices and $(\mu^k)$ of aggregate membership choices.  Passing to a
subsequence if necessary, we may assume that $p^k \rightarrow p^*$,
$\bar{q}^k \rightarrow q^*$, $\bar{\mu}^k \rightarrow \bar{\mu}^*$.  We may
now employ Schmeidler's version of Fatou's lemma (see Hildenbrand (1974))
to conclude that there is is a measurable mapping   $f^* : A \rightarrow
\R^N_+ \times \Mem$  such that   \begin{itemize}  \item for almost all $a \in
A$, $f^*(a) \in B(a,p^*,q^*) \subset X_a$   \item for almost  all $a \in A$,
$f^*(a)$ belongs to agent $a$'s quasi-demand set;  that is, there does not
exist 
   $(x', \ell') \in X_a$ such that
   $u_a(x', \ell') > u_a(f^*(a))$ and  
   $(p^*,q^*) \cdot (x', \ell') + p^* \cdot \tau(\ell') < p^* \cdot e_a$ 
  \item $\int_A f^*(a) \, d\lambda = (\bar{e}, \bar{\mu}^*)$ \end{itemize}    By
definition, $(f^*, p^*, q^*)$ is a pure transfer quasi-equilibrium for $\cal E$. 
Club irreducibility implies that $(f^*, p^*, q^*)$ constitute a pure transfer
equilibrium for $\cal E$, so the proof is complete.  \qed

\vfill
\newpage

%%%%%%%%%%%%%%%%%%%

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\end{document}