Repeated Moral Hazard with Worker Mobility
via Directed On-the-Job Search
Kunio Tsuyuhara∗
Preliminary Draft

This Version: January 29, 2009First Version: January, 2009

Abstract
This paper studies a repeated moral hazard problem in a general equilibrium framework.
I develop a model of dynamic employment contracts by integrating an optimal contracting
problem into an equilibrium search framework. The proposed framework enables us toanalyze the interaction between the contracting problem and endogenously evolving outsideenvironment via worker mobility, and I characterize the optimal long-term wage contract aswell as the optimal incentive compatible eﬀort-tenure proﬁle. The optimal contract exhibits
an increasing wage-tenure proﬁle for two reasons: 1) it induces the workers to stay more likely
in their current contracts, and 2) it induces the workers to make eﬀorts when the current
up-front wages cannot. The optimal incentive-compatible eﬀort has also an increasing proﬁledue to an interaction between 1) the workers’ fear of loosing their jobs, and 2) their incentive
to obtain better outside oﬀers. I then show the existence of an equilibrium. The equilibrium
inherits the “block recursivity” developed by Shi (2008) and Menzio and Shi (2008); that is,
individuals’ optimal decisions and optimal contracts are independent of the distribution ofworkers.
JEL Codes: D8, E24, J6.
Keywords: Repeated Moral Hazard, Directed On-the-Job Search, Worker Mobility∗

Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario, Canada M5S

3G7 (e-mail: kunio.tsuyuhara@utoronto.ca). I would like to express my sincere gratitude to Shouyong Shi for his
extensive help and encouragement in writing this paper. I am especially grateful to Li, Hao, and Ettore Damiano
for beneﬁcial discussion and suggestions. I would also like to thank Yuliy Sannikov, Andrzej Skrzypacz, andVasiliki Skreta for helpful comments.

1

1

Introduction

Structures of long-term contractual relationship in the economy are intrinsically related to themobility of the contracting party. For example, worker motivation inside a ﬁrm changes over timedepending on alternative employment opportunities outside the ﬁrm. Workers may ﬁnd outside
opportunities more appealing and choose to leave their current employer. This mobility, in turn,aﬀects long-term contractual structure inside the ﬁrm. Understanding this interaction between
the structure of long-term contracts and mobility of workers within a dynamic environment is
the central issue of this paper.
I study a repeated moral hazard problem in a dynamic general equilibrium framework. I
develop a model of long-term employment contracting by integrating a moral hazard problem
into an equilibrium search framework. The proposed framework enables us to examine propertiesof the optimal long-term contract, especially a wage-tenure proﬁle, in relation to worker mobility
via directed on-the-job search and endogenously evolving outside environment. It is also possible
to investigate how workers’ eﬀorts evolve over tenure; i.e., are senior workers make more eﬀort
or less than junior workers?
In my model, there are risk-neutral ex-ante homogeneous ﬁrms and risk-averse ex-ante homogeneous workers in a labor market. Firms oﬀer contracts with diﬀerent values each other toattract workers, and each value forms a submarket. Workers are allowed to search for a newcontract both on- and oﬀ-the-job. They direct their search for a job, i.e., choose which submarket to enter, depending on the value of contract oﬀered in each submarket and the probability ofobtaining the job there. The numbers of ﬁrms and workers entering a submarket determine how
tight the submarket is (tightness of submarket), and a free-entry condition adjusts the tightness
so that ﬁrms are indiﬀerent among entering any submarkets (zero-proﬁt condition).
Once a ﬁrm and a worker form a match and sign a contract, they perform a series of projects:
one in each period. Each project takes only two outcomes: success or failure, and the probabilityof success depends stochastically on the worker’s unobservable eﬀort. The series of projects
continues as long as the project keeps succeeding, but if a project fails at any time, the match
separates and the worker needs to search for a new job.
Firms can commit to a long-term contract, while employed workers can quit their jobs in any
period if they ﬁnd new and better contract through on-the-job search. A long-term contract
can depend only on the tenure length of the worker on the current contract. This implies that2

a ﬁrm cannot respond to the worker’s outside oﬀer and that the period wage cannot depend on
the current outcome of the project. Therefore, ﬁrms need to design a long-term contract so thatit can induce the worker to stay on the contract as well as to make an appropriate level of eﬀort.
I ﬁnd that the optimal long-term contract exhibits an increasing wage-tenure proﬁle. It isso for two reasons: 1) it can induce the workers more likely to stay in their current contracts,
and 2) it can induces the workers to make eﬀorts when the current up-front wages cannot. The
former is a standard result in the labor search literature with on-the-job search (e.g., Burdettand Coles (2003) and Shi (2008)). Given the worker mobility, a ﬁrm has an incentive to backload
wages to entice the worker to stay; as wages rise with tenure, it is more diﬃcult for the worker
to ﬁnd a better oﬀer elsewhere, and so the worker’s quit rate falls. On the other hand, the latteris new in the literature. Since the current wage does not depend on outcomes of the current
project, it cannot give the worker incentives to make eﬀort for the current period project. The
ﬁrm, however, can promise to give higher wage in the next period, and thus higher continuation
value, if the current project succeeds. Hence, the worker’s motivation for staying employed toreceive higher continuation value in the ﬁrm and obtain better outside oﬀer can give incentive
for the worker to make eﬀort for the current project. Overall, an increasing wage proﬁle is lesscostly to the ﬁrm than a constant proﬁle that promises the same value to the worker.
I ﬁnd that the optimal incentive-compatible eﬀort has also an increasing proﬁle. This is due
to a combined eﬀect between 1) the workers’ fear of loosing their jobs and 2) their incentive
to obtain better outside oﬀers. Given an increasing wage-tenure proﬁle, a worker’s value of thecurrent contract increases, and thus cost of loosing the job after a failure increases over time.
Also, given theincreasing value of the current contract, the worker’s value of optimal search also
increases. They both increase the expected beneﬁt of eﬀort making over time, and the worker
keeps making more and more eﬀort over the tenure.
I then show that an equilibrium with these feature indeed exists. The equilibrium inherits
the “block recursivity” developed by Shi (2008) and Menzio and Shi (2008); that is, individuals’
optimal decisions and optimal contracts are independent of the distribution of workers.
This paper makes contribution in such literature as dynamic contracting, contracting in general equilibrium, and equilibrium labor search. Literature on dynamic contracting is enormous,
but this paper is especially related to Rogerson (1985), Holmstrom and Milgorm (1987), Spearand Srivastava (1987), and Sannikov (2008). These papers focus on contracting between a single
ﬁrm and a single worker, and the outside environment is kept constant over time. Since con-3

tracting agents are not allowed to look for another contracts in the market, the theory does notaddress the issue of mobility which is the key aspect of this paper.
Key papers in the literature on contracting in general equilibrium theory are Prescott andTownsend (1984), and Phelan (1995). They consider contracting problems in which agents canmove across contracts in competitive equilibrium frameworks. Common feature of these models
is that contracting agents are perfectly mobile in the markets. On the other hand, the present
paper focuses on frictional mobility in a market and shows how it aﬀects the resulting optimal
contract.
Closely related papers in the equilibrium labor search literature are Burdett and Coles (2003),
Menzio and Shi (2008), and Shi (2008). They consider labor market matching models and shed
new lights on interactions between employment contracting problems and frictional mobility
via on-the-job search in labor markets. However, they fail to incorporate issues of asymmetric
information, and incentive problems within the contractual relationship are abstracted fromthese analysis.
There are several recent papers to integrate these areas of research to obtain novel insights.
Shimer and Wright (2004) study a static employment contracting problem with search frictions
and bilateral asymmetric information. Manoli and Sannikov (2005) studies a dynamic employment contracts in an environment with competition between ﬁrms and with informational
frictions. Board (2007) characterizes the optimal self-enforcing contracts in a large anonymouslabour market which allows for on-the-job search by workers. They all try to examine further
implications of contract theory by placing it in more realistic contexts, and the current papershares this spirit.
This paper, however, presents some distinct features. First, in my model, a worker’s outsideenvironment has deﬁnite interaction with his behavior inside the ﬁrm, and it evolves endogenously via worker mobility as an optimal outcome. Then, the directed search framework enable
me to characterize the equilibrium contract without referring the distribution of workers in themarket and to prove the existence of an equilibrium with such a contract. Because of thesefeatures, I can obtain very clear predictions on the optimal contract.

4

2

A Model of Labor Market with Search Friction
and Moral Hazard

2.1

Physical Environment

I consider a labor market with a continuum of ex-ante homogeneous, inﬁnitely lived workers
with measure 1 and a continuum of ex-ante homogeneous ﬁrms whose measure is determined bycompetitive entry. Time is discrete and continues forever. Each worker has a utility function
u(w) where w is income in a period. I assume u : R → R is twice continuously diﬀerentiable,
strictly increasing, weakly concave, and the ﬁrst derivative is bounded from both below andabove, i.e., u0 (w) ∈ [u0 , ū0 ] for all w. When employed, each worker exerts costly eﬀort, e, for¯
the project of the ﬁrm in each period where e ∈ R+ . I assume cost function c : R → R is
twice continuously diﬀerentiable, strictly increasing, weakly convex. Each worker maximizes the
expected sum of lifetime utilities minus costs of eﬀort discounted at the rate β ∈ (0, 1), that is,∑∞ t
t=0 β (u(wt ) − c(et )).
Firm’s project outcome takes only two values: {0, y}. When outcome is y the worker’s
performance is a “success,” and when 0 it is a “failure.” The probability of success depends
on an eﬀort level by the worker and is given by r(e) where r : R → R is twice continuously
diﬀerentiable, strictly increasing and weakly concave in e. I also assume that r0 (0) < ∞ andlime→∞ r(e) = 1. Each ﬁrm maximizes the expected sum of proﬁts discounted at the rate β.
There is a continuum of labor markets indexed by x, where x denotes the value of contractoﬀered to a worker in that submarket, i.e., whenever a vacant ﬁrm meets a worker in thatsubmarket it has to oﬀer him an employment contract that gives him the lifetime utility x. I
assume that x ∈ X ⊆ R+ . Let G a cumulative distribution of employed workers over X and ua fraction of unemployed workers.The ratio of vacant ﬁrms to searching workers in submarket x
is denoted by θ(x) and refered to as the tightness of submarket x.
In each period, there are two stages: search and matching, and production. In the search
and matching stage, ﬁrms post a vacancy at a ﬂow cost k > 0 and announce long-term contract
to recruit a worker. Firms can direct the workers’ search by oﬀering in diﬀerent submarket fromeach other. Each worker, independent of the employment status, receives the opportunity ofsearching for a job with a probability λ ∈ [0, 1]. If a worker has the opportunity of searching thelabor market and choose to visit submarket x, he meets a vacant ﬁrm with probability p(θ(x)),
where p : R+ → [0, 1] is twice continuously diﬀerentiable, strictly increasing and strictly concave
5and such that p(0) = 0, p0 (0) < ∞. On the other hand, a vacant ﬁrm ﬁnds a worker withprobability q(θ(x)), where q : R+ → [0, 1] is twice continuously diﬀerentiable, strictly decreasing
and strictly convex and such that θ−1 p(θ) = q(θ), q(0) = 1, and p(q −1 (·)) is concave.
If an employed worker receives the opportunity of searching and matches with a ﬁrm, and if heaccept the oﬀer, he must leave his previous employment position before entering the production
stage with a new ﬁrm. If he rejects the oﬀer, he returns to his previous employment positionand enters the production stage.
In the production stage, each unemployed worker produces at home and consumes b unitof output. Each employed worker receives the current period wage before the production takes
place. Therefore, the wage cannot depend on the project outcome as in standard moral hazard
literature. Then the worker chooses how much eﬀort to make, and the project outcome is realized
depending on the worker’s eﬀort level.
At the end of the period after the project outcome is realized, if project fails, the match is
destroyed and the worker looses his employment position. If the project succeeds, the matchstays together and the worker keeps his employment position in the next period. Projects
continue forever as long as they are succeeding. There is no exogenous separation of the matchand failure is the only reason of separation.

2.2

Contractual Environment

I assume ﬁrms can commit to long-term contracts but workers can quit a job at any period. In
particular, when an employed worker receives an opportunity to search for an outside contract,
the ﬁrm cannot respond to the employee’s outside oﬀers. Firms must design contracts thatspeciﬁes worker’s wage and the recommended eﬀort level in each period. Employment contracts
can specify only the worker’s wage as afunction of his tenure on the job conditional on thatthe worker stays with the ﬁrm and -possibly-of the outcome of a randomization device. As
mentioned, wage in each period does not depend on the realization of the project outcome.
Workers receive up-front payments at the beginning of each production stage. Since the worker’s
eﬀort level is unobservable to the ﬁrm, the recommended eﬀorts need to be incentive compatible
to the worker.

6

2.3

Equilibrium Conditions

In the following, I describe the equilibrium conditions for this economy. I will set up the optimal
contracting problem as a recursive problem following the approach taken by Spear and Srivastava(1987) and much of the recent dynamic game literature.
First, I will explain a market tightness from ﬁrms free entry condition. Second, given themarket tightness function, I will illustrate worker’s problems in terms of optimal search andoptimal eﬀort making when employed and present the value of optimal search for employedworkers. Third, I will give the value of unemployed worker. Finally, I will present the ﬁrm’s
optimal contracting problem as a recursive problem.
2.3.1

Market Tightness and Free Entry Condition

During the search stage, ﬁrms choose how many vacancies to create and where to locate them.
Let J(x) denote a ﬁrm’s value of meeting a worker in submarket x. Then, the ﬁrm’s expected
beneﬁt of creating a vacancy in submarket x is given by q(θ(x))J(x), the product of the probability and the value of meeting a worker in the submarket. Given the market tightness function
θ(x), if the cost k of creating a vacancy is strictly greater than the expected beneﬁt, then ﬁrms
do not create any vacancies in submarket x. If k is strictly smaller than the expected beneﬁt,then ﬁrms create inﬁnitely many vacancy in x. When they are equal, then ﬁrms expected proﬁts
are zero and independent from the number of vacancies they create in submarket x. Therefore,
in any x, θ(x) is consistent with ﬁrms’ proﬁt maximization if
q(θ(x))J(x) − k ≤ 0,(1)

and θ(x) ≥ 0, with complementary slackness. The complementary slackness condition ensuresthat if the expected beneﬁt is less than the cost of creating vacancy, then no ﬁrms create vacancy
in submarket x; that is θ(x) = 0. Also, free entry and exit condition drives the maximized
expected proﬁt to zero.
2.3.2

Worker’s Problems

Optimal Search of the Worker
Suppose a worker’s current value of employment position is V . If he receives the opportunity tosearch for a new contract and visits submarket x, he ﬁnd an employer with probability p(θ(x))
7and yields the additional value of x − V . The worker chooses which submarket to visit tomaximize the expected value of search. I will denote the worker’s value of search as
D(V ) = max p(θ(x))(x − V ),
x∈R

(2)

given the current value of employment position V . I denote with m(V ) the worker’s optimal
search policy of this problem, and denote the composite function p̂(V ) = p(θ(m(V ))).
Optimal Eﬀort Choice of the Worker
Consider a worker in the production stage. If the project succeeds, the current contract will
provide a continuation value to the worker, W , and it will be the worker’s reservation value
from the current contract. In the next search stage, if he receives the opportunity to searchwith probability λ, he will obtain the expected beneﬁt of W + D(W ) through the optimal search
explained above. If he does not receive the opportunity to search, he keeps the reservation value
W . On the other hand, if the project fails, the worker will loose the job in the next period andspend as an unemployed receiving the value of unemployed U as explained below. Hence, thenet expected beneﬁt of success with the eﬀort level e isr(e)(λ(W + D(W )) + (1 − λ)W ) + (1 − r(e))U = r(e)(W + λD(W )) + (1 − r(e))U.
The cost of eﬀort level e is c(e). Therefore, given the continuation value W , the worker will
choose his eﬀort level to solve:(())
max − c(e) + β r(e)(W + λD(W )) + (1 − r(e))U .
e∈R

Given the structure of the contract, current period wage is independent of the project outcome
and does not aﬀect the worker’s optimal eﬀort choice. Note that the expected beneﬁt from theeﬀort is given in the next period and the worker discount its value at β.
Worker’s Value of Unemployment
Finally, consider an unemployed worker at the beginning of the production stage. The worker
obtains u(b) from the home production. Let U be the value of unemployed. If he receives
opportunity to search for a job, he will obtain the expected beneﬁt of U + D(U ) through the
optimal search. If he does not receive the opportunity to search, he will stay unemployed andreceive U . Therefore, the value of unemployed is expressed recursively as()
U = u(b) + β U + λD(U ) .
8(3)

2.3.3

The Firm’s Optimal Contracting Problem

Consider a ﬁrm that promises to provide a continuation value V in this period. Let J(V ) denotethe current value of contract for the ﬁrm. The ﬁrms chooses: i) w, how much wage to pay
in this period, ii) e, how much eﬀort to induce, and iii) W , how much continuation value toprovide to the worker in the next period conditional on the survival of the relationship. I also
allow for randomization over these choices, that is the ﬁrm oﬀer two sets of contract and aprobability distribution, {πi }i=1,2 , over these subcontracts. Denote by ξ = ({wi , ei , Wi , πi }i=1,2 )the contract oﬀered by the ﬁrm at the beginning of a period.
.
The ﬁrm’s optimal contracting problem is given byJ(V ) = maxξ∑

πi {r(ei )y − wi + βr(ei )(1 − λp̂(Wi ))J(Wi )}

(4)

i=1,2

subject toV =∑πi {u(wi ) − c(ei ) + β[r(ei )(Wi + λD(Wi )) + (1 − r(ei ))U ]} ,

i=1,2

(
(
))
ei ∈ arg max − c(e) + β r(e)(W + λD(W )) + (1 − r(e))U , for i = 1, 2,e∈Randπ1 + π2 = 1.
By inducing an eﬀort level e and paying w, the ﬁrm’s expected current period payoﬀ is r(e)y −w.
In the next period, by oﬀering continuation value W , the current worker will stay in the current
ﬁrm with probability r(e)(1 − λp̂(W )): product between a probability the project succeeds anda probability the worker will not leave for the outside option, and the ﬁrm will enjoy the valueJ(W ) of remaining contract.
To design the contract, the ﬁrm faces three constraints. The ﬁrst is the promise-keeping
or consistency constraint, that is, the contract has to provide the worker with the promisedvalue V . Since oﬀer of the contract is evaluated ex-ante, only the expected value over thevalues of subcontracts must provide the promised continuation value; either subcontract may
fail to provide the promised value. Then, the eﬀort that the ﬁrm wants to induce must be
incentive compatible, that is, given the contract oﬀered the worker voluntarily choose to makethat level of eﬀort. Since realization of subcontract occurs before the worker chooses his eﬀort
9

level, both subcontracts the ﬁrm prepares need to meet the incentive compatibility constraint.
Finally, when the ﬁrm randomizes the contracts, the probabilities assigned to each subcontract
must sum to one. I denote with ξ(V ) the optimal policy functions given V associated with thiscontracting problem.

2.4

Block (Distribution-free) Recursive Equilibrium

Deﬁnition: A Recursive Equilibrium is a set of functions {J ∗ , θ∗ , D∗ , m∗ , U ∗ , ξ ∗ } and a distribution of workers (G∗ , u∗ ) such that
1. θ∗ satisﬁes condition (1),
2. D∗ and m∗ satisﬁes the functional equation (2),
3. U ∗ satisﬁes the functional equation (3),4. J ∗ and ξ ∗ satisﬁes the functional equation (4), and
5. (G∗ , u∗ ) is consistent with m∗ and ξ ∗ .
Deﬁnition: A Block (Distribution-free) Recursive Equilibrium is a recursive equilibrium suchthat functions {J ∗ , θ∗ , D∗ , m∗ , U ∗ , ξ ∗ } are independent from the distribution of workers (G∗ , u∗ ).

3Equilibrium for the Optimal Long-Term Contract

In this section, we will ﬁrst describe general properties of equilibrium and characterize theoptimal long-term contract. Then I will show the existence of equilibrium in this environment.
Proofs of each lemma is almost identical to that in Menzio and Shi (2008) or requires only minor
modiﬁcations for considering worker’s eﬀort choice. I collect all the proofs for those requiringextra steps or modiﬁcations in the appendix.
We start with specifying a set of functions with certain properties. Then, we take an arbitrary
function as a ﬁrm’s value function and characterize the equilibrium objects and the optimal
contract given the properties of functions in the set. With all these results, we construct anoperator deﬁned over the set of functions and show that any ﬁxed point of the operator is a
Block Recursive Equilibrium.

10

3.1

General Properties

We deﬁne the set J (X) of functions J : X → R such that:(i) J(V ) is strictly decreasing and Lipschitz continuous with respect to V,(ii) J(V ) is bounded both from below and above, and(iii) J(V ) is concave.
We can show that the set J (X) is non-empty, bounded, closed, and convex subset of the spaceof bounded, continuous functions on X with the sup norm.
3.1.1

Market Tightness and Free Entry Condition

First, we take an arbitrary ﬁrm’s value function J ∈ J (X) and solve the equilibrium condition(1) with respect to the market tightness function θ. We getq −1 (k/J(x)) if J(x) ≥ k
θ(x) =
0otherwise.

(5)

Because q is a probability, q −1 is deﬁned only on [0, 1], that is J(x) ≥ k. Since J(x) is strictly
decreasing, there exists a unique x̂ ∈ R such that J(x) > x̂ for all x < x̂ and J(x) < x̂ for allx > x̂. That is, oﬀering any contract with the value more than x̂ provide negative proﬁts even if
the ﬁrm can hire a worker with probability one. Therefore, no ﬁrm will enter submarkets withx > x̂, and the market tightness takes nonnegative value only if x ≤ x̂. We call this thresholdvalue an eﬀective bound given J. Since J is bounded from above, θ is also bounded from above.
Now we have the following properties of market tightness functions.
Lemma 3.1. If x < x̂, the market tightness function θ(x) is strictly positive and strictly decreasing, and Lipschitz continuous.θ(x) = 0 for all x ≥ x̂

3.1.2

Worker’s Problem

Optimal Search of the Worker
In this section, we insert the market tightness function, θ(x), obtained from an arbitrary ﬁrm’s
value function J ∈ J (X) into the worker’s search problem:
D(V ) = max p(θ(x))(x − V ).
x≥V

Now, we have the following series of properties which enable to compute the optimal contracts.
11

Lemma 3.2. For all V ∈ X, the worker’s objective function f (x; V ) = p(θ(x))(x − V ) is strictly
concave with respect to x.
The concavity of the problem implies that, given a continuation value of the current contract,
the worker will ﬁnd unique submarket that he optimally visits.
Lemma 3.3. The worker’s optimal search strategy m(V ) ∈ arg max p(θ(x))(x − V ) is unique,weakly increasing and Lipshitz continuous.
The lemma also implies that a worker searches in a submarket which oﬀers a higher value thehigher your continuation value from the current contract. This can be seen as a version of thesingle crossing property.
Lemma 3.4. If V < x̂, the worker’s value of searching D(V ) is strictly positive and weakly
decreasing and Lipschitz continuous. If V ≥ x̂, D(V ) = 0.
As long as a worker’s current continuation value is less than the eﬀective bound x̂, there aresubmarkets that oﬀer higher values and they provide positive expected values. However, once
his continuation value hits the bound, there are no ﬁrms oﬀering better oﬀer and the market
tightness becomes zero, D(V ) = 0 since p(0) = 0.
Lemma 3.5. p̂(V ) is weakly decreasing and Lipschitz continuous.Moreover p̂(x̂) = 0
Remember that the composite function p̂(V ) is a probability that a worker with a continuation
value V from the current contract meets a ﬁrm in the optimally targeted submarket. Since a
worker with higher V applies to a submarket with higher tightness, he is less likely to meet aﬁrm in an optimally targeted submarket.
Optimal Eﬀort Choice of the Worker
We now turn to a worker’s optimal eﬀort choice problem:(())
max − c(e) + β r(e)(W + λD(W )) + (1 − r(e))U .
e∈R

Under the assumption on c(·) and r(·), it is a concave problem with respect to e, and we applies
the ﬁrst-order approach to characterize the optimal eﬀort.
Lemma 3.6. Given a continuation value W , there is a unique level of optimal eﬀort. The
worker’s optimal eﬀort function is increasing in the continuation value of contract, W .
12

Proof : First, for a given continuation value W , the worker’s optimal eﬀort is implicitly andgiven by−c0 (e) + βr0 (e)(W + λD(W ) − U ) = 0.
0c (e)Deﬁne Ω(W ) = W + λD(W ) − U . Then we can write βr
0 (e) = Ω(W ). Under the assumptions
0c(c(·): convex, continuous, r(·): concave continuous, and r0 > 0 everywhere), βr
0 is continuousand monotone increasing, and thus invertible. Therefore, there exists a unique level of e tosatisfy the equality.
Moreover the inverse function is also continuous. Hence, we write e(W ) = g(Ω(W )) where g0cis the inverse function of βr
0 . Diﬀerentiating the composite function,∂e(W )= g 0 (·)Ω0 (W )
∂W
((
) )0
c0 −1
=
(1 − λp̂(W ))βr0( 00 0)c r − c0 r00 −1=(1 − λp̂(W ))
βr02
βr02
(1 − λp̂(W )),= 00 0c r − c0 r00
where the ﬁrst equality is by the chain rule and the third is by the inverse function theorem.
Since the denominator of the right hand side is strictly positive by assumption, and thus thederivative is positive, the worker’s optimal eﬀort is increasing in W .

3.2¤

Characterization of the Optimal Contracts

We now characterize the optimal contract. Given all the previous results, a ﬁrms problem isgiven by the following recursive form:J(V ) = maxξ∑

πi {r(ei )y − wi + βr(ei )(1 − λp̂(Wi ))J(Wi )}

i=1,2

13

subject to{ξ ∈ Ξ = {wi , ei , Wi , πi }i=1,2 : Wi ∈ [0, x̂] for i = 1, 2∑V =πi {u(wi ) − c(ei ) + β[r(ei )(Wi + λD(Wi )) + (1 − r(ei ))U ]}i=1,20− c (ei ) + βr0 (ei )(Wi + λD(Wi ) − U ) = 0, for i = 1, 2}
π1 + π2 = 1, πi ∈ [0, 1] for i = 1, 2First, we consider only the promise-keeping constraint and the incentive compatibility constraint
to obtain the following result.

Proposition 3.1. Under the optimal contract, the current period wage is independent of therealization of lottery.
Proof. Let η(V ) be the Lagrange multiplier for the promise-keeping constraint. The ﬁrst ordercondition for wi implies
η(V ) =

1
u0 (wi )for i = 1, 2.

This implies that w1 = w2 . Therefore, the current wage, w, does not depend on the realization¤of the lottery.

Given the above proposition, the ﬁrm’s problem can be greatly simpliﬁed. It essentiallymakes it as a one-variable, unconstrained problem, at least for our characterization purpose.
Lemma 3.7. (Reduction of the Problem) Let γ = ({Wi , πi }i=1,2 ). The optimal contracting
problem can be reduced to the following optimization problem with respect to γ.
{J(V ) = max
γ

(
− u−1

V − βU −

∑

[
πi

]
− c(e(Wi )) + βr(e(Wi ))[Wi + λD(Wi ) − U ]

i

+

∑

[
]
πi r(e(Wi )) y + β(1 − λp̂(Wi ))J(Wi )

}

i

where
{
}
γ ∈ Γ = {Wi , πi }i=1,2 : Wi ∈ [0, x̂], πi ∈ [0, 1] for i = 1, 2 and π1 + π2 = 1

14

)

(6)

Proof. Substituting the optimal eﬀort function into the objective function and the ﬁrst constraint
eliminate the incentive compatibility constraint. Then, using the result that w1 = w2 makes thepromise-keeping constraint as
V = u(w) + βU +∑[]πi − c(e(Wi )) + βr(e(Wi ))[Wi + λD(Wi ) − U ] .

iSince u is strictly concave and thus invertible, this constraint can be solved for w. Substituting
resulting expression for w into the objective function gives the desired form of unconstrained¤

optimization problem.

3.2.1

The Optimal Wage and Continuation Value Processes

We are interested in a shape of long-term wage-tenure proﬁle in this environment. The following
result seems to be a standard result of backloaded wage proﬁle in the dynamic on-the-job search
literature, but gives and additional insight into the mechanism. Moreover, from the dynamic
moral hazard perspective, this kind of characterization seems to be a novel result.
Proposition 3.2. Under the optimal contract, wages are nondecreasing over the tenure if theworker stays on the contract, independent of the realization of lottery. Especially, if V < x̂
they are strictly increasing, and they stay constant once the continuation value hits the eﬀective
bound x̂.
Proof : First, ignoring the eﬀective bound constraint, the ﬁrst order condition of the reduced
form of the problem (6) with respect to Wi is−1· (−πi βr(e(Wi ))(1 − λp̂(Wi )))u0 (w)+ πi r0 (e(Wi ))[y + β(1 − λp̂(Wi ))J(Wi )]g 0 (·)(1 − λp̂(Wi ))[]+ πi r(e(Wi ))β (1 − λp̂(Wi ))J 0 (Wi ) − λJ(Wi )p̂0 (Wi ) = 0)
where we use that ∂Ω(W= 1 − λp̂(W ) and e0 (W ) = g 0 (·)Ω0 (W ). Then, dividing through by∂Wβ, πi , r(e), (1 − λp̂(Wi )) gives
1
r0 (e(Wi ))
− 0
+
u (w)
r(e(Wi ))()
yλJ(Wi )p̂0 (Wi )+ (1 − λp̂(Wi )) g 0 (·) + J 0 (Wi ) −= 0.
β
1 − λp̂(Wi )151Now, from the ﬁrst order condition of the constrained problem, we have η(V ) = u0 (w(V)) .

The theorem of Lagrangian multiplier implies that J 0 (V ) = −η(V ), so we have1
J 0 (V ) = − 0.
u (w(V ))(7)

1
Shifting one period forward gives J 0 (Wi ) = − u0 (w(Wwhere w(Wi ) is the wage in the nexti ))period when the current lottery realization is i.
Substituting it into the previous equation and rearranging the terms gives)(
1
1
λJ(Wi )p̂0 (Wi ) r0 (e(Wi )) y
−
=
−
+
+
(1
−
λp̂(W
))
g 0 (·).
iu0 (w(Wi )) u0 (w)1 − λp̂(Wi )r(e(Wi )) β(8)The right hand side is positive since p̂0 (Wi ) is non-positive. Therefore, we have1
u0 (w(Wi ))−1
u0 (w)> 0.

(9)

Since u is concave function, this implies w(Wi ) > w for i = 1, 2. Hence, the next period wage ishigher than the current wage.
However, given the eﬀective bound constraint, ever increasing wage proﬁle is not optimal;
after some point, the ﬁrms is paying more than the project produce. Once a worker’s value ofcontract hits the eﬀective bound x̂, in equilibrium, there is no other ﬁrms oﬀering higher value
and p̂(x̂) = 0, so the worker has no incentive to search on-the-job. The ﬁrm keeps the value andpays the wage()
ŵ = −u−1 x̂ − βU + c(e(x̂)) − βr(e(x̂))(x̂ − U ) .
¤The equation (8) in the proof gives explicit mechanics of backloaded wage-tenure proﬁle; theﬁrst term considers the worker’s mobility whereas the second term considers the moral hazard
problem. Given the worker’s mobility, a ﬁrm has an incentive to backload wages to entice theworker to stay; as wages rise with tenure, it is more diﬃcult for the worker to ﬁnd a better oﬀer
elsewhere, and so the worker’s quit rate falls (see Shi 2008). On the other hand, the second
term is new in the literature. Since the current wage does not depend on outcome of the current
project, the current period wage cannot give a worker incentives to make eﬀort for the current
period project. The ﬁrm, however, gives incentives to promise higher continuation value in thenext period. Overall, a rising wage proﬁle is less costly to the ﬁrm than a constant proﬁle thatpromises the same value to the worker.

16

Corollary 3.1. Under the optimal contract, the next period promised continuation values arenondecreasing over the tenure if the worker stays on the contract, independent of the realizationof lottery. Especially, if V < x̂, they are strictly increasing, and they stay constant once they hit
the eﬀective bound x̂.
Proof : If V < x̂, with (7), the inequality (9) implies J 0 (Wi ) − J 0 (V ) < 0. Then, concavity of J
implies Wi > V for i = 1, 2. If V ≥ x̂, optimal oﬀer stays constant at x̂.
3.2.2

¤

The Optimal Incentive Compatible Eﬀort Process

Proposition 3.3. Under the optimal contract, the worker’s optimal eﬀorts are nondecreasing if
the worker stays on the contract, independent of the realization of lottery. Especially, if V < x̂,
they are strictly increasing, and they stay constant once the continuation value hits the eﬀective
bound x̂.
Proof : As shown in lemma 3.6, the optimal eﬀort function is increasing in the continuation
value of the contract. Since the continuation value of the contract is higher than the current
value of the contract (Corollary (3.1)), the worker will make higher eﬀort next period than the¤current period eﬀort if stays on the contract.

This is due to a combined eﬀect between 1) a worker’s fear of loosing the job and 2) his
incentive to obtain the better outside oﬀer. Given rising wages, a worker’s value of the current
contract increases over time and cost of loosing the job is increasing. Also, given increasing valueof the current contract, worker’s value of optimal search increases. Both of these factors give
the worker incentives to make higher and higher eﬀorts. These, in turn, sustain the increasing
wage-tenure proﬁle oﬀered by the ﬁrm. In some sense, moral hazard problems are ameliorated
through the market forces.

3.3

Existence of a Block Recursive Equilibrium

So far, we have examined properties of equilibrium objects and characterized the optimal contracts given an arbitrary ﬁrm’s value function J ∈ J (X). In this section, we establish that suchan equilibrium, especially a Block Recursive Equilibrium exists.
Proposition 3.4. A Block Recursive Equilibrium exists.
17

Detailed proof is given in the appendix. I will only outline the argument in the proof. First,
by inserting all the previous equilibrium objects, given an arbitrary value function J ∈ J (X),
into a ﬁrm’s optimal contracting problem, we construct the following operator that maps from
J (X) to some space of functions:(T J)(V ) = maxξ∈Ξ∑

πi {r(ei )y − wi + βr(ei )(1 − λp̂(Wi ))J(Wi )} .

i=1,2

We show that T is a self-map, that is, T maps from J (X) to itself (Lemma 5.8). Then, we showthat T is a continuous map (Lemma 5.9). These suﬃce to show that the operator T satisﬁes theassumptions of Schauder Fixed Point Theorem (Stokey and Lucas with Prescott, 1989, Theorem
17.4); that is, (i) T is continuous, (ii) the family of functions T (J ) is equicontinuous, and (iii)
T maps the set J (X) into itself. Therefore, there exists a ﬁrm’s value function J ∗ (V ) ∈ J (X)such that T J ∗ = J ∗ . Denote with {θ∗ , D∗ , m∗ , U ∗ , e∗ , ξ ∗ } the respective functions associated toJ ∗ . By construction, the functions {J ∗ , θ∗ , D∗ , m∗ , U ∗ , e∗ , ξ ∗ } satisfy conditions in the deﬁnitionof a Recursive Equilibrium and do not depend on the distribution of workers. Therefore, they
constitute a Block Recursive Equilibrium.

4

Conclusion

In this paper, I developed a search theoretic general equilibrium model of an employment contracting with repeated moral hazard. The optimal contract exhibits an increasing wage-tenure
proﬁle as well as an increasing incentive-compatible eﬀort proﬁle. Moreover, I showed that an
equilibrium exists and is independent of the distribution of workers.
I am currently analyzing how the presence of moral hazard problem aﬀects the distributionof workers. Since the equilibrium is independent from the distribution, it is possible to analyze
the evolution of the distribution out of the steady state. I am also examining impacts of somepolicies, such as a minimum wage law or change in unemployment beneﬁt, on the optimal
contract and on the distribution of workers. While, in this paper, I focused exclusively on
positive aspects of the benchmark model, it may also be interesting to investigate eﬃciency ornormative properties of the equilibrium.
There are several potential extensions and future research areas. First, it is important toinvestigate workers’ saving behavior since, in my model, there are always risks of loosing their18

job. Allowing the workers to access to a credit market is an important extension. Other morechallenging extensions include 1) introducing heterogeneous ﬁrms (sectors) to examine workers’
inter-sectorial mobility, 2) allowing ﬁrms to choose either ﬁre or retain the worker after a project
fails, 3) introducing costly search on workers’ side, and 4) considering delayed eﬀects of worker’s
eﬀort on the project outcome.
Lastly, as a companion part of my ongoing research project, I am studying a case of repeatedadverse selection in which workers repeatedly receive unobservable productivity shocks or consumers repeatedly receive unobservable endowment shocks as in Phelan (1995). Incorporatingother cases of asymmetric information seems to be an important research area.

19

5

Appendix

5.1

Other Properties of Equilibrium

Note that, in the text, all the objects in the characterizations depend on a speciﬁc J ∈ J (X).
To show the existence of a ﬁxed point of the operator T J(V ), we need to show that they arecontinuous with respect to J ∈ J (X). The following series of lemmas imply the continuity ofthem. Almost all are shown in Menzio and Shi (2008) and proofs are omitted.
5.1.1

Free Entry Condition and Market Tightness

Lemma 5.1. Consider Jm , Jn ∈ J (X). Let θj (x) be the market tightness function implied by
Jj for j = m, n. If ||Jm − Jn || < ρ, then ||θm − θn || < εθ ρ.

5.1.2

Worker’s Search Problem

Lemma 5.2. Consider Jm , Jn ∈ J (X). Let Dj (V ) be the worker’s value of searching impliedby Jj for j = m, n. If ||Jm − Jn || < ρ, then ||Dm − Dn || < εD ρ.
Lemma 5.3. Consider Jm , Jn ∈ J (X). Let mj (V ) be the optimal search strategy implied by Jj
for j = m, n. If ||Jm − Jn || < ρ, then ||mm − mn || < εm ρ.
Lemma 5.4. Consider Jm , Jn ∈ J (X). Let p̂j (V ) = p(θj (mj (V ))) be the conposite function
implied by Jj for j = m, n. If ||Jm − Jn || < ρ, then ||p̂m − p̂n || < εp ρ.

5.1.3

Worker’s Value of Unemployment

Lemma 5.5. Consider Jm , Jn ∈ J (X). Let Uj be the worker’s unemployment value implied by
Jj for j = m, n. If ||Jm − Jn || < ρ, then ||Um − Un || < εU ρ.

5.1.4

Worker’s Optimal Eﬀort

Lemma 5.6. Consider Jm , Jn ∈ J (X). Let Ωj (W ) = W + λDj (W ) − Uj be the worker’s net
continuation value implied by Jj for j = m, n. If ||Jm − Jn || < ρ, then ||Ωm − Ωn || < εΩ ρ.

20

Proof of Lemma 5.6:|Ωm (W ) − Ωn (W )|= |λ(Dm (W ) − Dn (W )) − (Um − Un )|≤ |λεD − εU |ρ.
Lemma 5.7. Consider Jm , Jn ∈ J (X). Let ej (W ) = g(Ωi (W )) be the worker’s optimal eﬀort
function implied by Jj for j = m, n. If ||Jm − Jn || < ρ, then ||em − en || < εe ρ.
Proof of Lemma 5.7:
Let g¯0 = | sup g 0 (·)|. Given the assumptions about c(·) and r(·), g¯0 < ∞. Then,|em (W ) − en (W )| = |g(Ωm (W )) − g(Ωn (W ))|≤ g¯0 |Ωm (W ) − Ωn (W )|≤ g¯0 εΩ ρ.

5.2

Omitted Lemmas and Proofs

These lemmas extend the results shown in Menzio and Shi (2008) to show the existence ofˆ ) be a ﬁrm’sa Block Recursive Equilibrium when there is a moral hazard problem. Let J(Vˆ ) = (T J)(V ).
updated value function by the operator T , i.e., J(Vˆ ) belongs to the set J (X). That is,
Lemma 5.8. The ﬁrm’s value function J(Vˆ ) is strictly decreasing, and Lipschitz continuous with respect to V .
(i) J(Vˆ ) is bounded both from below and above.
(ii) J(Vˆ ) is concave.
(iii) J(VProof of Lemma 5.8: The operator is a self-mapping.
(i) From the characterization result, let F be the objective function of the reduced problem,i.e.,{
F (V, γ) =

(
− u−1

V − βU −

∑

[
πi

]
− c(e(Wi )) + βr(e(Wi ))[Wi + λD(Wi ) − U ]

i

+

∑

[
]
πi r(e(Wi )) y + β(1 − λp̂(Wi )J(Wi )

i

21

}

)

where γ is a contract. Let Γ be the set of feasible contracts as deﬁned in the text. By the Inverse
Function Theorem,

[
]
1
1
1
F (V, γ) = − 0∈ − 0,−
u
u (w)
ū0
¯
Now, for any Va , Vb ∈ X, such that Va ≤ Vb , we have0ˆ b ) − J(Vˆ b )| ≤ max |F (Vb , γ) − F (Va , γ)|
|J(V
γ∈Γ
¯
¯∫ V
¯
¯ b 0
¯
F (V, γ)dV ¯¯= max ¯
γ∈Γ

≤ max
γ∈Γ

Va

∫ Vb

¯ 0
¯
¯F (V, γ)¯ dV

Va1≤ 0 |Vb − Va |u¯ˆ ) is Lipschitz continuous in V. From this result, J(Vˆ ) is absolutely continuous andTherefore J(Vthus almost everywhere diﬀerentiable (Folland, 1999). Moreover, at any point of diﬀerentiability,
we have Jˆ0 (V ) = F 0 (V, γ(V )) where γ(V ) is the optimal contract given V (Milgrom and Segal,
2002). Then,[]1
1
F (V, γ(V ))dV ∈ − 0 (Vb − Va ), − (Vb − Va )
u
ū0
Va
¯
ˆ ) is strictly decreasing and the diﬀerence is bounded.
Hence, J(Vˆ ). Let w be the lowest possible wage under the(ii) Next, we estimate the bounds of J(V¯
feasible contract. That is∑
w = min u−1 V −πi (−c(e(Wi )) + β[r(ei )(Wi + λD(Wi )) + (1 − r(ei ))U ])
γ∈Γ¯ˆ b ) − J(Vˆ a) =
J(V

∫ Vb

0

i=1,2

Since u0 is increasing function and the expected continuation value for the worker is boundedˆ ) isby x̄ = supJ∈J x̂J , which is ﬁnite, we have w ≥ u0 (x + c(e) − β x̄). Using the fact that J(V¯¯¯strictly decreasing in V , we haveˆ ) < J(
ˆ x)
J(V
¯
¯
≤ r(ē)y − u0 (x + c(e) − β x̄) + β J¯ ≡ J.
¯¯0ˆ ) ≤ J¯ = y−u (¯x+c(¯e)−β x̄) .
Then, J(V1−β
Similarly to the previous argument,ˆ ) > J(x̄)ˆ
J(V≥ r(e)y − u0 (x̄ + c(ē) − βU ) + β J ≡ J.
¯
¯ ¯
22

0

ˆ ) ≥ J = r(¯e)y−u (x̄+c(ē)−βU ) . Hence, J(Vˆ ) is bounded both from below and above.
Then, J(V1−β¯ˆ ) can be shown with two-point convexiﬁcation result developed by(iii) Concavity of J(VMenzio and Shi (2008) and omitted in this paper.

¤

Lemma 5.9. Consider Jm , Jn ∈ J (X). Let Jˆj (W ) be the ﬁrm’s value implied by Jj for j = m, n.
If ||Jm − Jn || < ρ, then ||Jˆm − Jˆn || < εT ρ.
Continuity of the operator with respect to J ∈ J (X).
Proof of Lemma 5.9: Continuity of the operator.
Let Fj be the objective function of the ﬁrms optimal contracting problem implied by Jj :Fj : Γ × X → R. Consider Jm , Jn ∈ J (X) such that ||Jm − Jn || < ρ. Take V ∈ X such thatJˆm (V ) − Jˆn (V ) > 0. Let γj be the maximizer of Fj and wj (γ) be the wage function given by
Jj . Then, we have0 ≤ |Jˆm (V ) − Jˆn (V )|= |Fm (γm , V ) − Fn (γn , V )|≤ |Fm (γm , V ) − Fn (γm , V )|¯∑¯≤ ¯ − wm (γm ) +πi,m r(em (Wi,m )) [y + β(1 − λp̂m (Wi,m ))Jm (Wi,m )]+ wn (γm ) −

∑

i

¯
¯
πi,m r(en (Wi,m )) [y + β(1 − λp̂n (Wi,m ))Jn (Wi,m )] ¯i

≤ |wm (γm ) − wn (γm )|¯∑¯
πi,m ¯r(em (Wi,m )) [y + β(1 − λp̂m (Wi,m ))Jm (Wi,m )]+i¯¯− r(en (Wi,m )) [y + β(1 − λp̂n (Wi,m ))Jn (Wi,m )] ¯.

We want to estimate a bound for |Jˆm (V ) − Jˆn (V )|. We will consider a bound for each partof the last expression separately as follows.
1. |wm (γm ) − wn (γm )| :Since u is concave function, for any w1 and w2 , |w1 − w2 |u0 < |u(w1 ) − u(w2 )|. Also,∑u(wm (γm )) = V − βUm −πi,m [−c(em (Wi,m )) + βr(em (Wi,m ))Ωm (Wi,m )iu(wn (γm )) = V − βUn −∑πi,m [−c(en (Wi,m )) + βr(en (Wi,m ))Ωn (Wi,m )

i

23

Then
|u(wm (γm )) − u(wn (γm ))|{∑≤ β|Um − Un | +πi,m |c(em (Wi,m )) − c(em (Wi,m ))|i}+ β|r(em (Wi,m ))Ωm (Wi,m ) − r(en (Wi,m ))Ωn (Wi,m )|Now, consider the last part: |r(em (Wi,m ))Ωm (Wi,m ) − r(en (Wi,m ))Ωn (Wi,m )|.
|r(em (Wi,m ))Ωm (Wi,m ) − r(en (Wi,m ))Ωn (Wi,m )|≤ |r(em (Wi,m ))Ωm (Wi,m ) − r(en (Wi,m ))Ωm (Wi,m )|+ |r(en (Wi,m ))Ωm (Wi,m ) − r(en (Wi,m ))Ωn (Wi,m )|= |r(em (Wi,m )) − r(en (Wi,m ))|x̄ + r(en (Wi,m ))|Ωm (Wi,m ) − Ωn (Wi,m )|≤ r0 (e)|em (Wi,m ) − en (Wi,m )|x̄ + |Ωm (Wi,m ) − Ωn (Wi,m )|¯0≤(r (e)εe x̄ + εΩ )ρ¯I use the fact that Ω(·) is bounded by x̄. Collecting them together, we have|u(wm (γm )) − u(wn (γm ))|≤ (βεU + c0 (ē)εe + β(r0 (e)εe x̄ + εΩ ))ρ.
¯
Hence
|wm (γm ) − wn (γm )| ≤ u0−1 · (βεU + c0 (ē)εe + β(r0 (e)εe x̄ + εΩ ))ρ.
¯
2.
∑
i

¯
¯
πi,m ¯r(em (Wi,m )) [y + β(1 − λp̂m (Wi,m ))Jm (Wi,m )]¯¯− r(en (Wi,m )) [y + β(1 − λp̂n (Wi,m ))Jn (Wi,m )] ¯ :This expression can still be divided into subcomponents after expanding the brackets and collecting terms:

24

(ii) |r(em (Wi,m ))Jm (Wi,m ) − r(en (Wi,m ))Jn (Wi,m )| :|r(em (Wi,m ))Jm (Wi,m ) − r(en (Wi,m ))Jn (Wi,m )|≤ |r(em (Wi,m ))Jm (Wi,m ) − r(en (Wi,m ))Jm (Wi,m )|+ |r(en (Wi,m ))Jm (Wi,m ) − r(en (Wi,m ))Jn (Wi,m )|= |r(em (Wi,m )) − r(en (Wi,m ))|Jm (Wi,m )+ r(en (Wi,m ))|Jm (Wi,m ) − Jn (Wi,m )|≤ r0 (e)|em (Wi,m ) − en (Wi,m )|J¯¯
+ |Jm (Wi,m ) − Jm (Wi,m )|≤ (r0 (e)εe J¯ + 1)ρ.
¯(iii) |p̂m (Wi,m )Jm (Wi,m ) − p̂n (Wi,m )Jn (Wi,m )| :|p̂m (Wi,m )Jm (Wi,m ) − p̂n (Wi,m )Jn (Wi,m )|≤ |p̂m (Wi,m )Jm (Wi,m ) − p̂n (Wi,m )Jm (Wi,m )|+ |p̂n (Wi,m )Jm (Wi,m ) − p̂n (Wi,m )Jn (Wi,m )|= |p̂m (Wi,m ) − p̂n (Wi,m )|Jm (Wi,m )+ p̂n (Wi,m )|Jm (Wi,m ) − Jn (Wi,m )|≤(εp J¯ + 1)ρ.
(iv) |r(em (Wi,m ))p̂m (Wi,m )Jm (Wi,m ) − r(en (Wi,m ))p̂n (Wi,m )Jn (Wi,m )| :|r(em (Wi,m ))p̂m (Wi,m )Jm (Wi,m ) − r(en (Wi,m ))p̂n (Wi,m )Jn (Wi,m )|≤ |r(em (Wi,m ))p̂m (Wi,m )Jm (Wi,m ) − r(en (Wi,m ))p̂m (Wi,m )Jm (Wi,m )|+ |r(en (Wi,m ))p̂m (Wi,m )Jm (Wi,m ) − r(en (Wi,m ))p̂n (Wi,m )Jn (Wi,m )|= |r(em (Wi,m )) − r(en (Wi,m ))|p̂m (Wi,m )Jm (Wi,m )+ r(en (Wi,m ))|p̂m (Wi,m )Jm (Wi,m ) − p̂n (Wi,m )Jn (Wi,m )|≤ ((r0 (e)εe J¯ + 1)J¯ + (εp J¯ + 1))ρ¯

25

Finally, putting everything together gives that|Jˆm (V ) − Jˆn (V )|()≤ u0−1 · βεU + c0 (ē)εe + β(r0 (e)εe x̄ + εΩ ) ρ¯{}¯ 0 (e)εe J¯ + 1) + r(ē)(εp J¯ + 1) ρ+ (y + β + βλJ)(r¯≡εT ρ.
¤
Proof of Proposition 3.4: Existence of a Block Recursive Equilibrium.
Given Lemma 5.8 and 5.9, we can show that the operator T satisﬁes the assumptions ofSchauder’s Fixed Point Theorem (Stokey and Lucas with Prescott, 1989, Theorem 17.4); thatis, (i) T is continuous, (ii) the family of functions T (J ) is equicontinuous, and (iii) T maps theset J (X) into itself. Therefore, there exists a ﬁrm’s value function J ∗ (V ) ∈ J (X) such that
T J ∗ = J ∗ . Denote with {θ∗ , D∗ , m∗ , U ∗ , e∗ } the respective functions associated to J ∗ . By construction,the functions {J ∗ , θ∗ , D∗ , m∗ , U ∗ , e∗ } satisfy conditions in the deﬁnition of a Recursive
Equilibrium and do not depend on the distribution of workers. Therefore, they constitute a
Block Recursive Equilibrium.

26

References
Atkeson, Andrew. 1991. “International Lending with Moral Hazard and Risk of Repudiation.”
Econometrica, 59(4): 1069-1089.
Board, Simon. 2007. “Relational Contracts with On-the-Job Search.” mimeo.
Burdett, Ken, and Melvyn Coles. 2003. “Equilibrium Wage-Tenure Contracts.” Econometrica,
71(5): 1377-1404.
Folland, Gerald B. 1999. Real Analysis: Modern Techniques and Their Applications, New York:John Wiley and Sons.
Helpman, Elhanan, and Jean-Jacques Laﬀont. 1975. “On Moral Hazard in General Equilibrium
Theory.” Journal of Economic Theory, 10: 8-23.
Holmstrom, Bengt. 1983. “Equilibrium Long-Term Labor Contracts.” Quarterly Journal of
Economics, 98: 23-54.
Holmstrom, Bengt, and Paul Milgrom. 1987. “Aggregation and Linearity in the Provision ofIntertemporal Incentives.” Econometrica, 55(2): 303-328.
Manoli, Dayanand, and Yuliy Sannikov. 2005. “Competitive Contracting and Employment
Dynamics.” mimeo.
Menzio, Guido, and Shouyong Shi. 2008. “A Tractable Model of Search On the Job andAggregate Fluctuations.” mimeo.
Milgrom, Paul, and Ilya Segal. 2002. “Envelope Theorems for Arbitrary Choice Sets.” Econometrica, 70(2): 583-601.
Phelan, Christopher. 1995. “Repeated Moral Hazard and One-Sided Commitment.” Journal of
Economic Theory, 66: 488-506.
Prescott, Edward C., and Robert M. Townsend. 1984. “Pareto Optima and Competitive Equilibria with Adverse Selection and Moral Hazard.” Econometrica, 52(1): 21-45.
Rogerson, William P. 1985. “Repeated Moral Hazard.” Econometrica, 53(1): 69-76.
Sannikov, Yuliy. 2008. “A Continuous-Time Version of the Principal-Agent Problem.” Review
of Economic Studies. 75(3): 957-984.
Shi, Shouyong. 2008. “Directed Search for Equilibrium Wage-Tenure Contracts.” forthcoming
in Econometrica.
Shimer, Robert, and Randall Wright. 2004. “Competitive Search Equilibrium with Asymmetric
Information.” mimeo.

27

Spear, Stephen E., and Sanjay Srivastava. 1987. “On Repeated Moral Hazard with Discounting.” Review of Economic Studies, 54: 599-617.
Stokey, Nancy L., and Robert E. Lucas with Edward Prescott. 1989. Recursive Methods in
Economic Dynamics. Cambridge, MA: Harvard University Press.
Thomas, Jonathan P., and Tim Worrall. 2007. “Dynamic Relational Contracts with Consumption Constraints.” Keele Economics Research Papers.

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