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Working Paper Series

Firms as Clubs in Walrasian Markets
with Private Information: Technical
Appendix

WP 05-11

Edward Simpson Prescott
Federal Reserve Bank of Richmond
Robert M. Townsend
University of Chicago
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

Firms as Clubs in Walrasian Markets
with Private Information: Technical Appendix
Edward Simpson Prescott¤
Federal Reserve Bank of Richmond

Robert M. Townsend¤
University of Chicago
Federal Reserve Bank of Chicago

Working Paper 05-11
November 1, 2005

Abstract
This paper proves the Welfare Theorems and the existence of a competitive equilibrium for the club economies with private information in Prescott and Townsend
(2005). The proofs cover lottery economies with a …nite number of goods and without
free disposal. A mapping based on Negishi (1960) is used.

JEL Classi…cation: D51, D71, D82, L11
Keywords: competitive equilibrium, clubs, private information

¤

Townsend would also like to thank the NSF and the NIH for …nancial support. The views expressed in this paper are solely those of the authors and do not necessarily re‡ect the views of the
Federal Reserve Banks of Chicago and Richmond or the Federal Reserve System. Contact information: Prescott, Federal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261, Edward.Prescott@rich.frb.org; Townsend, Dept. of Economics, University of Chicago, 1126 E. 59th St.,
Chicago, IL 60637, rtownsen@midway.uchicago.edu.

1

1

Introduction

This paper is the technical appendix to Prescott and Townsend (2005). It contains proofs
of the Welfare theorems and of the existence of competitive equilibria for their model. The
proofs are based on Negishi (1960) who proved existence by …nding a …xed point in the
space of Pareto weights that is a competitive equilibrium with zero transfers. While the
linear structure of Prescott and Townsend (2005) simpli…es the problem relative to his,
there are two other di¤erences that complicates the problem. Because of these di¤erences,
some modi…cations to his proof are required.
The …rst di¤erence is that we do not assume free disposal. Free disposal makes prices
positive and in our economies, with its hedonic properties, this is entirely inappropriate.
Negishi’s proof, as it stands, cannot handle negative prices. In his mapping, the n prices
are mapped into an n ¡ 1 dimensional simplex. We modify his approach to incorporate

negative prices by requiring prices to lie in the closed unit ball. The unit ball includes
the zero price point. In proofs based on the excess demand correspondence this can cause
problems because that mapping is not upper hemi-continuous. As we shall see, this is not
a concern for Negishi’s mapping in our context.
The second di¤erence between Negishi’s economy and ours is that Negishi assumes
that endowments are interior to consumption sets. There are two reasons he makes this
assumption. First, it satis…es Slater’s condition, a su¢cient condition for the application
of the Kuhn-Tucker theorem. Assuming an interior endowment is clearly inappropriate
in our economies. Instead, we obtain the existence of Kuhn-Tucker vectors directly from
the linear structure of our economy. A Kuhn-Tucker vector, the dual variables in a linear
program, will exist if a solution to the linear program is …nite.
The second reason that Negishi assumes that endowments are interior to consumption
sets is that when combined with positive prices (implied by free disposal and a non-satiation
assumption) there exists a cheaper consumption point for each agent. The existence of a
cheaper consumption point implies that the Pareto weights at the …xed point are strictly
positive. Strictly positive Pareto weights are important because they imply that the …xed
point of Negishi’s mapping is a competitive equilibrium rather than just a compensated
equilibrium. In our economies, it is natural to endow agents with zero units of almost
2

every commodity. Consequently, we cannot make the interiority assumption. Instead, we
will guarantee the existence of a cheaper point by making reasonable assumptions on the
production technology in our economy.

2

A Short Review of Prescott and Townsend (2005)

Prescott and Townsend (2005) study a model in which agents purchase memberships and
jobs in a variety of …rms. Each type of a …rm is de…ned by its contractual terms. These
include the inputs it uses, how hard participants work, and the output-contingent compensation of all members of the …rm. The joint consumption and joint production features of
these multi-agent contracts are what make them club goods.
Private information of agents’ e¤orts is a critical feature of the environment. We assume
that these contracts are exclusive in the sense that an agent cannot side trade away from
the contractual terms. This assumption is important because an agent who is not fully
insured by a contract has an incentive to purchase additional insurance, which undoes the
original contract.
Participating in a …rm, or club or contract, gives an agent utility, and creation of a …rm
uses resources. In this appendix we do not specify the details of these contracts; they are
described in the main paper. Instead, we de…ne utility and resource usage directly in terms
of the contracts and jobs because that is all that matters for the decentralization theory.
There is a continuum of agents but only a …nite number of types. Each type i is a
positive fraction ®i of the population. In most of Prescott and Townsend (2005) the only
di¤erence between agents is their wealth. For consistency, we lay out the model with this
assumption. For the proofs, however, the notation is expanded to be applicable to the case
where agents di¤er in their abilities or preferences.
Type-i agents are endowed with capital ·i. Agents sell their capital for income that they
use to purchase membership in single-agent or multi-agent …rms. An incentive compatible
self-employment …rm is a b1 2 B1. An agent’s utility from participation in a b1 contract is
u(b1 ): Each contract or …rm produces output and uses consumption. The net amount of
such consumption is r(c¡q)(b1). Similarly, each contract uses rk (b1) of the capital input.
Multi-agent …rms are similar. The main paper distinguishes among several di¤erent
3

classes of …rms: a supervisor and a worker, relative performance, and team production.
In this section, to simplify the exposition we only consider the supervisor-worker …rms.
Let b2 2 B2 be the contract describing one such …rm. The utility an agent receives from
participation in a b2 …rm also depends on his job within the …rm. We write his utility as
u(b2 ; j), j = w; s, where j = w indicates that the agent is the worker and j = s indicates
that the agent is the supervisor. Resource usages for a two-person …rm are r(c¡q)(b2) and
rk (b2 ).
We assume that there is a …nite number of elements in B1 and B2 so the commodity
space is L = <#B1 +2(#B2)+1 , where #Bi denotes the number of elements in set Bi . The
commodity space re‡ects a choice among the #B1 of self-employment …rms, the #B2 of

worker-supervisor …rms along with a position in that …rm, and the level of the capital
endowment sold. Let p(b1 ) denote the price of a b1 …rm and let p(b2; j) be the price of a b2
…rm joint with position j = w; s in that …rm. We let pk denote the price of a unit of the
capital endowment.
Agents purchase membership in one of several types of …rms plus, if appropriate, a
job in that …rm. Let xi(b1) denote the measure of type-b1 …rms purchased by a type-i
agent, and let xi(b2; j) denote the measure of job-j in type-b2 …rms purchased by a type-i
agent. In equilibrium, these measures will indicate the fraction of type-i agents assigned to
a particular type of …rm, and, if appropriate, to a particular job within a …rm. A type-i’s
consumption set Xi is
Xi = fxi (b1 ) ¸ 0; xi(b2; j) ¸ 0; j = w; sj

X
b1

xi(b1) +

X

xi (b2 ; j) = 1g:

b2;j

This set guarantees that (xi (b1 ); x(b2 ; j)) is a probability measure. 1
The problem of a type-i consumer is
max

X

xi (b1)u(b1) +

b1

X

xi(b2; j)u(b2 ; j)

b2 ;j

subject to the budget constraint
X
b1

1

xi(b1)p(b1) +

X
b2 ;j

xi (b2 ; j)p(b2; j) · pk ·i ;

Note that Xi ½ L and that, in particular, there is no choice corresponding to the last component, the
capital endowment, as that is supplied inelastically.

4

and xi 2 Xi .

The production sector forms …rms. Its input is capital and its outputs are memberships

in …rms, or clubs. To do this, it converts the capital stock into the capital input and it
creates …rms by supplying the capital input, state-contingent consumption, and people to
sta¤ the positions. Each multi-agent …rm needs an agent to purchase the worker position
and an agent to purchase the manager position. Because there is constant returns to scale
in setting up …rms, only one pro…t-maximizing entity is needed. To distinguish it from the
…rms of our theory, we refer to it as the production sector.
Let ±(b1) be the number of b1 …rms produced by the sector and let ±(b2) be the number
of b2 …rms produced. The sector purchases capital y· from consumers and then uses it to
create the capital input used by the …rms that it creates. As is conventional, the input yk
is negative in sign, and outputs, the number of …rms of each type, are positive in sign. The
capital constraint is

X
b1

±(b1 )rk (b1 ) +

X
b2

±(b2)rk (b2) + y· · 0;

(1)

so capital sold as part of …rms is less than or equal to capital purchased.
The sector also transfers contingent consumption between agents. It collects premia and
distributes indemnities to …rms experiencing low output, and these premia and indemnities
need to balance in order to sum to zero. With the resource requirements of the consumption
good rc¡q (bi ), written in expectation, the consumption resource constraint is
X
b1

±(b1)r(c¡q)(b1 ) +

X
b2

±(b2)r(c¡q)(b2) · 0:

(2)

With a continuum of agents the left-hand side is a …xed number.
In selling supervisor-worker …rms, b2, the production sector also sta¤s positions in them.
Each one of these b2 …rms requires two members, so for each level of b2 produced there
needs to be one worker for each supervisor. We write this club or matching constraint by
creating notation y(b2; w) for the number of multi-agent …rms with a worker and y(b2 ; s)
for the number of multi-agent …rms with a supervisor, then requiring
8b2; ±(b2) = y(b2; w) = y(b2; s):

(3)

There is also a trivial constraint for the self-employment …rm:
8b1 ; ±(b1 ) = y(b1 ):
5

(4)

The production set is
#B2
1
Y = fy 2 Lj9±(b1) 2 <#B
3 (1); (2); (3); (4) holdg:
+ ; ±(b 2) 2 < +

Y is the aggregate production set.
Given prices, the production sector maximizes
max

y2Y;±(b1);±(b2 )

X

p(b1 )y(b1 ) +

b1

X

p(b2 ; j)y(b2; j) + pk y·:

b2 ;j

Market clearing requires that the number of …rms of each possible type purchased by
agents be equal in number to the those sold by the production sector, and similarly that
the capital good sold by agents equal that bought by the production sector:
X

8b1;

X

8b2; j;

®ixi(b1) = y(b1);

(5)

®i xi(b2; j) = y(b2; j);

(6)

i

i

X

(7)

®i ·i + y· = 0:

i

Let xi denote the vector of purchases xi(b1) and xi (b2 ; j) for type i and let x be the
vector of consumptions across the I types of agents. Similarly, let y denote the vector of
output vectors y(b1); y(b2; j); and yk , and let p denote the vector of prices p(b1) and p(b2; j):
De…nition 1 A competitive equilibrium for this economy is an (x¤ ; y¤; p¤) such that
1. 8i, x¤i maximizes
P

b2;j

P

b1

xi(b1)u(b1)+

xi(b2; j)p¤ (b2; j) · p¤k ·i :

P

b2 ;j

xi(b2; j)u(b2; j) subject to xi 2 Xi and

P

b1

xi(b1)p¤ (b1 )+

2. y¤ maximizes p¤ y subject to y 2 Y:
3. Markets clear, that is, (5), (6), and (7) hold.
Pareto optimum can be found by solving the Pareto program. Let ¸i denote the Pareto
weight on type-i agents and ¸ the corresponding vector across types. The Pareto program
is:
max

x;y;±;k

X
i

0

¸i®i @

X

xi(b1)u(b1) +

b1

X
b2;j

1

xi (b2 ; j)u(b2; j) A

subject to xi 2 Xi , y 2 Y , (5), (6), and (7). Note that combining market clearing with the
production set gives the more traditional representation of the resource constraints. Also
note that this program is a linear program.
6

3

Proofs

In this section we provide proofs for a linear economy with lotteries that cover the economy
described above. The proofs also cover the important case where agents are heterogeneous
in ability and preferences. Because of the additional generality, we will modify the above
notation and make it more abstract. First, we de…ne the consumption set by a …nite number
of linear inequalities, that is,
Xi = fxi 2 <n+jgi xi ¡ bi · 0g:
As before xi is a vector. There may be any …nite number of constraints so the object gi is a
matrix. These constraints may hold as an equality or as an inequality. (There is no need to
worry about ¸ constraints. Just multiply these by negative one to put them in the above

form.) For expositional convenience, we will use the above notation to re‡ect that either
case is possible and then examine each case in the proofs as needed. In the economy laid
out above, the only consumption set constraint is to guarantee that the agent chooses a
probability measure. In that case, g i is a vector of ones, bi = 1; and the constraint holds at
equality. If agents are heterogeneous in their abilities or preferences, as in the extension of
the above model sketched out in Prescott and Townsend (2005), then there are constraints
that require xi = 0 for allocations that assign an agent to a contract that is infeasible for
his type. In general, however, the only requirement for the proofs below is that there be a
…nite number of linear constraints, xi ¸ 0; and that Xi be bounded.

In the above model agents inelastically supply the capital endowment. Consequently,

Xi would be in a lower-dimensional space then Y . Rather than distinguishing between
the endowment and consumption goods we simply expand the dimension of Xi , here, by
the number of inelastically supplied endowment goods. Consumption of these goods are
then constrained to equal zero. Of course, agents get no direct utility from supplying these
goods so the utility coe¢cient on them is zero.
Production is de…ned similarly. Again, we represent the production set by a …nite
number of linear inequalities. The production set is
Y = fy 2 <njfy · 0g;
7

where f is a matrix. As with the consumption set, the notation represents the possibility
that the constraints are either inequalities or equalities. In the model economy, the club
constraints hold at equality while the production constraints on consumption and capital
are inequalities. The production set is a convex cone.
Finally, we let »i denote the vector endowment of a type i. In the economy described
above, the natural endowment is for »i to be zero everywhere except for a ·i in the last
component. We do not impose that restriction here for two reasons. First, there is no
need to limit the applicability of the proofs below. Second, in the above economy there
are Pareto optima that cannot be supported as competitive equilibria with a natural endowment distribution, that is, where agents’ endowments only di¤er in their holding of
the capital (and they cannot be endowed with a negative quantity of capital). These optima can be supported if income transfers are allowed, and by considering the full set of
endowments we can study these income transfer cases as well.
Using dot-production notation we have the following de…nitions:
De…nition 2 A compensated equilibrium in this economy is an (x¤; y ¤; p¤) such that
1. 8i, x¤i minimizes p¤xi subject to xi 2 Xi and ui xi ¸ ui x¤i .
2. y¤ maximizes p¤ y subject to y 2 Y .
3.

P

¤
i ®i (xi

¡ » i) = y ¤ .

De…nition 3 A competitive equilibrium in this economy is the same as a compensated
equilibrium except that 1. above is replaced by
1. 8i, x¤i maximizes uixi subject to xi 2 X and p¤xi · p¤ »i .
Our proofs are based on solving the Pareto problem so we rewrite it in the general
notation. Goods are indexed by j. Consequently, the jth coe¢cient of the vector xi will be
denoted xi;j . For matrices like f or gi , let fj and g i;j denote the jth column of that matrix.
The Pareto program is
max

X

fx i g¸0;y i

8

¸ i® i u ix i

s.t.

X
i

®i (xi ¡ »i ) ¡ y = 0;
fy · 0;
8i; g ixi · bi :

4

Proofs

The economy described above is linear. In this section, we provide proofs of the Welfare
Theorems and of the existence of a competitive equilibrium for linear economies with a
…nite number of goods and without free disposal.
Our commodity space is Euclidean and consumption is bounded from below by zero.
The only additional assumptions we make are
Assumption 1 8i, Xi is bounded,
Assumption 2

P

i ®i (Xi

+ » i ) \ Y 6= ;;

Assumption 3 8i; for any xi component of a feasible allocation there exists x0i 2 Xi such
that uix0i > ui xi.

The …rst assumption is satis…ed when the consumption set has a probability measure
constraint. Because the consumption set is represented by a …nite set of linear inequalities,
it is convex and closed. The boundedness assumption then makes this set compact. No
additional assumptions are needed on the production set because it is a closed, convex
cone. The second assumption just assumes that a feasible allocation exists. The third
assumption is a no-satiation assumption and is easy to satisfy by simple assumptions on
the underlying economy. Furthermore, this assumption plus the linearity in the economy
implies that there are no local satiation points in the consumption set. Finally, note that
the linear structure of this economy means that the utility function and constraints are all
di¤erentiable.
We will use Kakutani’s …xed point theorem to prove the existence of a competitive
equilibrium. We state it without proof.

9

Theorem 1 Let Z be a compact convex set in <n and let h(z) be a non-empty, compactvalued, convex-valued, upper hemi-continuous correspondence mapping Z into Z. Then,
there is a …xed point zb such that zb 2 f(zb).

There are no features of our model that require modi…cation of the standard proofs of

the First Welfare Theorem so we state it without proof:
Theorem 2 (First Welfare Theorem) If (x; y; p) is a competitive equilibrium and no
agents are satiated at x; then the allocation (x; y) is Pareto optimal.
The Second Welfare Theorem as well as the existence proof use properties of solutions
to the Pareto program.
Theorem 3 Assuming that a feasible solution to the Pareto program exists, then for any
set of weights ¸ 2 SI¡1, an optimal feasible solution to the Pareto program exists.
Proof: The constraint set is closed. It is also bounded because for each i, Xi is bounded.
Furthermore, the set of feasible y is bounded by the resource constraint in conjunction with
the bounds on Xi . Therefore, x and y are chosen from compact sets. The objective function
is continuous. Therefore, a maximum exists. Q.E.D.
We now characterize a solution to the Pareto program with the …rst-order conditions.
Let pbj be the dual variable (or multiplier) on the market clearing constraint for the jth
b be the vector of dual variables on the production set constraints, and let °
bi
good, let ¹

be the dual variables on agent i’s consumption set constraints. Dual variables for equality
constraints can be of any sign while those for less-than-or-equal constraints are restricted
to be non-negative. When the notation is designed to capture constraints of either form,
the sign restrictions on the dual variables will not be explicitly written out.
Theorem 4 An allocation (x; y) with x ¸ 0 is a solution to the Pareto program if and only
b ¹
b; b
if it and the dual variables (p;
° ¸ 0) satisfy

8i; j; ¸i ®iui;j ¡ pbj ®i ¡ °b i gi;j = 0; (· 0 if xi;j = 0)
b fj = 0;
8j; pbj ¡ ¹

10

(8)
(9)

X

pb(

i

®i (xi ¡ » i) ¡ y) = 0;
b f y = 0;
¹

8i; °b i (gi xi ¡ bi ) = 0;

X
i

®i (xi ¡ »i ) ¡ y = 0;

(10)

fy · 0;

(11)

gi xi · bi:

(12)

Proof: The Pareto program is a linear program so the Kuhn-Tucker conditions are
necessary and su¢cient. Q.E.D.
At this point, it is worth proving a preliminary result.
b y) to be part of a solution to the Pareto program, p
by = 0.
Lemma 1 For (p;

b fy. By (11), ¹
bf y = 0
Proof: Multiplying (9) by yj and summing over j gives pby = ¹

so pby = 0. Q.E.D.

The other lemma we need is that with non-satiation pb 6= 0.

Lemma 2 With non-satiation, there does not exist a solution to the Pareto program such
that pb = 0.

Proof: By non-satiation there exists for each i, x0i 2 Xi such that uix0i > ui xi. If pb = 0,

then by (8) and (12), ¸i ®iui xi = °b ig ixi = °b i bi. By (12), b° ibi ¸ °b i gi x0i . (If °b i = 0 it holds

trivially, while if °b i ¸ 0 it holds because gi x0i · bi.) Therefore, ¸ i®i uix0i > b° ig ix0i. Then
because x0i ¸ 0; for some j, ¸i ®iui;j > °b i gi;j . But that contradicts (8) so pb =
6 0. Q.E.D.

We express the compensated equilibrium conditions in terms of the necessary and su¢-

cient conditions. For the consumer’s minimization problem let ¯ i ¸ 0 be the dual variable

on the constraint ui xi ¸ ui x¤i , and let À i be the vector of dual variables on the consumption
set constraints. For the production sector, let ¹ be the vector of dual variables on the
production constraints.
Lemma 3 Conditions 1 and 2 in the de…nition of a compensated equilibrium can be rewritten in the following form:
1. 8i, x¤i ¸ 0 and dual variables (¯ i ¸ 0; À i) satisfy condition 1 of a compensated
equilibrium if and only if they satisfy

8j; ¯ i ui;j ¡ pj ¡ À ig i;j = 0; (· 0 if xi;j = 0)

(13)

¯ i (uixi ¡ uix¤i ) = 0; uixi ¡ ui x¤i ¸ 0;

(14)

À i (gi xi ¡ bi ) = 0; gi xi ¡ bi · 0:

(15)

11

2. y and dual variable ¹ satisfy condition 2 of a compensated equilibrium if and only if
8j; pj ¡ ¹fj = 0
¹fy = 0; fy · 0:

(16)
(17)

Proof: Both problems are linear programs. Q.E.D.
Theorem 5 (Second Welfare Theorem) Any solution to the Pareto program for some
Pareto weights ¸ ¸ 0 can be supported as a compensated equilibrium.
Proof: Given a solution to the Pareto program, necessary and su¢cient condition (10)
implies that condition three of a compensated equilibrium, market clearing, holds. Let
b . Comparison of necessary and su¢cient conditions (9)
p = pb, ¯i = ¸i , º i = °b i =®i , and ¹ = ¹

and (11) with the necessary and su¢cient conditions (16) and (17) shows that the condition
two of a compensated equilibrium holds. Comparison of (8) and (12) with (13) and (15)
shows that these two necessary and su¢cient conditions of a compensated equilibrium
holds. The remaining part of condition one, (14), holds trivially because xi = x¤i . Q.E.D.
The conditions for a competitive equilibrium are nearly identical. The only change is
the necessary and su¢cient conditions for the Consumer’s problem. Let µ i ¸ 0 be the

dual variable on agent i’s budget constraint and let !i be the dual variable on agent i’s
consumption set constraints. Then, xi ¸ 0 and dual variables (µ i ¸ 0; !i) satisfy condition
one of a competitive equilibrium if and only if they satisfy

8j; ui;j ¡ µ i pj ¡ !ig i;j = 0; (· 0 if xi = 0)

(18)

µi p(xi ¡ »i ) = 0; p(xi ¡ »i ) · 0;

(19)

!i (gi xi ¡ bi ) = 0; g ixi ¡ bi · 0:

(20)

Theorem 6 Any solution to the Pareto program for some Pareto weights ¸ > 0 can be
supported as a competitive equilibrium.
b and ¹ = ¹
b . In addition, set 8i, »i = x¤i ,
Proof: As in the proof of Theorem 5, set p = p,

µ i = 1=¸i and ! i = °b i=(¸i ®i). Market clearing and the Production sector’s problem are

satis…ed by the same arguments used in the proof to Theorem 5. The Consumer’s problem
12

condition (19) holds trivially because » i = x¤i . Substituting into (8) gives us that condition
(18) holds. Finally, substituting into (12) and dividing by the scalar (¸i ®i) demonstrates
that condition (20) holds. Q.E.D.
In Debreu (1954), the existence of a cheaper point is used to show that a compensated
equilibrium is a competitive equilibrium. This argument is commonly referred to as Arrow’s
Remark. Here, we provide a similar result by demonstrating that the existence of a cheaper
point means that the Pareto weights are strictly positive. Later, this result will be used to
employ Theorem 6 when proving the existence of a competitive equilibrium.
Theorem 7 Take a solution to the Pareto program for ¸ ¸ 0. If, at the corresponding
compensated equilibrium, there exists for all i, a cheaper point satisfying xi 2 Xi , then
¸ > 0.

Proof: Let x be the solution to the Pareto program. From (8), for each i; ¸i ®iui xi ¡

b i ¡°
b i gi xi = 0. Take any x0i 2 Xi: Also by (8), ¸i ®iuix0i ¡ ®ipx
b 0i ¡ °
b i gi x0i · 0. Therefore,
®ipx
b i¡°
b i gi xi ¸ ¸i ®iui x0i ¡ ®ipx
b 0i ¡ °
b ig ix0i ; 8x0i 2 Xi:
¸ i®i ui xi ¡ ®ipx

Assume there exists a cheaper point x0i 2 Xi and consider the ¸i = 0 case. Then,
b 0i ¸ p
bxi + °
b i gi xi =®i ¡ °
b ig ix0i =®i ; 8x0i 2 Xi:
px

By (12), °b i (gi xi ¡ bi ) = 0 and °b i (gi x0i ¡ bi) · 0; 8x0i 2 Xi : Therefore, pbx0i ¸ pbxi, 8x0i 2 Xi ,
which contradicts the assumption that a cheaper point exists. Q.E.D.

4.1

Existence Theorem

A Negishi-style proof works by showing that at a …xed point of Negishi’s mapping, consumers’ budget constraints are satis…ed by a solution to the Pareto program. As discussed
above, we do not have free disposal so prices may be negative. To allow for negative prices,
we restrict prices to the closed unit ball, that is,
p
P = fp 2 <n j p ¢ p · 1g:
p
We can not use as our set the unit sphere because fp 2 <n j p ¢ p = 1g is not convex.

However, we will use this normalization in our mapping. In proofs based on excess demand
13

correspondences, a complication with using the unit ball is that the excess demand correspondence is not upper hemi-continuous at p = 0. As we will see, this is not a problem for
Negishi’s mapping here.
To show that a competitive equilibrium exists, we use the following mapping of (¸; x; y; p) !

(¸; x; y; p). The mapping consists of the following two parts
¸ ! (x0; y 0; p0 )
(¸; x; y; p) ! ¸0 :

Each variable is restricted to lie in a compact, convex set. Remember that I is the
number of types so we restrict ¸ 2 SI¡1 (the unit simplex), x 2 X, y 2 ¡, p 2 P , where X

is the cross-product over I of the Xi, and ¡ = fy 2 Y jy is feasibleg. The set P is compact
and convex. Market clearing and the compactness of the Xi guarantee that ¡ is a compact
set. The set ¡ is also convex.
For any ¸ 2 SI¡1 there is a solution to the Pareto program (x¤; y¤; p¤ ; ¹¤; ° ¤ ). Then we

normalize p to lie in the unit circle with

pe =

(

p¤
p ¤ ¤:
p ¢p

This normalization will not work if prices are zero but by Lemma 1, p¤ 6= 0: Finally, we
add one more step to the mapping by including in the correspondence the convex hull of
prices calculated from the normalization, that is,
p0 2 coPe ;

where Pe is the set of normalized prices. This step is important because it preserves convexity of the mapping while keeping prices in P . Furthermore, the set of pe is convex so

normalizing it to lie on the unit circle and then taking the convex hull does not add any
new set of relative prices to the mapping.
The …rst part of the mapping gives us (x0 ; y0 ; p0 ). This mapping is non-empty, compactvalued, convex-valued, and upper hemi-continuous. Non-emptiness, boundedness, and convex valued are trivial. Closed valued and upper hemi-continuity are not too di¢cult to
show. For example, the normalization of prices is a continuous function, which preserves
14

upper hemi-continuity. Furthermore, the convex hull of an upper hemi-continuous correspondence is itself upper hemi-continuous. (See, for example, Proposition 11.29(a) of
Border (1985).)
0

The second part of the mapping calculates the new Pareto weights, ¸ , as a function of
the transfers needed to support an allocation x from prices p. Because the sets P and X
are bounded, there exists a positive number A such that
X
i

jp(»i ¡ xi )j < A;

for any (x; p) 2 X £ P . Then, for any ¸ 2 SI¡1, let
^
^i = maxf0; ¸i + p(»i ¡ xi ) g; and ¸0i = P¸i :
¸
^
A
i ¸i

(21)

This mapping is a continuous function so it is trivially a non-empty, compact-valued, convex
valued, and upper hemi-continuous correspondence.2
To summarize, we have de…ned a mapping from SI¡1 £ X £ ¡ £ P ! SI¡1 £ X £ ¡ £

P . Each of these sets is convex and compact and cross-products of convex and compact

sets are convex and compact. Each part of the mapping is non-empty, convex-valued,
compact-valued, and upper hemi-continuous. Cross-products of correspondences preserve
these properties. Therefore, we have proven the following theorem.
Theorem 8 A …xed point (¸; x; y; p) of the mapping exists.
Proof: Apply the conditions of Kakutani’s …xed point theorem.
The …xed point is a solution to the Pareto program so by Theorem 1 it is also a
compensated equilibrium. (Note that prices can be scaled by a constant without a¤ecting
the solution to compensated or competitive equilibria.) The second part of the mapping,
subject to one condition, will provide us with a proof that the compensated equilibrium is
a competitive equilibrium.
Theorem 9 (Existence) For any given distribution of endowments, if the Pareto weights
at a …xed point of the mapping are non-zero, then a competitive equilibrium exists.
2

Note that y is not explicitly used in this portion of the mapping. It is not needed because by Lemma
1 any solution to the Pareto program satsi…es py = 0 and scaling p does not change this.

15

Proof: By Theorem 8 a …xed point to the mapping exists. The …rst part of the mapping
is a solution to the Pareto program so by Theorem 5 the solution can be supported as a
compensated equilibrium. Now assume that ¸ > 0. From the second part of the mapping,
(21), at a …xed point p(» i ¡ xi) must be the same sign for all i. From the necessary and
P

su¢cient conditions to the Pareto program, p(
py = 0. Therefore, p(»i ¡ xi ) = 0, 8i.

i ®i (xi

¡ » i) ¡ y) = 0 and from Lemma 1

This satis…es one part of the necessary and su¢cient conditions of the Consumer’s

problem. The remaining conditions simply require setting µ i = 1=¸i and !i = ° i =(¸i®i ).
Since ¸i > 0 we can make this substitution. Q.E.D.
The term µi = 1=¸i is consumer i’s marginal utility of income.

4.2

Cheaper point

The existence proof only guarantees the existence of a competitive equilibrium if ¸ > 0.
Because our …xed point is a compensated equilibrium, Theorem 7 implies that if there is a
cheaper point then ¸ > 0. Consequently, if a cheaper point exists then our …xed point is a
competitive equilibrium.
In Prescott and Townsend (2005) a self-employment contract b1 2 B1 is an output

dependent consumption schedule c(q), an e¤ort level a, and a capital input level k for which
e¤ort is incentive compatible. Production is f(qja; k), which is the probability distribution
of output q given the e¤ort level and capital input. The price of one of these contracts is
p(b1) = (E(c) ¡ E(q))¹(c¡q) + kpk ;
where ¹(c¡q) > 0 is the shadow price on consumption and the expectation is taken with
respect to the a and k in the contract.
Fortunately, only weak conditions are needed in this environment to guarantee the
existence of a cheaper point. For example, when each agent is endowed with a positive
amount of capital, that is, 8i; ·i > 0, and capital is scarce, that is, pk > 0, and each
agent has positive income. Because of the non-satiation assumption the shadow price of

consumption is positive (¹(c¡q) > 0), so if E(qja = 0; k = 0) ¸ 0, then a contract where

the agent gets zero consumption, works zero, and uses no capital for production is cheaper
than the agent’s equilibrium allocation.
16

In the important case where there is no capital input into production, each agent’s
income is zero. In this case, p(b1) = (E(c) ¡ E(q))¹ (c¡q) with ¹ (c¡q) > 0 because of the

non-satiation assumption. Then, as long there exists (c(q); a) 2 B1 such that E(c) · E(q),
there will be a cheaper point than each agent’s equilibrium allocation. Since the moralhazard literature is usually concerned with situations where the agent produces a surplus
for the principal, this condition seems mild indeed.

17

References
[1] Border, Kim. Fixed Point Theorems with Applications to Economics and Game Theory.
Cambridge: Cambridge University Press, 1985.
[2] Debreu, Gerard. “Valuation Equilibrium and Pareto Optimum.” Proceedings of the
National Academy of Sciences 40 (1954): 588-92.
[3] Negishi, Takashi. “Welfare Economics and the Existence of an Equilibrium for a Competitive Economy.” Metroeconomica 12 (1960): 3-8.
[4] Prescott, Edward Simpson and Robert M. Townsend. “Firms as Clubs in Walrasian
Markets with Private Information.” Manuscript, October 2005.

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