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Working Paper Series
A "Coalition Proof" Equilibrium for a
Private Information Credit Economy
WP 90-08
Jeffrey M. Lacker
Federal Reserve Bank of Richmond
John A. Weinberg
Purdue University
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Working Paper 90-8
A "COALITION PROOF" EQUILIBRIUM FOR A PRIVATE INFORMATION CREDIT ECONOMY
Jeffrey M. Lacker
Research Department, Federal Reserve Bank of Richmond
P.O. Box 27422, Richmond, VA 23261
and
John A. Weinberg
Department of Economics, Krannert Graduate School of Management,
Purdue University, West Lafayette, IN, 47907
November 5, 1990
BBSTRACT: This paper examines an economy in which agents with private
information about their own productive capabilities seek to raise capital to
fund their investment projects. We employ an equilibrium concept which is
closely related to Coalition Proof Nash Equilibrium. In equilibrium, all
agents who succeed in raising capital (entrepreneurs) are pooled; they all
receive the same contract or consumption schedule. Entrepreneurs, however,
are separated from those who fail to raise capital. This separation results
in productive efficiency for the economy. If the economy has no viable
alternative investment opportunity (other than agents' projects) then
equilibrium allocations can be supported by a (non-intermediated) securities
market. If there is a viable alternative, the equilibrium allocations can
only be supported through the formation of a form of financial intermedary
coalition.
A preliminary draft of this paper was presented at the 1990 meeting of the
WEA. We would like to thank Paul Fisher for his comments on that earlier
draft. We have also benefited from extensive discussions with Charles Kahn.
His comments, and those of Nicholas Yannelis and other participants in the
University of Illinois Microeconomic Theory workshop and the Purdue
Macroeconomics workshop are greatly appreciated. As the authors retain mutual
responsibility for all errors, past, present and future, comments continue to
be appreciated. The views expressed are those of the authors and do not
represent the views of the Federal Reserve Bank of Richmond or the Federal
Reserve Board.
-l-
btroduction
The basic question asked in this paper has been asked many times before:
how will a productive resource be allocated among potential users who are
privately informed about their abilities in using that resource? Such a
private information problem has been suggested by many as a source of market
failure. Following Stiglitz and Weiss (1981), private information has been a
key ingredient in many models of credit rationing and other types of market
imperfection. Another line of research seeks to characterize efficient
allocation mechanisms for private information environments and to find
institutional arrangements which support efficient allocation rules. This
"efficient mechanism" literature is heavily influenced by the seminal work of
Hurwicz (1971).
We shall refer to the environment examined in this paper as a "private
information credit economy." This environment is very similar to that
examined in the "market failure" literature. We, however, examine this
environment in a way which more closely follows the efficient mechanism
literature. Efficient allocations are defined relative to the set of
resource and incentive feasible allocations. A key question then becomes:
can the economy be "decentralized" in an efficient way?
Others who have
addressed this question for a similar environment include Boyd and Prescott
(1986), and Kahn (1987). In both of these papers, the central task is the
development of a notion of equilibrium for the economy; in both cases, the
chosen equilibrium concept is an adaptation of the core. The motivation for
such an equilibrium concept is that, if agents in the economy are free to
communicate and propose alternative arrangements, they will settle on an
arrangement which is sustainable in the sense of being unblocked by any
-2-
possible coalition. In Boyd and Prescott's work, the coalition structure
which emerges as part of the core is interpreted in terms of the institutions
which support efficient allocations.
In this paper, we follow a similar path. We propose, however, an
equilibrium concept which incorporates a different notion of sustainability.
We suggest that some potential deviations by coalitions may not be
"credible"; if the coalition sought to form, its proposed allocation would be
subject to further deviation by some of its members. Accordingly, we require
that an equlibrium allocation be unblocked only by "credible" deviations.
Our focus on the credibility of deviating coalitions closely follows
Bernheim, Peleg and Whinston's (1987) development of the notion of CoalitionProof Nash Equilibrium (CPNE). For a deviating coalition to be credible, it
must be immune from further credible deviations by sub-coalitions. The
necessary adaptation involves the adjustment of the notion of a blocking
coalition to respect the requirements of private information. We require
that any individual that is not intended to be part of a blocking coalition
have no incentive to gain admittance by misrepresenting his type.
In
addition, we do not fully articulate a game which yields the allocations we
consider as CPNE.
Instead, we describe the set of "Coalition-Proof"
allocations. That is, the "equilibrium" concept we employ bears the same
relationship to CPNE as does the core to Aumann's (1959) Strong Nash
Equilibrium.
For the environment we consider, there is an (essentially) unique
coalition proof allocation. This allocation divides the set of agents into
those who receive capital for productive purposes and those who simply invest
their endowed capital in the productive efforts of others. Agents' state
-3
-
contingent consumption schedules differ only according to whether an agent is
allocated any capital, and the "marginal" agent is indifferent between
receiving the consumption schedule of an investor and that of an
"entrepreneur." Since the set of coalition proof allocations contains the
set of core allocations, the core either coincides with the allocation
described above or is empty. We show that for an important range of
parameter values the core is empty.
In doing so, we demonstrate the appeal
of the credibility restriction on deviating coalitions. The coalition proof
allocation can be blocked only by a coalition which is willing to allocate
its capital among its members inefficientlv.
Following Boyd and Prescott (1986), we ask whether the proposed
equilibrium allocations can be achieved by a fully decentralized securities
market. We find that our economy has two important cases, depending on the
existence and quality of "outside" risk-free investment opportunities. In
one case, securities markets "work," and in the other case they don't. When
securities markets fail, we argue that equilibrium allocations can be
supported by coalitional arrangements resembling financial intermediaries.
These coalitions could be interpreted as organizations which contract with
lenders, promising a fixed return. The capital raised is then allocated
efficiently among borrowers and the risk-free investment, with borrowers
making the appropriate state-contingent repayment promises.
Intermediation
arises without a technology for producing information; such an information
technology was found essential for intermediaries to arise in the Boyd and
Prescott framework. We feel that these results shed some light on the role
of intermediation in private information economies.
-4-
1. The Environment
The economy is populated by a continuum of agents who produce and
consume a single consumption good using human resources and a single physical
resource. An agent is indexed by his type, 'cE T s [O,l], and the population
distribution of types is given by the finite, non-atomic measure VTon B(T),
the Bore1 sets of T.
maximize
Agents have identical preferences; they seek to
the expected value of their consumption of the single good.
Each agent has an endowment of one unit of the physical resource
(capital), but agents have access to diverse technologies for transforming
the resource into the consumption good.
Specifically, each agent is endowed
with a project which can produce a random output, y, per unit of capital, up
to a fixed capacity, x > 1.
< yg.
Unit output takes one of two values, yg and yb
The probability of the good outcome (yg) depends on an agent's type
and is denoted P(T). We assume that the function P(T) is continuous and
strictly increasing in 'Iand that 0 <-p(0)
and p(1) I 1. To summarize,
expected output of a type 'cproject in which x 5 x units of capital have been
invested is written
XV(T) : X[P(r>Yg + (1-P(T))Yb]
In addition to the investment projects of individual agents, there is a
risk free, constant returns to scale technology available to all agents.
This technology delivers r units of consumption good per unit of investment.
For
most
of what follows, it is assumed that yb < r -Cyg.
It is worth
-5-
noting, here, that a type 'cindividual's autarkic consumption is the maximum
of r and u(r).
Although all agents know the distribution of types, n(r), each
individual's type is her private information. There is no technology
available for verifying or evaluating agents' types. The output from an
individual's project is publicly observed, as are any contracts into which an
agent might enter. Hence, the only possible method of signalling one's type
is through one's choice of contracts. An agent seeking to raise capital can
issue state contingent claims, since output is public information. An agent
whose 'cis high might seek to signal that fact by offering
claims
which imply
a big difference between her good state and bad state consumptions.
Finally, a semantic note is in order. Except in the case of autarky,
some
agents in this economy will put capital to productive use while others
will
invest their capital in the projects of others or in the risk free
technology. Any agent who uses capital in the operation of her own project
will be called an entrepreneur. Those who do not operate projects will be
called investors.
2. Allocations: Feasibilitv and Efficiency
An allocation for this economy must describe both the distribution of
capital among projects and the risk free technology and the distribution of
consumption goods across agents. The allocation will implicitly describe the
division of agents into entrepreneurs and investors; any agent whose project
is assigned a positive amount of capital is an entrepreneur. We will
consider only allocations which treat all agents of a given type identically,
since agents are only different in a meaningful way through their types.
-6Formally, an allocation consists of four functions and a scalar: the
function x(r) (for all 'ce T) is the capital assigned to a type 'cagent; the
functions cg('c)and cb('l)(for 'csuch that x(r) > 0) are the consumptions of
a type T entrepreneur contingent on good and bad output, respectively; a
function, co(r) (for 'csuch that x(r) = 0) is the consumption of a type 'I
investor; and k is the per-capita amount of capital invested in the risk free
technology. For notational simplicity, we define
C(T) E
for
(Cg(T),Cb(T))
both investors and entrepreneurs and impose a consistency condition on the
consumption of investors:
X(T) = 0
*
Cg(‘c)
=
Cb(‘c)
=
cO(‘c)-
(2.1)
An allocation is denoted a = (x,c,k>.l
We will use the notation E,c(r') for the expected consumption of a type
'Iagent receiving the consumption vector c('c'). Hence, we have that
=
P(~>Cg(~‘>
+ (l-P(T))Cb(T'). If x('c')= 0,
E,c(r')
then Erc(~') = cO(r'). An
obvious constraint on allocations is that agents can be no worse off than
under autarky:
E,c(T)
1
max[R(r),r].
(2.2)
In what follows we will want allocations and feasibility to be defined
for subsets of the population. Accordingly, we will consider arbitrary
measures $ on B(T) such that e(S) 5 n(S) for all S E B(T).2
We will define
feasibility for an allocation-coalition pair (a,$) to encompass resource and
incentive feasibility. Resource feasibility, in turn, is defined by two
-7
-
conditions. First, capital allocated to entrepreneurs cannot exceed the
aggregate capital endowment less capital invested in the alternative
technology.
s
x(r)#(dr) I
g(T) - k.
(2.3)
T
In addition, aggregate consumption cannot exceed aggregate production.3
5sx(T>u(T)$(dT>
+rk.
sE,cWCW
(2.4)
T
T
The careful reader will have noticed that the evaluation of aggregate
consumption on the left hand side of (2.4) assumes that each type 'cactually
consumes the intended consumption: co(r) if x(r)=0 and the state contingent
bundle (cg(r),cb(T)) if x(r)>O. Under private information, the type T agent
could claim to be some other type, T'.
consumption would be
In that case, type T'S expected
E,c(T') = p('dC,(T'> + (l-P(d)Cb(T').
In order to
assure that the evaluation of aggregate consumption in (2.4) is valid, the
consumption allocation must be incentive feasible:4
ETc(~)
>
E,c(T') for all T, 'c'E T.5
(2.5)
The set of feasible allocations A($) for coalition Q is the set of all
allocations satisfying (2.1) through (2.5).6
If this were a full information economy, the matter of determining
efficient allocations would be particularly simple. Consumption allocations
-8-
would be a matter of (Pareto) indifference, so long as all output was
consumed. The standard for the allocation of capital would simply be the
maximization of aggregate output subject to the resource constraint (2.3).
The solution to this problem would involve a marginal ~0.
types greater than ~0 would be funded up to capacity,
X,
All projects with
with any remaining
capital going to the alternative technology.
The marginal type in a productive efficient allocation and the amount
invested in the alternative technology will depend on the alternative
return, r, and on the coalition's measure, I$,over the types of agents.
If r
is large enough relative to the composition of the coalition, then the
marginal entrepreneur would be type 'crdefined by u('cr)= r.
If r is small;
then it is possible for all of a coalition's capital to be used up funding
projects with 'I> rr.
Define M(g) as the support of 4.
The definition of
productive efficiency is complicated by the fact that M(g) might not equal T.
That is, there may be gaps in M(e). Formally, production efficiency is
defined as follows
Definition 2.1
An allocation-coalitionpair, (a,$) is production
efficient if s is feasible for '#I
and:
a>
if X6([rr,11) 2 e(T), then
to E (T 1 x#([~,ll)
= #(T)l = T&#J);
X(T) = x for all 'I:
e [rO,l]flM($);and
X(T) = 0 for all T e [O,~o)flM(#);and
b)
if X$([Tr,ll> < +(T), then
to f 1~ 1 @(I~,111 = $([~r,ll)l
k = $U)
’ TO(@);
- X'#w01ll);
X(T) = x for all 'Le [TO,l]nM(#); and
x(r) = 0 for all 'I[O,rO)nM($).
- 9 -
For this economy, it will turn out that production efficiency is a
characteristic of (some of) the Pareto optimal allocations. In private
information economies, it is generally possible for incentive feasibility to
be incompatible with production efficiency as defined for the full
information benchmark.
An allocation a is Pareto optimal if there is no other feasible
allocation a' such that Erc'('c)2 ETc(r) for all 'IE T, with strict
inequality on a set of positive measure. We could, extending Prescott and
Townsend (1984), show that Pareto optimal allocations are solutions to
problems of the form
1
max
s
6(r)E,c(r)dr
subje:t to (2.1) through (2.5),
for some arbitrary (measurable) welfare weights
6(r) mapping T into I&.
Solutions to such programs correspond to what Holmstrom and Meyerson (1983)
call "interim incentive efficient allocations" and is in keeping with
treating all agents of a given type identically.
The set of Pareto optimal allocations is a big set.
It includes a wide
variety of production efficient allocations. These could include various
combinations of pooling and separation among entrepreneurs of varying types.
The task of the next section is to consider which, if any, of the Pareto
optimal allocations we might expect to be achieved through the rational
interaction of agents.
- 10 -
3. Eauilibrium
How might we expect agents in this economy to solve the resource
allocation problem? One can think of a number of institutional arrangements
which would dictate the way in which agents interact. Capital could be
exchanged for promised payments of consumption good in a decentralized,
competitive securities market. Alternatively, one can imagine large
intermediaries contracting separately with investors and entrepreneurs.
Either one of these institutional arrangements would bring with it a natural
notion of equilibrium.
Rather than imposing a particular institutional structure, we propose to
allow the agents in the economy considerable lattitude in seeking out
bilateral or multilateral contractual arrangements. Other authors who have
taken this sort of approach to private information economies, including Boyd
and Prescott (1986), Kahn (1987), Boyd, Prescott and Smith (1988), and
Marimon (1988) have employed equilibrium concepts which were adaptations of
the core.
In these works, an equilibrium is an allocation which is resource
and incentive feasible and cannot be blocked by any coalition with a feasible
allocation for that coalition. These equilibria differ from the standard
notion of the core by imposing additional restrictions on blocking coalitions
and their allocations. These restrictions are made necessary by private
information and the anonymity associated with "large" economies.
The equilibrium concept employed in this paper is similar to those in
the works cited above. We wish to imagine that the agents in this economy
have an unlimited ability to consider and discuss alternative arrangements.
Once adopted, however, an arrangement must be "self-enforcing." Following
- 11 -
Boyd and Prescott (1986) and Kahn (1987), one could define an equilibrium as
a feasible allocation which is not blocked by any feasible coalitionallocation pair.
As has been pointed out by Bernheim, Peleg and Whinston
(1987) and Kahn and Mookherjee (1988), such a definition treats the candidate
equilibrium and potential blocking allocations in an asymmetric fashion.
Consequently, Bernheim, Peleg and Whinston (1987) have proposed the notion of
a "Coalition-Proof Nash Equilibrium." This equilibrium concept has been
applied to a moral hazard economy by Kahn and Mookherjee (1988). The
equilibrium concept we will develop below is closely related to the CPNE.
As
in the adaptations of the core to private information settings, the notion of
a blocking coalition must be adjusted. Our equilibrium also differs from a
CPNE in that we do not explicitly define a game to which the full definition
of a CPNE can be applied. Instead, we simply seek allocations which are
"coalition-proof" in the same sense as in the CPNE.7
The set of possible sub-coalitions to a coalition # is
+(@Ii
$':B(T)+iR+1 for all S
l
B(T), g'(S) I #(,S),g' + 0
>
For a coalition, 9, let A($) be the set of feasible allocations for that
coalition. The entire population is described by the coalition n.
We will
denote the set of all possible coalitions as P,and the set of feasible
allocations for the population as A.
Our definition of an equilibrium makes use of the following notion of a
blocking coalition.
Definition 3.1:
An allocation for coalition 4,
- 12 -
a=(x,c,k)
l
A(#), is blocked by a coalition
4' s +(I$)and allocation 8' = (x',c',k') if:
(i>
a' E A($'):
(ii>
E,c'(r) 2 E,c(T) almost everywhere with respect to $';
(iii)
there exists SeB(M(g)), with g'(S) > 0 such that
E,c'(r) > E,c(r) for all 'cE S;
(iv>
if SeB(M($)) and S'eB(M(@)) are such that
$(S) > g'(S) and $'(S') > 0, then
E,c(T) 2 Erc'(r') for all 'ce S and T' E S'.
The second and third conditions in the above definition are the usual
requirements of a blocking coalition. The fourth condition is added to
respect the constraints of private information. It states that if a deviant
coalition intends to leave any agents of type 'cin its complement, then those
left behind cannot strictly prefer to join the coalition (including by
claiming to be another type).
Using the above definition of a blocking coalition, the cure is defined
as consisting of all unblocked allocations for the entire economy. We,
however, require an equilibrium allocation to be immune only to deviations by
coalitions which are themselves immune to further deviations by similarly
immune sub-coalitions. We will call this the requirement that an allocation
- 13 -
be "coalition-proof." The set of coalition-proof (CP) allocations can be
described by the Hcoalition-proof correspondence."
Definition 3.2: The coalition-proof correspondence (CPC) is a
mapping, a:&A, with the following properties:
(i>
u(g) E A(@) for all # e 4; and
(ii)
a E a($) if and only if there does not exist
g' E O(g) and 8' e ~(9') which block 8.
Definition 3.2 states that any allocation-coalitionpair which is not
coalition-proof can be blocked by a sub-coalition with an allocation which is
coalition-proof. On the other hand, if an allocation is coalition-proof,
then any "threat" by a coalition to deviate is deterred by a credible threat
of further deviations from the deviant coalition. We can now define an
equilibrium as a coalition-proof (CP) allocation for the full population.
Definition 3.3:
An allocation a* is an equilibrium for this
economy if and only if a* E a(n).
The requirement that an allocation be coalition-proof is weaker than the
requirement that an allocation be unblocked. Accordingly, the set of
equilibrium allocations will contain any core allocations. If the core is
empty, as it is in some cases for the present economy, there will typically
still exist coalition-proof allocations.
Except for the distinction in our definition of blocking necessitated by
the information structure of our economy, Definition 3.3 corresponds to the
- 14 -
definition of a CPNE used in Kahn and Mookherjee (1988) and follows the
definition of a coalition proof correspondence given in Greenberg (1989).
Greenberg shows that, given a notion of "blocking," a definition of a CPNE in
terms of what we have termed the coalition-proof correspondence is fully
equivalent to the recursive definition originally formulated by Bernheim,
Peleg and Whinston (1987). This equivalence allows the extension of the
concept to an environment with an infinite number of agents.8
4. Existence of Eauilibrium
We begin proving the existence of an equilibrium by describing a
candidate CPC.
First, for any g+(n)
select any '~oeTo(#). Set x and k
according to:
X
x(r)
‘kWW[ro,
(4.1)
VreM($)n[O,rG)
0
=
k
Note that
0.
11
=
g(T)
-
(4.2)
X$([q),ll)
x and k satisfy production efficiency, and that when ro > rr, k =
Next set c(r) to satisfy:
c(r)
CO
wrsM($)n[r()Jl
(O,cg)
VreM(#)n[O,ro)
=
(4.3)
where CO = p(rC)cg, and cg satisfies
P(~OM([O,~O>)
+
s1
'10
- 15 -
Define a(#) as the set of allocations &=(x,c,k) that satisfy (4.1), (4.2),
and (4.3) for some rOeTO(
Note that
~(4)
E A(#). We can now state our
central result.
Pronosition 1:
u is a CPC.
Thus the set of equilibrium
allocations, U(9), is nonempty.
Proof.
3.2.
Since
U(g)
s A(#) V#e#(lr),we need to prove (ii) in Definition
Take an arbitrary feasible (a,$). We will construct the set of
allocation-coalitionpairs (a',$'), for $'e9($), that might block (a,#) and
such that &'~a($'). We will then prove that if (a,$) can be blocked by such
a pair,
a$~($),
and that if (a,#) cannot be blocked by such a pair, then
a-f($).
We state without proof the following easily verified implications of
incentive and resource feasibility: E,c(r) is nondecreasing and is strictly
increasing
when c,(r) f cb(r); E,c(r)/p(r) is nonincreasing and is stricly
decreasing if cb('c)# 0; cb(') is nonincreasing; cg('l)is nondecreasing;
XV(~)-Erc(r) is nondecreasing; and cb('c)I c,(r).
We begin with an arbitrary (a,$) such that @O(n), aeA(rj)and g(T) > 0.
We will denote the set of potential blocking allocation-coalition pairs by
(a', #'> with
a’=~(#‘).
The subscript z indexes the amount (measure) of
-
investors' capital to be used.
Let z be the amount (measure) of investors in
the original coalition, if it were to allocate capital according to
production efficiency. If X#([rr,11) 2 e(T), then & =
#(T)(X-1)/X.
If
X#([Tr,l]) < 9(T), then i = 4(T) - @([rr,l])* The first case is that in
which the original coalition does not have sufficient capital to fully fund
- 16 -
all projects with expected output no less than the risk free alternative
output. The second is the case in which the original coalition's endowment
is sufficient to fund all type above 'cr. We will call z the "investment
size" of the coalition $;1and say that z is the efficient investment size of
the original coalition 4.
For each ze(O,i), define 'CO(Z)and rl(z) as follows:
if (X - l)@([rr,l]) <
if (X - I>$([~r,ll>
-q(z)
:
Z,
then
2 z9 then
('I1 (x-l)$([T,l]> = z).
In constructing production efficient allocations, one makes investors out of
all 'cup to some threshold. Hence TO(Z) is the set of T such that one can
extract exactly z units of capital by making all types up to T investors. If
one seeks to make some 'c> suprO(z) an investor in a production efficient
allocation, then one needs a coalition with investment size greater than z.
Similarly, rl(z) is the set of r such that all r'2r can be fully invested
with capital in an efficient allocation of z units of capital.
Each of the correspondences TO(Z) and rl(z) is single valued except at a
finite number of points in (O,z]; they can be set-valued only where there are
- 17 -
gaps in M(g). Furthermore, r0 is strictly increasing, and '~1is strictly
decreasing, except when r,erl(z). Note that 'rO(i)= rl(z) = TO(#), as
defined in the definition (2.1) of production efficiency.
The sets TO(Z) and rl(z) characterize production efficient capital
allocations for coalitions of investment size z.
In particular, we will
to be the set of entrepreneurs in the
define X(z) 1 [suprl(z),l]flM($)
coalition $;1. Similarly, let Xc(z) Z [O,infrO(z)]nM($)be the set of
investors. Specification of efficient capital allocation is completed by
noting that investment in the alternative is k(z) = z - (x-1)4(X(z)).
We now want to charaterize consumption allocations, consistent with
a'(z) E
a(#;),
which might block (a,'$). Hence, all investors will be given a
constant consumption, CO. This consumption must be preferred to the original
allocation by all types below 'CO(Z)and not preferred by any type above
q)(Z). Accordingly, define CO(Z) by
co(z)
E
{cOlE,c(r) I CO, for .reX'(z)
and E,c(r) 1 CO, for reM($)\Xc(z)]
for ze(O,z), and
co(Z)
5
[Elc(x),+*), where r=infrO(z).
The definition of co(z) allows for the possibility that the original
allocation "wastes" some consumption good.
Once one has the efficient sets
of investors and and entrepreneurs, if their consumptions don't exhaust
available output, consumption can rise. Clearly, there will be some level
- 18 -
above which CO(~) will be infeasible. It is also clear that, as CO(~) rises,
the consumption given to entrepreneurs must
rise.
Otherwise, too many
investors would be drawn to the consumption CO(~). Entrepreneurs get a
consumption schedule, (O,cg), which must be preferred to the status quo by
all r above Al
and not preferred by all below Al.
The set of cg which
can serve this purpose can be expressed as
J+(T)
c,(z)
-= fcglpr
g, for ~W#)\X(z),
E,dQ
IC
and P(T)
5
cg,
for reX(z))
for ze(O,i), and
c,(Z) :
[E;c(;),+m), where i = suprl(i)
Let Xc (with no z-index) be the set of investors in the original (a,$).
Recall that, in the original allocation, all investors received a constant
consumption. Therefore, CO(Z) is constant (and single valued) for z < #(Xc).
For z 2 9(X'), CO(Z) is single valued except, possibly, at (a finite number
of) points in [#(Xc),;] which correspond to gaps in M(Q)). For z > $(Xc),
co(z) is strictly increasing. Note that the set [(cO,z) 1 cOecO(z), ze(O,i]l
is an unbroken curve in lR2.
Recall, now, that feasibility of (a,#) implies that E,c(r)/p(r) is
nonincreasing, and strictly decreasing when cb(r) > 0.
nonincreasing, Erc(r)/p(r) is constant on at
most
Since
q,(T)
iS
a right hand interval of
[O,l], and is strictly decreasing elsewhere. Discontinuities can only occur
- 19 -
at gaps in M(#).
Hence, c,(z) is set valued only at gaps in M($).
that, as we raise z, we draw in entrepreneurs from the top down.
Note
Therefore,
c,(z) is constant on, at most, a left-hand interval in [O,z] (corresponding
to the right-hand interval in [O,l] over which E,c(~)/p(r) is constant).
Elsewhere, it is strictly increasing.
The sets X(z), Xc(z), CO(Z), and c,(z) describe the set of all
allocations for coalitions of investment size z which: make all members at
least as well off as in the original allocation; give nonmembers (r E
(suprO(z),infrl(z))no incentive to join, even by misrepresenting their
types; and have (internally) incentive feasible consumption schedules which
have the "right form" (some CO for investors and some (0,~~) for
entrepreneurs). To complete our specification of potential blocking (a,#),
we need to find investment sizes for which the allocations given by these
correspondences are resource feasible.
To do so, we begin by defining the
'correspondence l'(z)by:
[xv(r)-p(r)cg]$(dr)+ rk(z)
and
r(z)
E
Note that
ircz,cgl
XV(T)
1 cg E CgWl
- p(r)c, is strictly increasing in 'c. This fact is most
clearly seen by observing that xyb<r, and that, for an investment to be
preferred to the alternative, xv(r)-p(r)cg2r. As z increases, X(z) grows by
- 20 including lower types (rI(z) nonincreasing). Therefore, the average residual
paid to investors, T(z,cg), is nonincreasing in z.
Further, the average
residual can only be constant in z if X(z) is empty. This would occur if all
members of the original coalition had types below r,.
In that case, the
-
average residual would be identically equal to r, for all z in [O,z]. As a
result, we have that I'(z)is either constant everywhere, or strictly
-
decreasing everywhere. Note that the set ((a,~) I xsI'(z),ze(O,z]) is an
unbroken curve in 178'.Given the definition of c,(i), T'(i)extends to -00.
For any z s (O,;], define $i as follows: #L(S) = g(S) for
SeB(X(z)UXc(z)); and @L(S) = 0 for SeB[T](X(z)UXc(z))]. Let a(z)%(z) be
the set of allocations that satisfy: if T e X'(z), then x(r) = 0 and C(T) =
(co,co), co E q)(z); if r e X(z), then x(r) =
x
and c(r) = (O,cg), cg s
c,(z)- The set of feasible (a',$') which satisfy $'EQ(#I)and a'~($')
and
*
which potentially block (a,$) can be derived from the set Z :
z*
E
(Z I CO(Z) n
r(z)
is nonempty].
To see that Z* is nonempty, we need to show that supl'(z)linfco(z)for
some z>O (since l'(z)is decreasing and CO(Z) is nondecreasing). To find such
z, note, first, that for all z I #(Xc), cO(z)=cO, a constant. Now consider
the limit of supl'(z)as z goes to zero.
If X (the set of investors in the
original (a,$)) is nonempty, then this limit is [Xv(i) - E;c(;)]/(X-l),
where, T = supM(@). If X is empty, T is identically r everywhere, including
in the limit. From the feasibility of the original allocation, this limit is
greater than co when X is nonempty and greater than or equal to co when X is
- 21 -
empty. The existence of a z such that I'(z)> CO(Z) implies that there is
some z such that l'(z)=cO(z>.
We have constructed a set of allocation-coalitionpairs, D* g ((a',$;) 1
zez*, and a'scr(z)]. Pairs in D* satisfy conditions (i), (ii) and (iv) of
definition 3.1 of a blocking allocation-coalition. The remaining condition
for blocking (condition (iii) of Definition 3.1) is the condition that some
types be made strictly better off for a successful block. Hence, we can
complete our proof of the proposition by proving that a E o(#) (where (a,$)
is the original coalition-allocationpair) if and only if E,c'(r) = E,c(r)
for all 'IEM(
for all (a',$') s D".
First, suppose that 8 E a($).
Then, for all z E (O,;], CO(Z) = CO, and
c,(z) = cg (where CO and cg are investors' and entrepreneurs' consumptions
from the original allocation). Therefore, Erc'(r) = Erc(r) for all 'ce
M(#L), for all allocation-coalitionpairs in D".
Now, suppose that Erc'(r) = E,c(r) for all reM($'), for all (a',$') s
D*.
We will prove that a s a(#) for three cases:
empty for all z;
2)
1)
T I Tr, so X(z) is
X(z) is nonempty (i > rr) and Z* = (i); and 3)
is nonempty and Z" = (z"], z*<i
X(z)
(recall that, when X(z) is nonempty, I'is
strictly decreasing, so that, in both of the last two cases, Z" is a
singleton).
Case 1:
Since 'cI 'cr,preduction efficiency requires no entrepreneurs
(all investment goes to the alternative technology). For all
co'.
Z,
I'(z)= r =
The assumption that E,c'(r) = E,c(r) implies that, for all r s Xc, CO =
co' = r.
Feasibility of (a,#) requires
- 22 -
[X(T)V(T) - E,c(r)]$(dT) +
The right hand side of the above inequality is less than or equal to r, with
equality only if g(X) = 0.
Thus feasibility requires that (a,'#~)
is
production efficient, in order for CO = r to be satisfied. For this case,
production efficiency and investors' consumption equal to r yield B e u(g).
then the only possible blocking coalition is the
IfZ" = {z),
-
Case 2:
Since Erc(r) = E,c(r) for all 'ce M(g) = M(@'), we
coalition of the whole.
C(T) = (c6,cb) for all r e Xc(Z); and C(T) = (0,cg) for all 'cs
know that:
X(Z).
Therefore, X(Z) s:X; only entrepreneurs can have output-contingent
consumption. Feasibility of (a,$) implies
C
co9(X > +
E,c(OCW
s
X
s
x(~>v(~)#(d~l + rk.
s
X
The left hand side of the above can be written
co9(XC)
+ c()~(XC(Z)\XC)
+ J pWcgWO
X(Z)
C
=
q)@(X
(2,
+
s
pWcg’CW
G>
=
xu(r)$(dr) + rk.
s
X(2
Therefore, resource feasibility of (a,'#)requires that B be production
efficient and, thus, X=X(i), X(T) = x for 'cE X, and k = k(i).
These
- 23 I
, and, therefore, that a' e
8=8
equalities imply that
a(#)
by
construction.
Case 3:
In this case, the potential blocking coalition has investment
size z* less than i, so that
(X(z*)UXc(z*)>
c
(XUXc).
As in case 2, we have
that c(r) = (0,~~') for all 'cs X(z*). This equality implies that X(z*)nX.
For all 'Ie X'(z*), we have c(r) = (cO',cO'). This, in turn, implies that
C(T) = (c6,cb) for all 'ce Xc.
Define X, Xc, k as the production efficient
capital allocation for 4 (the entire original coalition), From the fact that
#Iis strictly bigger (in investment size) than $', together with the fact
that xv(r) - E,c(r) is increasing in 'I(from incentive feasibility), we have:
[Xv(T)-ETc(T)l@(dT)+ rk
1
#aC)
'
[Xv(r)-ETc(r>]$(dr)+ r-k .
2
The last inequality follows from the fact that the left hand side is l'(z*),
while the right hand side is l'(i). From this inequality, we have:
s
x(r)u(T)$(dr) + rk
5
X
s
x
C
<
COG
>
+
Er40CW
s
x
<
co
+
E,c(d$(W.
s
X
xu(r)@(dr) + rk
- 24 -
The last inequality, here, follows from the fact that E,c(r) 2 CO for all r e
MC@), and the fact that X E X.
This chain of inequalities implies that
(a,$), the original allocation-coalitionpair, was not feasible, a
'c> rr (so that
contradiction. Hence, case 3) is not possible. That is, if X(z) is nonempty for all z), and if E,c'(T) = E,c(r), then z* = z.
Then, by
case 2), B s u(g). Q.E.D.
Equilibrium allocations, U(Y), have a straightforward interpretation.
Take a production efficient allocation of capital and support it with a
feasible consumption allocation which: does not waste any consumption good;
pools all entrepreneurs at a single consumption bundle (0,~~); and pools all
investors at a single consumption point, co, such that the marginal type '10
is just indifferent between the investor's and the entrepreneur's
consumption. The marginal type, '0, is determined by the selection of a
production efficient allocation of capital. For any coalition, including the
coalition of the whole, it is possible for there to be more than one possible
marginal type (that is, for TO(#), as defined in Definition 2.1, to contain
more than a single point). For any given efficient marginal type, the
procedure described by (4.1), (4.2) and (4.3) yields a unique allocation.
Hence, a sufficient condition for the uniqueness of equilibrium is that the
support of 71be the entire interval [O,l].
It will be useful, here, to examine the relationship between the
coalition proof allocations derived above and the core of the economy. The
core can be defined, using definition 3.1, as any allocation on the entire
economy which is unblocked. Clearly, any unblocked allocation is, in
particular, unblocked by coalition proof deviations. Hence, our set of
I
- 25 -
equilibrium allocations contains the core. Since our equilibrium is unique
(except when TO contains more than a single element), it follows that either
the core is the same single allocation, or
the core is empty. We show below
that, if production efficiency requires that some capital go to the
alternative investment, the core is empty.
Pronosition 2:
Let k* denote the allocation of capital to the
alternative technology in an allocation sea(n).
the set of entrepreneurs in a.
Let X" denote
If k" > 0, and n(X*) > 0, then
the core is empty.
Proof: The inclusion of the core in
u(n)
means that to prove that the
core is empty we need only construct, for any CP allocation A, a blocking
(a',$'). All other allocations are blocked by something in the CPC. Note that
in the CP allocation 8, with k* > 0 and n(X*> > 0, two types of "investments"
are being made;
some capital is being allocated to the alternative, and the
rest is being allocated to entrepreneurs in the form of a pooled contract
ULcg).
Since all entrepreneurs have r 2 rr, the expected return on this
pooled contract is strictly greater than r.
Consider a coalition which takes
all of the entrepreneurs from the original allocation and "some" of the
investors. Suppose that this coalition invests all of its capital in
entrepreneur's projects (none in the alternative). That is, the coalition's
composition can be given by:
#'(Xl = IT(X);and #'(Xc) = n(XC) - k* (where X
and Xc are the entrepreneurs and investors, respectively, in the CP
allocation).
Now consider the following allocation for #':
X' = X, and Xc' = Xc;
C(T) = (0,~~) for z e X; and c(r) = (cO,cO) for 'Iin Xc.
This is not a
- 26 blocking allocation, since all types have the same consumption as in a.
Note, however, that average output will be strictly greater than in the
original allocation. None of this surplus output can be given to investors
without drawing in all investors (in violation of $'(Xc) < n(XC) ).
Give
some of the surplus to entrepreneurs by setting cg' = cg + E, for E close to
zero. This increase makes entrepreneurs strictly better off.
Note, however,
that the increase in cg also makes some investors near the margin ('Ijust
below rr) switch to being entrepreneurs. This switch lowers average output.
For small E, though, average output will stay high enough so that the
remaining investors can be paid the original co.
In fact, for E small
enough, there will still be surplus output left after paying the investors co
and the entrepreneurs cg + E.
For some E near zero, then, (a',$'), as constructed above, successfully
blocks (a(n>,~>. Since any allocation a # a(n) can be blocked by (a',#') for
some @' e iP(n)and a' e a(@'), the core is empty. Q.E.D.
We feel the proof is suggestive of the value of the credibility
restriction on potential deviations. All of the allocations which can be
used to block allocations in o(v) are inefficient. They necessarily attract
"sub-marginal" entrepreneurs. In order to avoid investing in the alternative
technology, a blocking coalition must invest some capital in projects which
are even worse than the alternative technology. In addition, a blocking
allocation may have to throw away some surplus output. The less it throws
away, the more llinferior"entrepreneurs are attracted. One can easily
imagine that the coalition described above, if it formed, would realize that
it was using its capital inefficiently. If one so imagines, however, it is
- 27 -
hard to imagine that such a coalition would ever form as an effective
blocking coalition.
If the coalition proof allocation does set k=O (because r is "low"),
then even an inefficient (non-credible) coalition cannot effectively deviate.
In this case, there is only one "type" of investment being made.
All capital
is invested in the single, pooling, entrepreneur's contract. Hence there is
no gain to entrepreneurs from forming a coalition which invests nothing in
the alternative; nothing is already invested in the alternative. In this
case, it seems as though the CP allocation is also the (unique) core
allocation.
Note that the comparison between CP allocations and the core depends on
the value of the return to the alternative investment. In the next section,
we argue that the value of this return is also crucial in determining whether
CP allocations can be supported as the competitive equilirium of a
decentralized securities market. We might suggest, here, that the case in
which r is "high" (and k > 0) is the more "natural" case.
If we wish to
imagine an economy with an "entrepreneurial sector" and a sector of more well
established productive activity (the alternative investment), then the
typical case would seem to be that in which some capital goes to each of the
sectors. Such reasoning is somewhat self-serving, since it tends to
strengthen the case for the coalition proof equilibrium concept we employ.
Even though the core is empty, we are able to describe an arrangement with
desirable sustainability properties. The remaining task is to search for
institutions which may support this arrangement.
- 28 -
5. Alternative institutional arrangements
The equilibrium concept employed above imposes very little structure on
the way in which agents interact in this economy. A natural question to ask
is whether the essentially unique coalition-proof allocation described in
section 4 can be supported by a competitive securities market.
If not, what
other institutional arrangements might allow the economy to achieve the
proposed allocation? This line of questioning follows in the spirit of Boyd
and Prescott (1986). For a somewhat different private information credit
economy, they show cases in which a securities market "works" and others in
which such an institution "fails." In the latter cases, they argue that one
should expect alternative institutions to arise. In the case of their
economy, the alternative institution which does the trick is a form of
"large" financial intermediary. What follows is an informal discussion of
the effectiveness of and alternatives to securities markets
in our economy.
For this economy, we propose to view a securities market as one in which
entrepreneurs offer contingent claims contracts in order to raise capital
from investors. Given their beliefs about what types of entrepreneurs are
offering what contracts, investors will purchase claims paying the highest
rate of return. Taking the beliefs of investors and the "market rate of
return" as given, entrepreneurs will seek to offer contracts which
maximize
their own expected consumption. Clearly, in equilibrium, all projects which
receive financing must pay the
same
rate of return under investors' beliefs.
In addition, it cannot be possible for any entrepreneur (or would be
entrepreneur) to offer a contract which increases her own expected
- 29 -
consumption and which is expected by investors to pay a greater than market
rate of return.
A securities market equilibrium will typically exist in our economy. '
Whether a securities market equilibrium fails or succeeds to support the
coalition-proof allocation seems to depend on the return to the alternative
technology. Consider, first, the case in which this return, r, is small
enough so that production efficiency implies no investment in the alternative
(this is case (a) of Definition 2.1).
In this case C(T) = (0,~~) for
all 'I
2 To. This allocation corresponds to a market in which each entrepreneur
offers output contingent payments r(y) such that
= xyb.
r(y,) = xyg - cg
and r(yb)
Since all entrepreneurs are offering the same (pooling) contract,
investors' expected return is averaged over [~0,1]. This average return is,
of course, exactly the consumption CO assigned to investors by the
coalition-proof allocation. With the pooled contract, higher r entrepreneurs
will find themselves paying a higher than average return while lower T
entrepreneurs pay lower than average (as evaluated by the entrepreneurs
themselves, not by investors). Hence, high r entrepreneurs would like to be
able to separate themselves by offering a contract with a higher cg and a
lower
Cb.
This is not possible, however, since
Cb
= 0.
Any feasible
deviation which a high type would find attractive would also be attractive to
lower type entrepreneurs. The coalition-proof allocation is supported by a
securities market equilibrium in this case.
The case of low r in our model corresponds to one of the cases discussed
in Boyd and Prescott (1986). Their general model allows for the costly
production of publicly observed signals of agents' types. One special case
arises when the signal is uncorrelated with the true type. This is
- 30 -
equivalent to there being no technology for information production, as in our
model.
As in our low r case, there is no alternative investment technology.
For this case, Boyd and Prescott show that a securities market equilibrium
supports their core allocation. Except for the pooling of many types of
entrepreneurs on one consumption point
(in the Boyd and Prescott model there
are but two types of agent), our coalition-proof allocation is very similar
to their core allocation. In both cases, those agents who become investors
are paid enough to keep them from trying to mimic better types
(entrepreneurs). They earn rents
in that they earn a rate of return which
is strictly greater than their autarkic consumption.
Our case of high r does not correspond to any case considered by Boyd
and Prescott. A number of other authors, however, have produced market
failure results for private information credit markets with non-trivial
alternative investment opportunities. Stiglitz and Weiss (1981), among
others, have suggested the existence of credit rationing in such environments
while Meza and Webb (1987) have produced an over-investment result for an
environment very similar to our own.
The notion of equilibrium used in these
models is very similar to the securities market equilibrium sketched above.
It should not be surprising, then, that, in our case of high r, a securities
market cannot support the coalition-proof allocation. The reason is quite
intuitive. There are two types of investments that investors can make; they
can invest in the pooled contract offered by the entrepreneurs or they can
invest in the alternative technology. Since, in the coalition-proof
allocation, all entrepreneurs are "infra-marginal" (u(r) 2 r), the expected
rate of return from entrepreneurs' projects is greater than r.
This is
incompatible with a securities market equilibrium. An equilibrium must have
- 31 -
the return averaged over entrepreneurs' projects equal to r.
This is
achieved by having investors' consumption smaller than in the coalition-proof
allocation and entrepreneurs' good state consumption (c,) greater than in the
coalition-proof allocation. These consumptions draw "sub-marginal" (U(T) <
r) types out of investing and into entrepreneurship. This yields an "overinvestment" result analogous to that of de Meza and Webb; too much is
invested in entrepreneurial projects and not enough in the alternative
technology.
If a securities market fails to be efficient in the case of high r, then
is there
some
other form of institution which succeeds? A natural
possibility is some form of intermediated financial market.
One can imagine
some agents acting as intermediaries (one might wish to assume here that
acting as an intermediary does not interfere with an agent's other role as an
investor or entrepreneur); an intermediary would try to attract agents into
its organization or coalition by offering contracts for both entrepreneurs
and investors which imply an allocation for that coalition. Once a coalition
is formed, its members might be free to renegotiate contracts; if so, the
intermediary's initial proposal must be coalition proof for the coalition it
is seeking to attract. If there is "free entry"
into intermediation, then
intermediaries must make zero profit.
The intermediaries in this scenario have four of the five
characteristics of financial intermediaries highlighted by Boyd and Prescott;
they borrow from one subset of agents (investors) and lend to another
(entrepreneurs); both subsets are "large"; would-be borrowers have private
information concerning their own credit risk; and
claims
issued by the
intermediaries (payments to investors) have different state-contingent
- 32 -
payoffs from the
claims
issued by the ultimate borrowers. The one
characteristic missing is that intermediaries do not spend resources
producing information on the attributes of would-be borrowers. While
expenditures on evaluation (and monitoring) are certainly an important
component of the activities of real-world intermediaries, it is interesting
that we find a role for intermediaries in an environment where no such
expenditure is possible. In the Boyd and Prescott environment, the
existence of an informative, costly signal was essential for intermediaries
to have an important role in resource allocation.
Although the environments are different, the reason why intermediaries
may
be needed is essentially the same in the present model as in Boyd and
Prescott; intermediaries allow the economy to avoid the "excessive
signalling" that occurs in a securities market.
In Boyd and Prescott,
excessive signalling takes the form of over-investment in evaluation. In our
case of high r, signalling amounts to claiming to have a high enough T to be
an entrepreneur; this claim is backed up by a willingness to accept the
state-contingent entrepreneur's consumption schedule. When agents with u(r)
< r seek to be entrepreneurs (as occurs in the securities market if r is
high), then excessive signalling is occuring. Hence, the need for
intermediaries is brought about not by the existence of a costly evaluation
technology but by the existence of any signalling behavior which might be
over-utilized in a securities market setting.
6. Conclusion
Our understanding of the allocation of resources under private
information is still in its infancy as compared, for instance, to our
- 33 understanding of "classical" (Arrow - Debreu) environments. Key questions
deal with how and why observed institutions arise and with the role of
institutions in solving resource allocation problems. We have contributed to
the effort to answer these questions by applying an equilibrium concept which
puts relatively few restrictions on the way in which economic agents
interact. There may be conditions under which more restrictions are natural.
For instance, limited communication may restrict agents' abilities to propose
and discuss alternative arrangements. Similarly, limited commitment or ex
post
private information can restrict agents' abilities to make state-
contingent arrangements. One goal of the theory of mechanism design should
be to map out how various combinations of such frictions affect the types of
bilateral or multilateral arrangements into which economic decision makers
will enter. This paper has attempted a modest step in that direction.
- 34 -
NOTES
1. Actually, we will regard as equivalent any two functions on T that
differ only on sets of measure zero. Thus, an allocation is an equivalent
class of functions.
2. Note that, since ITis nonatomic, all measures Qion B(T) such that
$(S)ln(S) for all SeB(T) are also nonatomic.
3. The resource constraint (2.4) is, as written, requires that exnected
aggregate consumption not exceed exnected aggregate production. However, with
a continuum of independent random variables, realized aggregate values are
not necessarily equal to expected aggregate values. We, however, do not need
the law of large numbers. All we need is for entrepreneurs to get their
state-contingent (cb(r),cg(T)), and for the realized residual to be
distributed among investors in
some
(possibly random) way so that expected
consumption is co(r) for investors. That is, given risk neutrality, co can
be an expected value of a consumption lottery. It is, nevertheless,
convenient to speak of CO as if it were deterministic (that is, as if a law
of large numbers did apply). Nothing in our analysis would change if we
treated CO explicitly as a random variable.
4. In order to avoid cumbersome statements of conditions regarding r, we
adopt the following
semantic
convention: if g(T,T') is some measurable
expression (such as E,c(T) - ETc(~')), then the statement, ))g(r,T')2 0 'd&.,
- 35 -
‘VT%$‘,”
means that, for all S'%(i) such that g(S)>0 and for all S'eB(S')
such that $(S')>O,
['#(S)'#(S')l-1!sfs,g(r,~%'(d~')#4dr)
2
0.
Equalities are defined similarly.
5. Imposing incentive compatibility as a feasibility constraint amounts
to invoking the revelation principal in some form.
In a multilateral
setting, this requires showing that any attainable allocation can be achieved
by a direct mechanism that indices truthful revelation as a (Nash)
equilibium. With an infinite number of agents, and a fixed "number"
(measure) of agents of each type, an individual agent's message has no effect
on the "aggregate" nature of messages sent. This allows a straightforward
application of the revelation principal. Note that, although we are allowing
for some degree of "cooperation" in the choice of allocations, the
implementation of an allocation must satisfy the noncooperative requirement
of incentive compatibility.
6. We also require that x and c be $-measurable. Recall, from note 1,
that x and c are equivalence classes of functions which differ only on a set
of measure zero with respect to $.
7. We could, following Kahn and Mookherjee (1988), define a contracting
game whose CPNE outcome is an allocation satisfying our definition of an
equilibrium. Our equilibrium bears the same relationship to CPNE as do the
core-type equilibria to the Strong Nash Equilibrium concept of Aumann (1959).
- 36 8. Kahn and Mookherjee (1990) show that, with infinite sets of agents
and infinite strategy spaces, what they call a consistent set, analagous to
our CPC, may fail to exist even though CPNE, by the recursive definition, do
exist.
If, however, a consistent set does exist, then it exactly corresponds
to the set of CPNB.
9. Because of the dependence on investors' (off-equilibrium) beliefs,
there may in fact be multiple equilibria for the securities market
institution. We are confident that the equilibria on which we focus would
survive the usual type of refinements based on beliefs (as, for instance, in
Cho and Kreps (1987)).
- 37 -
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