OPTIMAL AND EQUILIBRIUM GROUPS
Suzanne Scotchmer*
Myrna Holtz Wooders**
Discussion Paper 1251
July 1, 1986
Abstract
We discuss economies with complementarities and/or crowding in
production or consumption. We de…ne a price system with the property that core states of the economy are the same as equilibrium states,
provided the core has equal treatment. Prices for agents (wages in the
production examples, or club admission fees in the club examples) internalize intra-group “externalities” generated by agents. These prices
depend on relative scarcities of types of agents and can be positive or
negative.
We thank members of the L.S.E. Theory Workshop, Bob Anderson,
Karl Dunz and David Pines for helpful discussion. The National Science
Foundation of the USA, Grant 8420289, and the Social Science Research
Council of Canada supported this research.
*Harvard University
**University of Toronto
Harvard Institute of Economic Research
Littauer Center
Cambridge, Massachusetts 02138
1. Introduction
Potential gains from cooperation and potential con‡ict over how
these gains will be shared are pervasive in economics. Aumann and
Shapley (1974, p. 3) write,
Interaction between people - as in economic or political activity - usually involves a subtle mixture of competition and
cooperation. Thus bargaining for a purchase is cooperative,
in that both sides want to consummate the transaction, but
also competitive, in that each side wants terms that are
more favorable to itself, and so less favorable to the other
side. People cooperate to organize corporations, then compete with other corporations for business and with each other
1
for positions of power within the corporation. Political parties compete for the voter’s favor, but cooperate in forming
ruling coalitions and in “log-rolling”. Often it is impossible
to draw a clear border between “cooperation” and “competition”.
An ongoing line of research, beginning with Edgeworth (1881), investigates conditions under which cooperation (the core) and competition
lead to the same outcomes. The intuitive reason that cooperative and
competitive outcomes are the same is that both involve “no surplus”
conditions; if e¢ciency gains are available, cooperative agents will agree
to achieve them, while competitive agents will turn e¢ciency gains to
pro…t. Although Aumann and Shapley and most other investigators have
studied private goods economies without externalities, there is an emerging literature on economies with externalities and endogenous formation
of groups of agents that applies much more broadly.1
Bene…ts to forming groups accrue because agents have di¤erent endowments to trade, because production may require cooperation between
di¤erent kinds of labor, or because the group can pool its private resources to produce shared public goods. This paper discusses economies
in which bene…ts to group formation arise from jointness in production and/or consumption. Our models relate to club theory and, in the
language of club theory, we admit nonanonymous (or discriminatory)
complementarities and crowding. This means that utility and/or productivity depends on the numbers of agents of each type in a group
rather than simply on the total number of agents in the group (except
as a special case). For simplicity we say that there is nonanonymous
crowding.
There seems to be no natural application of the Arrow-Debreu price
system to cover intragroup externalities of the type discussed here. In
our view, a “natural” price system requires admission prices to groups,
which may be interpreted as (positive or negative) prices for agents
themselves. That such a price system di¤ers from the Arrow-Debreu
price system is evident in that every core state of the economy is an
equilibrium.2
This paper is illustrative and expository, and we therefore give the
analysis with respect to several illuminating examples meant to indicate
the potential breadth of our approach. We discuss production, where
a coalition is a …rm using workers whose productivity depends on the
pro…le (the number of agents of each type) of labor within the …rm.3 We
also discuss club examples, with preferences de…ned on consumption of
2
private good and of public good provided by the club, and on the pro…le
of agents in the club.
The development of our argument is as follows. First, we show the
equivalence between the core and competitive equilibrium4 outcomes
when groups of agents (organizations) consistent with the core are small
enough relative to the economy so that no group contains all agents of
any type. In our context, this equivalence turns out to be a particularly
easy and straightforward way to characterize competitive equilibrium.
Second, in coalition economies with nonanonymous complementarities or crowding, one cannot de…ne “the optimal group” without reference to the relative abundances of agents of di¤erent types.5 This is true
even when there is no “integer problem” and no income e¤ects.6 There
is no analogous phenomenon in the case of a homogeneous population
since (when there is no “integer problem”) the optimal sharing group,
which maximizes per capita utility, is well de…ned and independent of
population size.
Third, one expects competitive prices to be determined by relative
abundances of primary resources. When there are complementarities in
production or nonanonymous crowding costs in consumption, abundance
of primary resources means abundance of agents of di¤erent types. The
relative abundances of agents should determine how relatively well o¤
agents are in equilibrium and this is exactly what we discover. For …xed
preferences, the utility achieved in the core by each type of agent depends
on the abundance of that type relative to other types (and it is best to
be scarce!). Since utilities in the core become wages in competitive
equilibrium7 , the wage of each type depends on relative abundances.8
Fourth, the competitive prices can be “disaggregated” in several
ways, as appropriate to the particular example. In our production model,
core utilities are wages and there is no further disaggregation. But in the
club model, we can disaggregate the competitive price in several (equivalent) ways. First, an analogy to the production coalition is that the …rm
“buys” an agent for a wage and appropriates the agent’s endowment.
In return, the …rm provides the agent’s consumption bundle: private
good, club good, and crowding. A second more familiar price system
(to club theorists) reinterprets these prices as individualized admission
prices. The consumer chooses his preferred club (pro…le of members,
and amount of club good), pays a nonanonymous admission price,9 and
spends the rest of his income on private good. Pro…t-maximizing …rms
choose the pro…les of members which they will permit, and the amount of
club good they will provide, taking the nonanonymous admission prices
as …xed. Finally, we present a third disaggregation in which there are
admission fees and Lindahl prices (times quantities) for a public good.
3
In most of the situations we consider, it is clear that a competitive
equilibrium state is in the core. However, in our …nal example, with
public goods and exclusion and with possible complementarities in production, the proof that an equilibrium state is in the core is not obvious.
Thus for this case a proof is provided.
By the time the reader reaches the end of our examples, we hope that
he/she will be convinced that economies with endogenous groups and
intragroup externalities can be viewed as perfectly competitive, similarly
to private goods exchange economies. Taking as given a complete price
system which includes admission prices to groups, agents optimize, and,
in equilibrium, demands for private goods and admissions to groups
equal supplies. Equilibria are optimal and equivalent to the core.
All of the points exposited in this paper are transparent in economies
with transferable utility, and thus we con…ne attention to such economies.
Transferable utility greatly simpli…es the analysis, while enabling us to
capture concepts which should apply in the more general case. These
economies are easy to analyze because all e¢cient states of the economy10
maximize the unweighted sum of payo¤s to agents, and thus the unweighted sum of payo¤s to groups in the partition. An e¢cient partition
of the population into groups is independent of how utility is divided up
within or across groups.
Section 2 discusses e¢cient allocations in an economy with transferable utility. Section 3 de…nes and characterizes the core. Section 4 shows
that the core and competitive equilibrium states of the economy are the
same. Section 5 further develops the production and club examples.
The purpose of this paper is not to discuss existence of the core
and competitive equilibrium, but rather to develop and exposit the
relationship between them in club economies. We turn brie‡y to existence in Section 6. Section 7 summarizes and concludes our work.
2. What is an Optimal Group
Before introducing our framework, it may be useful to discuss the
concept of an optimal group. In contrast to the situation with anonymous crowding, the concept of an optimal group with nonanonymous
crowding cannot be de…ned without reference to the abundance of agents.
Since we con…ne attention to economies with transferable utility, it is
useful …rst to de…ne such economies as those for which e¢cient states of
the economy maximize the sum of an appropriately de…ned “value” for
groups in the partition.11
Suppose there are three types of agent, a, b and c, and the pro…le of a
coalition t is nt = fnat ; nbt ; nct g, where nit is the number of type-i agents
4
in group t. If nct equals zero, it is a type-ab group, and so on. A partition
of the population is a set of coalitions fnt g such that for i = a; b; c, we
P
have nit = N i , where N i is the total number of type-i available in
6t
the economy. Suppose we can de…ne a value function V [n], specifying
the “total utility” which the coalition n can produce. Transferability is
present if we can distribute this utility among agents on a one-to-one
basis in any way we wish. Section 5 below gives examples of production
and consumption club economies which satisfy this de…nition.
De…nition: An economy has transferable utility if there is a value
function V [¢] such that every e¢cient state of the economy partitions
P
agents into coalitions in a way that maximizes V [nt ]. (The product
of the groups,
P
t
t
V [nt ], can be distributed in any way.)
We begin the discussion with an example.
Example 1: V [¢] is the private good produced by a group. Type-a is
complementary with everyone in production, whereas type-b and type-c
get in each other’s way.12
(1)
(2)
(3)
(4)
(5)
V [na ; nb ; 0] = 5
for na = nb = 1,
0 otherwise
a
c
a
c
V [n ; 0; n ] = 5
for n = n = 1,
0 otherwise
b
c
b
c
V [0; n ; n ] = 0
all (n ; n )
a
b
c
V [n ; n ; n ] = 9
for na = nb = nc = 1,
0 otherwise
a
a
V [n ; 0; 0] = 1
for n = 1,
0 otherwise
(Similarly for b and c.)
Given the mix of types in a group, one of fa; b; c; ab; ac; abcg, the
optimal group has one unit of each type of agent. The following examples show that for di¤erent population sizes, the groups in the e¢cient
partition di¤er. The di¤erence is due to neither the integer problem nor
to income e¤ects, since neither is present. Rather it is due to the fact
that optimal groups depend on the relative abundances of types.
Example 1a: Let N a = 40; N b = N c = 10. Then e¢cient allocations
require ten groups of type-ab, ten groups of type-ac, and twenty groups
of type-a, where a type-ab group has one member of type-a and one
member of type-b, a type-ac group has one member of type-a and one
member of type-c, and a type-a group has one member of type-a.
Example 1b: Let N a = N b = N c = 20. Then e¢cient allocations
require twenty groups of type-abc.
Example 1c: Let N a = 20; N b = 10; N c = 30. Then e¢cient allocations require ten type-abc groups, then type-ac groups and ten type-c
groups.
5
It is important to notice that in Example 1a, type-abc groups (with na
= nb = nc = 1 members) were possible. Similarly, in Example 1b, typeab and type-ac groups were possible, and in Example 1c, type-ab groups
were possible.
Of course, an optimal group cannot be de…ned without knowing preferences and technology, summarized by the value function V [¢]. This example has shown that it also cannot be de…ned without knowing relative
abundances. In the following section we discuss the core. An optimal
group is a group which appears in the core and therefore achieves core
utilities in the sense of equations (7) and (8) below.
3. The Core
We now show the relationship of the core to e¢cient states of the
economy. Recall that the core of an economy consists of those states of
the economy with the property that no group of agents, using only their
own initial endowments, can do better for themselves.13 In general,
core states of the economy are a small subset of e¢cient states, since
the core is de…ned with respect to …xed initial endowments of goods,
while e¢cient states are de…ned independently of initial endowments.
The point here is that utilities achieved in the core depend on relative
abundances of types of agents, even with the endowment of goods, to
each agent of each type …xed.
In the transferable utility economy we have described, an e¢cient
partition of the population (which depends on the relative abundances
of types), is independent of the distribution of utility and thus one would
expect the same e¢cient partition to characterize the core as well.
As a reminder, we include Figure 1, depicting a utility envelope for a
homogeneous population in a standard club economy. Suppose there is a
shared facility G, which can be produced at cost C(G; n) and the e¢cient
amount of G for a size-n group is G(n). Each agent has endowment w.
If private good is consumed in amount x, utility can be represented
as U[x; n; G(n)] = x + h[G(n); n].14 Then V (n) = nw ¡ C[G(n); n] +
nh[G(n); n] is the total available utility within a group of size n and
V (n)=n is per capita utility. An e¢cient partition of the population
P
maximizes V (nt ) over all possible partitions fnt g. This is true whether
t
or not groups of size n¤ can be accommodated; that is, whether or not
there is an “integer problem.” If the population size is greater than n¤ ,
the core requires groups of size n¤ and if, in addition, the population is
not an integer multiple of n¤ , then groups of n¤ are not feasible and the
core is empty.
In both a homogeneous population, and in a heterogeneous population, core coalitions will give the same utility to people of the same type,
6
provided the population is large enough that all of type-i is not in one
coalition. The content of the following lemmas is well known, but they
are included here for completeness.
Lemma 1: Suppose that a core state does not have all type-i agents
in one coalition.15 Then all type-i agents achieve the same utility.
Proof:
Suppose not. First, if the high-utility type-i agent is in
a group alone, it is immediate that such utility is also available to the
low-utility agent. Suppose then that the high-utility type-i agent is in
a coalition with other agents. A low-utility person in another coalition
could o¤er to replace the high-utility person, giving epsilon more utility
to the other members and achieving an improvement for himself. The
conclusion of the Lemma follows. Q.E.D.
Assuming that all agents of type-i get the same utility, if utilities
u = [ua; ub ; uc ] are core utilities then
(6)
ua ¸ V [n] ¡ (na ¡ 1)ua ¡ nb ub ¡ nc uc for all n = [na ; nb ; nc ]
with equality for core coalitions.
(Symmetric expressions apply for b and c.)
The right-hand-side is the utility that can be achieved by type-a in a
group [na ; nb ; nc ], provided core utilities [ua; ub ; uc ] must be o¤ered to the
other members in order to induce them to join the group. Expression
(6) says that if [ua ; ub ; uc ] are core utilities, type-a cannot achieve more
utility than ua in any group. Symmetric expressions must be satis…ed
for types b and c.
If (6) holds, and if in addition there is a partition of the population
t
fn g which allows utilities u to be achieved (that is, u is feasible), then
we have a core state of the economy:
Lemma 2: Utilities [ua ; ub ; uc ] and the partition fnt g of agents into
core coalitions describe a core state of the economy if and only if (7) and
(8) hold. 16
(7)
u ¢ n ¸ V (n)
for all n17
(8)
u ¢ nt = V (nt )
for nt in the core partition
Equation (7) is a restatement of (6) and (8) says that utilities u are
achieved in a feasible partition fnt g.
A slightly di¤erent interpretation of this characterization of the core
shows the link between core coalitions with nonanonymous crowding,
and the familiar optimal group with homogeneous population. For a
coalition to be in the core, it must be preferred by types a, b and c
simultaneously in the sense that no agent of any type could suggest
another coalition which gives at least as much utility to the others and
more to himself. Let the core utilities be [ua ; ub ; uc ]. In any group
7
suggested by an agent of type-a, he must give to types b and c their
core utilities. Otherwise, he could not get them to join his group. Thus,
the utility envelope for type-a, which is graphed in Figure 2, takes as
parametric the core utilities of b and c.
(9) ua (na ; ub ; uc ) = max
b c
(n ;n )
V [na ;nb ;nc ]¡nb ub ¡nc uc
na
If a coalition of a; b and c is in a core partition, then this coalition
achieves the maximum in Figure 2 and ua is the utility achieved by typea in the core. If the maximum is less than ua , then no such coalitions are
in the core. Type-a also has a utility envelope for coalitions composed
of only a and of a; b. Analogously, type-b has such utility envelopes for
coalitions consisting only of b, of a; b, and of a; b; c. Similarly for type-c.
This shows that the “optimal” group for type-a is a partial equilibrium concept in which type¡a assumes that other members can be
“hired” by paying them their core utilities, as if they were productive
inputs.
Lemma 3: Core states are e¢cient; i.e., a core state of the economy
P
partitions the population in a way which maximizes V [nat; nbt ; nct ]:
t
E¢ciency of the core usually follows immediately from the observation that the coalition of the whole cannot improve the utility of each
member. This implies e¢ciency if V (n) re‡ects the fact that groups can
always subdivide into smaller units.18 Such is not assumed in Example
1, where V [2; 2; 0] = 0 < 10 = 2V [1; 1; 0]. Thus it is not “obvious” that
allocations in the core are e¢cient.
Proof: Let fntg be a core partition.
P
P
Suppose the Lemma were false. That is, suppose V (nt ) < V (~
nj )
t
j
where (~
nj ) partitions the population. Let [ua ; ub ; uc ] be a core utility
P
P
vector. Since u ¢ nt = V (nt ) for each nt , and since u ¢ nt = u ¢ n
~j
then
P
t
u ¢ nt =
P
j
u¢n
~j <
P
j
t
j
V (~
nj ). For some n
~ j , we must have u ¢ n
~j <
V (~
n ), which contradicts (7).
j
Q.E.D.
We return now to the discussion of Examples 1a, 1b, and 1c.
In each case, the endowment of an agent of type-i does not vary for
the three cases, since his only endowment is his labor. Nevertheless, the
utility achieved by type-i in the core varies. In Example 1a, type¡a gets
one unit of utility. This is the utility a type¡a agent achieves in optimal
groups with only type¡a, and by Lemma 1, type¡a agents in all groups
get the same utility. Thus, the type¡ab groups, which give a total of 5
8
units of utility, give 4 units to type-b. The type¡ac groups give 4 units
to type-c.
Reasoning analogously for each example, we see that:
Example 1a: ua = 1: ub = 4: uc = 4:
Example 1b: Any distribution of utilities such the ua + ub + uc =
9; ua + ub ¸ 5; ua + uc ¸ 5; ua ¸ 1; ub ¸ 1; uc ¸ 1, is a core allocation.
Example 1c: ua = 4: ub = 4: uc = 1.
Even though the per capita endowment does not vary in an “obvious”
sense, these examples make it clear that in economies with nonanonymous crowding, an important part of the endowment is the scarcity of
the type relative to complementary types. In Example 1a, type¡a is
abundant, and even though type¡a is complementary with both other
types, type¡a does not get much utility. Compare with Example 1c, in
which type¡a is scarcer relative to the agents with whom type-a could
productively for groups.
4. Equivalence of the Core and Competitive Equilibrium
Competitive equilibrium can be constructed from the core, and thus
every core state of the economy with equal treatment is a competitive
equilibrium. The conceptual framework presented here is as simple as
possible, and we then give examples in Section 5 which illustrate its
broad applicability.
We normalize the price of product, V (n), to be one. The price paid
for an agent entitles the …rm to the agent himself (his labor) and his
endowments of private goods if he has any.
De…nition: A partition (nt ) and a set of prices p = [pa ; pb ; pc ] is a
competitive equilibrium if V [n] ¡p ¢ n · 0 for all n, and V [nt ] ¡ p ¢ nt = 0
for nt in the partition.
Lemma 4: Provided states of the economy in the core give the same
utility to agents of the same type, core states are competitive equilibrium
states and conversely. The competitive prices p are equal to core utilities
u.
Proof: With p = u, the characterization of the core, conditions (7)
and (8) say that the only pro…table …rms are those which correspond
to core coalition. All other production schemes make negative pro…t.
Hence, a core state of the economy is a competitive equilibrium. That
competitive equilibrium utilities (prices) are core utilities follows from
the “if” part of Lemma 2.
Q.E.D.
It follows that competitive equilibria are e¢cient.
9
The focal aspect of competitive equilibrium is that the goods of interest are the agents themselves. The …rm must pay for agents.19
Furthermore, since the utilities which are achieved in the core depend
on relative abundances, it follows that competitive wages depend on
relative abundances. If one type is scarce relative to the types with
whom it is complementary, its equilibrium wage will be high.
This simple competitive equilibrium becomes considerably more complex in the examples pursued below.
With the equilibrium concept de…ned above, every core allocation is
an equilibrium. Since complete equivalence does not occur with ArrowDebreu equilibrium, we comment on how our prices di¤er from ArrowDebreu prices.
Our equilibrium concept has prices for agents themselves. In the
context of an Arrow-Debreu private goods exchange economy, admitting prices for agents in addition to prices for goods has the consequence of enlarging the set of competitive equilibrium states to the entire
core.20 Prices for agents could be interpreted as admission fees to “market places” or “trading groups,” and these fees must be paid before an
agent is allowed to trade goods at Arrow-Debreu prices.
Consider an Edgeworth box with two consumers and two private
goods. As one agent’s admission fee is increased, in equilibrium the
other agent’s fee must decrease (otherwise the admission fees would not
sum to zero) and this e¤ectively ”moves” the initial endowments in the
Edgeworth box. Since admission prices sum to zero, they serve as a
mechanism to redistribute income. For appropriate admission fees, every
core allocation is an equilibrium. The Arrow-Debreu equilibrium has no
mechanism to redistribute income and hence the core is larger than the
set of equilibrium states.
In contrast to Arrow-Debreu private goods exchange economies where
admission fees are ”arti…cial” and converge to zero if the economy is replicated, admission fees (or prices for agents) in our context will generally
not equal zero, even in a large economy.
5. Examples
Example 2: Production
Firms produce a homogeneous output, sold at price one, according to
the production function V [n]. Let wages for workers equal core utilities:
[ua ; ub ; uc ] = [pa ; pb ; pc ], or u = p.
A …rm’s production set is f(x; n) j x · V (n)g.21 Pro…t is V [n] ¡ p ¢ n.
It follows from (7) and (8) that pro…t is nonpositive for all coalitions
n (all points in the production set), and pro…t equals zero for the core
allocation.
10
Example 3: Production with U-Shaped Average Cost
The resource cost of producing output q in a coalition n is C(q; n).
For example, let C(q; n) = K + A(n)q2 , where A(n) is an “e¢ciency”
parameter not equal to zero for any n. C(q; n) excludes payments to
workers. The idea is that the average cost including payments to workers,
[C(q; n)+p¢ n]=q, is U-shaped. Suppose that the only consumption good
is the net amount of q produced. “Net” means that the cost C(q; n) must
be paid out of the product, so that net production in a coalition n is
q ¡ C(q; n). Then
(10) V (n) = max
q ¡ C(q; n)
q
P
t
An e¢cient partition of the population fnt g maximizes the sum
V [nt ], which is the total net product.
Now consider a core partition. If u is a vector of core utilities, a
coalition in the core partition solves
(11) max
V (n) -u ¢ n = max q ¡ C(q; n) ¡ u ¢ n
n
(q;n)
The maximum is zero, and V (n) ¡ u ¢ n = q ¡ C(q; n) ¡ u ¢ n < 0 for
all (q; n).
Consider a competitive equilibrium in which p = u are the wages
which the …rm must pay to workers. For all production plans (q; n),
q ¡ C(q; n)¡ p ¢ n · 0, which means pro…t is nonpositive. For production
plans which correspond to core coalitions, pro…t is zero. Hence the core
is a competitive equilibrium.
Since cost C(q; n) + u ¢ n is minimized by choice of n, we have @C(q; n)/@na = ua. That is, each type-a agent is paid his ”e¢ciency
wage”, which is the reduction in resource cost that his labor entails.
In the example give above, @C(q; n)/@na = q 2@A(n)/@na. The zeropro…t condition (price equals minimum average cost) entails that these
e¢ciency wages exhaust the total product. Otherwise, competitive equilibrium does not exist.22
Example 4: Pure Crowding
We consider entertainment clubs in which all bene…ts accrue from the
interaction of agents. No public goods are produced. Type-a agents are
dancers, type-b agents are spectators and type-c are technical assistants.
Type-a like to dance for spectators, and also like to have props. Type-b
like to watch dancers and …nd technical assistants a nuisance. Type-c
like to set up the props required by dancers, but …nd that spectators get
in the way.
This example is formally identical to Example 2, except that we must
ensure there is a mechanism to transfer utility. Suppose endowments are
fwi g and type-i’s utility is U i [xi ; nt ] = xi +hi [nt ], where xi is the amount
11
of private good consumed by type-i. Such an economy has transferable
utility within the range of utilities for which each agent consumes some
private good. V [n] is the total (transferable) utility produced by group
n.
P
(12) V [n] + ni [wi +hi (n)]:
i
tal social endowment of private good is X =
P
i
N i wi . An e¢cient allo-
cation partitions the population so as to maximize the sum of utilities
P
P P it i t
P P it it
V [nt ] = X +
n h [n ] =
n [x + hi [nt]]:
t
t
t
i
i
We present a competitive equilibrium which shows the similarities of
this example to Example 2. In Example 2, the …rm assumed it could
always get as many workers as it wanted at wages ui and the agent
could then spend his income on private good produced by any …rm. The
…rm from which he purchased could be di¤erent from the …rm at which
he worked. The pure crowding example is slightly more subtle in that,
when a …rm hires agent i, part of agent i’s consumption takes place
inevitably within the …rm. He gets utility hi [nt ] from being in the …rm.
If that utility is too low, the agent will not be willing to work at the
…rm. Therefore, it is not necessarily true that the …rm can get as many
”workers” as it wants by paying wages ui .
We …nesse this problem by “pretending” that agent i’s employment
can be separated from his consumption. This pretense disappears in
equilibrium, when the agent is happy to purchase his consumption from
the …rm where he works. Suppose then that a …rm can get as many typei agents as it wants by paying the price pi = ui . The …rm purchases the
agent’s endowment wi as well as his labor.23 Out of his wages, agent i
can purchase private good x at price one, and can purchase admission in
one …rm. Such admissions and private good are what the …rms produce.
Since …rms sell admissions, we must specify a set of prices for these
admissions. De…ne pi [n] as the price that agent i must pay for admission
in a type-n …rm.24 We claim that equilibrium admission prices are pi [n]
= hi [n]. At these prices, each agent is indi¤erent to purchasing from all
…rms, since the utility he gets from being in a type-n …rm is xi + hi [n] =
ui ¡hi [n] + hi [n] = ui . In particular, he will be happy to purchase from
the …rm that employs him.
Now we must show that at these prices, the core is a competitive
equilibrium. By choice of n the …rm maximizes pro…t, which is equal to
P i
P
n [p ¡ [n] ¡ (ui ¡ w i )] = ni [hi [n] + wi ¡ ui ] = V (n) ¡ u ¢ n. The
i
i
only pro…table …rms are core coalitions.
In the equilibrium we have presented, the …rm pays the agent for his
productive input, and the agent pays the …rm for admission. We did this
12
to show the unity with the production example. The more usual way of
describing prices in a “club economy” is to de…ne a composite price paid
by the agent to the …rm for admission. We now present a second set of
competitive prices as net (composite) prices which the agent pays to the
…rm. Let the net price be ui [n] = wi - ui + hi [n]. At these prices, each
agent is again indi¤erent among all …rms, and gets utility ui by choosing
P
admission in each. Pro…t in a type-n …rm is ni [wi ¡ ui + hi [n]] =
i
V (n) - u ¢ n · 0 for all n, with equality for core coalitions.
P
In any …rm, n, the sum of admission prices ni ui [n] is less than
or equal to zero and equals zero in …rms which appear in equilibrium.
That is, the admission prices sum to zero. In general some agents will
pay negative prices and other agents will pay positive prices.
We might be inclined to think of dancers as being paid and of spectators as paying. Such a view misses the reciprocal nature of crowding.
The product of the group is produced mutually by all members through
their complementarities. Who pays and who gets paid-whether an agent
has a positive or negative price-depends on relative scarcities of types.
The utility achieved in competitive equilibrium, ui , depends on relative scarcities, as pointed out in Example 2. Another expression of
this fact is that net admission prices ui [n] are lower if core utility is
higher. Lower admission prices (which might be negative) are a reward
for scarcity, provided agent i is complementary with other agents in
producing utility.
Example 5: Club Economy
This example is the same as Example 4, except that each coalition can
produce some facility or ”club good”, G. Suppose agents have endowments fwi g and preferences can be represented U i [x; G; ng = x+hi [G; n].
A state of the economy is described by the collection fGt; xit ; nit g, where
i indexes types of agents, and tindexes groups; that is, it is described as
a partition of the population into coalitions, an amount of club good in
each coalition, and a distribution of the remaining private good.
Suppose the resource cost of providing the club facility is C(G; n).
Consumption of private good is …xed by the aggregate endowment, after
the costs of producing club goods have been subtracted away. That is
P
X = C[Gt; nt ] is available for consumption, where X is the aggregate
t
social endowment. Then within the range of e¢cient states of the economy which permit each agent to consume some private good, e¢cient
allocations maximize
P
P P it i t t
(13) X ¡ C[Gt ; nt] +
n h (G ; n )
t
t
i
The …rst two terms are the net amount of private good available,
13
which adds the same amount to total utility irrespective of how it is
divided among agents. De…ne
P
(14) H[n] = max ¡C[G; n] + ni hi [G; n]:
G
i
And let G(n) be the optimizer. E¢cient states of the economy maxP
imize H[nt ] and utility is distributed by distributing consumption of
t
private good. De…ne
P i i
(15) V [n] =
n w + H[n]
i
Then one can see from (13) that e¢cient states of the economy parP
tition the population so as to maximize V [nt ].
t
We now turn to the core. If u is a vector of core utilities, it follows
from (7) and (8) that V [n] ¡ u ¢ n · 0 for all n, with equality for core
coalitions.
There are two available price systems for this economy. The …rst
is the one discussed above for pure crowding, which we shall now call
Composite Admission Prices. The second has the public good produced
within the coalition by pro…t-maximizing competitive …rms and sold at
personalized “per unit” prices. These we shall call Lindahl prices. In
the second case there are, in addition, nonanonymous admission prices,
which may again be negative or positive depending on relative scarcities.
(i) Composite Admission Prices: A club is a [G; n]. That is, an agent
who joins the club gets crowing n, and facility G. The club chooses [G; n]
to maximize pro…t. A competitive equilibrium is a set of admission prices
for clubs of di¤erent types [G; n], and a set of clubs, such that no club
could enter with positive pro…t and each existing club makes zero pro…t.
We claim that the following prices and state of the economy constitute a competitive equilibrium. Let u be the core utilities. Then ui
[G; n] = wi ¡ui + hi [G; n] are competitive admission prices. As in Example 4, each agent is indi¤erent among all existing or potential clubs
and the prices are nonanonymous. Pro…t to a type-[G; n] club is
P i i
P
P i i
(16)
n u [G; n] ¡C[G; n] = ¡ ni ui +
n w + ¡ C[G; n]] +
P
i
i
i
ni hi [G; n]
i
It follows that
P
(17) ni ui [G; n] ¡C[G; n] · V (n) - u ¢ n · 0
i
for all [G; n]
The …rst inequality is an equality if G = G(n). Then, since V (n) ¡
u ¢ n · 0 for all n with equality for core coalitions, the only pro…table
clubs are core coalitions, each with an optimal facility. Hence, the core
state of the economy is a competitive equilibrium with individualized
admission prices.
14
Again, the competitive admission prices ui [G; n] depend on core utilities. If type-a is scarce, so that his core utility is high, then equilibrium
admission prices for type-a will be low and perhaps negative. There is
high demand for type-a individuals.
(ii) Lindahl Prices: A complete price system consists of an admission price for each agent to each type of coalition,25 n, and personalized
prices-per-unit of public good in each coalition. We shall call the admission prices ui (n) and the prices-per-unit pi (n). Within each coalition
there are pro…t maximizing …rms which produce the public good G to
make pro…t. Their presumption is that the pro…le of members in the
coalition, n, is …xed, and that each member will purchase the number
of units of G produced. These …rms take price pi (n) as …xed. Each
consumer chooses a coalition, n, and how much public good to purchase.
At equilibrium, no consumer wants to move and no consumer prefers a
di¤erent quantity of public good in his coalition than the amount produce. Coalitions in equilibrium make zero pro…t, and no coalition could
earn positive pro…t.
De…nition: A Lindahl equilibrium is a set of endowments and prices,
i
fw ; ui (n); pi (n)g, and a state of the economy fGt ; ntg26 such that (i),
(ii) and (iii) are satis…ed:
(i) In each coalition, nt, …rms producing public goods are at a pro…t
maximum: For each nt, and for all G,
P
P
(18) ¡C[Gt ; nt ] + nit Gt pi (nt ) ¸ ¡C[G; nt ] + nitGpi (nt)]
i
i
(ii) At the speci…ed prices, each consumer belongs to a club, nt , and
consumes an amount of public good, Gt , which he prefers. That is, if nit
> 0, then for all [G; n],
(19) wi ¡pi (nt )¡Gt pt(nt )+hi [Gt ; nt ] ¸ wi ¡pi (n)¡Gpi (n)+hi [G; n]:
P
(iii) No coalition could make positive pro…t. For all n, ni ui (n) ·
0, with equality for coalitions in the Lindahl equilibrium.
For the core state of the economy to be a Lindahl equilibrium, we
need two additional assumptions. In order to avoid the possibility that
…rms make positive pro…t, we assume that C[G; n] = c(n)G.27 In order
to ensure that in each coalition the consumer’s preferred G is the e¢cient
amount of G, we need to assume that hi [G; n] is concave in G.28
We now show how equilibrium prices can be constructed from the
core state of the economy, and show that with these prices the core state
of the economy is a Lindahl equilibrium. Let29
(20) ui (n) = w i ¡ ui + hi [G(n); n] ¡ G(n)hiG [G(n); n]
(21) pi (n) = hiG [G(n); n]
15
We notice that the admission prices depend on core utilities, while
the per-unit prices do not. If an agent is in high demand, due to the
fact that he is complementary with other agents and is scarce, his core
utility will be high and in each coalition, n, his admission price will be
low and perhaps negative.
To verify that with these prices the core allocation is a Lindahl equilibrium, we must verify (i), (ii) and (iii) in the de…nition. Since Gt solves
(14), we have
P
P
C(nt) = nit hiG [G(nt ); nt] = ni pi (nt).
i
i
Multiplying both sides of this equation by G, we see that for all G,
pro…t is zero. Since Gt is pro…t maximizing, (i) is satis…ed. To verify (ii),
we will show that the amount of public good chosen by i is G(n) and thus
Gt = G(nt ). We then show that all coalitions yield the same utility ui.
Therefore consumers cannot do better than to choose coalitions assigned
to them by the core. But then all coalitions make nonpositive pro…t and
core coalitions make zero pro…t. Hence (iii) is satis…ed.
If any type-i agent chooses coalition n, and consumes public good in
amount G, the utility he achieves is
(22) w i ¡ ui (n) ¡ Gpi (n) + hi [G; n] = ui ¡ [hi [G(n); n] ¡ G(n)pi (n)] +
[hi [G; n] = Gpi (n)]:
Since the consumer chooses his consumption of G to maximize the
amount of utility he gets in coalition n, he maximizes the last bracketed
term. But from the de…nition of pi (n) and concavity of hi [¢] in G, G(n)
maximizes the last bracketed term, and therefore maximizes the utility
achieved in a type-p coalition. The last two terms net to zero, and the
maximum utility achievable in each coalition is ui .
P
The sum ni ui (n) is less than or equal to zero for every n, as can be
i
seen by writing it out and using (20), and substituting G(n)
G(n)
P
i
P
i
ni hiG [G(n); n] =
ni pi (n) = c(n)G(n). Therefore, the only pro…table coalitions are
those in the core.
In the previous examples, it is clear that an equilibrium state is in
the core. For pure public goods (without exclusion and without complementarities in production) it has been shown (Foley (1970)) that a
Lindahl equilibrium state is in the core. For nonpure public goods, a
Lindahl equilibrium state may no be in the core (see Ellickson(1973)).
For our example, equilibrium states are in the core. The argument is not
obvious so we present it in detail, concluding our equivalence results.
Suppose the Lindahl equilibrium fwi ; ui (n); pi (n)g and fGi ; nt g yields
utilities ui = wi ¡ui (nt )¡Gt pi (nt )+hi [Gt ; nt ] and the Lindahl state of the
16
economy is not in the core. Then there is a coalition, say p = (na ; nb ; nc ),
an amount of private good for each type, say x¹ = (¹
xa ; x¹b ; x¹c ), and an
¹ such that
output of the public good, say G,
P i i
P
¹=
(23)
n x + c(n)G
ni wi , and
i
i
(24) x¹i + hi ui
for all i such that ni > 0
Equation (23) says the consumption in coalition n is feasible, and
equation (24) says the coalition gives more utility to its members than
they achieve in a Lindahl coalition.
P
Multiplying (24) by ni , summing over i, and substituting ni x¹i =
i
P i i
¹ from (23), we obtain
n w ¡ c(n)G
i
(25)
P
i
¹ + P ni hi [G;
¹ n] > 0
ni (wi ¡ ui ) ¡ c(n)G
i
It follows that
P
P
(26) ni (wi ¡ ui ) ¡ c(n)G(n) + ni hi [G(n); n] > 0
i
Substituting c(n) +
P
i
i
ni pi (n), and using expression (20) and (21)
for Lindahl prices,
P
P
(27) ni ui (n) = ni (wi ¡ ui ¡ G(n)pi (n) + hi [G(n); n]) > 0
i
i
This contradicts the fact that we have a Lindahl equilibrium.
6. Remarks on Existence
This paper has exposited the relationship between the core and competitive equilibrium in economies with transferable utility, but has not
addressed the question of when the core exists. Whenever the economy
is ”large enough” for more than one core coalition to contain each type
of agent so that the core has equal treatment, existence of the core is
equivalent to existence of competitive equilibrium, and thus, to know
when competitive equilibrium exists, it is enough to understand when
the core exists.
In private goods exchange economies, existence of the core and competitive equilibrium follow from convexity of production sets and preference sets. In coalition economies such as in this paper, convexity does not
guarantee existence of core states of the economy or competitive equilibrium. Existence of the core requires that it is feasible to partition agents
into clubs of optimal sizes.30 A game is ”balanced” if such partitioning
is feasible, and some authors have is simply assumed balancedness.31
Shaked (1982) shows that when V (n) is homogeneous the core exists.
Homogeneity means that, with the ratios of workers …xed, doubling the
workers will double the output. In the case of homogeneous population, described by Figure 1, Shaked’s assumption means that the percapita-utility curve V (n)/n is ‡at. Such an assumption may describe
17
production technologies, but is less appropriate when the advantage to
group formation is due to sharing a public good, as in club examples.
In club examples, there may be increasing returns to group size until
some critical size, which is usually called the ”minimum e¢cient scale”.
It is often assumed that beyond the minimum e¢cient scale crowding
becomes severe and per-capita product declines. Groups which are too
large are ine¢cient, just as groups which are too small are ine¢cient.
The return to group size may be homogeneous over some domain,
but not over the entire domain. If homogeneity occurs over a domain
in which maximum average product is achieved, then the core will be
nonempty for all su¢ciently large population sizes, keeping the initial
ratios of types …xed. This occurrence would be illustrated by Figure 1
for the case of homogeneous population if maximum average utility were
achieved on an interval rather than at the isolated point n¤ .
The analogous condition with nonanonymous crowding is this. Fix
the relative abundance of types as s, where s is a point in the simplex
of dimensionality equal to the number of types. Consider all population
sizes rs which preserve the relative abundances, r is a real number. Then
rs can never be a core coalition unless it maximizes V (rs)=r.32 Suppose
that there is an interval of real numbers, r, which maximize V (rs)=r. If
this is true for every s in the simplex, then there is a su¢ciently large
multiple of any arbitrary population such that the core exists, and the
core exists for all larger populations.
In general the condition under which the core exists, namely, balancedness, is very restrictive. ”Most” club economies will not satisfy
balancedness. In large economies (ones with many agents) with some
”friction”, it is appropriate to consider approximate equilibrium. For
economies with transferable utility, Wooders (1979) showed that for any
prespeci…ed population, there exists an integer multiple of that population for which the core exists. The core also exists for all replications of
the integer multiple.33 This is shown with continuously divisible agents
by Scotchmer (1986). Wooders also shows that for all su¢ciently large
economies, an approximate core exists, where ”approximate” means
that, although a deviating coalition might be able to improve its utility,
the improvement is less than a prespeci…ed bound.34 The prespeci…ed
bound becomes small as the economy becomes large. This result can be
reinterpreted to mean that an approximate competitive equilibrium exists. As the economy is replicated, a convergent sequence of approximate
core partitions converges to an exact core partition.35
The intuition behind existence of the approximate core is as follows.
Take as …xed the initial population, with its relative abundances of types.
It follows from a result in number theory that for a large enough popu18
lation size which preserves the relative abundances, the core exists. The
core also exists for all integer replications of this population size. For
any other population size there will be ”leftover” consumers who cannot
be placed in groups which are optimal in the sense of equation (7). As
a percentage of total population size, these leftovers become negligible
as the population becomes large. Thus, if they are distributed among
other groups, the distortion away from optimality which they introduce
becomes small. This is the argument in Wooders (1979, 1983).
Transferable utility greatly facilitates discussion of the core, but is
not required. Wooders (1983) shows that the approximate core exists
for all su¢ciently large economies in a game without side payments.
Wooders and Zame show that the approximate core exists or su¢ciently
large economies with transferable utility, without restricting agents to a
…nite set of types. A model related to Example 5 is in Wooders (1985).
7. Conclusion
Through a series of examples, we have attempted to suggest that
equivalence of the core and competitive equilibrium arises quite naturally
in coalition economies with complementarities. Agents are themselves
”goods”, and competition will establish their prices. Which agents are
”goods” (get paid to join groups) and which are ”bads” in the sense
that they must pay to join groups is determined endogenously from the
relative scarcities. Whether an agent’s price is positive or negative is
determined partly by the bene…ts he confers on the group (the other
members’ hi [¢] functions) and partly by his scarcity.
Our model is intended to capture the ideas expressed by Aumann
and Shapley: ”People cooperate to organize corporations (groups), then
compete with other corporations (groups) for business, and with each
other for positions of power within the corporation (group) . . . it is
impossible to draw a clear border between ’cooperation’ and ’competition”’. In the coalition economies we have described, cooperative and
competitive outcomes coincide. Moreover, competition for agents from
outside a group ensures o only that the groups in equilibrium will be
optimal, but also determines how surplus within each group is divided
among member.
19
FOOTNOTES
1. Some work in this line includes Buchanan (1965), Boehm (1974)
and Shubik and Wooders (1983b) on coalition production economies,
and Tiebout (1956), McGuire (1974), Ellickson (1973), Wooders (1979,
1980, 1985), Scotchmer (1985a,b) and Shubik and Wooders (1986) on
economies with club goods. Berglas (1976) and Berglas and Pines (1981)
are also in the spirit of these results. Some other related research is
discussed in Sandler and Tschirhart 1980).
2. We elaborate on the connection between our price system and
Arrow-Debreu prices in Section 5.
3. While this is a familiar phenomenon, our equilibrium treatment
of it di¤ers from the Arrow-Debreu treatment. In the latter treatment,
wages are per unit time and workers decide how many hours to work.
In our treatment, workers have a …xed amount of labor to o¤er and
must supply all of it to the employer for a lump sum compensation.
This is what it means to ”have a job.” This example is related to the
job matching literature. See Crawford and Knoer (1981) and related
literature.
4. In our competitive equilibrium the goods are the agents themselves
(their labor, or their involvement in the group), a private good and
possibly a public good available to members of the group. As discussed
below, the prices for agents may be interpreted as admission fees.
5. This point is also discussed by Shaked 1982).
6. The ”integer problem” in club theory typically refers to the fact
that the set of agents in the economy may not be partitionable into clubs
of the ”optimal” size. When crowding is nonanonymous, the integer
problem does not have an established meaning. Here, we mean that the
core is empty; our motivation for this will become clear.
7. In the production model, core utilities becomes wages. In the
club model, core utilities are related to club–admission prices rather
than wages.
8. This point is not at all surprising. In a model where the agents
themselves become goods (or bads) we would expect scarcity to determine prices; that it does so is gratifying. We show below that in club
models, core utilities determine club admission prices. Thus, the admission prices re‡ect scarcity. Although there is a strong intuition that this
should occur, we notice that it does not occur with anonymous crowding,
in which competitive admission prices are independent of the population
sizes (c.f. Scotchmer and Wooders (1986)).
9. Nonanonymous prices are reasonable in this context, since the
premise is that agents of di¤erent types can be distinguished. If agents
20
of di¤erent types were indistinguishable, subjective crowing costs would
have to be anonymous.
10. A ”state of the economy” is a partition of agents into coalitions
(groups), and an allocation of private and public goods within coalitions.
11. For more discussion of this idea, see Shapley and Shubik (1986).
12. As long as there is some mechanism to make utility ”transferable”, production can be interpreted quite loosely. Suppose type-b
are neoclassical theorists, type-c are radical economists and type¡a are
econometricians. One could also interpret this example in the congestion
sense. Type¡a likes to be in groups containing both (or either) b or c.
Type-b likes being in groups with type-a, but dislikes being in groups
with type-c.
13. We note that the core of a game consists of those feasible payo¤s
(utilities) that cannot be improved upon by any subset of agents (i.e.,
no coalition, acting by itself, can do better for its members).
14. The restriction to membership in only one club is inessential to
the model and results discussed here. See the appendix to Shubik and
Wooders (1986) for an example illustrating this point.
15. This assumption means that coalitions are “small” relative to
the economy
16. This relationship between core states of economy and payo¤s
(utilities) is well known and was …rst pointed out for Edgeworth’s contract curve (the core of an economy) and the core of a game by M.
Shubik (1959).
17. For vectors u = (ua ; ub ; uc ) and n = (na ; nb ; nc ), the expression
u ¢ n means ua na + ub nb + uc nc .
18. In the language of game theory, if the game is superadditive.
19. If we had constructed a model with divisible agents, rather than
discrete, and if V [¢] were then di¤erentiable, it would be the case that ui
= @V [¢]/@ni , for all i. That is, each agent is paid his marginal product.
20. The authors are preparing a paper which shows how this would
work.
21. This applies both to existing and ”potential” …rms.
22. There may be a core state of the economy which is not a competitive equilibrium if for some type on core coalition contains all agents
of that type.
23. Of course, one could think of the wage as being the net amount
i
u ¡ wi , and the agent keeps his endowment. See below.
24. Unlike club economies with anonymous crowding, competitive
prices must be nonanonymous.
25. These admission prices di¤er from the Composite Admission
Prices above, since they sum to zero in equilibrium. The prices above
21
sum to the cost of the facility.
26. Of course, the state of the economy also speci…es the amount of
private good consumed by each consumer in each equilibrium coalition.
This is implied by the endowment and the prices.
27. If pro…t were not zero, we would have to de…ne ownership shares
of …rms in di¤erent coalitions and distribute the pro…t. Thus, the incomes wi would be endogenous. We prefer to avoid this additional notation.
28. Otherwise, although the e¢cient G(n) will satisfy pi (n) = @hi [G(n); n]=@G,
there may be another G, preferred by consumer i, which also satis…es this
condition. In that case, the Lindahl equilibrium will not exist and the
e¢cient (core) state of the economy cannot be decentralized by Lindahl
prices. Whenever the Lindahl equilibrium does exist, it will be e¢cient.
29. Recall that G(n) is the e¢cient amount of G for coalition n.
30. This is easily seen in the homogeneous case, as in Figure 1. The
core is empty when the population size is not an integer multiple of n¤ ,
a circumstance unrelated to whether preferences and coats are convex.
For a speci…c numerical example, see Shubik and Wooders (1983a).
31. See Ichiishi (1977) and Greenberg (1979).
32. This is provided the population size is larger than required to
achieve the maximum on r of V (rs)=r.
33. Wooders calls this a “subsequence” of economies.
34. Such as equilibrium seems natural when there are coalition formation costs, for example, moving costs.
35. The percentage of agents not in their preferred coalitions goes to
zero.
22
Figure 1:
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26