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Working Paper Series
Firms as Clubs in Walrasian Markets
with Private Information
WP 00-08
Edward S. Prescott
Federal Reserve Bank of Richmond
Robert M. Townsend
University of Chicago
Federal Reserve Bank of Chicago
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Firms as Clubs in Walrasian Markets
with Private Information
Edward S. Prescott∗
Federal Reserve Bank of Richmond
Robert M. Townsend∗
University of Chicago
Federal Reserve Bank of Chicago
September 8, 2000
Federal Reserve Bank of Richmond Working Paper 00-8
Abstract
Using private information and club theories, this paper develops a theory of firms
in general equilibrium. Firms are defined to be assignments of technologies and
agents to clubs. In equilibrium, firms form endogenously and multiple types may coexist. We formulate the general equilibrium problem as both a Pareto program and
as a competitive equilibrium. Welfare and existence theorems are provided. In the
competitive equilibrium, club memberships are priced and purchased, so the market
determines which organizations exist as well as who is a member. Pareto optima and
competitive equilibria of several examples are computed. In our examples, elements
of limited commitment and monitoring capabilities are important factors that affect
both the internal organization of firms and their equilibrium distribution. Furthermore, equilibrium organizational structure varies with the aggregate endowment of
capital and the distribution of wealth.
JEL Classification: D51, D71, D82, L20
Keywords: private information, general equilibrium, clubs, theory of the firm
∗
The authors would like to thank Bryan Ellickson, Lionel McKenzie, Suzanne Scotchmer, William Zame
and seminar participants from Chicago, Princeton, Philadelphia Fed, Venice, the 2000 Winter Econometric
Society meetings, the 2000 NBER GE conference, and the 2000 SITE GE workshop for helpful comments.
The second author would like the NSF and the NIH for financial support. The views expressed in this
paper are solely those of the authors and do not necessarily reflect 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 incorporates multi-agent private information models into classical general equilibrium theory. We do this by studying a model where there is risk and moral hazard in
production, but where multi-agent firms may form to mitigate these incentive problems
through supervision. In our theory, not only does the market set the prices of credit,
insurance, and production inputs, it also determines which kinds of firms form, their internal organization, their compensation structure, and who joins them in what capacity. A
byproduct of our theory is that the distribution of wealth affects the industrial organization
of the economy, as well as the economy-wide distribution of labor effort and consumption.
Agency, or private information, theory has been enormously influential in the study of
the firm. In early work, Alchian and Demsetz (1972) used it to develop a theory of the firm
based on the supervision of team production.1 Other work has used agency theory to study
internal features of the firm. For example, Holmström and Milgrom (1990, 1991) used it
to study job design and task assignment. In labor economics, Lazear and Rosen (1981)
used it to study tournaments, while Mirrlees (1976) and Calvo and Wellicz (1978) used it
to study incentives in hierarchies.2 In growth and development, Bannerjee and Newman
(1993) and Aghion and Bolton (1997) used agency models to study organizational structure,
occupational choice, and growth. In corporate finance, agency theory has been extensively
used to study capital structure by analyzing the strategic interaction between managers,
equity holders, and debtors.3
In these examples there are two common features that we want to emphasize. First,
more than one person participates in each arrangement, and second, private information
causes an incentive problem. Because of these features, contracts in these environments
are characterized by joint consumption and joint production. For example, in a career
tournament, promotion depends on relative performance. In team production and in a
hierarchy, production depends on all of the member’s efforts.
Arrangements with joint production and joint consumption are precisely what club
theory, developed originally by Buchanan (1965), was designed to study. A club’s members,
1
A more formal treatment is contained in Holmström (1982).
Also, see the survey in Lazear (1996).
3
See the survey in Harris and Raviv (1992).
2
2
be it through their characteristics or activities, can affect the production of the club good or
their enjoyment of it. Common examples of clubs include swimming pools and marriages.
Our strategy in this paper is to treat firms as clubs. We work with a private information model where firms form to supervise production. These multi-agent supervisory
arrangements require multi-agent contracts with the joint consumption and joint production features of a club good. Like Alchian and Demsetz (1972), we identify these supervisory
production arrangements as firms.
We decentralize the economy using recent developments in general equilibrium club
theory by Cole and E.C. Prescott (1997) and Ellickson, Grodal, Scotchmer, and Zame
(1999). Unlike partial-equilibrium treatments of multi-agent principal-agent theory, our
decentralization does not take reservation utility levels, input prices, or even assignments
of individuals to principal-agent relationships as exogenous. Instead, club memberships
(the multi-agent contracts or firms) and other goods are priced and traded in markets and
the industrial organization of the economy is solely a function of the primitives.4
Our identification of firms as clubs departs from the classical treatment of the firm in
competitive equilibrium, where a firm is an exogenously specified entity that maximizes
profits.5 Instead, our identification is closer in spirit to that of McKenzie (1959, 1981).
In McKenzie’s formulation, the aggregate production set is a convex cone and firms are
identified with entrepreneurial factors supplied by individuals. In this paper, the aggregate
production set is also a convex cone but instead of individuals supplying an entrepreneurial factor they supply their participation in a contract, participation that we identify as
membership in a firm. The formation of firms thus appears as another linear activity in
the aggregate production set.6
Section 2 contains a prototype that is used to illustrate our theory and techniques.
Production requires capital and labor effort. Individuals may work alone, as in a sole
proprietorship, or they may work in a two-person firm. A sole proprietor’s output is public
4
Legros and Newman (1996) is an important paper on extending multi-agent principal-agent models
to general equilibrium. They studied incentive-based organizations in a general equilibrium economy with
risk neutral agents. Instead of using competitive analysis, they used a core equilibrium concept and took
the price of the capital input as exogenous.
5
See Arrow and Debreu (1954).
6
For more on this perspective, see the discussions contained in McKenzie (1981) and Hornstein and
E.C. Prescott (1993).
3
but his effort is private information so there is a moral hazard problem. The two-person
firm consists of a worker and a supervisor. The worker operates a technology identical
to that of a sole proprietor. All the supervisor does is monitor the worker’s effort; the
supervisor’s effort is not an input into production. The organizational tradeoff is between
a sole proprietorship limited by moral hazard versus a supervisor-worker firm that requires
costly labor monitoring.
Following Cole and E.C. Prescott (1997) and Ellickson, et al. (1999), we work in a
continuum economy with a finite number of types of agents, where only clubs that consist
of a finite number of agents may form. We make the minor assumption that points of
relevant sets like consumption and output lie in finite sets and the important economic
assumption that allocations are lotteries over these sets.7 Our commodity space contains
the set of incentive-compatible contracts and the corresponding jobs. Agents then choose
a probability distribution, which may be deterministic, over these assignments.8 This
commodity space allows us to properly add up memberships in firms, ensuring that in
equilibrium there is one person for each job in each firm.9
After describing the environment and defining the contracts as club goods, we formulate
the general equilibrium problem as a Pareto program. In addition to the usual resource
constraints on consumption and capital usage, there are also resource constraints on firm
formation. These constraints ensure that for each firm created, there is one person assigned
to each job in the firm. The Pareto program is linear but its dimensions are too large to
practically solve using standard linear programming methods. Instead, we describe how
the Pareto program can be computed using the Dantzig-Wolfe decomposition algorithm.
7
Private information environments naturally utilize such lotteries. Consumption sets constrained by
incentive compatibility conditions may not be convex, or even if they are, commodity spaces without
lotteries may preclude possible gains to trade. See Cole (1989).
8
This assignment feature of our models is related to occupational choice models. Sattinger (1993)
discusses these models in his survey on assignment models of the earnings distribution. Like our paper,
assignment models in the tradition of Koopmans and Beckmann (1957) formulate the allocation problem
as a linear program and use the dual to generate prices. Unlike our paper, this literature does not use
lotteries, restricting solutions to be integer valued.
9
This decentralization uses a different commodity space then the one used by E.C. Prescott and
Townsend (1984a, 1984b) to decentralize their private information economies, which corresponded to single
agent principal-agent theory. Instead of directly incorporating incentive and related constraints in the set
of feasible commodities, they placed these constraints in agents consumption sets and used as a commodity
space the space of contract characteristics. We did not use this commodity space because it cannot be used
to keep track of the numbers of clubs.
4
Details of this implementation are contained in Appendix A. The last subsection of Section 2
provides a numerical example for illustrative purposes.
Section 3 contains the decentralization. A competitive equilibrium is defined. The
Welfare theorems and an existence theorem are provided. The proofs are based on Negishi’s
mapping from the space of Pareto weights to the space of transfers. These proofs are
contained in Appendix B.
Section 4 reports solutions to several additional numerical examples for the prototype.
In the examples, wealth distribution and the aggregate capital stock determine the equilibrium industrial organization of the economy. Section 5 demonstrates how our approach
can be used to study other incentive problems. It contains the following extensions to the
prototype: limited commitment, incentive problems internal to a multi-agent firm, agents
who differ intrinsically in their preferences and abilities, and hierarchies with incentives.
Finally, Section 6 provides some concluding comments.
2
A Prototype
In this prototype, we consider a continuum economy where agents can be assigned to two
different types of firms. The first type of firm is sole production, or self-employment.
Output of this firm is a stochastic function of the capital input and effort. While output
and the capital input are publicly observed, effort is not, thus creating a version of the
classic moral hazard problem. The second type of firm is a supervisor-worker pair. Like
the self-employment firm, there is a capital input and an effort input into production. The
supervisor monitors and observes the worker’s effort. To avoid difficult issues involving
collusion we assume that since effort is observed by more than one person, it can costlessly
be made public, as in Harris and Townsend (1981). Supervision is not costless because
we require the supervisor to extend the same amount of effort as the worker he monitors,
though we do assume that supervisory effort is less onerous. Furthermore, in several of
the economies that we study, capital is relatively abundant and labor is relatively scarce.
Consequently, assigning a person to supervise rather than to work alone can lead to foregone
output.
Our strategy is first to describe the environment and then to describe each type of
5
possible firm. For each type of firm, there is a corresponding contract as in the agency
literature. The contract will specify objects such as the capital input, effort taken, and
output dependent consumption. In the two-agent firm, the contract will specify these
objects for both members of the firm. This multi-agent contract is the club good. After
specifying the contract for a firm, we will represent a contract as a single good. Utilities and
resource constraints will be expressed in terms of the contracts rather than the components
of the contracts. This representation is an essential step in our decentralization of these
club economies with private information, though in subsequent analysis, we retrace our
steps and analyze prices in terms of the items in the contract.
2.1
Environment and notation
There is a continuum of agents of measure one. All agents have the same preferences over
consumption c ∈ C, effort a ∈ A, and job j ∈ J = {w, s}. Job j = w means the agent
is a worker and job j = s means the agent is a supervisor. When referring to effort and
consumption of the worker and supervisor in a two-person firm, we will index them by the
job. We assume that a member of a self-employed club is a worker for utility and production
purposes. We express the utility function over consumption, effort, and job as U(c, a, j).
Even though we do so in our examples, it is not necessary for us to assume separability
or concavity of this function. We will be choosing probabilities over consumption and the
other variables and expected utility is linear in lotteries.
There is a production technology freely available to all agents that produces output
q ∈ Q as a stochastic function of worker’s effort a and a capital input k ∈ K. We
represent this function as a conditional probability distribution p(q|a, k). For convenience,
we assume that p(q|a, k) > 0, for all q, a, k. Furthermore, production shocks are identically
and independently distributed over technologies.
We assume that the sets C, Q, A, and K contain finite numbers of elements. This
assumption will greatly facilitate computation and proofs. For some variables, like consumption, this assumption should be viewed as an approximation to a continuum. For
other variables, like the capital input, this assumption can be viewed as fundamental and
not an approximation. In practice, we often want to study technologies with an indivisibility in inputs such as with a plant or on a smaller scale a machine.
6
There is a finite number, I, of agent types in the economy. We denote an agent’s type
by i ∈ {1, ..., I}. The number of each type i of agents is a positive fraction, αi > 0,
of the population. In the Pareto program, types will only differ in their Pareto weights
λi ≥ 0. In the decentralization, types will only differ in their non-negative endowment of
capital κi . The total endowment of capital is κ =
P
i
αi κi . This capital is divisible and is
the fundamental ingredient into creating the capital input k. We assume that it can be
converted at a constant rate of return of one.10
2.1.1
Self-employment firms
The contract
A self-employment firm or club consists of one unsupervised agent. The agent’s effort a
is private information but his output q and consumption c are public information. The
contract between this agent and the rest of the economy is a probability distribution over
an assigned capital input k, a recommended effort a, an output q, and consumption c. The
probability distribution is defined over the set C × Q × A × K. We assume that there
are n1 elements in this grid. Given that an agent is in a self-employment firm, we denote
π(c, q, a, k) as the probability of receiving a (c, q, a, k) grid point. There is no explicit
reference in self-employment clubs to the job of the agent since by construction the agent
is the worker. It should be noted that this distribution does not preclude a deterministic
assignment of capital, effort, and output-contingent consumption. Many of our examples
will contain this feature. An example of such a contract would be
probability
0.20
0.80
cw
0.00
0.45
q
0.00
1.00
a
0.40
0.40
k
2.00
2.00
In this contract, the agent uses 2.00 units of the capital input and supplies 0.40 units
of labor effort, both with probability one. He produces the low output (q = 0.00) twenty
percent of the time and the high output (q = 2.0) eighty percent of the time. If he produces
the low output he receives 0.00 units of consumption, and if he produces the high output
he receives 0.45 units of consumption.
10
We distinguish between the capital endowment and the capital input in order to not restrict potential
values of the distribution of wealth and to allow transactions in the credit market of arbitrary size.
7
The expected utility an agent receives from being in a self-employment club is
X
π(c, q, a, k)U(c, a, w).
c,q,a,k
Not all probability distributions over the grid are feasible. A feasible π must be incentive
compatible, be consistent with the technology p(q|a, k), and be a probability measure.
Incentive constraints take the following form,
X
c,q
π(c, q, a, k)U (c, a, w) ≥
X
π(c, q, a, k)
c,q
p(q|â, k)
U (c, â, w), ∀k, â, a.
p(q|a, k)
(1)
The next set of constraints ensure that π be consistent with the exogenous probability
p(q|a, k). These “mother nature” constraints are
∀q̄, ā, k̄,
X
π(c, q̄, ā, k̄) = p(q̄|ā, k̄)
c
X
π(c, q, ā, k̄).
(2)
c,q
Finally, the probability measure constraint is that π be non-negative and that
X
π(c, q, a, k) = 1.
(3)
c,q,a,k
The set of feasible contracts is Π1 = {π ∈ <n+1 |π satisfies (1), (2), (3)}. We will refer to each
element of Π1 as a different type of self-employment firm.
The contract as a commodity
If we were to restrict ourselves to self-employment firms, we could follow E.C. Prescott and
Townsend (1984) and decentralize this economy by making the commodity space Euclidean with the same number of dimensions as there are grid points. Agents would choose
quantities of each grid point and the incentive, mother nature, and probability measure
constraints would be placed in agents’ consumption sets.
As we will see in our discussion of the supervisor-agent firm, that strategy does not
work in our club environment. Instead, we use a commodity space that is based on an
alternative representation of the contracts. Rather than letting agents choose components
of the contract, we let agents choose quantities of feasible contracts and base our commodity
space on this alternative representation.
Let B1 ⊂ Π1 be the set of basic feasible solutions to the linear inequalities defined by
Π1 , with the notation b1 ∈ B1 . The cardinality of B1 is m1 . As is well known, this set
8
is the set of extreme points of the convex polyhedron defined by Π1 . This set is bounded
because Π1 is bounded. Furthermore, because there are a finite number of variables and a
finite number of linear inequalities, the set B1 contains a finite number of elements.
Our strategy is to allow agents to choose a lottery only over the set of contracts B1 .
There is no loss in generality from this assumption since the lotteries make the convex hull
of B1 feasible, and the convex hull of B1 is Π1 . To implement our strategy, we first define
utility in terms of these contracts. Preferences are
u(b1 ) =
X
b1 (c, q, a, k)U (c, a, w).
c,q,a,k
Next, we define the expected amount of resources used by a type b1 self-employment firm.
Consumption use is the excess of consumption over output,
r(c−q) (b1 ) =
X
c,q,a,k
b1 (c, q, a, k)(c − q),
and capital usage is
rk (b1 ) =
X
b1 (c, q, a, k)k.
c,q,a,k
2.1.2
Multi-agent firms as supervisor-worker pairs
The contract
There are two agents in this type of firm or club. The contract between one of these
firms and the rest of the economy is a probability distribution over an assigned capital
input k, an effort level for the worker aw , an effort level for the supervisor as , an output
q, consumption for the supervisor cs , and consumption for the worker cw . We write the
grid as C × C × Q × A × A × K and denote n2 as the number of elements in this grid.
Given a supervisor and worker in a supervisor-worker firm, let π(cw , cs , q, aw , as , k) be the
probability of the members receiving point (cw , cs , q, aw , as , k). At this point, we do not
identify who is the worker and who is the supervisor. That decision will be incorporated
later. An example of one of these supervisor-worker contracts is
cs
q
aw = as
probability cw
0.50
0.05 0.35 0.00
0.40
0.05 0.35 1.00
0.40
0.50
9
k
1.00
1.00
With probability one, the supervisor-worker firm uses 1.00 units of the capital input and
both the worker and supervisor supply 0.40 units of effort. The low (0.00) and high (1.00)
outputs are produced with equal probability, while the worker receives a constant wage of
0.05 and the supervisor receives a constant wage of 0.35.
The expected utility that a worker receives from being in a supervisor-worker firm is
X
π(cw , cs , q, aw , as , k)U(cw , aw , w),
cw ,cs ,q,aw ,as ,k
while the expected utility of a supervisor is
X
π(cw , cs , q, aw , as , k)U(cs , as , s).
cw ,cs ,q,aw ,as ,k
As we said earlier, we assume that the use of a supervisor makes the worker’s effort
public. Consequently, there are no incentive constraints. The mother nature constraints
are
∀q̄, āw , ās , k̄,
X
π(cw , cs , q̄, āw , ās , k̄) = p(q̄|āw , k̄)
cw ,cs
X
π(cw , cs , q, āw , ās , k̄).
(4)
cw ,cs ,q
We also assume that to supervise a worker, a supervisor must spend an equal amount of
effort as the worker. For example, if a worker works an 8-hour shift, the supervisor needs
to be there the whole time. We write these constraints by putting zero probability on other
effort configurations,
∀(cw , cs , q, aw , as , k) 3 aw 6= as , π(cw , cs , q, aw , as , k) = 0.
(5)
Finally, the probability measure constraint is that π be non-negative and that
X
π(cw , cs , q, aw , as , k) = 1.
(6)
cw ,cs ,q,aw ,as ,k
The set of feasible contracts is Π2 = {π ∈ <n+2 |π satisfies (4), (5), (6)}. Again, we refer to
each element of this set as a particular type of supervisor-worker firm.
In this two-agent firm, the object π is a contract that specifies joint usage of capital,
each member’s effort, and each member’s output dependent consumption. The contract
contains joint production features as well as joint consumption features. For these reasons,
the contract is a club good.
10
The contract as a commodity
As with the self-employment firm, we express the contract in units rather than in terms
of components. Let B2 be the set of basic feasible solutions to Π2 , with notation b2 ∈ B2 .
The cardinality of B2 is m2 . Preferences defined over these contracts are
u(b2 , j) =
X
b2 (cw , cs , q, aw , as , k)U (cj , aj , j), j = w, s,
cw ,cs ,q,aw ,as ,k
while the expected resource usage for each b2 ∈ B2 is
r(c−q) (b2 ) =
X
cw ,cs ,q,aw ,as ,k
rk (b2 ) =
X
b2 (cw , cs , q, aw , as , k)(cw + cs − q), and
b2 (cw , cs , q, aw , as , k)k.
cw ,cs ,q,aw ,as ,k
2.2
The Pareto program
Our strategy is to choose lotteries over the grid of basic feasible solutions (which themselves
are lotteries). As we mentioned earlier, there is no loss in generality from this strategy.
The sets of feasible contracts, Π1 and Π2 are both convex sets, and any element in a convex
set can be represented by a convex combination of its extreme points. The advantage of
this strategy is that our problem remains in Euclidean space, greatly facilitating proofs.
Furthermore, we will interpret the lotteries as fractions of the population in equilibrium.
The commodity space is <m1 +2m2 +1 , where mi is the number of elements in Bi , i = 1, 2,
where m2 is multiplied by two because there are two jobs in supervisor-worker clubs, and
where the last dimension is for the capital input. One of the choice objects is the fraction
of each agent type i assigned to each firm type, that is over B1 and B2 , and assigned to
each possible job within a firm. We refer to these probabilities as πi , but the support is the
types of firms, not the grid used earlier. It is useful to explicitly express some constraints
in terms of the number of clubs. Denote δ(b1 ) as the number of b1 clubs in the economy
and δ(b2 ) as the number of b2 clubs in the economy.
We can now proceed to write out the Pareto program.
11
Program 1
max
πi ,δ≥0,k
X
i
λi αi
X
πi (b1 )u(b1 ) +
b1
X
b2 ,j
πi (b2 , j)u(b2 , j)
subject to the probability measure constraints
∀i,
X
πi (b1 ) +
b1
X
πi (b2 , j) = 1,
(7)
b2 ,j
the club or matching constraints
∀b1 , δ(b1 ) =
∀b2 , δ(b2 ) =
X
X
αi πi (b1 ),
(8)
i
αi πi (b2 , 1) =
i
X
αi πi (b2 , 2),
(9)
i
and the resource constraints
X
δ(b1 )r(c−q) (b1 ) +
b1
X
b1
δ(b1 )rk (b1 ) +
X
b2
X
b2
δ(b2 )r(c−q) (b2 ) ≤ 0,
(10)
δ(b2 )rk (b2 ) − k ≤ 0, and
(11)
k ≤ κ.
(12)
The program assigns a fraction πi (b1 ) of each agent type i to each b1 -type of selfemployment firm and, similarly, it assigns a fraction πi (b2 , j) to each job j in each b2 -type
of supervisor-worker firm. Incentive compatibility and mother nature constraints have
already been incorporated through our definitions of B1 and B2 . Constraints (7) are the
probability measure constraints, one for each agent type i. The resource constraints are
(10), (11), and (12), with the coefficients r defined earlier and the number of firms δ defined
as above. The consumption resource constraint (10) requires that no more consumption is
consumed than produced by the economy. Constraints (11) and (12) ensure that the use
and production of the capital input does not exceed the capital endowment κ.
12
The club or matching constraints are (8) and (9). The first set of these constraint (8),
for the self-employment firms, simply defines the number of these firms in the economy.
The second set of club constraints (9) defines the number of supervisor-worker firms, as
well, but it also ensures that for each supervisor-worker club in the economy with contract
b2 ∈ B2 , there is one agent assigned to be a supervisor of a b2 club and one agent assigned
to be a worker of that same type of club.11
This program is linear. Despite the apparent simplicity of the notation, however, it is
impractical to directly compute solutions to it. The sets B1 and B2 contain astronomical
numbers of elements, so the number of variables and the number of club constraints (8)
and (9) are huge.12 Enumerating this set is clearly not practical. To compute solutions, we
instead rewrote the program in terms of the components, that is, the grids of consumption,
output, effort, and capital. The details are contained in Appendix A.
2.3
An example
In this subsection, we provide a simple example that illustrates the Pareto Program. We
use the following grids
C = {0.00, 0.05, 0.10, ..., 1.20},
Q = {0, 1},
A = {0.0, 0.4},
K = {0, 1, 2}.
There are two types of agents with λ1 = 0.2 and λ2 = 0.8. The aggregate capital endowment
is κ = 0.4. (Later we will vary the Pareto weights and the capital endowment in several
experiments.) The two types of agents are equal fractions of the population, so α1 = α2 =
0.5.
11
If we had followed E.C. Prescott and Townsend (1984) and used components of contracts as our
commodity space, it is not clear how we could have designed analogous constraints that would ensure that
the club memberships properly add up.
12
For example, if there are 10 constraints and 100 variables in Π1 then there are at most 100 choose 10
possible candidates for the set of basic feasible solutions (over 13 trillion).
13
Preferences
Preferences are
U (c, a, w) = c0.5 + (1 − a)0.5 , and
U (c, a, s) = c0.5 + (1 − 0.25a)0.5 .
Agents are risk averse and dislike effort but supervisory effort is less painful than worker
effort.
Technology
There are two inputs into production, the capital input and the worker’s effort. When
capital is 0, effort is unproductive and output is virtually zero.13 Formally,
a q=0 q=1
p(q|a, k = 0.0) = 0.0 .99
.01
0.4 .99
.01
When capital is at its intermediate value of one, effort becomes productive. The production
function is
a q=0 q=1
p(q|a, k = 1.0) = 0.0 0.8
0.2
0.4 0.5
0.5
Finally, when capital is at its highest level, effort is even more productive. The production
function is
a q=0 q=1
p(q|a, k = 2.0) = 0.0 0.6
0.4
0.4 0.2
0.8
In expected terms, the marginal product of capital for the low effort is practically linear
while it is diminishing for the high effort.
Experiment: κ = 0.4, λ1 = 0.2
We report a solution to the case where the aggregate capital stock is κ = 0.4 and the Pareto
weight on type-1 agents is λ1 = 0.2 (so λ2 = 0.8). In the Pareto optimum we found two
types of supervisor-worker firms and two types of self-employment firms. The levels of b2
for the two supervisor-worker firms are as follows. The first one consists of two low Pareto
13
For programming convenience we did not set any of these probabilities to zero or one.
14
weight type-1’s. There are number δ(b2 ) = 0.10 of these clubs in the economy, and these
clubs employ forty percent of the type-1’s (π1 (b2 ) = 0.40), which is twenty percent of the
population. The contract is
cs
q
aw = as
probability cw
0.50
0.05 0.05 0.00
0.40
0.50
0.05 0.05 1.00
0.40
k
1.00
1.00
Because there is no moral hazard in supervisor-worker firms and agents are risk averse,
consumption is constant over output.
The second type of supervisor-worker firm consists of a type-1, who is supervised by a
type-2. There are number δ(b2 ) = 0.30 of these firms in the economy, employing 60 percent
of the population. They employ the remaining 60 percent of type-1’s (π1 (b2 ) = 0.60) and
they also employ 60 percent of the type-2’s (π2 (b2 ) = 0.60). This contract is
probability cw
cs
q
aw = as
0.50
0.05 0.35 0.00
0.40
0.50
0.05 0.35 1.00
0.40
k
1.00
1.00
Supervisors are paid more in this type of firm than in the previous type because these
supervisors are type-2’s, who have the higher Pareto weight. Also, notice that there is
full-risk sharing for the entire type-1 population since there is no incentive problem for any
of their efforts, that is, none of them are self-employed.
While the remaining two firms are technically self-employment firms, they really correspond to inactivity for the residual 40 percent of the type-2 population. In these firms,
both zero units of capital and effort, a = k = 0.0, are supplied. Members of these firms
are essentially the idle rich. There are number δ(b1 ) = 0.16 clubs in which the member
receives 0.35 units of consumption with conditional probability one and there are number
δ(b1 ) = 0.04 clubs in which 0.40 units of consumption are received.14 These firm distributions imply that type-2 agents are randomly assigned with probability 0.40 to inactive
self-employment b1 firms. Finally, note that the total number of firms in this economy is
14
Despite the curvature of the utility function, type-2 agents did not always receive the same consumption.
This is not the result of any inherent non-convexity in the economy but, instead, it is solely a function of
the coarseness of the consumption grid. Since preferences are separable and utility from consumption is
concave, if we made the consumption sufficiently fine the program would put mass one on some consumption
point between 0.35 and 0.40.
15
0.60 but because supervisor-worker firms use two people, the entire population of size 1.0
is assigned to a firm.
3
Decentralization
The commodity space is Euclidean with L = <m1 +2m2 +1 . Consequently, prices p lie in this
set as well.
Consumer’s problem
The consumption set for type-i agents is
Xi = {xi ∈ <m1 +2m2 ⊂ L+ |
X
xi (b1 ) +
b1
X
xi (b2 , j) = 1}.
The endowment of a type-i agent is an ξi ∈ L. Expected utility for an agent is
P
b2 ,j
(13)
b2 ,j
P
b1
x(b1 )u(b1 )+
x(b2 , j)u(b2 , j). Sometimes we write it more compactly as ui xi . A type-i’s problem is
to maximize expected utility,
max
x
i
X
x(b1 )u(b1 ) +
b1
X
x(b2 , j)u(b2 , j)
b2 ,j
subject to xi ∈ Xi and a budget constraint
X
b1
xi (b1 )p(b1 ) +
X
b2 ,j
xi (b2 , j)p(b2 , j) ≤ ξi p.
In our economies, the natural endowments are those where agents of type-i have a nonnegative amount of capital, κi and zero quantities of the other goods. In these cases we will
write the endowment as ξi = (0, 0, ..., 0, κi ) so income will equal pk κi . Income from their
capital endowment is used by agents to purchase a firm type, a job, and a distribution over
the capital input, output, and consumption.
Before continuing with the decentralization, it might be helpful to examine the prices
and income that support the Pareto optimum reported in the previous example as a competitive equilibrium. In this experiment we found that the type-1 consumers had negative
income. It will be shown later that the price of capital is non-negative so this optimum
cannot be supported solely by a distribution of the capital good. Instead, some taxes and
transfers are needed as well. Normalizing the price of capital to be one, per capita incomes
16
needed to support this optimum were -0.217 for type-1 agents and 1.017 for type-2 agents.
This could correspond to a capital distribution of 0.0 to type-1’s with a tax on them of
-0.217 (all expressed in per capita terms) and a distribution of 0.8 to type-2’s with a transfer
to them of 0.217.
The prices of a job and its associated contract b2 in the two types of supervisor-agent
firms, that is, the p(b2 , j), are
Worker
first type of b2 firm
-0.253
second type of b2 firm -0.253
Supervisor
-0.076
0.921
Note that the price of being a worker in either firm is identical. If the price was not
identical, the type-1 agent would choose the cheaper firm since both firms give him identical
consumptions and efforts, that is, identical utilities from membership. Note again, that the
type-1 worker does not care about the type of his partner. The negative price for the
workers (and a supervisor in the first firm) means that the worker or the supervisor is
being paid to join the firm. For type-1 agents, being paid to join a firm is how they earn
“income” to overcome the deficit in their budget constraint. Note that the payment for
being a supervisor is less than it is for being a worker because the supervisor gets more
utility.
The prices of the two types of self-employment firms, p(b1 ), are
price
first type of b1 firm
1.129
second type of b1 firm 1.295
The price paid to be in the first type of firm is slightly lower than the price paid to be in the
second type because members consume a lower amount in the first one, while producing
the same expected amount in both.
Intermediary’s problem
Returning to the decentralization, we label the Arrow-Debreu firm of standard competitive
analysis as an intermediary, partly because it trades in insurance and credit contracts, but
also to avoid confusion with the self-employment and supervisor-worker firms that are the
focus of this paper. Because there is constant returns to scale, we assume without loss of
generality that there is only one intermediary. The intermediary produces a y ∈ L that is
17
in the intermediary’s production set. For convenience, we denote δ(b1 ) as the number of b1
clubs produced by the intermediary and δ(b2 ) as the number of b2 clubs produced by the
intermediary.
The production set consists of several constraints. The first one is a resource constraint
on the capital input. The intermediary purchases capital yκ from consumers and then uses
it to create as its output the capital input k used in creating goods (b1 ) and (b2 , j). As
is conventional, the input yk is negative in sign and outputs are positive in sign. This
constraint is
X
δ(b1 )rk (b1 ) +
b1
X
b2
δ(b2 )rk (b2 ) + yκ ≤ 0,
(14)
so capital sold is less than or equal to capital purchased.
The intermediary also produces insurance contracts that are contingent consumption
transfers. It collects premia and distributes indemnities, and these need to balance to zero.
This constraint is
X
δ(b1 )r(c−q) (b1 ) +
b1
X
b2
δ(b2 )r(c−q) (b2 ) ≤ 0.
(15)
In selling supervisor worker firms, b2 , the intermediary is also staffing positions in them.
Each one of these b2 firms requires two members so for each level of b2 produced there needs
to be one worker for each supervisor. We write this matching constraint (and the trivial
one for the self-employment firm) as
∀b2 , δ(b2 ) = y(b2 , w) = y(b2 , s),
(16)
∀b1 , δ(b1 ) = y(b1 ).
(17)
The intermediary’s production set is Y = {y ∈ L, δ(b1 ) ∈ <m1 , δ(b2 ) ∈ <m2 |(14), (15), (16), (17) hold}.
The intermediary’s maximization problem is to solve
max
y,δ(b1 ),δ(b2 )
X
p(b1 )y(b1 ) +
b1
X
p(b2 , j)y(b2 , j) + pk yκ
b2 ,j
subject to (y, δ(b1 ), δ(b2 )) ∈ Y .
Market clearing
Market clearing is as follows
∀b1 ,
X
i
αi (xi (b1 ) − ξi (b1 )) = y(b1 ),
18
(18)
∀b2 , j,
X
i
αi (xi (b2 , j) − ξi (b2 , j)) = y(b2 , j),
−
X
αi κi = yκ .
(19)
(20)
i
As we said earlier, in our economies we will usually set ξi (b1 ) = 0, and ξi (b2 , j) = 0.
As is apparent, we have mapped our example into the standard formulation. Unlike
our example, however, in the definitions and theorems that follow we allow heterogeneity
in preferences and consumption sets. We provide this generality because we are interested
in extending our economies in these directions and these extensions do not require any
substantive modifications to the proofs.
Definition 1 Let x be the vector of consumptions across the I types of agents. A compensated equilibrium in this economy is a (x∗ , y ∗ , p∗ ) such that
1. ∀i, x∗i minimizes pxi subject to xi ∈ Xi and ui xi ≥ ui x∗i .
2. y ∗ maximizes py s.t. y ∈ Y .
3.
P
∗
i αi (xi
− ξi ) = y ∗ .
Definition 2 A competitive equilibrium in this economy is the same as a compensated
equilibrium except that 1. above is replaced by
1. ∀i, x∗i maximizes ui xi s.t. xi ∈ X and pxi ≤ pξi .
Notice that substitution of the market clearing constraints, 3 in Definition 1, into the intermediary’s production set delivers the resource constraints in the Program 1. Alternatively,
we can write Program 1 in terms of xi and y if we substitute the production set and market
clearing constraints for the resource constraints.
3.1
Theorems
In this section we state the Welfare Theorems as well as an existence theorem. Aside from
the bounded linear structure of the economy, the only assumption we use is the following
Assumption 1 (Non-satiation) For any feasible allocation x, there exists for all i, x0i ∈
Xi such that ui x0i > ui xi .
19
In our economies non-satiation is relatively easy to satisfy. For example, one can include
a very high utility consumption grid point that cannot be offered to any agent type with
probability one.
Our proofs are based on Negishi (1960) and are contained in the Appendix B. The
Negishi mapping finds a fixed point in the space of Pareto weights at which consumers’
budget constraints are satisfied. The mapping solves the Pareto program for each set of
Pareto weights so the fixed point is also a solution to the Pareto program. We adopted
Negishi’s strategy because his programming approach is relatively straightforward in our
linear economies. Furthermore, the proofs will demonstrate that as long as we restrict ourselves to analyzing Pareto optima generated by non-zero Pareto weights, we are guaranteed
that a competitive equilibria exists that supports that Pareto optimum. This is a useful
result since we will be computing Pareto optima and then finding prices and incomes that
support them as competitive equilibria.
Theorem 1 (First Welfare Theorem) Given Assumption 1, every competitive equilibrium is Pareto optimal.
As usual, the proof is straightforward so we do not include it in Appendix B. However,
note that a competitive equilibrium is a fixed point of Negishi’s mapping at λ > 0, that
a fixed point of his mapping is a solution to the Pareto program with λ > 0, and thus is
necessarily Pareto optimal.
The Second Welfare Theorem is usually stated in terms of all Pareto optima. We now
state a slightly stronger version of it.
Theorem 2 (Second Welfare Theorem) Any solution to the Pareto program for some
Pareto weights λ ≥ 0 can be supported as a compensated equilibrium.
This theorem is slightly stronger than the usual formulation because all Pareto optima are
solutions to the Pareto program but not all solutions to the Pareto program — those where
λi = 0, for some i — need be Pareto optimal.
The next theorem provides a sufficient condition that can be checked to see if a Pareto
optimum can be supported as a competitive equilibrium
20
Theorem 3 Any solution to the Pareto program for some Pareto weights λ > 0 can be
supported as a competitive equilibrium.
The standard approach for showing that a compensated equilibrium is a competitive
equilibrium is to assume there exists a cheaper point. The next theorem demonstrates the
connection between this strategy (Arrow’s Remark) and Theorem 3. It shows that a cheaper
point at a compensated equilibrium implies that the corresponding Pareto weights are nonzero so a competitive equilibrium exists. In a sense, then, the Pareto weight condition is
more general.
Theorem 4 Take a solution to the Pareto program for λ ≥ 0. If, at the corresponding
compensated equilibrium, there exists for all i, a cheaper point satisfying xi ∈ Xi , then
λ > 0.
We now prove existence of a competitive equilibrium using the Negishi mapping.
Theorem 5 (Existence) For any given distribution of endowments, if the Pareto weights
at a fixed point of Negishi’s mapping are non-zero, then a competitive equilibrium exists.
In the next subsection, we discuss different assumptions that may be used to guarantee
that a competitive equilibrium exists.
3.2
Analysis of equilibrium prices
Let µ(c−q) be the dual variable on the consumption resource constraint and µk be the dual
variable on the capital input resource constraint in Program 1. At a competitive equilibrium, these variables are equal to the dual variables in the intermediary’s maximization
problem (see the proofs to Theorems 2 and 3). Using this substitution and the first-order
conditions to the intermediary’s problem, prices are
p(b1 ) = µ(c−q) r(c−q) (b1 ) + µk rk (b1 ),
p(b2 , 1) + p(b2 , 2) = µ(c−q) r(c−q) (b2 ) + µk rk (b2 ),
pk = µk .
The first equation is the price of purchasing a unit of the b1 point, which is a probability
distribution over the (c, q, a, k) grid. The term r(c−q) is the expected compensation to the
21
self-employed person above and beyond what he produces and rk is the expected amount of
the capital input used in production. The price of a b1 good is simply the shadow price of
net consumption times the quantity of consumption plus the shadow price of capital times
the quantity of capital used. In order for an agent to purchase a self-employment firm at
level b1 , he must pay the resource cost required to operate the firm at that level.
The second equation is the total sum of purchase prices paid by a worker and a supervisor in a b2 firm. Just as with the self-employment firm, the total cost of a b2 supervisor-agent
firm is the value of the resources used by that level of b2 .
Remark 1 Different strategies can be used to guarantee existence of a competitive equilibrium. One strategy that can be used when agents are endowed with non-negative quantities
of the capital endowment and zero units of other goods is the following. Assume that the
zero consumption point is in the consumption grid and that expected output is positive for
all levels of effort and the capital input. We know from the resource constraint that µk ≥ 0
so agents will have at least zero income and we know that µ(c−q) > 0 by non-satiation. For
many standard utility functions, a contract that assigns zero capital, the lowest effort, and
distributes the zero consumption grid point is incentive compatible and thus in our commodity space. Assigning this contract with probability one is in the agent’s consumption set.
By assumption, such an allocation will produce positive output and therefore have a negative price. Since with non-negative endowments of the capital input, agents’ incomes are
non-negative, a cheaper point exists in their consumption set and a competitive equilibrium
exists.
To give a more specific interpretation of the pricing functions p(b1 ) and p(b2 , j), we
substitute in the r’s to obtain
p(b1 ) = µ(c−q)
X
c,q,a,k
p(b2 , 1) + p(b2 , 2) = µ(c−q)
b1 (c, q, a, k)(c − q) + µk
X
cw ,cs ,q,aw ,as ,k
+µk
X
X
b1 (c, q, a, k)k,
(21)
c,q,a,k
b2 (cw , cs , q, aw , as , k)(cw + cs − q)
b2 (cw , cs , q, aw , as , k)k.
(22)
cw ,cs ,q,aw ,as ,k
By marginally varying the components of b1 and (b2 , j) one at a time, we can see how these
pricing functions behave.
22
For example, suppose that we vary b1 , the probability of a self-employment firm receiving (c, q, a, k), in such a way as to keep the determination of the inputs a and k in
the firm constant (though these maybe deterministic or randomly determined) but allow
consumption compensation c to vary. In this contract, the joint probability of consumption
and output is
P
a,k b1 (c, q, a, k)
≡ prob(c, q), so the first term on the right-hand side of the
pricing equation (21) reduces to
µ(c−q)
X
c,q
prob(c, q)(c − q) = µ(c−q) E(c − q),
(23)
where the expectation is defined under the contract b1 (c, q, a, k). This expectation is the
expected deficit of compensation over output stipulated by the contract, multiplied by
the shadow price of consumption. Note that if consumption were a constant c under the
contract, then (23) simplifies even further to
µ(c−q) E(c − q) = µ(c−q) c − µ(c−q) E(q).
In this case the consumption compensation to the owner is purchased at price µ(c−q) , and
the output of the firm is sold at this same price. More generally, of course, c will vary with
q for incentive reasons, but the price of contract c(q) is still its actuarial fair value.
Similarly, we can vary marginally the capital input k. Here the easiest case for interpretation is a comparison of two contracts b1 (c, q, a, k) which vary with a discrete, nonrandom choice of capital k. In this case, again reverting to probabilities, b1 (c, q, a, k) ≡
prob(c, q, a, k), so if a particular input combination (a, k) receives weight or probability one
and all other input combinations receive probability zero, then for that (a, k) combination
prob(c, q, a, k) ≡ prob(c, q|a, k)prob(a, k) ≡ prob(c, q|a, k)
Taking the derivative with respect to k in the first term in (21) delivers
µ(c−q)
X
c,q
(c − q)
∂prob(c, q|a, k)
.
∂k
(24)
Condition (24), of course, is the marginal revenue product effect, here on the expected
net deficit. Again, in the special case of constant, output-invariant consumption c, (24)
reduces to
−µ(c−q)
X
q
q∂prob(q|a, k)
,
∂k
23
which is the expected marginal product of k in production times minus the shadow price
of consumption, as an increase in the expected surplus reduces the expected net deficit.
Likewise, a small marginal change in the recommended effort delivers
µ(c−q)
X
c,q
(c − q)
∂prob(c, q|a, k)
∂a
with the same interpretation.
We can gain additional insight into the behavior of prices by turning to the problem
confronting the household at these prices, and again considering special cases. Suppose
that xi (b1 ) puts mass one on at most one b1 and no other. Then the simplified problem
confronting the household is
max
X
U (c, a, w)b1 (c, q, a, k)
c,q,a,k
by choice of b1 in B1 satisfying the budget constraint
p(b1 ) ≤ pk κi .
(25)
Writing out that budget constraint more fully yields, as in (21),
µ(c−q)
X
c,q,a,k
b1 (c, q, a, k)(c − q) + µk
X
c,q,a,k
b1 (c, q, a, k)k ≤ pk κi .
(26)
Proceeding as before, suppose the household is choosing a contract or self-employment
firm which holds the (a, k) input pair fixed but varies in the consumption compensation
c(q), that is in prob(c, q). Then this special problem reduces still further to
max
X
U (c, a, w)prob(c, q)
(27)
prob(c, q)(c − q) ≤ Ir ,
(28)
c,q
subject to the first term in (26)
µ(c−q)
X
c,q,a,k
where Ir is residual of the household’s budget constraint left to purchase consumption, both
its level and its variability. Without incentive constraints the household would purchase
some level with no variation, that is, full insurance. For example, with two outputs q1 and
24
q2 and hence two states of nature, with exogenous probabilities determined by the input
combination (a, k), here held fixed, and with the choice of state contingent consumptions c1
and c2 , the ratio of probabilities in the budget line in (28) matches the ratio of probabilities
in the objective function (27). Thus the solution is full insurance, c1 = c2 unless one hits
a binding incentive constraint.
Likewise, a small change in the input choice k for the self-employment firm that the
household is buying implies both a negative income effect in the budget, in the term
µk (κi − k), and a change in the market odds line above, in the prices, as embodied in
prob(q|a, k).
Turning at last to the supervisor-worker firm, all of the above analysis has a straightforward extension. We focus only then on the decision as to whether to be a worker or
a supervisor in a particular b2 firm. Formally, let xi (b2 , w) denote the chosen probability
of being a worker and xi (b2 , s) be the probability of being a supervisor. Then the choice
problem of the consumer is
max xi (b2 , w)u(b2 , w) + xi (b2 , s)u(b2 , s)
subject to xi (b2 , w) + xi (b2 , s) = 1, and
xi (b2 , w)p(b2 , w) + xi (b2 , s)p(b2 , s) ≤ pk κi .
If for household i probability xi is to be interior, that is neither zero nor one, then the
first-order condition holds at equality for household i and
u(b2 , s) − u(b2 , w) = δi (p(b2 , s) − p(b2 , w)).
(29)
Define ∆U ≡ u(b2 , s) − u(b2 , w), and ∆P ≡ p(b2 , s) − p(b2 , w). As established in
Appendix B, δi is the Lagrange multiplier on household i’s budget constraint and it must
equal 1/λi in a competitive equilibrium. Thus if household i is indifferent to being a worker
or a supervisor, then
λi ∆U = ∆P.
Put differently, since household type i is indifferent in equilibrium to being a worker or a
supervisor, then he is willing to randomize, so the price differential is precisely the utility
differential, scaled by the Pareto weight.
25
More generally, there will not be an interior solution and equation (29) will not hold
at equality. In this case, with two types of agents and Pareto weights λ1 ≤ λ2 , for any b2
chosen with positive probability in equilibrium such that ∆U ≥ 0, we get the inequalities
λ1 ∆U ≤ ∆P ≤ λ2 ∆U,
and thus the price differential ∆P is bounded by the utility differences scaled by the Pareto
weights. For these inequalities to hold in equilibrium, the high Pareto weight agent must
be assigned the supervisory job, which is the higher utility job when ∆U ≥ 0. If for some
reason the equilibrium chooses a b2 where the high utility job is the worker’s job then
∆U ≤ 0, and the above inequalities would be reversed.
4
More Numerical Examples
In this section, we report solutions to different parameterization of the earlier example.
The only parameters we vary are the Pareto weights and the aggregate capital endowment.
All other features of the environment, the utility function, the grids, or the production
technology, were left unchanged. However, in Section 5.1 we add limited commitment to
the model. For each example, we calculated Pareto optimum by solving Program 1a in
Appendix A. Then, using this solution we solved the dual to Program 1 to calculate prices
and a distribution of the capital endowment that supported the Pareto optimum.
Experiment: κ = 0.4 and λ1 = 0.3 (Same aggregate capital but more equal Pareto
weights)
In this experiment, we leave the aggregate capital stock unchanged from the earlier example
but make the Pareto weights more equal. With this change, the distribution of clubs
changes. In this allocation there are 0.15 clubs consisting of two type-1’s that use 2 units
of capital and 0.10 clubs consisting of two type-1’s that use 1 unit of capital. Type-1’s
are gaining because now some of them get to do the less onerous supervisory effort while
at the earlier λ1 = 0.2 more of them were supplying the worker’s effort. All type-2’s are
now idle. This may appear surprising at first because type-2’s supply no effort, thus effort
disutility declines relative to what it was in the λ1 = 0.2 experiment. However, type-2’s
consumption also declines so that overall their total utility declines as is expected with a
26
declining Pareto weight. Again, supporting prices and incomes entail taxes on type-1’s and
transfers to type-2’s. However, at λ1 = 0.4 the allocation is similar to the λ1 = 0.3 one, but
incomes for the type-1’s arrive at zero so this allocation can be supported with a capital
distribution that gives zero capital to type-1’s and all of it to the type-2’s.
Experiment: Lower aggregate capital levels
If we lower the economy-wide aggregate level capital to κ = 0.2 then for all λ1 < 0.5
membership in supervisor-worker clubs consists of two type-1 agents. Because capital
is scarce in these economies, there is no shortage of low-Pareto weight agents to supply
supervisory effort.
Experiment: Higher aggregate capital levels
In the other direction, as economy-wide capital increases we see the persistence of (1, 2)
clubs, for relatively low λ1 . The reason for this persistence is that as capital becomes
abundant, supervisory effort becomes scarce and it is worth having high Pareto weight
individuals suffer a disutility of effort from supervision because their monitoring produces
a lot of consumption that they can consume.
We also observe an increased frequency of non-trivial self-employment clubs as κ increases from 0.4 and as long as λ1 is not too low. The next experiment contains these
characteristics.
Experiment: κ = 1.75, λ1 = 0.4 (Much higher aggregate capital)
In the Pareto optimum there are no supervisor-worker clubs. All production is being done
through self-employment, some of it trivial in a sense that will be explained below.
The first self-employment club consists of a type-1. There were 0.50 of these clubs in
the economy, which is the entire type-1 population. They are fully capitalized and effort
is being supplied at its maximum. As a consequence, moral hazard constraints bind and
consumption is a non-constant function of output.
probability
0.1040
0.0960
0.8000
cw
0.00
0.05
0.45
q
0.00
0.00
1.00
a
0.40
0.40
0.40
k
2.00
2.00
2.00
The type-2’s are split among three different self-employment clubs. There are 0.125
of the first type of club. In these clubs, the agents use zero units of capital, supply zero
27
units of effort and receive 0.75 units of consumption. There are 0.252 of the second type
of club. In these clubs, the agent uses 2.0 units of capital, supplies no effort, and receives
0.75 units of consumption. The last club is identical to the previous one except that the
agent receives 0.70 units of consumption. There are 0.123 of these clubs. These latter two
clubs have an expected level of output of 0.4 even though no effort is supplied.
The prices of the four type of self-employment clubs are
Type-1
First type-2
Second type-2
Third type-2
price
-0.232
3.795
3.795
3.538
The self-employment firm labeled Type-1 is the only club that type-1’s chose. The price
of this club is negative because net resource flows out are higher than the cost of inputed
capital. The other clubs are the three self-employment clubs chosen by type-2’s. For all
three of these clubs, the prices of membership are positive because net consumption inflows
far outweigh the capital input costs. Notice that the price of the first and second type-2
clubs are the same, despite using different levels of capital. They are the same because
both clubs supply the same consumption and effort. An agent’s utility is identical from
these two goods so if the price differed he would only choose the one that gave a higher
utility. Finally, the price of the third type-2 club is lower than the price of the other two
type-2 clubs since less consumption is paid out to members of this club.
4.1
Uniqueness
In our numerical experiments, we found that unequal Pareto weights and a smaller capital
stock led to more supervisor-worker firms. One concern with this statement, and similar
statements summarizing numerical examples, is that there may be multiple equilibria supported by different industrial organizational structures, that is, the mapping from Pareto
weights to industrial organization is a correspondence and the numerical solutions are picking a particular set of points from this correspondence.
In general, this is a potential problem but it did not seem to be a factor in the examples.
As we vary Pareto weights, we move along the Pareto frontier, from vertex to vertex of the
28
piecewise linear Pareto frontier, unless our hyperplane is coincident with a linear portion of
the Pareto frontier.15 At each vertex of the frontier, more than one separating hyperplane,
that is, set of Pareto weights, may support a particular Pareto optimal distribution of
utilities. As we discussed earlier, the Pareto weights equal the inverse of the marginal
utility of income so a Pareto optimum may be supported by more than one price system
and distribution of income.16 Indeed, we computed examples where this was the case
but the industrial organization of the economy was unchanged at these vertices. For this
reason, we do not think multiple equilibria affect our conclusions based on the numerical
examples.17 Finally, it is also worth mentioning that the piecewise linearity of the Pareto
frontier is not just a function of the consumption and action grids. Even with a continuum
of consumption and action levels and well-behaved preferences, incentive constraints or an
indivisibility in capital can create linear portions of Pareto frontiers.
5
Extensions
This section contains four extensions to the prototype. The first one adds a limited commitment problem. The second one restricts the supervisor to observe a signal that is only
partially correlated with the agent’s effort. The third extension adds heterogeneity in agents
abilities and preferences. The final one adds to the prototype by incorporating hierarchies
with incentives problems.
5.1
Limited commitment
In this section, we extend the previous example to incorporate limited commitment. We
keep the same supervisor-worker environment but with one modification. When a firm
receives the capital input, the supervisor may abscond with the capital and convert it into
15
Since we used a simplex-based algorithm to solve the linear programs, the numerical solutions were
always vertices of the Pareto frontier.
16
It is also possible that more than one set of dual vectors could support a solution to a particular
Pareto program. As was evident in Section 3.2, prices are unique up to the values of the dual variables.
Uniqueness of the dual variables is assured if the solution to the linear program is non-degenerate.
17
Multiple competitive equilibria are also possible for a given distribution of endowments and the general
equilibrium literature has long been concerned with this issue. In performing our Second Welfare Theorem
calculations, we did find an example where two different Pareto optima could be reached with the same
initial distribution of the capital stock. But, again, the industrial organization of the economy was not
affected, even though consumption levels were.
29
own consumption at some constant rate of return. If a supervisor does this he cannot
be punished or rewarded except to the extent that he receives utility from consuming the
capital. In other words, no consumption transfers to or from him may be made. In a
supervisor-worker firm, we assume that only the supervisor may run off with the capital.
We also allow the agent in the self-employment club to run off with any capital assigned
to him.
Incorporating this limited commitment feature requires an additional variable in the
grid spaces plus some additional constraints in the sets Π1 and Π2 . We let d ∈ D = {0, 1},
where d = 0 means the agent stays in the firm and does not run off with the capital.
Conversely, d = 1 means the agent runs off with the capital. If an agent runs off with the
capital, he converts it into consumption at some exogenous rate r with no effort supplied.
We modify the utility function to handle the departure decision. In particular, we write
V (k, d = 1) = U (rk, 0, j), while utility from staying is unchanged from before.
Self-employment firms
The grid space for the self-employment club is now C × Q × A × D × K. The addition of
limited commitment requires constraints that guarantees that it is optimal for an agent to
stay in the firm when recommended to do so. (Since the departure decision is public, we
will not require a limited commitment constraint on agents recommended to leave.)
The limited commitment constraint is
∀k,
X
c,q,a
π(c, q, a, d = 0, k)U (c, a, w) ≥
X
π(c, q, a, d = 0, k)V (k, d = 1).
(30)
c,q,a
On the left-hand-side of the constraint is the self-employed agent’s utility from staying and
on the right-hand-side is his utility from running off.
The incentive constraints are almost identical to the ones in the earlier problem. They
only apply if the agent stays, that is, if d = 0. The constraints are
X
X
p(q|â, k)
π(c, q, a, d = 0, k)U (c, a, w) ≥
π(c, q, a, d = 0, k)
U(c, â, w), ∀k, â, a. (31)
p(q|a, k)
c,q
c,q
The next set of constraints ensure that π is consistent with the exogenous probability
p(q|a, k). These “mother nature” constraints are straightforward if the agent stays. They
are simply
∀q̄, ā, k̄,
X
π(c, q̄, ā, d = 0, k̄) = p(q̄|ā, k̄)
c
X
c,q
30
π(c, q, ā, d = 0, k̄).
(32)
However, if the agent runs off, we assume that production is the same as it would have
been with no capital and no effort. Consequently, the mother nature constraints are
∀q̄, ā, k̄,
X
π(c, q̄, ā, d = 1, k̄) = p(q̄|a = 0, k = 0)
c
X
π(c, q, ā, d = 1, k̄).
(33)
c,q
Note that when d = 1, the only relevant specification of the production function is the one
with zero effort and zero capital. The grid allows non-zero specifications of effort along this
path but in equilibrium these grid points will receive zero weight since they give the agent
disutility without affecting production.
Finally there is the probability measure constraint, namely that π be non-negative and
X
π(c, q, a, d, k) = 1.
(34)
c,q,a,d,k
The set of feasible contracts in the self-employment firm with limited commitment is
Π1 = {π ∈ <2n1 |(30), (31), (32), (33), (34)}. The set of commodities is then the set of basic
feasible solutions, B1 , to Π1 .
Supervisor-worker firms
For supervisor-worker firms, we proceed much like we did for self-employment firms. We
assume for programming simplicity that if the supervisor runs off with the capital then the
remaining worker provides zero effort along with the zero capital input. In general, this is
not a desirable assumption since for some production functions effort may be productive
even for zero capital inputs. However, in the examples we study, effort is not productive
for a zero capital input so there are no adverse consequences from this assumption.
The grid space for the supervisor-worker firm is now C × C × Q × A × A × D × K.
The addition of the limited commitment requires a limited commitment constraint that
guarantees that it is optimal for an agent to stay in the firm when recommended to do
so. Again, since the departure decision is public we will not require a limited commitment
constraint on agents recommended to leave.
In the supervisor-worker firm, the limited commitment constraint applies only to the
supervisor. Therefore, the constraint is
∀k,
≥
X
π(cw , cs , q, aw , as , d = 0, k)U (cs , as , s)
cw ,cs ,q,aw ,as
X
π(cw , cs , q, aw , as , d = 0, k)V (k, d = 1).
cw ,cs ,q,aw ,as
31
(35)
The next set of constraints ensures that π be consistent with the exogenous probability
p(q|aw , k). These “mother nature” constraints are straightforward if the agent stays. They
are simply
∀q̄, āw , ās , k̄,
X
π(cw , cs , q̄, āw , ās , d = 0, k̄) = p(q̄|āw , k̄)
cw ,cs
X
π(cw , cs , q, āw , ās , d = 0, k̄).(36)
cw ,cs ,q
As before, if the agent runs off, we assume that production is the same as it would have
been with no capital and no effort. Consequently, the mother nature constraints are
∀q̄, āw , ās , k̄,
X
π(cw , cs , q̄, āw , ās , d = 1, k̄)
cw ,cs
= p(q̄|aw = 0, k = 0)
X
π(cw , cs , q, āw , ās , d = 1, k̄).
(37)
cw ,cs ,q
As before, we need constraints that ensure aw = as . We do this by
∀(cw , cs , q, aw , as , d, k) 3 aw 6= as , π(cw , cs , aw , as , d, k) = 0.
(38)
Finally there is the probability measure constraint, namely that π be non-negative and
X
π(cw , cs , q, aw , as , d, k) = 1.
(39)
cw ,cs ,q,aw ,as ,d,k
The set of feasible contracts in the supervisor-worker firm with limited commitment is
Π2 = {π ∈ <2n2 |(35), (36), (37), (38), (39)}. The set of commodities is then the set of basic
feasible solutions, B2 , to Π2 .
5.1.1
Numerical examples
The parameterization of our limited commitment examples are the same as in the previous
examples. In particular, if the limited commitment problem were formulated so that the
commitment constraints never bound, e.g., if r = 0, then the solution to the limited
commitment problem would be the same as the previous example. Again, we varied the
aggregate capital stock κ and the Pareto weights. The only additional parameter required
to run these experiments is the return on capital received by an agent who runs off, that
is, r. In the following examples, we set r = 0.25.
32
Experiment: κ = 0.4, λ1 = 0.2
In the Pareto optimum we found one supervisor-worker firms and three self-employment
firms. The supervisor-worker firm consisted of a type-1 worker supervised by a type-2.
There were 0.40 of these clubs in the economy, which employed eighty percent of the type1’s and eighty percent of the type-2’s. The b2 for these clubs was
cs
q
aw = as
probability cw
0.50
0.05 0.35 0.00
0.40
0.05 0.35 1.00
0.40
0.50
k
1.00
1.00
The remaining three clubs are inactive self-employment clubs. Two of them are clubs
for type-2’s and the other one is a club for type-1’s. The only thing to note about them is
that consumption of type-1’s is 0.05 and consumption of type-2’s is around 0.35 (it is not
always 0.35 for the consumption grid reasons discussed earlier.)
Comparison of this solution with the corresponding example in the first model illuminates the effect of limited commitment. In these examples, limited commitment does not
change the firm size distribution. In both cases, there are 0.4 numbers of supervisor-worker
firms. Limited commitment, however, does change firm membership. In the previous example, there were 0.1 numbers of (1, 1) clubs, where the notation (i, j) refers to a type-i
agent being the worker and a type-j agent being the supervisor. In these clubs, type-1’s
both worked and supervised. The advantage of this for type-2’s was that they did not
have to expend very much effort. All the effort they expended was in the 0.3 numbers
of (1, 2) clubs in which they supervised. With limited commitment, (1, 1) clubs of the
previous specification are not feasible. In that example, type-1 agents received 0.05 units
of consumption, regardless of whether they were a supervisor or an agent. With limited
commitment the supervisor in one of those clubs would run off with the one unit of capital,
giving him 0.25 units of consumption and a lower effort level. Consequently, the program
decides to eliminate (1, 1) clubs and replace them with (1, 2) clubs. In these club, higher
consumption levels are given to the supervisors to keep them from running off with the
capital as well as because they have a high Pareto weight.
A supporting competitive equilibrium with transfers
Below we report prices that support the above Pareto optimum. In this experiment we
found that the type-1 consumers had negative income. Consequently, this optimum cannot
33
be supported solely by a distribution of the capital good. Instead, some taxes and transfers
are needed as well. Normalizing the price of capital to be one, per capita incomes needed
to support this optimum were -0.044 for type-1 agents and 0.844 for type-2 agents. This
could correspond to a capital distribution of 0.0 to type-1’s with a tax on them of -0.044
(all expressed in per capita terms). Similarly, type-2’s would be endowed with 0.8 units
of capital and receive a transfer 0.044. Note that taxes and transfers are less than in the
environment without limited commitment. Here there is more equality.
The price of the supervisor-agent club is as follows
Worker
-0.083
Supervisor
0.796
Cost of club inputs
0.713
Notice that the price of membership sums to the cost of club inputs. Prices of the
self-employment clubs are not reported but since they are inactive, that is, use 0.0 units of
capital, the price is simply the expected net resource flow of the consumption good.
Other experiments
We find similar patterns for nearby levels of capital. For lower κ = 0.2, all non-trivial
production is done by each (1, 2) supervisor-worker firms. Interestingly, in the less unequal
range of the Pareto weights, we find that the Pareto optima can be supported with positive
distributions of capital, that is, there is no need for tax and transfers.
As capital increases, we continue to see the (1, 2) firms. Of course, there are more of
them, because capital is less scarce. Interestingly, at κ = 0.6 we observe the existence of
some (1, 2) supervisor-worker firms that use 2.0 units of capital, even while some type-1’s
and type-2’s are idle (and others are in (1, 2) firms that use 1.0 units of capital). This
occurs despite the assumed diminishing marginal returns to production. The reason for
this pattern is that idle workers suffer less disutility. Apparently, the program is balancing
this advantage with the loss of output.
At higher κ = 1.0 we observe extensive use of highly capitalized (1, 2) firms. At unequal
Pareto weights, all active production is done by this type of club with a capital input of 2.0.
At λ1 = .4, however, we see the arrival of another kind of inactive self-employment firm.
These firms with a high-weight individual are being given two units of capital and that
individual is told to run off with the capital. Apparently, an allocation with more numbers
of (1, 2) clubs would give type-2 agents too much disutility from supervising. Interestingly,
34
the consumption of type-2’s who run off is close to the level of type-2’s who do not run off
but supervise in (1, 2) firms. In general, if the program assigns a fraction of agents of a given
type to stay and the rest to runoff, the program wants to fully insure these agent against
consumption variability but cannot because an agent who runs off with the capital is barred
from making or receiving ex post transfers. For this parameterization, consumption from
running off is close enough to consumption for those who are not assigned to runoff that
the extra consumption variability has a relatively minor effect on the value of the objective
function.
Solutions to examples with higher levels of aggregate capital look much like the κ = 1.0
case. At unequal weights, (1, 2) firms predominate and at more unequal weights we see selfemployment firms in which the agent is told to run off with the capital. The only difference
between the experiments is that as the capital level increases, the self-employment firms
show up at more unequal Pareto weights.
5.2
Incentive problems internal to a multi-agent firm
In this section, we demonstrate how the problems can be modified to include incentive
concerns internal to the firm. We modify the first example by allowing the supervisor to
observe a signal that is correlated with the agent’s effort, as in Holmström (1979). The
first model can be viewed as a special case of this example in which the principal observes
a perfectly correlated signal of the worker’s effort. For simplicity, we drop the limited
commitment portion of the problem, but it is easy enough to retain.
The grid space for the self-employment club is C × Q × A × K as in the original
formulation. Just as in that model we are assuming that a supervisor needs to be physically
present to observe the signal. Therefore there is no supervision in a self-employment firm.
Consequently, the sets Π1 as well as B1 are unchanged from the first model.
Let Z be the set of signals. We assume that like output, the signal is publicly observed
and is determined by a combination of effort, the capital input, and a hidden shock. The
grid space for the supervisor-worker firm is now C × C × Q × Z × A × A × K. The
addition of the signal means that we need to add the signal to the production function.
We allow the supervisor’s effort to effect the informativeness of a signal. We write the joint
probability of the output and the signal as a function of effort and the capital input as
35
p(q, z|aw , as , k). We no longer assume that the worker’s and supervisor’s effort must be the
same. We also assume that all (q, z) pairs are possible for all (aw , as , k) triplets. Of course,
the probability distribution of output alone is then p(q|aw , k) =
P
z
p(q, z|aw , as , k), ∀as ,
and as the notation suggests the output does not depend on the supervisory effort.
With the addition of a correlated signal there is now an incentive constraint on the
worker’s effort. We still assume that the supervisor’s effort is public. The worker’s incentive
constraints are
X
π(cw , cs , q, z, aw , as , k)U (cw , aw , w)
cw ,cs ,q,z
≥
X
π(cw , cs , q, z, aw , as , k)
cw ,cs ,q,z
p(q, z|âw , as , k)
U (cw , âw , w), ∀k, as , aw , âw , (40)
p(q, z|aw , as , k)
where the worker takes an action knowing the supervisor’s effort. The addition of a correlated signal means the worker’s consumption will depend on the signal as well as the output.
Because there is no supervisor to observe effort in a self-employment firm, consumption of
an agent in that firm will depend on output alone.18
The next set of constraints ensures that π be consistent with the exogenous probability
p(q, z|aw , as , k). These “mother nature” constraints are straightforward. They are simply
∀q̄, z̄, āw , ās , k̄,
X
π(cw , cs , q̄, z̄, āw , ās , k̄) = p(q̄, z̄|āw , ās , k̄)
cw ,cs
X
π(cw , cs , q, z, āw , ās , k̄).(41)
cw ,cs ,q,z
Finally, there is the probability measure constraint, namely that π be non-negative and
X
π(cw , cs , q, z, aw , as , k) = 1.
(42)
cw ,cs ,q,z,aw ,as ,k
The set of feasible contracts in the supervisor-worker firm with limited supervision is
Π2 = {π ∈ <n2 ∗nz |π satisfying (40), (41), (42)}, where n2 was the number of elements
in the grid of the original problem and nz is the number of signals. Again, B2 is the set
of basic feasible solutions to Π2 and the problem is thus mapped into the framework of
Program 1.
18
Supervisory incentives can be examined as well by assuming that the supervisor’s effort is private
information. In this case, an incentive constraint on the supervisor’s effort would be needed, but as in the
multi-agent firms of E.S. Prescott and Townsend (1999), this can be easily done.
36
5.3
Heterogeneity
If agents are heterogenous in ability or preferences, it may be necessary to expand the
commodity space by indexing club memberships by the member’s intrinsic type. A type is
intrinsic if it affects the set of feasible allocations. To incorporate this kind of heterogeneity
we proceed as follows. There are still i = 1, ..., I types of agents, but now each type is also
identified with an intrinsic type t(i) ∈ <+ . Two types i1 and i2 may be of the same intrinsic
type, that is, t(i1 ) = t(i2 ). Indeed, the earlier examples can be thought as a special case,
where there was only one intrinsic type. In those examples, utility did not depend on type
nor did the production function.
The utility function is Ut (c, a, j). Heterogenous ability is incorporated by indexing
the production function by the intrinsic type of the worker, tw , and the intrinsic type
of the supervisor, ts , (if any). The production function is p(tw ,ts ) (q|aw , as , k). For a selfemployment firm, we write p(tw ) (q|aw , k).
We now index the firms not only by type of organization but also by the composition of
intrinsic types. If, for example, there are two intrinsic types, then the set of self-employment
firms, Nse is
Nse = {(t1 ), (t2 )},
and the set of supervisor-agent firms, Nsa , is
Nsa = {(t1 , t1 ), (t1 , t2 ), (t2 , t1 ), (t2 , t2 )},
with the total set of firms being N = Nse ∪ Nsa . For each n ∈ N, there is a set of basic
feasible solutions that we write as B n with bn as an element of this set. Sometimes we
0
write an element of this set as b(t) or b(t,t ) . The sets B n are constructed exactly as before
except that the appropriate type-dependent utility and production functions are now used
for each firm. As before, we use j = {s, w} to index an individual’s job within a firm.
Finally, because utility may be type dependent we write the utility function over basic
feasible solutions bn and the job as ut (bn , j).
In Program 2, we now index the choice variable by the type of the firm, that is, we
choose πin (bn ), πin (bn , j) and δ n (bn ). The Pareto program is
37
Program 2
max
πin ,δn ≥0,k
X
i
λi αi
X X
X X
πin (bn )ut(i) (bn , w) +
n∈Nse bn
n∈Nsa bn ,j
subject to the probability measure constraints
∀i,
X X
n∈Nse
X X
πin (bn ) +
bn
n∈Nsa
πin (bn , j)ut(i) (bn , j)
πin (bn , j) = 1;
(43)
bn ,j
the matching constraints for self-employment firms
∀n ∈ Nse, ∀bn , δ n (bn ) =
X
αi πin (bn ),
(44)
i
and the matching constraints for supervisor-agent firms
∀n ∈ Nsa , ∀bn , δ n (bn ) =
the resource constraints
X
αi πin(bn , w) =
i
X
αi πin (bn , s),
X n n
n,bn
X
n,bn
(45)
i
δ (b )r(c−q) (bn ) ≤ 0,
(46)
δ n (bn )rk (bn ) − k ≤ 0,
(47)
k ≤ κ;
(48)
and constraints that restrict agents to choose membership in firms that are feasible for
their basic type. We do this by requiring that zero probability be allocated to assignments
that put an intrinsic type in a firm and job that the intrinsic type cannot be assigned to.
For self-employment firms, the restrictions are
0
0
0
(t )
∀i, ∀t0 6= t(i), ∀b(t ) , πi (b(t ) ) = 0.
(49)
For supervisor-worker firms, the restrictions are
(t,t0 )
0
∀i, ∀(t, t0 ) 3 t 6= t(i), ∀b(t,t ) , πi
0
(t,t0 )
∀i, ∀(t, t0 ) 3 t0 6= t(i), ∀b(t,t ) , πi
0
(b(t,t ) , w) = 0,
0
(b(t,t ) , s) = 0.
(50)
To decentralize this economy we use the commodity space and add to agent’s consumption sets the restrictions that they cannot choose membership in a firm that does not
contain their type. This condition is represented in the program by constraints (49) and
(50), though of course, it would be written in x notation.
38
5.4
Hierarchies
More complicated organizational structures can be incorporated into our framework as well.
Incentives can be added to the Rosen (1982) hierarchy model.19 To incorporate hierarchies,
we would proceed as follows.
Production may be done by a variety of firms. Each firm consists of z levels of hierarchy.
A level in a hierarchy is denoted by h. A firm is indexed by the number of people at each
level in its H levels of hierarchy. Denote the type of firm by n = (n1 , n2 , ..., nz ), where nh
(a non-negative integer) is the number of people working at the hth level in the hierarchy.
The number of people working a given level in a hierarchy may be 0. For example, a selfemployment firm would be identified by n = (1, 0, 0, ..., 0), while a supervisor-worker pair
would be identified with n = (1, 1, 0, 0, ..., 0). It will usually make sense to assume that if
there is no one assigned to a given level of a hierarchy then no one can be assigned to a
higher level in the hierarchy. Finally, we define J n as the set of (h, l) hierarchy-position
assignments associated with an n-type firm.
For simplicity, we do not allow types to differ in any intrinsic way. Preferences are
defined as U (c, a, h), so utility depends on the level within the hierarchy but not the position
within that level. Let ah,l be the action of an agent in the lth position at level h in the
hierarchy. Consumption may differ across positions since we want to allow it to depend
on performance measurements. Let ch,l refer to the consumption of the lth agent in the
hth level of a hierarchy. Sometimes ah will refer to the vector of efforts at level h, while c
and a will refer to the vector of consumptions and efforts, respectively, across and within
hierarchical levels. Also, define a variable m that is a vector of performance measurements.
These measurements may be portions of output attributable to an individual or the result
of internal evaluations. We denote by q the joint output of the hierarchy. The technology is
then written as p(m, q|a1,1 , a1,2 , ..., a1,n1 , a2,1 , ..., az,nz , k). A contract for an n-type hierarchy
is
π n (c, m, q, a) = π n (c1,1 , c1,2 , ..., c1,n1 , c2,1 , ..., cz,nz , m, q, a1,1 , a1,2 , ..., a1,n1 , a2,1 , ..., az,nz , k)
19
Early papers that looked at incentives in hierarchies include Mirrlees (1976) and Calvo and Wellisz
(1978).
39
Incentive constraints would look something like
∀n, ∀h, l, ∀ah,l ,
≥
X
π n (c, m, q, a, k)U(ch,l , ah,l )
c,m,q,a−(h,l) ,k
X
π n (c, m, q, a, k)
c,m,q,a−(h,l) ,k
p(m, q|a−(h,l) , âh,l , k)
U (ch,l , âh,l ), ∀k, âh,l .
p(m, q|a−(h,l) , ah,l , k)
(51)
Depending on the monitoring technology these constraints would need to be modified. For
example, if actions taken by a higher level in the hierarchy make lower level actions public
then the set of deviating actions might need to be altered.
Technology constraints are
∀n, ∀m̄, q̄, ā, k̄,
X
π n (c, m̄, q̄, ā, k̄) = p(m̄, q̄|ā, k̄)
c
X
π n (c, m, q, ā, k̄).
(52)
c,m,q
and the probability measure constraint is
X
π n (c, m, q, a, k) = 1.
(53)
c,m,q,a,k
Thus we can define the set
Πn = {π n ∈ <l |(51), (52), (53), hold},
and let B n being the set of basic feasible solutions to this set. We write the utility of
someone in the lth position of the hth level of a hierarchy as u(bn , (h, l)). Resource usage
is written r(c−q) (bn ) and rk (bn ).
The choice variables are πin (bn , (h, l)) and δ n (bn ). The first term is the probability that
a type i is in a type-n firm, assigned to level h of the hierarchy, assigned position l in that
level of the hierarchy, and the firm’s grand contract is bn .
The Pareto program is
Program 3
max
X
αi λi
i
s.t. ∀i,
X
X
πin (bn , (h, l))u(bn , h))
X
X
πin (bn , (h, l)) = 1,
n bn ,(h,l)∈J n
n bn ,(h,l)∈J n
40
∀n, ∀(h, l), (h0 , l0 ) ∈ J n , ∀bn ,
δ(bn ) =
X
αi πin (bn , (h, l)) =
i
n
XX
n
bn
αi πin (bn , (h0 , l0 )),
i
XX
bn
X
δ n (bn )r(c−q) (bn ) ≤ 0,
δ n (bn )rk (bn ) − k ≤ 0,
k ≤ κ.
The matching constraints ensure that for each occupied position at each level of a type-n
firm hierarchy, there is one and only one person assigned to it.
To decentralize this economy we proceed as before.
6
Conclusion
Our framework can address other questions in organizational theory. For example, the
models studied in E.S. Prescott and Townsend (1999) can be adapted. One of their models
considered the tradeoff between relative performance and collusion (as also studied by
Holmström and Milgrom (1990), Itoh (1993), and Ramakrishnan and Thakor (1991)). To
map that problem into this paper’s framework would require allowing two types of clubs
to form: one would be a relative performance club and the other would be a collusion
club. Agents would choose which type of club to enter. Collusion models are particularly
interesting, challenging, and a relatively unexplored area of the theory of the firm, not to
mention incentive theory. For more on these models see Tirole (1992).
E.S. Prescott and Townsend (1999) also studied a model in which two workers are assigned to work a set of technologies. This problem is related to the task assignment problem
studied by Holmström and Milgrom (1991) and more generally to the multi-agent principal
agent models studied in papers like Demski and Sappington (1984) and Mookherjee (1984).
For a given set of technologies, the workers assigned to them may work each of them jointly
or individually. Efforts on individually worked technologies are private information while
efforts on jointly worked technologies are public, though there are some costs to this type
41
of assignment. We found examples of individually and jointly worked technologies. To map
this model into the framework of this paper, agents would match into pairs that simultaneously decide which technologies to operate jointly and which separately. If joint operation
is itself interpreted as an organization then in the sense of this paper, we have allocations
that resemble organizations within clubs.
For both models in the earlier paper we solved a version of Program 1 in which pairs
of agents were a priori matched together. Within these matches, however, the type of
organization was endogenous. We found that the internal distribution of Pareto weights was
a critical factor in determining organizational structure. The extension in this paper makes
endogenous both the internal distribution of Pareto weights as well as the net resource flow
from the rest of the economy to a matched pair of agents.
42
A
Computing
Because of the large number of variables and constraints, Program 1 requires too much
memory to compute using standard linear programming algorithms. In this appendix, we
describe an algorithm that computes solutions to Program 1, despite its large dimensions.
It is essentially a version of the Dantzig-Wolfe decomposition algorithm as developed in
Dantzig and Wolfe (1961). We will provide only the briefest description of their algorithm. More information on it can be found in advanced linear programming textbooks
like Bertsimas and Tsitsiklis (1997).
First, we take advantage of the structure of the matching constraints to change Program 1 from a linear program with many variables and many constraints to one with many
variables but only a few constraints. Consider the trivial club constraint in (8) associated
with a particular b1 contract. The only variables with non-zero coefficients for this constraint are π1 (b1 ), π2 (b1 ), and δ(b1 ). We can think of these variables and the club constraint
as blocks of non-zero coefficients in the constraint matrix. The values of these variables do
not affect the club constraints for other b1 and b2 , just as the values of variables involving
other b1 and b2 do not affect the feasibility of this b1 constraint. Writing this block in
isolation gives
h
α1 α2
π1 (b1 )
h
i
−1
π2 (b1 ) = 0 .
δ(b2 )
i
(54)
When combined with the non-negativity constraints on π1 (b1 ), π2 (b1 ), and δ(b1 ), this system
of equations defines a polyhedral cone with its vertex at the origin. As such, it can be
represented as the set of all non-negative linear combinations of its extreme rays. The two
extreme rays of this cone are
[ 1/α1
0
1 ], and
[
0
1/α2 1 ].
(55)
The scale of these rays is indeterminate so we scale them to δ(b1 ) = 1. The interpretation
of the first ray (the first row in (55)) is that it takes 1/α1 units of π1 (b1 ) to make one unit of
δ(b1 ). In particular, one unit of the first ray corresponds to one self-employment firm with a
b1 contract and a type-1 as the member. The second ray is interpreted similarly for type-2
43
agents. Because the rays have this type-specific interpretation we label the quantity of them
with the notation δ (i) (b1 ), i = 1, 2. In the program, these quantities will be constrained by
the non-club constraints.
The exact same steps can be done for each of the b2 matching constraints in (9). For
a given b2 contract, the only non-zero variables for its matching constraints are π1 (b2 , w),
π1 (b2 , s), π2 (b2 , w), π2 (b2 , s), and δ(b2 ). Again, we think of these matching constraints and
corresponding variables as an isolated block because the values of these variables do not
affect the feasibility of any other matching constraints, and the values of other variables do
not affect the feasibility of these matching constraints. Writing this block in isolation gives
"
α1 0 α2 0 −1
0 α1 0 α2 −1
#
π1 (b2 , w)
π1 (b2 , s)
π2 (b2 , w)
π2 (b2 , s)
δ(b2 )
=
"
0
0
#
.
(56)
Along with the non-negativity constraints, this system of equations is a polyhedral cone
with a vertex at the origin. As before, it can be represented by the set of all non-negative
linear combinations of its extreme rays. The four extreme rays of this cone are
0
0
[ 1/α1 1/α1
[ 1/α1
0
0
1/α2
[
0
1/α1 1/α2
0
[
0
0
1/α2 1/α2
1
1
1
1
],
],
], and
].
(57)
The scale of rays is indeterminate and, as before, we scale them to δ(b2 ) = 1 so that each
ray can be interpreted as one unit of a type-pair specific match. For example, the second
row of (57) corresponds to one firm with a b2 contract that consists of a type-1 as the worker
and a type-2 as the supervisor. Because the rays represent type-pair specific matches we
0
introduce the notation δ (i,i ) (b2 ) to indicate the number of b2 firms that consist of a type-i
worker and a type-i0 supervisor.
Our strategy is to rewrite Program 1 in terms of these extreme rays, the type-specific
club goods. Each ray will correspond to a variable in the new linear program and the program will choose non-negative quantities of these variables, which means that the matching
constraints will be automatically satisfied. The objective function and the non-matching
constraints of Program 1 will be expressed in terms of the new variables δ (i) (b1 ) and
44
0
δ (i,i ) (b2 ). More specifically, the objective function and the constraints that connect the
blocks, that is (7), (10), (11), and (12), will be evaluated at (55) and (57) in order to determine the coefficients for the new variables. This means, for example, that αi πi (b1 ) = δ (i) (b1 ),
αi πi (b2 , w) =
P
i0
0
δ (i,i ) (b2 ), and αi πi (b2 , s) =
P
i0
0
δ (i ,i) (b2 ).
The new version of Program 1 takes the following form.
Program 1a
max
δ (i) ,δ(i,i0 ) ≥0,k
X
i
λi
X
δ (i) (b1 )u(b1 ) +
X
0
δ (i,i ) (b2 )u(b2 , w) +
i0 ,b2
b1
X
i0 ,b2
0
δ (i ,i) (b2 )u(b2 , s)
subject to the probability measure constraints
∀i,
X
b1
δ (i) (b1 ) +
X
0
δ (i,i ) (b2 ) +
i0 ,b2
X
0
δ (i ,i) (b2 ) = αi ,
i0 ,b2
the consumption resource constraint
XX
δ (i) (b1 )r(c−q) (b1 ) +
(i) b1
XX
(i,i0 ) b2
0
δ (i,i ) (b2 )r(c−q) (b2 ) ≤ 0,
and the capital resource constraints
XX
(i) b1
δ (i) (b1 )rk (b1 ) +
XX
(i,i0 ) b2
0
δ (i,i ) (b2 )rk (b2 ) − k ≤ 0, and
k ≤ κ.
With two types, Program 1a contains only five constraints. However, it still has an enormous number of variables, even more than in Program 1.
Despite the large number of variables in Program 1a, it is still practical to compute it.
The strategy is to work with a subset of the variables and then generate new variables as
they are needed. In linear programming terms, this is called a column-generating method.
The Dantzig-Wolfe algorithm is based on this idea.
We start by solving Program 1a restricted to a small set of the variables, namely, δ (i) (b1 )
0
and δ (i,i ) (b2 ) that correspond to a proper subsets of B1 and B2 , but which can still deliver a
feasible solution. The solution to this restricted program delivers a candidate solution which
is optimal with respect to the small set of variables, but not necessarily with respect to
45
the entire unrestricted set of variables. This restricted program also delivers dual variables
on the constraints in Program 1a, which can be interpreted as the value of relaxing these
constraints.
The next step in the algorithm, and the reason column-generating methods work for
this program, is based on our definition of b1 and b2 as the extreme points of the sets Π1
and Π2 , respectively. Recall that Π1 is defined over the consumption, output, action, and
capital inputs grids. Its dimensions are relatively small. For each (i)-type firm, we solve
a subprogram that uses as its constraint matrix, the set Π1 . The objective function of the
subprogram consists of the effect of the subprogram variables on the sum of the objective
function of Program 1a and the constraints of Program 1a, the latter weighted by the dual
variables from the restricted master program.
Just like in the simplex algorithm, the idea of this step is to see if there is another
δ (i) (b1 ), that is, another basic feasible solution to Π1 , which can be introduced into the
restricted Program 1a in order to improve upon the candidate solution. If the optimum to
the subprogram does not improve upon it then none of the other basic feasible solutions
to the subprogram, that is, none of the other δ (i) (b1 ), will improve upon it either. Since
in practice linear programming algorithms usually visit only a small set of vertices, solving
the subprogram is an inexpensive way of checking all the δ (i) (b1 ).
The same step is performed over Π2 for each (i, i0 )-type of supervisor-worker firm. Again,
0
the goal is to see if another variable — here a δ (i,i ) (b2 ) — can be introduced which will improve
on the candidate solution generated by the restricted version of Program 1a. If no such
0
δ (i) (b1 ) or δ (i,i ) (b2 ) exists then the candidate solution is indeed optimal. Otherwise, the
0
δ (i) (b1 ) and δ (i,i ) (b2 ) calculated from the subprograms are added to the set of variables in
the restricted Program 1a and the algorithm repeats. Once a solution to Program 1a has
been found, the corresponding solution to Program 1 is easily found.
The advantage of the algorithm is that the entire sets of B1 and B2 do not need to be
stored in memory. Instead, new elements are generated only as needed. As long as not too
many iterations are required, memory requirements and speed will not be an issue. In our
numerical examples, the algorithm converged very quickly, so the only memory limitation
was the size of the subprograms. Finally, it is worth pointing out that the subprograms
based on the Πi decompose with respect to action, capital-input pairs. If needed, this
46
property can be used to increase the size of the subprograms that can be computed.20
B
Proofs
B.1
Overview
The proofs in this appendix are based on Negishi (1960). Negishi proves existence by
finding a fixed point in the space of Pareto weights that is a competitive equilibrium with
zero transfers. While not explicitly stated in his paper, his mapping is based on the second
Welfare Theorem.21 Aside from our linear structure, there are two substantial differences
between Negishi’s economy and ours. Because of these differences, some modifications to
his proof are required.
The first difference between our economies and his 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. With negative
prices this mapping does not work. We modify his approach to avoid this problem in the
following way. We divide prices into positive and negative components. We then map the
positive and negative portions into a 2n − 1 dimensional simplex, and then subtract the
negative component from the positive one to recover the actual prices.
The second difference 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. The first reason is that this assumption satisfies Slater’s condition, a sufficient
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 finite.
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
20
See E.S. Prescott (2000) on how to use the Dantzig-Wolfe decomposition algorithm to compute solutions
to moral-hazard programs.
21
Takayama (1985) does emphasize this.
47
cheaper consumption point implies that the Pareto’s weights at the fixed point are strictly
positive. Strictly positive Pareto weights are important because they imply that the fixed
point of Negishi’s mapping is a competitive equilibrium rather than just a compensated
equilibrium.
This proof should be viewed as an existence proof for bounded linear economies in
Euclidean space without free disposal and without satiation.
We start by stating without proof several needed theorems.
Theorem 6 (Goldman-Tucker) There exists an optimal solution x∗ to a linear program
and an optimal solution µ∗ to its dual if and only if (x∗ , µ∗ ) is a saddle point of the Lagrangian L(x, µ).
See Takayama (1985) for example. The vector of dual variables µ∗ is a Kuhn-Tucker vector.
We also need a theorem connecting the Kuhn-Tucker vector to the saddle-point of the
Lagrangian plus the necessary and sufficient first-order conditions.
Theorem 7 In a concave program, for a vector to be a Kuhn-Tucker vector, it is necessary
and sufficient that it be part of a saddle point of the Lagrangian. Furthermore, this condition
holds if and only if the usual necessary and sufficient conditions hold.
See Rockafellar (1970) for example.
The remaining mathematical result we need is Kakutani’s fixed point theorem.
Theorem 8 (Kakutani’s Fixed Point Theorem) Let K be a compact convex set in
<n and let f (x) be a non-empty, compact-valued, convex-valued, upper hemi-continuous
correspondence mapping K into K. Then, there is a fixed point x̂ such that x̂ ∈ f (x̂).
The Pareto program for this economy is
Program B
max
{xi }≥0,y
s.t.
X
i
X
λi αi ui xi
i
αi (xi − ξi ) ≤ y,
48
−
X
i
αi (xi − ξi ) ≤ −y,
f y ≤ 0,
∀i, gi xi ≤ bi .
In terms of the problem in the text, the xi are the agents’ probabilities and y is the
intermediary’s production plan. The program is written differently than Program 1 in
the text. The economy-wide resource constraints have been rewritten in terms of market
clearing and the constraints on the intermediary’s production. This substitution brings y
explicitly into the Pareto program.
All variables are vectors in this program so there may be more than one constraint of
each type and more than one good. However, there are a finite number of variables and
constraints so the program is a linear program. An important feature of this program is that
the market clearing constraint have been rewritten into two sets of inequality constraints
(the first two sets of constraints in program). This alternative method for representing
equality constraints will be useful when the mapping is described. The last set of constraints
in the program gi xi ≤ bi define the consumption sets Xi (equation (13) in the text). By
the same logic used in describing the market clearing constraints, the notation gi xi ≤ bi is
general enough to incorporate any equality constraints in the consumption sets.
Theorem 9 Assuming that a feasible solution to the Pareto program exists, then for any
set of weights λ ∈ S I−1 , a solution to the Pareto program exists.
Proof: The constraint set is closed. It is also bounded because the probability measure
constraints in gi xi ≤ bi bound each xi and feasible y are then bounded by the resource
constraint in conjuction with the bounds on {xi }. Therefore, {xi } and y are chosen from
compact sets. The objective function is continuous. Therefore, a maximum exists. Q.E.D.
We will denote goods by the index j. Consequently, the jth coefficient of the vector xi
will be denoted xi,j . For matrices like f or gi we let fj and gi,j denote the jth column of
that matrix.
Theorem 10 A solution to the Pareto program is a saddle point of
L({xi }, y, pl , pg , µ, γ) =
X
i
λi αi ui xi − (pl − pg )
49
"
X
i
#
αi (xi − ξi ) − y − µf y −
X
i
γi (gi xi − bi ),(58)
where x ≥ 0 and y are maximizing variables and (pl ≥ 0, pg ≥ 0, µ ≥ 0, γ ≥ 0) are
minimizing variables. The necessary and sufficient conditions are
∀i, j, λi αi ui,j − (plj − pgj )αi − γi gi,j = 0, (≤ 0 if xi,j = 0)
(59)
∀j, (plj − pgj ) − µfj = 0,
(60)
αi (xi − ξi ) − y = 0,
(61)
X
i
µfy = 0,
∀i, γi (gi xi − bi ) = 0,
f y ≤ 0,
(62)
gi xi ≤ bi .
(63)
Proof: The Pareto program is a linear program. Use Theorems 6 and 7. Q.E.D.
At this point, it is worth proving a preliminary result.
Lemma 1 For ((pl − pg ), y) to be part of a saddle-point of (58), (pl − pg )y = 0.
Proof: Multiplying each (60) by yj and summing over j gives (pl − pg )y = µf y. By (62),
(pl − pg )y = 0. Q.E.D.
The other lemma we need is that with non-satiation there will not be a solution with
pl − pg = 0. Otherwise, if these two vectors are equal, then prices will be zero.
Lemma 2 With non-satiation, there does not exist a saddle point (x, y, pl , pg , µ, γ) to the
Lagrangian such that pl − pg = 0.
Proof: A saddle point solves
max
{x}≥0,y
X
i
l
g
αi λi ui xi − (p − p )
"
X
i
#
αi (xi − ξi ) − y − µf y −
X
i
γi (gi xi − bi ),
By non-satiation there exists for some i, x0i ∈ Xi such that ui x0i > ui xi . Furthermore,
γi (gi xi − bi ) ≥ γi (gi x0i − bi ), ∀x0i ∈ Xi . (The latter hold trivially if γi = 0. And if γi > 0
then gi xi = bi so if gi x0i ≤ bi then γi gi x0 ≤ γi bi = γi gi x.) Since µfy = 0 by (62), if pl = pg
then αi λi ui x0i − γi gi x0i > αi λi ui xi − γi gi xi , which, when summed over i, contradicts the
assumption that (x, pl , pg ) is a saddle point. Q.E.D.
B.2
Second Welfare Theorem
We first express the compensated equilibrium conditions in terms of the necessary and
sufficient conditions.
50
Lemma 3 Conditions 1 and 2 in the definition of a compensated equilibrium can be rewritten respectively in the following form:
1. ∀i, x∗i is part of a saddle point of
L(xi , βi , νi ) = pxi + βi (ui xi − ui x∗i ) − νi (gi xi − bi ),
where xi ≥ 0 is a minimizing variable, and βi ≥ 0, νi ≥ 0 are maximizing variables.
The necessary and sufficient conditions for it are:
∀j, βi ui,j − pj − νi gi,j = 0, (≤ 0 if xi,j = 0)
(64)
βi (ui xi − ui x∗i ) = 0, ui xi − ui x∗i ≥ 0,
(65)
νi (gi xi − bi ) = 0, gi xi − bi ≤ 0.
(66)
2. y is a saddle point of
L(y, µ̃) = py − µ̃f y,
where y is a maximizing variable and µ̃ ≥ 0 is a minimizing variable. The necessary
and sufficient conditions for it are:
∀j, pj − µ̃fj = 0
(67)
µ̃fy = 0, f y ≤ 0.
(68)
Proof: Both problems are linear programs. If they have finite solutions then Kuhn-Tucker
conditions hold. Q.E.D.
Theorem 2 (Second Welfare Theorem) Any solution to the Pareto program for some
Pareto weights λ ≥ 0 can be supported as a compensated equilibrium.
Proof: Given a solution to the Pareto program, necessary and sufficient condition (61)
implies that condition 3 of a compensated equilibrium, market clearing, holds. Let p =
(pl − pg ), βi = λi , νi = γi /αi , and µ̃ = µ. Comparison of necessary and sufficient conditions
(60) and (62) of Program B with the necessary and sufficient conditions (67) and (68) of
the Intermediary’s problem shows that the Intermediary’s problem is satisfied. Comparison
51
of (59) and (63) from Program B with (64) and (66) shows that these two necessary and
sufficient conditions from Consumer’s problems hold. The remaining condition from the
Consumer’s problem (65) holds trivially since xi = x∗i . Q.E.D.
The conditions for a competitive equilibrium are nearly identical. The only change is
the necessary and sufficient conditions for the Consumers’ problems. These conditions are
∀i, xi is part of a saddle point of
L(xi , δi , ωi ) = ui xi − δi p(xi − ξi ) − ωi (gi xi − bi ),
where xi ≥ 0 is a maximizing variable and δi ≥ 0, ωi ≥ 0 are minimizing variables. The
necessary and sufficient conditions for it are:
∀j, ui,j − δi pj − ωi gi,j = 0, (≤ 0 if xi = 0)
(69)
δi p(xi − ξi ) = 0, p(xi − ξi ) ≤ 0,
(70)
ωi (gi xi − bi ) = 0, gi xi − bi ≤ 0.
(71)
Theorem 3 Any solution to the Pareto program for some Pareto weights λ > 0 can be
supported as a competitive equilibrium.
Proof: As in the proof of Theorem 2, set p = (pl − pg ), and µ̃ = µ. In addition, set ∀i,
ξi = x∗i , δi = 1/λi and ωi = γi /(λi αi ). Market clearing and the Intermediary’s problem
are satisfied by the same arguments used in the proof to Theorem 2. Consumer’s problem
condition (70) holds trivially because ξi = x∗i . Substituting into (59) gives us that condition
(69) holds. Finally, substituting into (63) and dividing by the scalar (λi αi ) demonstrates
that condition (71) holds. Q.E.D.
We now state a version of Arrow’s remark.
Theorem 4 Take a solution to the Pareto program for λ ≥ 0. If, at the corresponding
compensated equilibrium, there exists for all i, a cheaper point satisfying xi ∈ Xi , then
λ > 0.
Proof: A solution to the Pareto program must be a saddle point of the Lagrangian. The
saddle point condition requires for all i,
0
0
0
0
λi αi ui xi − αi (pl − pg )xi − γi (gi xi − bi ) ≥ λi αi ui xi − αi (pl − pg )xi − γi (gi xi − bi ), ∀xi .
52
Assume a cheaper point exists. Now substitute p = pl − pg and consider the λi = 0 case.
Then,
0
0
0
pxi ≥ pxi + γi (gi xi − bi )/αi − γi (gi xi − bi )/αi , ∀xi .
0
0
We know, however, that γi (gi xi − bi ) = 0 and that γi (gi xi − bi ) ≤ 0. Therefore, pxi ≥ pxi ,
0
0
∀xi satisfying g(xi − bi ) ≤ 0, which contradicts the cheaper good assumption. Q.E.D.
B.3
Existence Theorem
A Negishi-style proof works by showing that at a fixed point of Negishi’s mapping, consumers’ budget constraints are satisfied by a solution to the Pareto program. To show that
such weights exist, we use the following mapping of (λ, x, y, pl , pg ) → (λ, x, y, pl , pg ). The
mapping consists of the following two parts
0
0
λ → (x0 , y 0 , pl , pg )
(λ, x, pl , pg ) → λ
Each variable is restricted to lie in a compact, convex set. Remembering that I is the
number of types, we restrict λ ∈ S I−1 (the unit simplex), x ∈ X, y ∈ Γ, (pl , pg ) ∈ S 2n−1 ,
where n is the number of goods, where X is the cross-product over I of the Xi , and
Γ = {y ∈ Y |y is feasible}. Market clearing and the compactness of the Xi guarantee that
Γ is a compact set. The set Γ is also convex.22
First, for any λ ∈ S I−1 there is a solution to the Pareto program (x∗ , y ∗ , pl∗ , pg∗ , µ∗ , γ ∗ ).
0
Then we normalize (pl∗ , pg∗ ) to be in the unit simplex with pji = pj∗
i /
0
0
P
i,j
pj∗
i . This gives
us (x0 , y 0 , pl , pg ). This mapping is non-empty, compact-valued, convex-valued, and upper
hemi-continuous.
The second part of the mapping calculates the new Pareto weights as a function of the
transfers needed to support the Pareto optimum. Recall that the sets S 2n−1 and X are
compact. Therefore, there exists a positive number A such that
X
i
22
|(pl − pg )(ξi − xi )| ≤ A,
There are two minor differences between this mapping and Negishi’s. The production plan is not
explicitly mentioned in the second part of the mapping because with our constant-returns-to-scale technology there are no profits that may affect agent’s income. The other difference is the splitting of prices into
negative (pl ) and positive (pg ) components.
53
for any (x, pl , pg ) ∈ X × S 2n−1 . Then, for any λ ∈ S I−1 , let
λ̂i = max{0, λi +
(pl − pg )(ξi − xi )
λ̂i
}, and λ0i = P .
A
i λ̂i
(72)
This mapping is a continuous function so it is trivially a non-empty, convex valued, and
upper hemi-continuous correspondence.
To summarize, we have defined a mapping from S I−1 × X × Γ × S 2n−1 → S I−1 × X ×
Γ × S 2n−1 . 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 11 A fixed point (λ, x, y, pl , pg ) of the mapping exists.
Proof: Apply the conditions of Kakutani’s fixed point theorem.
The fixed point is a solution to the Pareto program so by Theorem 2 it is a compensated
equilibrium. The second part of the mapping, subject to one condition, will provide us with
a proof that the compensated equilibrium is a competitive equilibrium.
The next step is to show that the fixed point is a competitive equilibrium. We know
that the intermediary’s problem and market clearing are satisfied. It remains to show that
the Consumer’s problem is satisfied. We will not be able to do this in general. However,
we will be able to show that in all but the most extreme cases the fixed point will also be
a competitive equilibrium.
Theorem 5 (Existence) For any given distribution of endowments, if the Pareto weights
at a fixed point of the mapping are non-zero, then a competitive equilibrium exists.
Proof: By Theorem 11 a fixed point to the mapping exists. The first part of the mapping
is a solution to the Pareto program so by Theorem 2 the solution can be supported as a
compensated equilibrium. Now assume that λ > 0. Set p = pl − pg . From the second part
of the mapping (72), at a fixed point, p(ξi − xi ) must be the same sign for all i. From the
P
necessary and sufficient conditions to the Pareto program, we know that p(
y) = 0. From Lemma 1, we know that py = 0. Therefore, p(ξi − xi ) = 0, ∀i.
54
i
αi (xi − ξi ) −
This satisfies one part of the necessary and sufficient conditions of the Consumer’s
problem. The remaining conditions simply require setting δi = 1/λi and ωi = γi /(λi αi ).
Since λi > 0 we can make this substitution. Q.E.D.
The term δi = 1/λi is consumer i’s marginal utility of income.
55
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