Adverse Selection in Competitive Search Equilibrium
Veronica Guerrieri
University of Chicago

Robert Shimer
University of Chicago

Randall Wright
University of Pennsylvania
February 2, 2009

Abstract
We extend the notion of competitive search equilibrium to an environment withadverse selection. Uninformed principals post contracts to attract informed agents.
Agents observe the contracts and apply for one, trading off the probability of matchingwith a principal against the terms of trade offered by the contract. We characterize
equilibria as the solution to a constrained optimization problem and show that inequilibrium principals offers separating contracts to attract different types of agents.
We then present a set of examples, including a workplace rat race, insurance against
layoff risk, and lemons in asset markets, to illustrate the usefulness of our model.

1Introduction

This paper studies equilibrium and efficiency in an economy with adverse selection. A large
number of uninformed principals compete to attract a large number of informed agents.
We extend the notion of competitive search equilibrium to allow for private information.
Principals post incentive-compatible contracts which specify an action profile if they matchwith a particular type of agent. Agents observe the posted contracts and direct their search
towards the most attractive ones. Matching is limited by the restriction that each principal
can match with at most one agent, and also possibly by search frictions. For example, if
fewer principals offer a particular contract than the number of agents who wish to obtain it,each agent matches only probabilistically. Principals and agents form rational expectations
about the market tightness of each contract—the ratio of principals posting that contract toagents who direct their search to that contract—as well as the composition of agents who
search for the contract.
Part of the contribution of this paper is technical: we develop a canonical extension to
the competitive search model (Montgomery, 1991; Peters, 1991; Moen, 1997; Shimer, 1996;
Acemoglu and Shimer, 1999; Burdett, Shi and Wright, 2001; Mortensen and Wright, 2002)
that allows for ex-ante heterogeneous agents with private information about their type. We
prove that, under mild assumptions, including a weak version of a single-crossing condition,
there exists an equilibrium where principals offer separating contracts: each contract posted
attracts only one type of agent, and different types of agents direct their search towards
different types of contracts. The expected utility of each type of agent is uniquely determined in equilibrium. Moreover, the set of competitive search equilibria is easily found by
sequentially solving a constrained optimization problem for each type of agent.
We also present a series of examples and applications of our model. These serve three
purposes: they illustrate the usefulness of our approach; they show that some well-known
results in either contract theory or search theory can change when we combine elements ofboth in one model; and they enable us to explore the role of our assumptions and discuss
what happens when they are relaxed.
The first example is a version of the classic costly signaling model (Spence, 1973; Akerlof,
1976). Workers are heterogeneous both in their productivity and in their cost of working
under unpleasant conditions, for example long workdays. More productive workers find long
workdays to be less costly. We focus on an extreme case where firms care about workers’
productivity but not about the length of the workday. In equilibrium, firms separate workers
by making more productive workers work longer days, a version of a rat race. Curiously,
we find that the probability that each type of worker gets a job—the unemployment rate—1

is unchanged from a version of the model with symmetric information, and hence with anefficient workday. We prove that a Pareto improvement is feasible if there are sufficiently few
unproductive agents or if the difference in the cost of working long days is small relative tothe difference in productivity levels. We also consider a version of this example where long
workdays are cheaper for less productive workers. In this case there exists an equilibrium
where workers are pooled and there is no distortion in the length of the workday, but wefind that other equilibrium may also exist.
Our second example is a modified version of the Rothschild and Stiglitz (1976) insurance
model.1 Risk-averse workers and risk-neutral firms match in pairs in order to produce output.
However, only some pairs are productive and workers differ in the probability that they canform a productive match. In equilibrium, firms separate workers by partially insuring themagainst the productivity shock. In particular, workers are worse off if they lose their job thanif they had never found a job. We interpret this example as providing an explanation for
why firms do not insure workers against the risk of a layoff. To do so would risk attracting
too many unproductive workers. We show that, even if a pooling contract does not Pareto
dominate the equilibrium, a partial pooling allocation—only pooling some types of workers—
may be Pareto superior.
This example shows that competitive search offers a resolution to the famous nonexistence
problem in Rothschild and Stiglitz (1976). When there are relatively few low productivity
workers, equilibrium may not exist in the original Rothschild-Stiglitz model, because any
separating contract is less profitable than a pooling contract that cross-subsidizes low productivity workers. In our model, such a pooling contract is infeasible, regardless of thecomposition of the worker pool. The key difference is that in our model, firms are small andso a deviation cannot attract the entire population. Suppose a firm posts a contract that is
supposed to attract a representative cross-section of the population. Because it can matchwith at most one worker, the more workers who try to obtain a contract, the less likely eachworker is to match. This drives some workers away from the contract. Critically, the mostproductive workers are the first to leave, because their outside option—trying to obtain aseparating contract—is more attractive. This means that only undesirable workers would be
attracted by such a deviation, which makes it unprofitable.
In the first two examples, our use of competitive search equilibrium affects the contracts
that are offered in equilibrium but asymmetric information does not affect the equilibrium
1This example builds on a discussion between Daron Acemoglu and one of the authors. We discuss therelationship between this example and the original Rothschild-Stiglitz model more in the body of the text.
We focus on a labor market application because we find the assumption that a firm can only hire a small
fraction of the available workers more reasonable than the assumption that an insurance company can only
cover a small fraction of the population.

2

frictions. Our third example reverses this: asymmetric information makes it harder for someagents to find a trading partner but does not affect the terms of trade conditional on findinga partner. We study a stylized model of asset trade (Akerlof, 1970). Agents want to sell aheterogeneous object, say apples that could be good or bad, to principals. Principals value
apples more than agents do, so there are gains from trade. However, bad apples are valued
less than good apples—they are lemons. There are no search frictions, so everyone on the
short side of the market will match.
We show that in equilibrium agents holding good apples only trade probabilistically.
The probability of a meeting, rather than the terms of trade within a meeting, screens out
the agents holding bad apples. This economizes on any cost that principals must incur inmeeting an agent. Once again, we find that the competitive search equilibrium is Pareto
dominated by a pooling allocation if there are few enough agents holding bad apples. We
also find that if there are no gains from trade in bad apples, adverse selection will entirely
shut down the market for good apples, an extreme version of the lemons problem.
Our paper is related to a growing literature exploring search models with private information. In particular, Faig and Jerez (2005), Guerrieri (2008), and Moen and Rosén (2006)
propose different extensions of competitive search models with one-side private information.
However, in all these papers agents are ex ante homogeneous and all heterogeneity is matchspecific. Inderst and Müller (1999) is an exception that extends the standard notion ofcompetitive search to an environment with ex ante heterogeneous agents. That model is a(slightly) special case of our first example. Inderst and Wambach (2001) explore a version of
the Rothschild and Stiglitz (1976) model with a finite number of principals and agents and acapacity constraint for each agent. This paper is related to our second example. Our paper
is the first to develop a general framework for defining and analyzing competitive search withadverse selection in a wide variety of applications.
Many papers have studied related economies with adverse selection but without search.
Driven by the nonexistence issue in Rothschild and Stiglitz (1976), Miyazaki (1977), Wilson(1977), and Riley (1979) propose alternative notions of equilibrium that offer possible resolutions. In contrast to these papers, we generally find that there is no cross-subsidization in
equilibrium. The key difference is our assumption that each principal is small relative to thenumber of agents. Since he cannot attract all the agents with a pooling contract, a principal
must deduce which agents are most attracted to such a contract, which we show eliminates
the incentive to pool.
Prescott and Townsend (1984) study adverse selection in competitive economies, concluding pessimistically that “there do seem to be fundamental problems for the operation ofcompetitive markets for economies or situations which suffer from adverse selection.” (p. 44)
3

More recently, Bisin and Gottardi (2006) propose a notion of Walrasian equilibrium withcompetitive markets, but agents are restricted to trade only incentive-compatible contracts.
They also find that there always exists a separating equilibrium. Although our notion ofequilibrium is more strategic, their equilibrium allocation has some features that are similar.
In particular, the incentive-compatibility condition that they impose on the set of admissible
trades is analogous to a condition that arises endogenously in our model.
The rest of the paper is organized as follows. In Section 2, we develop the generalenvironment, define competitive search equilibrium, and discuss the critical assumptions. In
Section 3, we show how to find competitive search equilibria by solving a relatively simple
constrained optimization problem. We prove that a separating equilibrium always exists andshow that the equilibrium vector of agents’ payoffs is unique. In Section 4, we define theclass of incentive-feasible allocations, and discuss whether equilibrium outcomes are efficient
within this class. In Sections 5-7, we explore the examples and applications discussed above;in each case we characterize equilibria and discuss efficiency. Section 8 concludes.

2

Model

We consider a static model. There is a measure 1 of agents, a fraction πi > 0 of whom are oftype i ∈ I ≡ {1, 2, . . . , I}. The type is the agent’s private information. There is also a large
measure of ex ante homogeneous principals. Principals and agents have a single opportunity
to match.
A principal may post a contract, at cost k > 0, which gives him an opportunity to matchwith an agent. We discuss the nature of contracts in the next paragraph. Let Y ⊂ Rn
denote the space of feasible action profiles for principals and agents who are matched, andassume Y is compact and nonempty. A typical action profile y ∈ Y may specify actions by
the principal, actions by the agent, and transfers between them, among other possibilities.
A principal who matches with a type i agent gets a payoff vi (y) − k if they undertake theaction profile y ∈ Y. A principal who does not post a contract gets payoff normalized tozero, while one who posts a contract but fails to match gets −k. A type i agent matchedwith a principal gets payoff ui(y) if they undertake the action profile y ∈ Y, while unmatchedagents get a payoff normalized to zero. Assume ui : Y 7→ R and vi : Y 7→ R are continuousand bounded for all i.
We use the revelation principle to assume without loss of generality that the contracts
are revelation mechanisms. More precisely, a contract is a vector of action profiles, C ≡{y1 , . . . , yi, . . . , yI } ∈ YI , specifying that if a principal and agent match, the latter announces
her type i, and they implement yi . A contract C ≡ {y1 , . . . , yI } is incentive compatible if
4

ui (yi) ≥ ui(yj ) for all i.2 Let C ⊂ YI denote the set of incentive compatible contracts.
Principals only post incentive compatible contracts.
We turn now to the matching process. We assume that all agents observe the set ofposted contracts and direct their search to the most attractive ones. Let Θ(C) denote theratio of principals offering contract C ∈ C to agents who direct their search towards that
contract, Θ : C 7→ [0, ∞]. Let pi (C) denote the share of these agents whose type is i, with
P (C) ≡ {p1 (C), . . . , pi (C), . . . , pI (C)} ∈ ∆I , the I-dimensional unit simplex. That is, P (C)P
satisfies pi (C) ≥ 0 for all i and i pi (C) = 1 and so P : C 7→ ∆I . The functions Θ andP are determined endogenously in equilibrium and are defined for all incentive compatible
contracts, not only the ones that are posted in equilibrium.

A type i agent seeking contract C matches with a principal with probability µ(Θ(C)),
independent of her type, where µ : [0, ∞] 7→ [0, 1] is nondecreasing. A principal offering
contract C matches with a type i agent with probability η(Θ(C))pi(C), where η : [0, ∞] 7→[0, 1] is nonincreasing. We impose that µ(θ) = θη(θ) for all θ since the left hand side is
the matching probability of an agent and the right hand side is the matching probabilityof a principal times the principal-agent ratio. Together with the monotonicity of µ and η,
this implies both functions are continuous. It is convenient to let η̄ ≡ η(0) > 0 denote thehighest probability that a principal can match with an agent, obtained when the principalagent ratio for a contract is 0. Similarly let µ̄ ≡ µ(∞) > 0 denote the highest probability
that an agent can match with a principal.
We summarize the setup by writing the expected utilities of principals and agents. The
expected utility of a principal who posts C = {y1 , . . . , yI } ∈ C isη(Θ(C))I
X

pi (C)vi (yi ) − k.

i=1

The expected utility of a type i agent who seeks contract C = {y1 , . . . , yI } ∈ C and reports
type j isµ(Θ(C))ui(yj ).
Incentive compatibility says that the agent is willing to report truthfully, ui (yi) ≥ ui (yj ).
We now generalize the notion of competitive search equilibrium to our environment with
ex ante heterogeneous agents and asymmetric information.
Definition 1 A competitive search equilibrium is a vector Ū = {Ūi }i∈I ∈ RI+ , a measure
λ on C with support C̄, a function Θ(C) : C 7→ [0, ∞], and a function P (C) : C 7→ ∆I
2Note that we are not concerned with moral hazard in this paper and so we assume that an action profiley ∈ Y can be implemented by any principal and agent i.

5satisfying(i) principals’ profit maximization and free-entry: for any C = {y1 , . . . , yI } ∈ C,
η(Θ(C))I
X

pi (C)vi (yi ) ≤ k,i=1

with equality if C ∈ C̄;(ii) agents’ optimal search: letŪi = max 0 ,max

C ′ ={y1′ ,...,yI′ }∈C̄′µ(Θ(C ))ui(yi′ );then for any C = {y1 , . . . , yI } ∈ C and i,Ūi ≥ µ(Θ(C))ui(yi),with equality if Θ(C) < ∞ and pi (C) > 0; moreover, if ui(yi ) < 0, either Θ(C) = ∞or pi (C) = 0;(iii) market clearing:pi (C)dλ({C}) ≤ πi for any i,C̄ Θ(C)Z
with equality if Ūi > 0.

In equilibrium, principals and agents take as given the expected utility of each type of
agent, Ū = {Ūi }i∈I . Notice that Ūi ≥ 0 for all i, as all agents can choose not to participate
and obtain their outside option 0. Moreover, principals and agents have rational expectations about the market tightness and the distribution of agents’ types associated with eachcontract, Θ and P .
Given these expectations, principals post contracts to maximize their expected profits.
Free entry ensures that profits are non-positive, and equal to zero for contracts that areP
posted in equilibrium. The restriction that η(Θ(C)) Ii=1 pi (C)vi (yi) ≤ k captures the ideathat if some contract C = {y1 , . . . , yI } offered positive profits, more principals would postit, driving down η(Θ(C)).
Also given their expectations, agents search optimally for contracts. The expected utilityof a type i agent is the highest utility that he can obtain from any contract that is posted inequilibrium, or zero—the value of the option not to search—if there are no contracts posted inequilibrium or if any equilibrium contract offers him negative utility. If an arbitrary contract
6

C offers type i agents less than Ūi , they will not direct their search towards that contract, norwill they direct their search towards a contract that gives negative utility should the agent
succeed in obtaining it.3 The restriction that Ūi ≥ µ(Θ(C))ui(yi ) captures the idea that if
type i agents anticipated that they could obtain higher utility from contract C = {y1 , . . . , yI },more would direct their search towards that contract, driving down µ(Θ(C)).
Finally, the market clearing condition guarantees that all type i agents direct their search
towards some contract, unless they are indifferent between matching and their outside option,Ūi = 0.
DefineȲi ≡ y ∈ Y | η̄vi (y) ≥ k and ui (y) ≥ 0 ,the set of action profiles for type i agents that deliver nonnegative utility to the agent while
permitting the principal to make nonnegative profits if the principal-agent ratio is equal tozero. Also define
[
Ȳ ≡
Ȳi .
i

In equilibrium, action profiles that are not in Ȳ are not implemented. For much of theanalysis, we make three assumptions on preferences over action profiles y ∈ Ȳ, which we
now present and discuss.
Assumption A1 Monotonicity: for all y ∈ Ȳ,v1 (y) ≤ v2 (y) ≤ . . . ≤ vI (y).
This says that, for any fixed action profile, principals weakly prefer higher types. Although
this is more than a normalization, the assumption could be relaxed at the cost of notation.
Assumption A2 Local non-satiation: for all i ∈ I, j < i, y ∈ Ȳi , and ε > 0, there exists ay ′ ∈ Bε (y) such that vi (y ′ ) > vi (y) and uj (y ′ ) ≤ uj (y).
Here Bε (y) ≡ {y ′ ∈ Y|d(y, y ′) < ε} and d(y, y ′) is the Euclidean distance between the twopoints, so Bε is a ball of radius ε. This is again a mild assumption, immediately satisfied inany example where the action profile y allows transfers. All of our results go through even
without this assumption if we restrict attention to strictly monotonic matching functions.
3
The restriction that if ui (yi ) < 0, either Θ(C) = ∞ or pi (C) = 0 rules out the possibility that type iagents search for contract C = {y1 , . . . , yI } only because they know that they have no chance of obtaining
the contract, Θ(C) = 0, and so get zero utility from this activity. Allowing for this possibility may enlarge
the set of equilibria by permitting off-equilibrium beliefs that seem unreasonable to us.

7Assumption A3 Sorting: for all i ∈ I, y ∈ Ȳi , and ε > 0, there exists a y ′ ∈ Bε (y) suchthat
uj (y ′) > uj (y) for all j ≥ i and uj (y ′ ) < uj (y) for all j < i.
This is an important assumption, guaranteeing that it is possible for principals to design
contracts that attract desirable but not undesirable agents. This is a generalized version ofa standard single-crossing condition. That is, assume that y ≡ (y1 , y2 ) has two dimensions,that we can ignore boundaries in the action profiles, e.g. because boundary points of Y are
not elements of Ȳ, and that agents’ utility functions are differentiable. Then A3 holds if themarginal rate of substitution between y1 and y2 is higher for higher types,∂ui (y1 , y2 )/∂y1∂ui (y1 , y2 )/∂y2
is monotone in i. Still, A3 is more general. For example, supposeui (y1 , y2) =
y12 + y22 1
1 ρi
1 ρi 1/ρi
y
+
y
−
− .
2 1
2 2
2

8

for some ρi ≤ 1. Also assume ρi is higher for higher types. Note that the elasticity of
substitution between y1 and y2 is 1/(1 − ρi ), increasing in i. Let Y = [0, 1]2 , a superset ofthe points where ui (y1 , y2) > 0. Indeed, by construction any point (y, 1, y2) on the boundaryof Y is not in Ȳ since ui (y1 , y2) < 0. It is easy to verify that this example fails the standard
single-crossing condition at action profiles of the form (y, y). However, it satisfies A3 since
it is always possible to increase the spread between y1 and y2 , attracting higher types andrepelling lower types. In Section 5.4 we consider an example where A3 fails, and show that
this can substantially change the nature of equilibrium.

3

Characterization

We characterize the equilibrium as the solution to a set of optimization problems. For any
type i, consider the following problem (P-i):maxθ∈[0,∞],y∈Yµ(θ)ui(y)s.t. η(θ)vi (y) ≥ k,and µ(θ)uj (y) ≤ Ūj for all j < i.

8

(P-i)

We say that a set I∗ ⊂ I and three vectors {Ūi }i∈I , {θi }i∈I∗ , and {yi}i∈I∗ solve problem (P)if:
1. I∗ denotes the set of i such that the constraint set of problem (P-i) is non empty andthe maximized value is strictly positive, given (Ū1 , . . . , Ūi−1 );2. for any i ∈ I∗ , the pair (θi , yi ) solves problem (P-i) given (Ū1 , . . . , Ūi−1 ) and Ūi =µ(θi )ui (yi)
3. for any i ∈/ I∗ , Ūi = 0.
Our main result is that we can find any equilibrium by solving problem (P) and converselythat any solution to problem (P) is an equilibrium. We first prove that problem (P) has asolution and provide a partial characterization of the solution.
Lemma 1 Assume A1-A3. There exists a set I∗ and vectors {Ūi }i∈I , {θi }i∈I∗ , and {yi}i∈I∗that solve problem (P). At a solution,
η(θi )vi (yi ) = k for all i ∈ I∗ ,µ(θi )uj (yi ) ≤ Ūj for all j ∈ I and i ∈ I∗ ,.
Proof. In the first step, we prove that there exists a solution to (P). The second and third
steps establish the two properties.
Step 1 Let us begin with i = 1. If the constraint set of problem (P-1) is empty, we simply
set Ū1 = 0. If the constraint set is non empty, then problem (P-1) is well-behaved: theobjective function is continuous in (θ, y) because µ and u1 are continuous, the set of (θ, y)satisfying the constraint η(θ)v1 (y) ≥ k is closed because η and v1 are continuous, and, since[0, ∞] × Y is compact, the constraint set is compact. Hence, (P-1) has a solution and aunique maximum Ū1 . If Ū1 > 0, let (θ1 , y1 ) be one of the maximizers.
We now proceed by induction. Fix i > 1 and assume that we have found Ūj for all j < iand (θj , yj ) for all j ∈ I∗ , j < i. We now focus on problem (P-i). If the constraint set is
empty, we again set Ūi = 0. If instead the constraint set is non-empty, then problem (P-i) is
well-behaved: the objective function is continuous in (θ, y) because µ and ui are continuousfunctions, the set of (θ, y) satisfying the constraints is compact because η, µ, vi , and uj
for any j < i are all continuous functions, and y ∈ Y which is compact by assumption.
Hence, (P-i) has a solution and a unique maximum Ūi . If Ūi > 0, let (θi , yi ) be one of the
maximizers.
9

Step 2 Suppose by way of contradiction that there exists i ∈ I∗ such that (θi , yi ) solves(P-i), but η(θi )vi (yi ) > k. This together with the fact that Ūi = µ(θi )ui(yi ) > 0, implies
that yi ∈ Ȳi and µ(θi ) > 0. Fix ε > 0 such that for all y ∈ Bε (yi), η(θi )vi (y) ≥ k. Then,assumption A3 ensures there exists a y ′ ∈ Bε (yi ) such thatuj (y ′) > uj (yi ) for all j ≥ i,uj (y ′) < uj (yi ) for all j < i.
Then the pair (θi , y ′) satisfies all the constraints of problem (P-i):
1. η(θi )vi (y ′ ) ≥ k, from the choice of ε;
2. µ(θi )uj (y ′) < µ(θi )uj (yi ) ≤ Ūj for all j < i, where the first inequality comes from theconstruction of y ′ and from µ(θi ) > 0, and the second inequality from the assumptionthat (θi , yi) solves (P-i).
Moreover, the pair (θi , y ′ ) achieves a higher value of the objective function in problem (P-i)
than does (θi , yi), µ(θi )ui (y ′) > µ(θi )ui (yi ); again the inequality comes from the constructionof y ′ and from µ(θi ) > 0. Hence, (θi , yi ) does not solve (P-i), a contradiction. This provesthat η(θi )vi (yi ) = k for all i ∈ I∗ .
Step 3 Fix i ∈ I∗ and suppose by way of contradiction that there exists j > i such thatµ(θi )uj (yi ) > Ūj . Let h be the smallest such j.
Note that since i ∈ I∗ , µ(θi )ui (yi) = Ūi > 0, which implies that µ(θi ) > 0 and ui(yi ) > 0.
Also, since (θi , yi ) solves (P-i), it satisfies the constraint η(θi )vi (yi) ≥ k, which ensuresη(θi ) > 0 and vi (yi) > 0. In particular, yi ∈ Ȳi .
Next, the pair (θi , yi) satisfies the constraints of problem (P-h) since
1. η(θi )vh (yi ) ≥ η(θi )vi (yi ) ≥ k, where the first inequality holds by assumption A1 because
h > i and yi ∈ Ȳi ⊂ Ȳ, and the second holds because (θi , yi ) solves (P-i);
2. µ(θi )ul (yi) ≤ Ūl for all l < h, which holds for(a) l < i because (θi , yi ) satisfy the constraints of (P-i),(b) l = i because Ūi = µ(θi )ui(yi ) by problem (P) and i ∈ I∗ ,(c) i < l < h by the choice of h as the smallest violation of µ(θi )uj (yi ) > Ūj .

10

Since by assumption, µ(θi )uh (yi ) > Ūh , we have not solved problem (P-h), a contradiction.
This proves that µ(θi )uj (yi) ≤ Ūj for all j ∈ I and i ∈ I∗ .
The next proposition proves that we can use any solution to problem (P) to construct
an equilibrium.
Proposition 1 Assume A1-A3. Consider a set I∗ and three vectors {Ūi }i∈I , {θi }i∈I∗ , and{yi }i∈I∗ that solve problem (P). There exists a competitive search equilibrium {Ū, λ, C̄, Θ, P }with Ū = {Ūi }i∈I , C̄ = {Ci}i∈I∗ where Ci = (yi, . . . , yi), Θ(Ci ) = θi , and pi (Ci ) = 1.
Proof. We proceed by construction.
• The vector of expected utilities is Ū = {Ūi }i∈I .
• The set of posted contracts is C̄ = {Ci }i∈I∗ where Ci ≡ (yi, . . . , yi).
• λ is such that λ({Ci}) = πi Θ(Ci ) for any i ∈ I∗ .
• For i ∈ I∗ and Ci = (yi, . . . , yi ), Θ(Ci ) = θi . Otherwise, for any incentive compatible
contract C ′ = {y1′ , . . . , yI′ } ∈ C, let J(C ′ ) = {j|uj (yj′ ) > 0} denote the types thatattain positive utility from the contract. If J(C ′ ) 6= ∅ and minj∈J(C ′ ) {Ūj /uj (yj′ )} < µ̄
thenŪjµ(Θ(C ′ )) = min′.
j∈J(C ) uj (yj′ )If this equation is consistent with multiple values of Θ(C ′ ), as may happen if µ is
not strictly increasing, pick one, e.g. the smallest such value for Θ(C ′ ). Otherwise, if
J(C ′ ) = ∅ or minj∈J(C ′ ) {Ūj /uj (yj′ )} ≥ µ̄, then Θ(C ′ ) = ∞.
• For i ∈ I∗ and Ci = (yi , . . . , yi), let pi (Ci ) = 1 and so pj (Ci ) = 0 for j 6= i. For any otherC ′ , define P (C ′) such that ph (C ′ ) > 0 only if h ∈ arg minj∈J(C ′ ) {Ūj /uj (yj′ )}. If there
are multiple elements of the arg min, pick one such value for P (C ′ ), e.g. ph (C ′ ) = 1 if
h is the smallest element of the arg min. If J(C ′ ) = ∅, again choose P (C ′ ) arbitrarily,e.g. set p1 (C ′ ) = 1.
Condition (i) For any i ∈ I∗ , the pair (θi , yi) solves problem (P-i) and in particular
Lemma 1 implies η(θi )vi (yi ) = k. This implies that profit maximization and free entry holdfor any posted contract {Ci}i∈I∗ .
Now consider an arbitrary incentive compatible contract; we show that principals’ profit
maximization and free-entry condition is satisfied. Suppose, to the contrary, that there exists
P
an incentive compatible contract C ′ = (y1′ , . . . , yI′ ) ∈ C with η(Θ(C ′ )) i pi (C ′ )vi (yi′ ) > k.
11

Note that this implies η(Θ(C ′ )) > 0 and so Θ(C ′ ) < ∞. In particular there exists some
type j with pj (C ′ ) > 0 and η(Θ(C ′ ))vj (yj′ ) > k. Since pj (C ′ ) > 0 and Θ(C ′ ) < ∞, ourconstruction of Θ(C ′ ) implies Ūj = µ(Θ(C ′ ))uj (yj′ ). Similarly, for all h,Ūh ≥ µ(Θ(C ′ ))uh (yh′ ) ≥ µ(Θ(C ′ ))uh (yj′ ),where the first inequality follows from the construction of Θ and the second from the requirement that C ′ is incentive compatible. The preceding inequalities prove that (Θ(C ′ ), yj′ )satisfies the constraints of problem (P-j). Since η(Θ(C ′ ))vj (yj′ ) > k, Lemma 1 implies that
there exists some pair (θ′′ , yj′′) that satisfies the constraints of problem (P-j) but delivers ahigher value of the objective function, µ(θ′′ )uj (yj′′ ) > Ūj ≥ 0. Since (θ′′ , yj′′) is in the constraint set of problem (P-j) and it delivers a strictly positive value for the objective, j ∈ I∗ .
But then the fact that Ūj is not the maximized value of (P-j) is a contradiction.
Condition (ii) By construction, the equilibrium functions Θ and P ensure that Ūi ≥µ(Θ(C ′ ))ui(yi′ ) for all contracts C ′ = {y1′ , . . . , yI′ }, with equality if Θ(C ′ ) < ∞ and pi (C ′ ) > 0.
Moreover, problem (P) ensures that, for any i ∈ I∗ , Ūi = µ(θi )ui (yi) where θi = Θ(Ci ) andCi = {yi , . . . , yi} is the equilibrium contract offered to type i agents.
Condition (iii) The market clearing condition is satisfied by the construction of λ.
The next proposition establishes the converse, that any equilibrium can be characterizedusing problem (P).
Proposition 2 Assume A1-A3. Let {Ū, λ, C̄, Θ, P } be a competitive search equilibrium. Let{Ūi }i∈I = Ū and I∗ = {i ∈ I|Ūi > 0}. For each i ∈ I∗ , there exists a contract C ∈ C̄ with
Θ(C) < ∞ and pi (C) > 0. Moreover, take any {θi }i∈I∗ and {yi }i∈I∗ such that for eachi ∈ I∗ , there exists a contract Ci = {y1, . . . , yi , . . . , yI } ∈ C̄ (so the ith element of Ci is yi )with θi = Θ(Ci) < ∞ and pi (Ci ) > 0. Then the set I∗ and vectors {Ūi }i∈I , {θi }i∈I∗ , and{yi }i∈I∗ solve problem (P).
Proof. From part (i) of the definition of equilibrium, any C ∈ C̄ has η(Θ(C)) > 0, henceΘ(C) < ∞. From part (iii) of the definition, Ūi > 0 implies pi (C) > 0 for some C ∈ C̄. This
proves that for each i ∈ I∗ , there exists a contract C ∈ C̄ with Θ(C) < ∞ and pi (C) > 0.
The remainder of the proof proceeds in five steps. The first four steps prove that for any
i ∈ I∗ and Ci = {y1, . . . , yi , . . . , yI } ∈ C̄ with θi = Θ(Ci) < ∞ and pi (Ci ) > 0, (θi , yi) solves
problem (P-i). First, we prove that the constraint η(θi )vi (yi ) ≥ k is satisfied. Second, we
prove that the constraint µ(θi )uj (yi ) ≤ Ūj is satisfied for all j. Third, we prove that such a12pair (θi , yi) actually delivers Ūi to type i. Finally, we prove that (θi , yi ) solves problem (P-i).
The fifth step proves that for any i ∈/ I∗ , either the constraint set of problem (P-i) is empty
or the maximized value is non-positive.
Step 1 Take i ∈ I∗ and Ci = {y1 , . . . , yi , . . . , yI } ∈ C̄ with θi = Θ(Ci ) < ∞ and pi (Ci ) > 0.
We prove that the constraint η(θi )vi (yi) ≥ k is satisfied in problem (P-i). Note first thati ∈ I∗ implies Ūi > 0. Since part (ii) of the definition of equilibrium implies Ūi = µ(θi )ui(yi ),this ensures µ(θi ) > 0.
To find a contradiction, now assume η(θi )vi (yi) < k. Part (i) of the definition ofP
equilibrium implies η(θi ) j pj (C)vj (yj ) = k, and so there is an h with ph (C) > 0 andη(θi )vh (yh ) > k. Since η(θi ) ≤ η̄, η̄vh (yh ) > k as well. Moreover, because θi = Θ(C) < ∞and ph (C) > 0, optimal search implies uh (yh ) ≥ 0. This proves that yh ∈ Ȳh .
Next, fix ε > 0 such that for all y ∈ Bε (yh ), η(θi )vh (y) > k. Then assumption A3 together
with yh ∈ Ȳ guarantees that there exists y ′ ∈ Bε (yh ) such thatuj (y ′) > uj (yh ) for all j ≥ h,uj (y ′) < uj (yh ) for all j < h.
Notice that y ′ ∈ Ȳh as well, given that uh (y ′) > uh (yh ) ≥ 0 and η̄vh (y ′) ≥ η(θi )vh (y ′) > k.
Next consider the contract C ′ = {y ′, . . . , y ′}. Let θ′ ≡ Θ(C ′ ). Note thatµ(θ′ )uh (y ′ ) ≤ Ūh = µ(θi )uh (yh ) < µ(θi )uh (y ′ ),where the first inequality uses the optimal search condition for contract C ′ , the second
equality holds by optimal search because θi < ∞ and ph (C) > 0, and the last inequality
holds by the construction of y ′, since µ(θi ) > 0. This implies µ(θ′ ) < µ(θi ); since µ is

nondecreasing, it also implies θ′ < θi .
Next observe that for all j < h, either uj (y ′) < 0, in which case pj (C ′ ) = 0 by part (ii)of the definition of equilibrium, orµ(θ′ )uj (y ′ ) < µ(θi )uj (yh ) ≤ µ(θi )uj (uj ) ≤ Ūj ,where the first inequality uses µ(θ′ ) < µ(θi ) and the construction of y ′ , the second uses incentive compatibility of C, and the third inequality follows from the optimal search condition
for contract C. This implies pj (C ′ ) = 0 for all j < h.

13

Finally, the profits from posting contract C ′ areη(θ′ )IX

pj (C ′ )vj (y ′) ≥ η(θ′ )vh (y ′ ) ≥ η(θi )vh (y ′) > k.

j=1

The first inequality follows because pj (C ′ ) = 0 if j < h and vh (y ′ ) is nondecreasing in h byassumption A1 together with y ′ ∈ Ȳh ⊂ Ȳ. The second follows because θ′ < θi and η isnonincreasing. The last inequality uses the construction of ε. Offering the contract C ′ istherefore strictly profitable, a contradiction.
Step 2 Again take i ∈ I∗ and Ci = {y1 , . . . , yi, . . . , yI } ∈ C̄ with θi = Θ(Ci) < ∞and pi (Ci ) > 0. Part (ii) of the definition of equilibrium implies µ(θi )uj (yj ) ≤ Ūj for allj while incentive compatibility implies uj (yi ) ≤ uj (yj ). This proves that the constraintµ(θi )uj (yi ) ≤ Ūj in problem (P-i) is satisfied for all j.
Step 3 Again take i ∈ I∗ and Ci = {y1 , . . . , yi, . . . , yI } ∈ C̄ with θi = Θ(Ci ) < ∞ andpi (Ci ) > 0. Part (ii) of the definition of equilibrium implies Ūi = µ(θi )ui(yi ), since θi < ∞and pi (Ci ) > 0. This proves that (θi , yi ) delivers utility Ūi to type i agents.
Step 4 Once again take i ∈ I∗ and Ci = {y1 , . . . , yi, . . . , yI } ∈ C̄ with θi = Θ(Ci) < ∞ andpi (Ci ) > 0. To find a contradiction, suppose there exists (θ′ , y ′) that satisfies the constraintsof problem (P-i) but delivers higher utility. That is, η(θ′ )vi (y ′) ≥ k, µ(θ′ )uj (y ′) ≤ Ūj for all
j < i, and µ(θ′ )ui(y ′ ) > Ūi .
We now use assumption A2. Note that µ(θ′ )ui (y ′) > Ūi > 0 implies µ(θ′ ) > 0 andui (y ′) > 0, while η(θ′ )vi (y ′) ≥ k > 0 implies η(θ′ ) > 0 and vi (y ′) > 0. In particular, y ′ ∈ Ȳi .
We can therefore fix ε > 0 such that for all y ∈ Bε (y ′ ), µ(θ′ )ui (y) > Ūi , and then choosey ′′ ∈ Bε (y ′) such that vi (y ′′ ) > vi (y ′) and uj (y ′′) ≤ uj (y ′ ) for all j < i. In particular, thisensures η(θ′ )vi (y ′′) > k, µ(θ′ )uj (y ′′) ≤ Ūj for all j < i, and µ(θ′ )ui (y ′′ ) > Ūi . Note that westill have y ′′ ∈ Ȳi .
We next use assumption A3. Fix ε′ > 0 such that for all y ∈ Bε′ (y ′′ ), η(θ′ )vi (y) > k andµ(θ′ )ui (y ′′) > Ūi . Choose y ′′′ ∈ Bε′ (y ′′ ) such thatuj (y ′′′ ) > uj (y ′′ ) for all j ≥ i,uj (y ′′′ ) < uj (y ′′ ) for all j < i.
In particular, this ensures η(θ′ )vi (y ′′′ ) > k, µ(θ′ )uj (y ′′′ ) < Ūj for all j < i, and µ(θ′ )ui (y ′′′ ) >Ūi . Finally, note that we still have y ′′′ ∈ Ȳi .
14

Now consider the contract C ′′′ ≡ {y ′′′, . . . , y ′′′}. From part (ii) of the definition of equilibrium, µ(θ′ )ui(y ′′′ ) > Ūi implies that µ(θ′ ) > µ(Θ(C ′′′ )), which in turn guarantees that
η(Θ(C ′′′ ))vi (y ′′′ ) > k.
We next claim that pj (y ′′′) = 0 for all j < i. First suppose µ(Θ(C ′′′ )) = 0 and pj (y ′′′ ) > 0
for some j < i. This implies Ūj = 0. But since µ(θ′ ) > 0, µ(θ′ )uj (y ′′) ≤ Ūj impliesuj (y ′′ ) ≤ 0, while by construction uj (y ′′′ ) < uj (y ′′ ). Part (ii) of the definition of equilibrium
therefore implies that either Θ(C ′′′ ) = ∞ or pj (y ′′′ ) = 0, a contradiction. So instead assumeµ(Θ(C ′′′ )) > 0. Recall µ(θ′ )uj (y ′′ ) ≤ Ūj , µ(Θ(C ′′′)) < µ(θ′ ), and uj (y ′′′ ) < uj (y ′′). These
inequalities jointly imply µ(Θ(C ′′′ ))uj (y ′′′ ) < Ūj , hence pj (y ′′′ ) = 0.
The profit from offering this contract is therefore′′′

η(Θ(C ))IX

pj (C ′′′ )vj (y ′′′ ) ≥ η(Θ(C ′′′ ))vi (y ′′′ ) > k,j=1

where the first inequality uses pj (C ′′′ ) = 0 for j < i and vj (y ′′′ ) is increasing in j, byassumption A1. This is a contradiction, which proves (θi , yi) solves (P-i).
Step 5 Suppose there is an i ∈/ I∗ for which the constraint set of problem (P-i) is nonempty
and the maximized value is positive. That is, there exists a (θi , yi) such that η(θ′ )vi (y ′) ≥ k,µ(θ′ )uj (y ′) ≤ Ūj for all j < i, and µ(θ′ )ui (y ′) > Ūi = 0. Replicating step 4, we can first finda y ′′ such that η(θ′ )vi (y ′′) > k, µ(θ′ )uj (y ′′) ≤ Ūj for all j < i, and µ(θ′ )ui (y ′′) > 0. Then wefind a y ′′′ such that η(θ′ )vi (y ′′′) > k, uj (y ′′) > uj (y ′′′ ) for j < i, and ui (y ′′) < ui(y ′′′ ). Finally,
prove that the contract C ′′′ = {y ′′′, . . . , y ′′′ } only attracts type i or higher agents and so must
be profitable and must deliver them positive utility, a contradiction.
It is worth stressing that we use assumption A2 in the proof of Proposition 2, but nowhere
else in the paper. We use the assumption because we need to establish that it is possible to
make the principal better off while not improving the well-being of agents. If η is strictly
decreasing, we could do this by reducing η; however, for our examples in Sections 5–7, it
is convenient to allow that η is only weakly decreasing, and so we introduce this additional
assumption.
The next proposition combines the previous results to prove existence of equilibrium anduniqueness of equilibrium payoffs.
Proposition 3 Assume A1-A3 hold. Then competitive search equilibrium exists. Moreover,the equilibrium vector Ū is unique.

15

Proof. Lemma 1 shows that, under A1-A3, there exists a solution for problem (P). Moreover, Proposition 1 shows that, under the same assumptions, if the set I∗ and the vectors{Ūi }i∈I , {θi }i∈I∗ , and {yi}i∈I∗ solve problem (P), there exists a competitive search equilibrium {Ū, λ, C̄, Θ, P } with the same Ū , C̄ = {Ci}i∈I∗ , where Ci = (yi , . . . , yi), Θ(Ci ) = θi ,and pi (Ci ) = 1. This proves existence.
Proposition 2 shows that in any competitive search equilibrium {Ū , λ, C̄, Θ, P }, Ūi is the
maximum value of the objective of problem (P-i) for all i ∈ I∗ and Ūi = 0 otherwise. Also,
Lemma 1 shows that there exists a unique maximum value Ūi for the objective of problem(P-i) for all i ∈ I∗ . This proves uniqueness.
When there are strict gains from trade for all types, we can prove that all types get
positive utility. We show by example in Section 7 below that the possibility of strict gains
from trade for type i is not enough to ensure positive utility for type i agents.
Proposition 4 Assume A1-A3 hold and that for all i, there exists y ∈ Y with η̄vi (y) > k
and ui (y) > 0. Then in any competitive search equilibrium, Ūi > 0 for all i and in particular
there exists a contract C ∈ C with Θ(C) < ∞ and pi (C) > 0.
Proof. We prove Ūi > 0 for all i using problem (P). Start with i = 1. Fix y satisfying
η̄v1 (y) > k and u1 (y) > 0. Then fix θ > 0 satisfying η(θ)v1 (y) = k. These points satisfy theconstraints of problem (P-1) and deliver utility µ(θ)u1 (y) > 0. This proves Ū1 > 0.
Now suppose we have proven that Ūj > 0 for all j < i. We prove Ūi > 0. Again
fix y satisfying η̄vi (y) > k and ui (y) > 0. Then fix θ > 0 satisfying η(θ)v1 (y) ≥ k andµ(θ)uj (y) ≤ Ūj for all j < i; this is feasible since Ūj > 0 and µ is continuous with µ(0) = 0.
These points satisfy the constraints of problem (P-i) and deliver utility µ(θ)ui(y) > 0, which
proves Ūi > 0 and establishes the induction step.

4

Incentive Feasible Allocations

This section sets the stage for studying the efficiency properties of equilibrium. In particular,
we define an incentive feasible allocation, which is necessary to look for Pareto improvements.
An allocation is a vector of expected utilities for the different types of agents, a set of posted
contracts, and the associated market tightness and composition of agents applying for theposted contracts.
Definition 2 An allocation is a vector Ū of expected utilities for the agents, a measure λover the set of incentive-compatible contracts C with support C̄, a function Θ̃ : C̄ 7→ [0, ∞],16

and a function P̃ : C̄ 7→ ∆I .4
It is incentive feasible whenever: (1) each posted contract offers the maximal expectedutility to agents who direct their search for that contract and no more to those who do not;(2) the economy’s resource constraint is satisfied; and (3) markets clear.
Definition 3 An allocation {Ū , λ, C̄, Θ̃, P̃ } is incentive feasible if
1. for any C ∈ C̄ and i ∈ {1, . . . , I} such that p̃i (C) > 0 and Θ̃(C) < ∞,Ūi = µ(Θ̃(C))ui (yi ),andŪi ≡ max µ(Θ̃(C ′ ))ui (yi′ )C ′ ∈C̄where C ′ = {y1′ , . . . , yI′ };2.
Z

η(Θ̃(C))I
X!

p̃i vi (yi ) − k dλ(C) = 0;i=1

3. for all i ∈ {1, . . . , I},Z

p̃i (C)dλ(C) ≤ πi ,Θ̃(C)with equality if Ūi > 0.
The set of incentive feasible allocations provides a good benchmark for what the economy
may be able to achieve through legal restrictions on the type of contracts that can be offered.

5

Rat Race

We now proceed through three examples that illustrate the usefulness of our model.

5.1

Setup

Our first example shows that in competitive search equilibrium principals can separate agentsby distorting directly the terms of the posted contracts in a version of a classic signaling
4

Notice that Θ̃ and P̃ are different objects than Θ and P because they are defined only over the set ofposted contracts.

17

model (Akerlof, 1976).5 We think of agents as workers who are heterogeneous both in termsof their productivity and their cost of working long hours. Principals can be thought ofas firms that are willing to pay more to hire more productive workers, but cannot observe
productivity, only hours worked. We show that if the cost of a long workday is lower for
more productive workers, firms will use hours worked to separate workers.
An action profile here consists of two elements, y = {t, x}, where t denotes a transferfrom the firm to the worker and x ≥ 0 denotes the length of the workday. The payoff of amatched worker of type i who undertakes action {t, x} isui(t, x) = t −x,aiwhere higher values of ai imply that x is less costly. The payoff of a firm matched with atype i agent who takes action {t, x} isvi (t, x) = bi − t,where bi is the productivity of the worker.
Assume µ is strictly concave and continuously differentiable. Also assume that I = 2
and, without loss of generality, that type 2 workers are more productive than type 1 workers:
b2 > b1 . We restrict the set of feasible action profiles to Y = {(t, x)|t ∈ [−ε, b2 ] and x ∈[0, b2 max{a1 , a2 }]} for some number ε > 0. We view the restriction that x ≥ 0 as technological, while the other restrictions ensure Y is compact but are otherwise without loss ofgenerality. A firm would never profit from offering a transfer higher than b2 and the worker
would never accept a negative transfer, although it is convenient to allow for the possibility ofa slightly negative transfer. Moreover, a worker would never provide set x > b2 max{a1 , a2 },given that the transfer is bounded by b2 ; she would prefer not to search.
The space of action profiles that provide nonnegative utility to a type i worker andnonnegative profit to a firm when the firm-worker ratio is zero areȲi = {(t, x) ∈ Y|x/ai ≤ t ≤ bi − k/η̄}.
The fact that b2 > b1 immediately implies that assumption A1 is satisfied. Assumption A2
holds because (t, x) ∈ Ȳi implies t ≥ 0 and so can be reduced to raise vi (t, x) and lower
uj (t, x); for this reason it is convenient to allow negative transfers.
The critical assumption is A3. Consider points (t, 0) ∈ Ȳ. There are nearby points (t′ , x′ )5

If we set ai = bi , this example is equivalent to a static version of Inderst and Müller (1999), although
the definitions of equilibrium are conceptually different.

18

with u1(t′ , x′ ) < u1(t, 0) and u2 (t′ , x′ ) > u2 (t, 0) if and only if a2 > a1 , so the more productive
worker finds it less costly to set higher x. At any other point {t, x} ∈ Ȳ, assumption A3
holds for all values of a1 and a2 . We therefore impose a2 > a1 in what follows and discuss
the nature of equilibrium if this assumption is violated at the end of this section.
Finally, we assume that η̄b1 > k so that there are gains from trade for both types ofworkers. By proposition 4, this implies that the equilibrium is characterized by both Ū1 > 0
and Ū2 > 0.

5.2

Competitive Search Equilibrium

Using the results in Section 3, we can characterize a competitive search equilibrium using
vectors (Ū1 , Ū2 ), (t1 , x1 , t2 , x2 ), and (θ1 , θ2 ) that solve problem (P). In this example, problem(P-i) is


x
Ūi =
max
µ(θ) t −θ∈[0,∞],(t,x)∈Y
ais.t. η(θ)(bi − t) ≥ k,xand µ(θ) t −≤ Ūj for j ≤ i.
aj
We claim the following result:
Result 1 There exists a unique competitive search equilibrium with µ′ (θi )bi = k, for i = 1, 2,
and so θ1 < θ2 ; x1 = 0,a1
x2 =µ(θ2 )

µ(θ2 )− θ2µ′ (θ2 )

and
ti =





−



µ(θ1 )− θ1µ′ (θ1 )1θi−′µ (θi ) µ(θi )k > 0;kfor i = 1, 2. Moreover,
Ū1 =




 
 


µ(θ1 )
a1 µ(θ1 )a1µ(θ2 )− θ1 k and Ū2 =− θ1 + 1 −− θ2 k.
µ′ (θ1 )a2 µ′ (θ1 )a2µ′ (θ2 )Proof. We prove this result by first solving problem (P-1), finding Ū1 and then using it tosolve problem (P-2).

19

Consider problem (P-1). Using η(θ) = µ(θ)/θ, we write it as


µ(θ)
x
Ū1 =
max
µ(θ) t −s.t.
(b1 − t) ≥ k.
θ∈[0,∞],(t,x)∈Ya1
θ
Lemma 1 implies that the constraint is binding so that we can eliminate t and reduce theproblem to


x
Ū1 =
max
µ(θ) b1 −− θk.
θ∈[0,∞],x∈[0,b2a2 ]a1
It is easy to see that at the optimum x = 0 and θ = θ1 , where θ1 solves the necessary andsufficient first order condition, µ′ (θ1 )b1 = k. Using this to eliminate b1 from the objectivefunction delivers the expression for Ū1 . Also use the constraint to solve for t1 .
Next, solve problem (P-2) using the solution for Ū1 , that is,


x
Ū2 =
max
µ(θ) t −θ∈[0,∞],(t,x)∈Ya2µ(θ)(b2 − t) ≥ k,
s.t.
θ 
 

x
µ(θ1 )and µ(θ) t −≤− θ1 k.
a1µ′ (θ1 )Again using Lemma 1, the first constraint binds and so we can eliminate t. It is easy toverify that the second constraint must be binding as well, so we can use it to eliminate x
and then check that at the solution x ≥ 0. The problem reduces toŪ2 = maxθ∈[0,∞]



a1
a2







µ(θ1 )a1− θ1 k + 1 −(µ(θ)b2 − θk) .
µ′ (θ1 )a2Then, at the solution θ = θ2 , where θ2 solves the necessary and sufficient first order condition,µ′ (θ2 )b2 = k. Note that concavity of µ implies θ1 < θ2 . Substituting into the objectivefunction gives the expression for Ū2 . Finally, use the constraints to compute t2 and x2 .
Again, concavity of µ ensures that µ(θ)/µ′(θ) − θ is increasing in θ, which in turn impliesx2 > 0.
In equilibrium, some firms post contracts to attract only type 1 workers and others postcontracts to attract only type 2 workers. In order to separate the two types of workers,
firms have to offer distorted contracts to workers of type 2, requiring a costly signal, while
the contracts offered to the type 1 workers are undistorted. Two features of this example
merit mention. First, the market tightness margin is equal to the first-best (symmetric
information) level for both types of contracts. This is because the worker bears the full cost20

of the distortion. Second, without additional assumptions, we cannot rule out the possibility
that the transfer to type 2 workers is lower than the transfer to type 1 workers. This is inaddition to the cost x > 0 that they must incur. Their reward for applying to firms offering
type 2 contracts is only a higher probability of trade, θ2∗ > θ1∗ .

5.3

Pareto Optimality

For some parameter values, the competitive search equilibrium is Pareto inefficient. Consider an allocation that treats the two types of workers identically, C̄ = {C}, where C =((t, 0), (t, 0)). Moreover, assume Θ̃(C) = θ∗ , with θ∗ solving µ′ (θ∗ )(π1 b1 + π2 b2 ) = k,p̃i (C) = πi , and λ({C}) = 1/θ∗ . Note that this defines θ1 < θ∗ < θ2 , an intermediatelevel of market tightness. Finally, choose t such that firms make zero profit, that is,
t=




1
θ∗
−
k
µ′ (θ∗ ) µ(θ∗ )The contract is incentive compatible since the types are treated identically. Moreover, allworkers are attracted to the posted contracts and get the same expected utility, so condition(1) of feasibility is satisfied. Also, the resource constraint and the market clearing condition
are satisfied by the choice of t and λ. Hence, it constitutes an incentive feasible allocation.
The expected utility of (both types of) workers is now
Ū =




µ(θ∗ )∗− θ k.
µ′ (θ∗ )Compare this with the equilibrium. Since θ∗ > θ1 , trivially Ū > Ū1 . On the other hand,Ū ≥ Ū2 if and only ifµ(θ∗ )a1∗−θ
≥µ′ (θ∗ )a2

µ(θ1 )− θ1µ′ (θ1 )





a1
+ 1−
a2




µ(θ2 )− θ2 .
µ′ (θ2 )This holds if a1 /a2 is sufficiently close to 1 (screening is very costly) or if π1 is sufficiently
close to zero (there are few type 1 workers). The reason is that in equilibrium, firms whowant to attract type 2 need to screen out type 1 agents. If a firm failed to do so, it would
be swamped by type 1 workers. This may not be optimal, however. If there are few type 1
workers or screening is too costly, it is preferable to subsidize type 1 workers and eliminate
costly screening.

21

5.4

Pooling

The sorting condition A3 plays a substantive role in the analysis of our model. To understandthe role of such a condition, consider now a variant of the same example where signaling
is cheaper for the less productive workers, a1 ≥ a2 while b1 < b2 . As we show below, thisviolates assumption A3. Firms would like to screen out low productivity workers, but the
only available screening technology works against them. If a firm posts a contract with thedesire to attract only the more productive workers, the less productive ones would report
to be more productive. We show that in this case there is a class of equilibria in whichfirms attract both types of workers and may prescribe the less productive workers to set
x ≥ 0. The most efficient (and arguably most natural) of these equilibria is the one wherefirms offer the same action profiles to all the workers and allow x = 0. Notice that, in thiscase, to characterize the equilibrium we cannot use the analysis in Section 3, which relies on
assumption A3, but we need to go back to the primitive Definition 1 of a competitive searchequilibrium.
The restriction that a1 ≥ a2 does not affect assumptions A1 and A2, but it violates
assumption A3. Fix t and set x = 0. For any nearby contract (t′ , x′ ),u1 (t′ , x′ ) − u1 (t, 0) = t′ − t − x′ /a1 ≥ t′ − t − x′ /a2 = u2(t′ , x′ ) − u2 (t, 0),
since x′ ≥ 0. It follows that there is no such value of (t′ , x′ ) with u1 (t′ , x′ ) < u1 (t, 0) andu2 (t′ , x′ ) > u2 (t, 0).
We claim that in this case, there exists a class of equilibria indexed by the amount ofaction x required from low productivity agents, x1 ∈ [0, a1 (b2 −b1 )(1−π1 )/π1 ]. In equilibrium,
all firms post the same contract C = {(t + x1 /a1 , x1 ), (t, 0)}, where t is chosen to ensure thatfirms make zero profits. Moreover, given that all the posted contracts are the same, all types
of workers look for the same contracts, and pi (C) = πi . A firm might consider offering acontract that attracts only one type of worker. If she tries to attract only type 1 workers, shewill lose the benefit of cross-subsidization and thus is unable to attract them while earning
positive profits. If she tries to attract type 2 workers, she will be unable to devise a contractthat will exclude type 1 workers, again making such a deviation unprofitable.
Result 2 Suppose a1 ≥ a2 . For any x1 ∈ [0, a1 (b2 −b1 )(1−π1 )/π1 ], there exists a competitive
search equilibrium where C̄ = {C} with C = {(t + x1 /a1 , x1 ), (t, 0)} andx1
θt = π1 b1 −+ π2 b2 −
k,
a1
µ(θ)

22

where θ solves

 


x1
µ (θ) π1 b1 −+ π2 b2 = k.
a1′

Moreover, the expected utility of both types of workers isŪ = µ(θ)t.
Proof. Fix x1 ≤ a1 (b2 − b1 )(1 − π1 )/π1 . Our proof proceeds by constructing an equilibrium.
Assume that C̄ = {C} where C = {(t + x1 /a1 , x1 ), (t, 0)}, Ū1 = Ū2 = Ū = µ(θ)t, andλ({C}) = 1/θ, where t and θ are defined above. Moreover, Θ(C) = θ and pi (C) = πi for
i = 1, 2. For any other incentive compatible contract C ′ = {(t′1 , x′1 ), (t′2 , x′2 )} ∈ C̄, C ′ 6= C,
suppose Θ(C ′ ) solves



x′1
′
Ū1 = µ(Θ(C )) t1 −a1′

if this defines Θ(C ′ ) < ∞; if Ū ≥ µ̄(t′1 − x′1 /a1 ), Θ(C ′ ) = ∞. Finally, suppose p1 (C ′ ) = 1
and p2 (C ′ ) = 0 for all such contracts.
By construction, profit maximization and free entry hold for the posted contract C. In
particular, t is chosen so that firms break even.
For any other incentive-compatible contract C ′ 6= C, the firm’s profit maximization andfree entry condition reduces toη(Θ(C ′))(b1 − t′1 ) ≤ k.
Since η(∞) = 0, this is obviously satisfied for contracts with Θ(C ′ ) = ∞. Otherwise, use
the construction of Θ(C ′ ) to eliminate t′1 from this requirement. We need to show that
x′1µ(Θ(C )) b1 −− Θ(C ′ )k ≤ Ū1 .
a1′

An upper bound on the left hand side is obtained by setting x′1 = 0 and choosing Θ(C ′ ) to
maximize the left hand side, µ′ (Θ(C ′ ))b1 = k. The restriction x1 ≤ a1 (b2 − b1 )(1 − π1 )/π1
implies that b1 ≤ π1 b1 − xa11 + π2 b2 , from which it follows that Θ(C ′ ) ≤ θ. That is,






x′1
µ(Θ(C ′ ))µ(θ)′′µ(Θ(C )) b1 −− Θ(C )k ≤− Θ(C ) k ≤− θ k = Ū1 ,
a1µ′ (Θ(C ′ ))µ′(θ)′

where the first inequality uses the preceding discussion, the second inequality holds becauseΘ(C ′ ) ≤ θ, and the third holds from the construction of Ū1 .
Next, workers’ optimal search for an incentive-compatible contract C ′ holds by construc23

tion for type 1 workers. For type 2 workers, we need to verify that


x′2
′
Ū2 ≥ µ(Θ(C )) t2 −.
a2′

To prove this, note that t′1 − x′1 /a1 ≥ t′2 − x′2 /a1 ≥ t′2 − x′2 /a2 , where the first inequality
comes from incentive compatibility of C ′ and the second from the assumption a1 ≥ a2 andthe feasibility restriction x′2 ≥ 0. This implies that




x′1
x′2
′
′
′
≤ µ(Θ(C )) t1 −= Ū1 ,µ(Θ(C )) t2 −a2a1′

which establishes the desired inequality since Ū1 = Ū2 .
Finally, the market clearing condition holds by construction.
This result shows that there exists a class of Competitive Search Equilibria parameterizedby x1 , the extent to which low productivity workers are forced to use the costly action. In
this example, firms would like to be able to screen the more productive type 2 workers, but
the assumption a1 ≤ a2 implies that they can use the action x only to attract less productive
type 1 workers. Still, there are equilibria where x is positive because firms fear that if they
did not require x > 0 from type 1 workers, they would be stuck exclusively with that typeof worker.
A positive value of x is socially wasteful, so these equilibria can be Pareto ranked. In
particular, the “pooling” equilibrium characterized by x1 = 0 is Pareto optimal, at least
within this class. In such an equilibrium, all the firms offer the same contract that prescribes
the same action profile to all the workers: x = 0 and a transfer which ensures that the firms
break even.
Next, we show that when a1 is strictly larger than a2 , then any competitive search equilibrium where all the firms post the same contract falls in the class of equilibria characterized
in Result 2.
Result 3 Suppose a1 > a2 . If there exists a competitive search equilibrium with C̄ = {C}
where C = ((t1 , x1 ), (t2 , x2 )), pi (C) = πi , and Θ(C) < ∞, then x2 = 0, and t1 − x1 /a1 = t2 .
Proof. Throughout this proof, we suppose there is an equilibrium characterized by thesingle incentive compatible contract C = ((t1 , x1 ), (t2 , x2 )). In the first step we prove thatx2 = 0 and in the second that t1 − x1 /a1 = t2 .

24

Step 1 Suppose x2 > 0. Given that pi (C) = πi > 0 for i = 1, 2 and Θ(C) < ∞, optimal
search requires that the expected utility of type i workers satisfies
xiŪi = µ(Θ(C)) ti −,ai
for i = 1, 2. Next, consider the contract C ′ = ((t2 −x2 /a2 , 0), (t2 −x2 /a2 , 0)). Optimal searchof type 2 workers requires that




x2
x2
µ(Θ(C )) t2 −≤ Ū2 = µ(Θ(C)) t2 −
,
a2
a2
′

which implies that Θ(C ′ ) ≤ Θ(C). Moreover, notice that
t1 −

x1
x2
x2
≥ t2 −> t2 − ,a1a1a2

where the first inequality follows from incentive compatibility of C, and the second from thefact that a1 > a2 and x2 > 0. This, together with Θ(C ′ ) ≤ Θ(C), implies that




x1
x2
′
Ū1 = µ(Θ(C)) t1 −> µ(Θ(C )) t2 −.
a1
a2
It follows that type 1 workers will never search for C ′ , that is, p1 (C ′ ) = 0. Hence, given that
Θ(C ′ ) ≤ Θ(C) < ∞, it must be that p2 (C ′ ) = 1 and Θ(C ′ ) = Θ(C). Then, the expected
profits for a firm posting C ′ arex2> η(Θ(C))(π1 (b1 − t1 ) + π2 (b2 − t2 )) = k.
η(Θ(C )) b2 − t2 +a2′

The first inequality follows from b1 < b2 ; the fact shown above that t1 > t2 − x2 /a2 + x1 /a1 ≥t2 − x2 /a2 ; and Θ(C ′ ) = Θ(C). The second equality follows from the fact that in the
proposed equilibrium firms post C and break even. Hence, contract C ′ represents a profitabledeviation, a contradiction.
Step 2 We now prove that t1 − x1 /a1 = t2 . Notice that incentive compatibility of C
and the result from the previous step that x2 = 0 imply that t1 − x1 /a1 ≥ t2 . To derive acontradiction, suppose that t1 −x1 /a1 > t2 . Consider a contract C ′ = ((t2 +x1 /a1 , x2 ), (t2 , 0)).
Thenµ(Θ(C ′ ))t2 ≤ Ū2 = µ(Θ(C))t2 ,25

where the first inequality follows from optimal search of type 2 workers for C ′ and the second
equality from optimal search of the same workers for C together with the assumption that
p2 (C) = π2 > 0. This implies that Θ(C ′ ) ≤ Θ(C). Hence,x1Ū1 = µ(Θ(C)) t1 −> µ(Θ(C ′))t2 ,a1
where the first equality follows from optimal search of type 1 workers for C and the assumption that p1 (C) = π1 > 0 and the second inequality comes from Θ(C ′ ) ≤ Θ(C) andthe assumption that t1 − x1 /a1 > t2 . Hence, it must be that p1 (C ′ ) = 0, and, given that
Θ(C ′ ) ≤ Θ(C) < ∞, p2 (C ′ ) = 1 and Θ(C ′ ) = Θ(C). It follows that the expected profits fora firm offering C ′ areη(Θ(C ′)))(b2 − t2 ) > η(Θ(C))(π1 (b1 − t1 ) + π2 (b2 − t2 )) = k,given that Θ(C ′ ) = Θ(C) and b2 − t2 > b1 − (t2 + x1 /a1 ) > b1 − t1 , where the last inequality follows from assumption. This shows that C ′ represents a profitable deviation, acontradiction.

6

Insurance

6.1

Setup

Our second example is closer to the original Rothschild and Stiglitz (1976) environment,
where risk neutral principals offer insurance contracts to risk averse agents who are heterogeneous in their probability of experiencing a loss. This example illustrates several featuresof our environment. We do not require that utility is quasi-linear. Nor do we require search
frictions, but can instead allow the short side of the market to match with certainty. Finally,
we show that even when a pooling allocation does not Pareto dominate the equilibrium,a partial pooling allocation, where only some types of agents are pooled together, may be
Pareto superior.
To be concrete, we again imagine worker-firm matches, where the productivity of a matchis initially unknown. Some workers (agents) are more likely to be productive than others, but
firms (principals) can only verify the ex post realization of productivity, not a worker’s type.
More precisely, a type i worker produces 1 unit of output with probability pi and 0 otherwise.
A contract specifies the worker’s consumption conditional on whether the pair can produce.
Workers are risk averse and firms risk neutral. In the absence of adverse selection, the26

marginal utility of consumption would be equalized across states. By incompletely insuring
workers against the risk of being unproductive, a firm can keep undesirable workers fromdirecting their search toward a particular contract. We believe that this model may provide
an explanation for why firms do not insure workers against layoff risk.
To illustrate that search frictions are not essential for our results, assume that the numberof matches is determined by the short side of the market, µ(θ) = min{θ, 1}. This assumption
allows us to focus on the risk of matches turning out to be unproductive. It also simplifies
our algebra considerably.
Now an action profile consists of a pair of consumption levels, y = {ce , cu }. The payoffof a matched type i worker who undertakes action profile {ce , cu } isui (ce , cu ) = pi U(ce ) + (1 − pi )U(cu ),where p1 < p2 < · · · < pI < 1 and U : [c, ∞) → R is increasing and strictly concave withlimc→c U(c) = −∞ for some c < 0 and U(0) = 0. The payoff of a firm matched with a typei worker who undertakes action profile {ce , cu } isvi (ce , cu ) = pi (1 − ce ) − (1 − pi )cu .
To ensure that assumption A1 is satisfied, we restrict the set of feasible action profiles to
Y = {(ce , cu )|cu + 1 ≥ ce ≥ c and cu ≥ c}. Moreover, since a reduction in cu raises vi (y) andlowers ui (y) and is feasible, A2 is satisfied. The assumption that limc→c U(c) = −∞ ensuresthat action profiles of the form {ce , c} yield negative utility for all types and so are not in Ȳ.
To verify A3, consider an incremental increase in ce to ce + dce and an incremental reduction
in cu to cu − dcu for some dce > 0 and dcu > 0. For a type i worker, this raises utility byapproximately pi u′ (ce )dce − (1 − pi )u′ (cu )dcu , which is positive if and only if
dce
1 − pi u′ (cu )>.
dcu
pi u′ (ce )Since (1 − pi )/pi is decreasing in i, an appropriate choice of dce /dcu yields an increase inutility if and only if j ≥ i, which verifies A3. Our propositions therefore apply.
Finally, assume p1 ≤ k < pI , which ensures that there are no gains from employing thelowest type, even in the absence of asymmetric information, but there may be gains fromtrade for higher types, for example by setting ce = cu = pI − k > 0. Let i∗ denote the lowest
type without gains from trade, so pi∗ ≤ k < pi∗ +1

27

6.2

Competitive Search Equilibrium

We again characterize a competitive search equilibrium using problem (P):Result 4 There exists a competitive search equilibrium where for all i ≤ i∗ , Ūi = 0; and for
all i > i∗ , θi = 1, Ūi > 0, and ce,i > ce,i−1 and cu,i < cu,i−1 are the unique solution topi (1 − ce,i) − (1 − pi )cu,i = k
and pi−1 U(ce,i ) + (1 − pi−1 )U(cu,i ) = pi−1 U(ce,i−1 ) + (1 − pi−1 )U(cu,i−1 ),where ce,i∗ = cu,i∗ = 0.
Proof. For i ≤ i∗ , consider the problem (P-i) without the constraint of keeping out lower
types. This relaxed problem should yield a higher payoff
Ūi ≤

max

θ∈[0,∞],(ce ,cu )∈Y

min{θ, 1} pi U(ce ) + (1 − pi )U(cu )s.t. min{1, θ−1 }(pi(1 − ce ) − (1 − pi )cu ) ≥ k.

At the solution, cu,i = ce,i = ci and so this reduces to
Ūi =

max

θ∈[0,∞],c≥cmin{θ, 1}U(c)s.t. min{1, θ−1 }(pi − c) ≥ k.
Either the constraint set is empty (if pi < k + c) or there are no points in the constraint set
that give positive utility. In any case, this gives Ūi = 0.
Turn next to a typical problem (P-i), i > i∗ :
Ūi =

max

θ∈[0,∞],(ce ,cu )∈Y

min{θ, 1} pi U(ce ) + (1 − pi )U(cu )s.t. min{1, θ−1 }(pi (1 − ce ) − (1 − pi )cu ) ≥ k
and min{θ, 1} pj U(ce ) + (1 − pj )U(cu ) ≤ Ūj for all j < i.

We claim first that the solution to this problem sets θi = 1. If θi > 1, reducing θi to 1
relaxes the first constraint without otherwise affecting the solution to the problem. If θi < 1,consider the following variation: raise θi to 1 and increase ce and reduce cu while keepingboth θ pi U(ce ) + (1 − pi )U(cu ) and pi ce + (1 − pi )cu unchanged. That is, the perturbed

28

consumption levels ce > ce,i and cu < cu,i are defined byθi pi U(ce,i ) + (1 − pi )U(cu,i ) = pi U(ce ) + (1 − pi )U(cu )and pi ce,i + (1 − pi )cu,i = pi ce + (1 − pi )cu .

By construction, this does not affect the value of the objective function nor the first constraint. Suppose it fails to relax one of the remaining constraints. Then it must be that forsome j < i,θi pj U(ce,i ) + (1 − pj )U(cu,i ) ≤ pj U(ce ) + (1 − pj )U(cu ).

Multiply this together with pi U(ce ) + (1 − pi )U(cu ) = θi pi U(ce,i ) + (1 − pi )U(cu,i ) andsimplify to obtain(pi − pj )(U(ce,i )U(cu ) − U(ce )U(cu,i )) ≥ 0.
But since pi > pj , ce,i < ce , and cu < cu,i , this is a contradiction. Thus the perturbation
relaxes each of these constraints. We may therefore without loss of generality focus on the
problem with θi = 1:Ūi = max(ce ,cu )∈Y

pi U(ce ) + (1 − pi )U(cu )s.t. pi (1 − ce ) − (1 − pi )cu ≥ k(P-i′ )and pj U(ce ) + (1 − pj )U(cu ) ≤ Ūj for all j < i.
We label this problem for use in the remainder of this proof.
Next we claim that the solution to problem (P-i′ ) has ce,i > ce,i−1 and cu,i < cu,i−1 . Weproceed by induction. For i = i∗ + 1, we write problem (P-i′ ) asŪi = max(ce ,cu )∈Y

pi U(ce ) + (1 − pi )U(cu )s.t. pi (1 − ce ) − (1 − pi )cu ≥ kand pj U(ce ) + (1 − pj )U(cu ) ≤ 0 for all j < i.
ce,i = cu,i = 0 satisfies the constraints but leaves the first one slack. Lemma 1 implies that
it is possible to do better, i.e. to obtain positive utility. On the other hand, consider any(ce,i, cu,i ) that delivers positive utility and satisfies the last constraint, sopi U(ce,i ) + (1 − pi )U(cu,i ) > 0 and pj U(ce,i ) + (1 − pj )U(cu,i ) ≤ 0.

29

Subtracting inequalities gives(pi − pj )(U(ce,i ) − U(cu,i )) > 0,
which proves ce,i > cu,i . Finally, if ce,i > cu,i ≥ 0, the last constraint is violated, while if
0 ≥ ce,i > cu,i , the objective is negative. This proves ce,i > 0 > cu,i when i = i∗ + 1.
Now suppose we have proven that ce,j > ce,j−1 and cu,j < cu,j−1 for all j < i, j > i∗ .
Setting ce,i = ce,i−1 and cu,i = cu,i−1 is again feasible because it satisfies all the constraints
in problem (P-i′ ). This follows because those values satisfy the constraints in problem (P(i − 1)′ ) and deliver value Ūi−1 . But this choice is not optimal because it leaves the first
constraint slack. So the solution to problem (P-i′ ) must be a policy (ce,i, cu,i ) that delivers
higher utility and satisfies the constraint of excluding type i − 1 agents:pi U(ce,i ) + (1 − pi )U(cu,i ) > pi U(ce,i−1 ) + (1 − pi )U(cu,i−1 )pi−1 U(ce,i ) + (1 − pi−1 )U(cu,i ) ≤ pi−1 U(ce,i−1 ) + (1 − pi−1 )U(cu,i−1 ).
Subtracting inequalities gives(pi − pi−1 ) U(ce,i ) − U(ce,i−1 ) − U(cu,i ) + U(cu,i−1 ) > 0.

This proves that U(ce,i ) − U(ce,i−1 ) > U(cu,i ) − U(cu,i−1 ). Finally, if U(ce,i ) > U(ce,i−1 )and U(cu,i ) > U(cu,i−1 ), the constraint of excluding type i − 1 agents is violated. If
U(ce,i ) < U(ce,i−1 ) and U(cu,i ) < U(cu,i−1 ), the objective function in problem (P-i′ ) is reduced compared to setting ce,i = ce,i−1 and cu,i = cu,i−1 . Thus the solution to problem (P-i′ )must have U(ce,i ) − U(ce,i−1 ) > 0 > U(cu,i ) − U(cu,i−1 ).
Finally, we claim that for all i > i∗ , the equilibrium consumption specified by type icontracts satisfies two binding constraints, firms earn zero profits and type i − 1 workers are
indifferent about applying for type i contracts,pi (1 − ce,i) − (1 − pi )cu,i = k
and pi−1 U(ce,i ) + (1 − pi−1 )U(cu,i ) = Ūi−1 .
In particular, these equations will typically exhibit two solutions; we look for the uniquesolution with ce,i > 0 > cu,i. It is straightforward to prove that these equations must bindfor i = i∗ + 1. For any i > i∗ + 1, suppose we have established this result for type i − 1 and30

proceed by induction. In particular, we know thatpi−1 U(ce,i−1 ) + (1 − pi−1 )U(cu,i−1 ) = Ūi−1 ,pi−2 U(ce,i−1 ) + (1 − pi−2 )U(cu,i−1 ) = Ūi−2 .
The first equation comes from the construction of Ūi−1 , the second from the assumption thattype i − 2 workers are indifferent about applying for type i − 1 contracts. Suppose that forsome j < i − 2
pj U(ce,i ) + (1 − pj )U(cu,i ) = Ūj .
Substituting for Ūj using the optimal search condition for j < i − 2 gives
pj U(ce,i ) + (1 − pj )U(cu,i ) ≥ pj U(ce,i−1 ) + (1 − pj )U(cu,i−1 ).
Combining this last inequality with the constraint for type i − 1,
pi−1 U(ce,i ) + (1 − pi−1 )U(cu,i ) ≤ Ūi−1 ,we obtain (pi−1 −pj )(U(ce,i−1 )U(cu,i ) −U(ce,i )U(cu,i−1 )) ≥ 0. Since pi−1 > pj for all j < i−2,
and given that ce,i > ce,i−1 , cu,i < cu,i−1 , we obtain a contradiction. Instead, the constraint
for type i − 1 must bind and the remaining constraints are slack, completing the proof.
An interesting feature of a competitive search equilibrium is that cu,i is negative, so aworker is worse off in a bad match than she would have been without a match at all. If one
thinks of a bad match as a layoff, an optimal contract must give a worker less utility if she
is laid off than if she never gets a job in order to keep out workers with still lower expectedutility.

6.3

Pareto Optimality

As in the first example, the equilibrium need not be efficient. Observe that a worker with pi
close to 1 suffers little from the distortions introduced by adverse selection. At the extreme,
if pI = 1, setting cu,I = c excludes all other workers without distorting the type I contract.
More generally, adverse selection has the biggest impact on the utility of workers with anintermediate value of pi . A Pareto improvement may therefore require only partial pooling.
To be concrete, suppose p1 = 1/4, p2 = 1/2, and p3 = 3/4 and there are equal numbersof type 1 and type 3 workers, so half of all matches are productive. Also assume U(c) =log(1 + c) and k = 1/8. Then in a competitive search equilibrium, ce,1 = cu,1 = 0.125
31and Ū1 = U(0.125); ce,2 = 0.786, cu,2 = −0.036, and Ū2 = U(0.312); and ce,3 = 0.858,
cu,3 = −0.073, and Ū3 = U(0.561). Pooling all three types, the best incentive-feasible
allocation sets ce = cu = 3/8 and Ūi = U(3/8), since half of all matches are productive. This
reduces the utility of a type 3 agent.
Instead consider an allocation that pools type 1 and type 2 workers. If there are sufficiently few type 1 workers, it is feasible to set consumption at ce = cu > 0.312, delivering
utility greater than Ū2 to both type 1 and 2. For example, suppose π1 = π3 = 0.1 andπ2 = 0.8. Then the utility of type 1 and 2 rises to U(25/72) = U(0.347). By raising theutility of type 2, it is easier to exclude them from type 3 contracts, reducing the requisite inefficiency of those contracts. This raises the utility of those workers, in this case toU(0.573).

6.4

Relationship with Rothschild-Stiglitz

This example is similar to the model of Rothschild and Stiglitz (1976, p. 630), where they“consider an individual who will have income of size W if he is lucky enough to avoid accident.
In the event an accident occurs, his income will be only W − d. The individual can insure
against this accident by paying to an insurance company a premium α1 in return for which
he will be paid α̂2 if an accident occurs. Without insurance his income in the two states,‘accident,’ ‘no accident,’ was (W, W − d); with insurance it is now (W − α1 , W − d + α2 )where α2 = α̂2 − α1 .” We can always normalize the utility of an uninsured individual to zeroand then express the utility of an individual who anticipates an accident with probability piasui (α1 , α2 ) = pi U(W − α1 ) + (1 − pi )U(W − d + α2 ) − κi ,where κi ≡ pi U(W ) + (1 − pi )U(W − d). Setting W = d = 1 and defining ce = 1 − α1 andcu = α2 , this is equivalent to our example, except for a level shift in the utility function.
Still, our characterization carries through to this environment, with one notable exception:
the cost of posting contracts may prevent trade for the highest types as well. For example, if
U(pi −k) < pi U(1) + (1 −pi)U(0), as may happen for pi close to 1, there gains from insurance
do not cover the contract posting cost.
Rothschild and Stiglitz (1976) prove that in any equilibrium, principals who attract typei agents, i > 1, offer incomplete insurance so as to deter type i − 1 agents. Under someconditions, however, such an equilibrium might not exist. Starting from this configurationof contracts, a principal may consider deviating by offering a pooling contract that attracts
multiple types of agents. This is profitable if the least cost separating contract is Pareto
inefficient.
32

Such a deviation is never profitable in our environment. In Rothschild and Stiglitz(1976), a deviating principal can attract and serve all the agents in the economy, or at leasta representative cross-section. In our model a principal cannot serve all the agents who arepotentially attracted to a contract. Instead, agents are rationed thorough the endogenous
movement in market tightness θ. Whether such a deviation is profitable depends on which
agents are most willing to accept a decline in market tightness. In this model, high typeagents will quickly give up on the pooling contract if it is too crowded with low type agents.
Low type agents, who have a lower outside option, Ūi−1 < Ūi , are more persistent. A
principal who tries to offer a pooling contract will end up with a long queue of type 1 agents,
the worst possible outcome.

7

Asset Markets

7.1

Setup

A feature of the previous examples is that market tightness is not distorted: θi is at thefirst-best level for any i. We now consider an example where principals may use tightness toscreen out undesirable types. Although all of our results hold more generally, to stress thepoint, we assume µ(θ) = min{θ, 1}, so matching is again determined by the short side of themarket and η̄ = 1. In this case, without any private information, θ would typically be equal
to 1.
Consider an asset market where sellers (agents) have private information about the valueof their asset, as in Akerlof (1970). Although buyers (principals) always value an asset morethan the seller does, some sellers’ assets are more valuable than others. Market tightness, orprobabilistic trading, is a useful screening device since sellers who hold a more valuable asset
are more willing to accept a low probability of trade at a given price. Thus, this example
shows how an illiquid market may serve as a useful screening device when asset holders haveprivate information about asset values.
We assume that each type i seller is endowed with one indivisible object, say an apple, oftype i, with value aSi > 0 for the seller and aB
i > 0 for the buyer, both expressed in units ofan outside good. The action profile in a contract for type i sellers consists of a pair {αi , ti },where αi is the probability that the seller gives the buyer the apple and ti is the expectedtransfer of the outside good that the buyer gives to the seller.6 The payoff of a matched type6

We assume that apples are indivisible and interpret α as the probability of trade. Equivalently, apples
may be divisible, with preferences linear in consumption. Then α may be interpreted as the fraction of fruit
traded.

33

i seller who reports to be a type j seller isui (αj , tj ) = tj − αj aSi ,while the payoff of a buyer matched with a type i seller who reports truthfully isvi (αi , ti ) = αi aBi − ti .
Note that we have normalized the payoff to zero in the absence of trade.
We set I = 2 and impose a number of restrictions on payoffs. First, both buyers andsellers prefer type 2 apples and both types of sellers like apples:B
aS2 > aS1 > 0 and aB
2 > a1 > 0.

Second, there are gains from trade, including the cost of posting, if the buyer is sure to trade:aSi + k < aB
i for i = 1, 2.
The available action profiles are Y = [0, 1]×[0, 1], with Ȳi = {(α, t) ∈ Y|αaSi ≤ t ≤ αaBi −k}.
Using these restrictions, we verify our four assumptions. As a preliminary step, note thatS
S
B
S
(α, t) ∈ Ȳi implies α ≥ k/(aBi − ai ) > 0 and t ≥ kai /(ai − ai ) > 0, so in any equilibrium
contract, trades are bounded away from zero. If α were too close to 0, buyers would be
unwilling to post contracts. But then t must be large enough for sellers to be willing totrade.
B
Since α > 0 whenever (α, t) ∈ Ȳi , the restriction aB
1 < a2 ensures A1 holds. A2 holds
because for any (α, t) ∈ Ȳi , a movement to (α, t − ε) with ε > 0 is feasible and raises thebuyer’s utility. The important assumption is again A3, here guaranteed by the restrictionthat aS1 < aS2 . Fix (α, t) ∈ Ȳ and γ ∈ (aS1 , aS2 ). For arbitrary δ > 0, consider the action
profile (α′ , t′ ) = (α − δ, t − γδ). Such an action profile is feasible for sufficiently small δ
because (α, t) ∈ Ȳ guarantees that α > 0 and t > 0. By construction,u2 (α′ , t′ ) − u2 (α, t) = δ(aS2 − γ) > 0andu1 (α′ , t′ ) − u1 (α, t) = δ(aS1 − γ) < 0.
p
Now for fixed ε > 0, choose δ ≤ ε/ 1 + γ 2 . This ensures (α′ , t′ ) ∈ Bε (α, t) and so assumption A3 holds.
34

7.2

Competitive Search Equilibrium

We use problem (P) to characterize the equilibrium.
Result 5 There exists a unique competitive search equilibrium with αi = 1, ti = aBi − k,BS
θ1 = 1, Ū1 = a1 − a1 − k,
θ2 =

S
aB
1 − a1 − k
< 1,
S
aB
2 − a1 − k

Sand Ū2 = θ2 (aB
2 − a2 − k).

Proof. Write problem (P-1) as
Ū1 =

max

θ∈[0,∞],(α,t)∈Y

min{θ, 1} t − αaS1

s.t. min{1, θ−1 } αaB
1 − t ≥ k.

Lemma 1 ensures that the constraint is binding, so that we can use it to eliminate t andrewrite the problem as
Ū1 =

max

θ∈[0,∞],α∈[0,1]S
min{θ, 1}α aB
1 − a1 − θk.

S
BS
Since aB
1 > a1 + k, it is optimal to set α = θ = 1. It follows that Ū1 = a1 − a1 − k.

Next we turn to problem (P-2):
Ū2 =

max

θ∈[0,∞],(α,t)∈Y

min{θ, 1} t − αaS2s.t. min{1, θ−1 } αaB
2 −t ≥ k
S
min{θ, 1} t − αaS1 ≤ aB
1 − a1 − k.

One can again prove that both constraints are binding. Eliminating t from the problem
using the first constraint gives
Ū2 =

max

θ∈[0,∞],α∈[0,1]S
min{θ, 1}α aB
2 − a2 − θkSBSs.t. min{θ, 1}α aB
2 − a1 − θk = a1 − a1 − k.

35Use the last constraint to eliminate α:SaB
1 − a1 − (1 − θ)k
B
S
a
−
a
− θk
2
2
S
θ∈[0,∞]aB
2 − a1aB − aS1 − (1 − θ)ks.t. 1∈ [0, 1].
S
min{θ, 1}(aB
2 − a1 )Ū2 = max

We include the last constraint to remember that α is a probability, lying between 0 and 1.
Since by assumption aS1 < aS2 < aB
2 , the objective function is decreasing in θ. We thus set θ
equal to the smallest value consistent with the two constraints, that is
θ2 =

S
aB
1 − a1 − k
< 1.
S
aB
2 − a1 − kThis implies α2 = 1, so the constraint binds. The value of the program, Ū2 , is then easy tocompute.
S
In the absence of private information, we would have θ2 = 1 and Ū2 = aB2 −a2 −k. Relative
to this benchmark, buyers post too few contracts designed to attract type 2 sellers, so thatsome of them fail to match. Since type 2 sellers hold better apples than type 1 sellers, they
are more willing to accept a lower matching probability in return for bigger transfers when
they do match. Note that the obvious alternative, setting θ2 = 1 but rationing though theprobability of exchange, α2 < 1, is more costly because it involves creating more contracts at
cost k per contract. Reducing the meeting rate is a more cost-effective rationing mechanism
than directly rationing trades in meetings.

7.3

Pareto Optimality

We again ask whether there is a feasible allocation that Pareto dominates the equilibrium.
Consider the allocation in which only a pooling contract is posted, with α1 = α2 = 1 andt1 = t2 = t. That is, Ȳ = {C}, where C = ((1, t), (1, t)). Moreover, Θ̃(C) = 1, p̃i (C) = πi ,Band λ({C}) = 1. Finally, set t = π1 aB
1 + π2 a2 − k. The choice of t ensures that the resource
constraint holds and the choice of λ ensures that markets clear. All the sellers apply to thesame contract so they receive trivially their maximum possible utility. Hence, the allocation
is incentive feasible. The expected payoff for type i sellers is
BSŪi = π1 aB
1 + π2 a2 − ai − k,36B
for i = 1, 2. Since aB
1 < a2 , type 1 sellers are always better off than in equilibrium. Type 2
sellers are better off if and only if

B
S
π1 aB
1 + π2 a2 − a2 − k >

S
B
S
(aB
2 − a2 − k)(a1 − a1 − k).
SaB
2 − a1 − kSince π2 = 1 − π1 , this reduces to
π1 <

S
aB
Ū2
2 − a2 − k=.
BSa2 − a1 − kŪ1

By assumption both the numerator and denominator are positive, but the numerator issmaller (the equilibrium gains from trade are smaller for type 2 sellers) because aS2 > aS1 .
Thus type 2 sellers prefer the pooling allocation only if there is sufficiently little cross subsidization, so π1 is small. The cost of cross subsidizing type 1 sellers then does not offset thebenefit, the increased efficiency of trade.

7.4

No Trade

So far, we have assumed that there are gains from trade for both types of apples. This section
S
shows that if there are no gains from trade for type 1 apples, aB
1 ≤ a1 + k, the market for
S
type 2 apples shuts down, even if there is still gains from trade, aB
2 > a2 + k. Intuitively, it is
only possible to keep agents holding bad apples out of the market by reducing the probabilityof trade in good apples. But if there is no market in bad apples, agents holding them will bewilling to accept an arbitrarily small probability of trade in the good apple market, shutting
it down.
Result 6 In any competitive search equilibrium, Ū1 = Ū2 = 0.
Proof. Write problem (P-1) as
Ū1 =

max

θ∈[0,∞],(α,t)∈Y

min{θ, 1} t − αaS1

s.t. min{1, θ−1 } αaB
1 − t ≥ k.

Lemma 1 ensures that the constraint is binding, so that we can use it to eliminate t andrewrite the problem as
Ū1 =

max

θ∈[0,∞],α∈[0,1]S
min{θ, 1}α aB
1 − a1 − θk.

SSince aB
1 ≤ a1 + k, the maximized value is Ū1 = 0, attained by setting θ = 0.

37

Next we turn to problem (P-2):
Ū2 =

max

θ∈[0,∞],(α,t)∈Y

min{θ, 1} t − αaS2

s.t. min{1, θ−1 } αaB
2 −t ≥ k
min{θ, 1} t − αaS1 ≤ 0.

One can again prove that both constraints are binding. Eliminating t from the problem
using the first constraint gives
Ū2 =

max

θ∈[0,∞],α∈[0,1]S
min{θ, 1}α aB
2 − a2 − θkSs.t. min{θ, 1}α aB
2 − a1 − θk = 0.

Use the last constraint to eliminate α:aS1 − aS2
θkS
θ∈[0,∞] aB
2 − a1θks.t.
∈ [0, 1].
S
min{θ, 1}(aB
2 − a1 )Ū2 = maxS
S
One can verify that the constraint set is nonempty. But since aB
2 > a2 > a1 , the fraction in
the objective function is negative and so the optimum is attained by setting θ = 0, giving

Ū2 = 0.

8Conclusion

This paper has developed a canonical model of adverse selection in a competitive searchequilibrium. Under a version of a single crossing property, we prove that there is a uniqueequilibrium in which principals offer separating contracts to agents. We characterize theequilibrium via the solution to a set of constrained optimization problems and illustrate theuse of our framework through three examples, including versions of the Akerlof (1976) rat
race model, the Rothschild and Stiglitz (1976) insurance model, and a simple model of asset
trade.
Given the tractability of our framework, we anticipate little difficulty in extending theresults to a dynamic environment with repeated rounds of contract posting and search. Thisis important for many applications. For example, a worker who fails to find a job today may
search again the following period.
38

It may also be interesting to study a framework where the informed party posts contracts.
In a standard competitive search model, the equilibrium allocation does not depend on
who posts contracts and who searches. With asymmetric information, contract postingby informed parties may introduce multiplicity of equilibrium through the usual signalingmechanism. While the equilibrium we study seems robust to this variant of the model, otherequilibria may arise.

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