Optimal Contracts with Random Auditing

Optimal Contracts with Random Auditing
Andrei Barbos
Department of Economics, University of South Florida, Tampa, FL.
April 16, 2015
Abstract
In this paper we study an optimal contract problem under moral hazard in a principal-agent
framework where contracts are implemented through random auditing. This monitoring instrument reveals the precise action taken by the agent with some nondegenerate probability r,
and otherwise reveals no information. We characterize optimal contracts with random perfect
monitoring under several information structures that allow for moral hazard and adverse selection. We evaluate the e¤ect of the intensity of monitoring, as measured by r, on the value of
the optimal contract. We show that more intense monitoring always increases the value of a
contract when the principal can commit to make payments even if the an evaluation reveals that
the agent took an action not allowed by the terms of the contract. When such commitment is
infeasible and in equilibrium the agent shirks under some realizations of his type, the value of a
contract may decrease in r.
JEL Classi…cation: D82, D86
Keywords: Optimal Contracts, Random Auditing, Commitment.
E-mail : [email protected]; Address: 4202 East Fowler Ave, CMC 206, Tampa, FL 33620-5500; Phone:
813-974-6514; Fax : 813-974-6510; Website: https://sites.google.com/site/andreibarbos/
1
Introduction
Previous literature that examined optimal contract problems under moral hazard considered situations where the principal observes some public signal that is imperfectly correlated with the agent’s
action (usually the e¤ort exerted). This signal is observed with probability 1 and is employed in the
contract design to provide incentives to the agent. In this paper we examine an alternative and,
to the best of our knowledge, novel scenario, where contracts are implemented through random
perfect monitoring. This monitoring instrument reveals the precise action taken by the agent with
some nondegenerate probability r, and otherwise reveals neither the agent’s action nor any signal
correlated with it, such as a measure of output.
Numerous real-world contracting environments can be captured by this speci…cation of monitoring technology. In many situations, it is costly or infeasible for the employer of a large workforce
to assess the contribution of each worker to the aggregate output or to its quality. Instead, the
employer can provide incentives with a monitoring scheme that evaluates the output of randomly
selected workers.1 Another typical example is that of an institution providing a service whose
quality is determined by its agents’actions. In many such situations, the service provider may not
have the capability to obtain feedback from all customers, but only from a sample of them.
The contribution of this paper is twofold. First, we characterize optimal contracts with random
auditing under several standard information structures that allow for moral hazard and adverse
selection. Second, we examine how the intensity of monitoring, as measured by the probability r,
impacts the value of an optimal contract. We show that a higher value of r increases the value of a
contract if the principal can credibly commit to not void the contract when the agent fails an audit
(we say that the agent fails an audit if the audit reveals that he exerted an action not allowed by
the terms of the contract; otherwise, the agent passes the audit). We show then that a higher value
1
For some real-world examples of speci…c …rms employing this type of monitoring, see Rahman (2012).
2
of r may decrease the value of a contract if the principal cannot make this commitment and the
contract induces shirking under some realizations of a state variable.
We investigate this optimal contracting problem in an agency framework built on several modeling choices. First, the cost incurred by the agent from performing his action (the e¤ort level)
depends on a state of nature which only the agent observes. Thus, the model combines features
of moral hazard, determined by the hidden action of the agent, with adverse selection, determined
by the private information the agent possesses about his cost type. Second, the contract is signed
at an ex-ante stage, before the agent learns his type. Contracts with random auditing can also
be characterized with a more typical interim contracting assumption; the ex-ante speci…cation we
adopt allows examining the impact of the principal’s ability to partially insure the agent against
unfavorable draws of his type that induce shirking, through a commitment to not void the contract
when this shirking is detected. Third, in our baseline speci…cation of the model, we assume that
the principal lacks the ability to make such a commitment. Finally, we assume away pre-play
communication and examine separately the optimal contract when communication is feasible.
We characterize the optimal contract in this setting and compare it with two benchmarks, the
…rst-best contract under full information, and the contract under pure moral hazard with no adverse
selection. Under full information, monitoring plays no role and the contract speci…es a constant
wage that perfectly insures the agent. If the agent is risk neutral over monetary transfers (more
precisely, if his utility function, which is everywhere assumed separable in monetary transfers and
cost of e¤ort, is quasilinear in the transfers), then the …rst-best contract can be implemented under
asymmetric information across all model speci…cations. This implies that with a risk neutral agent,
the intensity of monitoring has no impact on the value of the optimal contract. On the other hand,
if the agent is risk averse and there is moral hazard, the principal needs to solve the usual trade-o¤
between incentives and risk. Contracts under moral hazard specify a constant wage to be paid to
the agent when no audit is performed, which we refer to as a salary, and type-or-action-contingent
3
wages for situations when an audit is performed. When the agent’s cost type is also observable
ex-post with an audit, the action-and-type-contingent wage promised if the agent passes the audit
is higher than the salary for some types - these types receive a reward when they pass an audit
- while for the remaining types, it equals their salary. Thus, conditional on exerting e¤ort, the
agent prefers being audited.2 When the type is not observable ex-post, the action-contingent wage
is higher than the salary for some actions allowed by the terms of the contract, but lower than it
for certain allowed but low levels of e¤ort. Incentive provision with a one-dimensional allocation
space in the presence of moral hazard and adverse selection may thus require that the agent be
sometimes penalized relative to his salary even if he passes an audit.
A second objective of the paper is to examine how the intensity of monitoring impacts the value
of a contract and the role played in this context by a credible commitment of the principal to make
payments even when the agent fails an audit. While this type of commitment should be valuable,
i.e., it should result in a weakly higher value of the contract, it is less clear a priori how it a¤ects
the relationship between the intensity of monitoring and the value of the contract.
For all information structures that we consider, if it is optimal to induce the agent to exert e¤ort
under all potential cost types, implying that an audit is never failed, a higher value of r increases
the value of the contract. On the other hand, when an optimal contract induces shirking for some
types, more frequent monitoring can reduce the value of this contract if the principal cannot make
that commitment. This occurs when r is high and thus the agent is likely to be detected when
shirking, requiring the principal to pay a large risk premium ex-ante. If the principal can commit,
he avails of this tool to reduce the dispersion in the set of possible ex-post wages faced by the
agent and thus to lower the risk premium that needs to be paid. In this case, the increased power
of incentive determined by the higher probability of monitoring renders again the value of the
contract be everywhere increasing in r. In many employment situations, the cost of performing
2
Mookherjee and Png (1989) consider a model that exhibits the same preference of the agent for being audited.
4
observationally identical tasks by a worker may depend on circumstances which are not observed
by the employer or cannot be contracted upon.3 It is well known that in such cases, insuring the
risk-averse workers against high-cost realizations may improve the value of the employment contract
if the employer is approximately risk neutral.4 This article suggests that when such insurance is
not feasible, and thus an employee cannot respond to high-cost realizations by adjusting the e¤ort
level exerted or shirking, a high frequency of monitoring may be suboptimal.
As an extension, we also examine optimal contracts with random auditing when pre-play communication is feasible. In such situations, the principal can require the agent to declare his private
information after he learns it, but before it is determined whether an audit is performed or not. This
information can be employed to adjust the wage paid when an audit is not performed. Unlike the
case where communication is not feasible, the agent is never penalized when an audit is performed
provided that he passes it. Instead, when an audit is passed, some agent types receive a reward,
while other types, for which the salary is su¢ cient to provide incentives to exert e¤ort, receive only
their salary. Similarly to the case of pure moral hazard, the additional dimension on which the
contract terms can be speci…ed when communication is feasible allows providing incentives while
limiting the audit risk that the agent is subjected to.
Our paper contributes to the literature on optimal contracts by considering a novel type of
monitoring technology. The random nature of this technology relates it closest to the stream of
literature that studies the design of optimal contracts with costly state veri…cation. The seminal
paper in this literature is Townsend (1979) who examined deterministic state veri…cation in an
optimal insurance contract problem. Baiman and Demski (1980) allowed for a potentially random
acquisition of an additional informative signal of the agent’s action conditional on any particular
3
For instance, the activity of a …rm may require workers to perform tasks which are observationally identical to an
outside party, but which may incur di¤erent costs on the worker depending on the speci…cs of the particular situation
in which that task is performed. In other cases, the task may be identical across di¤erent situations, but the worker’s
ability may vary thus inducing a variance in the cost of performing that task across di¤erent workers. Finally, in
other cases, the cost of performing a particular task at a given time may depend on the physical or mental state of
the worker at that time or on his personal opportunity cost of the time required for ful…lling that task.
4
See Knight (1921) or Kihlstrom and La¤ont (1979).
5
observed outcome. Border and Sobel (1987) and Mookherjee and Png (1989) considered situations
where the state veri…cation is interpreted as an audit of a disclosure made by the agent regarding an
outcome (his income) which is determined in a stochastic manner by an unobservable action chosen
by the agent. The latter article shows that optimal monitoring requires random veri…cation and,
similarly to a …nding from our paper under certain information structures, that the agent should
not be penalized when an audit reveals that he reported thruthfully. Strausz (2005) studied the
strategic e¤ect of the timing of veri…cation in an agency model, distinguishing between veri…cation
performed during and after the agent takes his action.5 Our paper di¤ers from this literature in
that we consider that the audit reveals the action taken by the agent rather than the state.
The framework is introduced in section 2. In section 3 we characterize the optimal contracts with
random auditing, including the two benchmarks corresponding to the cases of complete information
and of pure moral hazard, respectively. This section is also where we examine the impact that the
intensity of monitoring has on the value of a contract and the role of commitment in this context.
In section 4, we study optimal contracts with communication. Section 5 concludes.
2
The Framework
There are two players, a principal (P) and an agent (A). P owns a …rm and o¤ers A a contract to
work for this …rm in exchange for monetary compensation. A can accept or reject the contract. If
A accepts it and exerts e¤ort e in service of the …rm during the period of the contract, he produces
an output whose value is y(e), where y 0 ( ) > 0 and y 00 ( )
0. This output is entirely appropriated
by P. P is risk neutral, and thus his payo¤ when A exerts e¤ort e and is paid a wage w is y (e)
A’s preferences are separable in wages and e¤ort, and are represented by a utility u(w)
w.
c(s; e).
The function u : R ! R captures A’s preferences over net monetary transfers; we assume u0 ( ) > 0
5
More recent contributions to this literature are Ben-Porath, Dekel, and Lipman (2014) and Mylovanov and
Zapechelnyuk (2014) who study optimal allocation problems with state veri…cation when no transfers are allowed.
6
and u00 ( )
0, and normalize u(0) = 0. The cost for A of exerting e¤ort e is c(s; e), where (i ) s
is a random variable that takes values in [s; s]
R, with a continuous density function f ( ) > 0,
and (ii ) c ( ; ) is a function with c (s; 0) = 0, ce > 0, cee > 0, cs > 0 and ces > 0, for all s 2 [s; s]
and e
0. In the following, as standard in the literature, we frequently refer to s as A’s type. A
does not know his type at the time when he is presented with the contract, but upon accepting the
contract, he observes it before choosing the e¤ort level. The utility of A’s outside option is u. The
functions y ( ), u ( ) and c ( ; ) are assumed to be twice continuously di¤erentiable. We also assume
that the set of feasible e¤ort levels is compact or otherwise that y( ) is bounded on R+ .
P does not directly observe the e¤ort e exerted by A or the output y(e).6 Instead, he owns
a monitoring instrument which allows observing e with probability r 2 (0; 1). With probability
1
r, P does not observe either e or any signal correlated with e. We consider the value of r to be
exogenous and public information.7 Monitoring is random and A does not know at the time when
he chooses the e¤ort level whether or not P will observe it. At the end of the contract period, it
is public information whether or not P performed the audit, and the e¤ort level e when an audit
was performed. Given a set of allowed e¤ort levels by a contract, if P acquires evidence through
an audit that A’s e¤ort level is not in this set, i.e., if A fails an audit, then P can void the contract
and no transfers are made. Unless speci…ed otherwise, we assume that P cannot credibly promise
ex-ante not to void the contract in such circumstances.
P can o¤er contracts with wage schedules that are de…ned contingent on all observables.8 More
precisely, P can o¤er a contract of the form
E; wn ; fw(e)ge2E , where (i ) E
R+ is a set of
allowed e¤ort levels, (ii ) wn is the wage paid to A if no audit is performed, and (iii ) w(e) is the
wage paid if an audit is performed and it reveals that A exerted e¤ort level e 2 E. One can think of
6
In line with the motivating examples from Introduction, we assume that P employs a large number of agents, and
that while he may observe an aggregate output, this carries virtually no information of an individual’s contribution.
7
The value of r can be easily endogenized by assuming a cost of monitoring for the principal that depends on r.
Since we do evaluate later the marginal increase in the value of a contract determined by an increase in r, the optimal
value of r would then be determined by setting this equal to the corresponding marginal cost of increasing r.
8
Two assumptions are made here. First, when an audit is performed, P obtains publicly veri…able evidence of A’s
e¤ort. Second, P can credibly promise di¤erent wages depending on whether or not an audit is performed.
7
wn as a salary, or base wage, o¤ered to the worker as long as he is not caught shirking, and of the
di¤erence w(e)
wn as a wage adjustment implemented when an audit is performed and A passes
it. As we show later, under certain information structures, this wage adjustment is nonnegative,
i.e, it constitutes a reward, but under others, it may be negative for low e¤ort levels.9
We complete the presentation of the framework with several observations. First, we note that
the contract de…ned above is designed on contingencies determined strictly by ex-post observable
outcomes. This implicitly assumes away pre-play communication. However, in principle, P could
also o¤er a contract of the type fe (s) ; w (s) ; wn (s)gs2[s;s] , which requires an explicit disclosure of
s after A learns it, but before he is informed whether or not an audit is performed. P would then
employ this message to adjust the wage paid when there is no audit and thus no observable action.
While situations with pre-play communication do frequently emerge in real world, in many other
employment situations, random monitoring is used precisely so as to reduce the administrative
burden. In this case, requiring all workers to disclose their private information, or equivalently to
select a contract out of a menu, may be administratively demanding and infeasible. We therefore
focus the analysis on contracts without communication, and characterize separately in section 4,
as an extension, the optimal contract when communication is feasible.10
Second, we assumed for simplicity that the lower bound is zero on the set of e¤ort levels that
A may exert and yet not be detected to be shirking in the absence of an audit. This modeling
speci…cation can be modi…ed at the cost of adding some slight complications to have a positive
lower bound on this set, and thus to allow capturing more realistic situations where workers cannot
"shirk in plain view".
Finally, we assumed that P can perfectly measure A’s e¤ort with an audit. This can be relaxed
to assume that an audit only reveals a signal correlated with the e¤ort, as in standard moral hazard
9
While not explicitly modeled here, one can think of this game as a stage play of a repeated game and of the wages
de…ned in the contract as promised continuation values to the agent under various contingencies. We are currently
working on a dynamic version of this model where these speci…cations are explicitly modelled.
10
See, for instance, Melumad and Reichelstein (1989) for a discussion on the value of communication in agencies.
8
problems. Thus, our model is a particular case of a generic principal-agent model where P observes
a signal informative of A’s action only with a nondegenerate probability.
3
Analysis
As benchmarks, we derive …rst the optimal contracts under two alternative scenarios to the richer
model introduced above. First, we elicit the e¢ cient outcome in this framework by examining the
case of full information, i.e., with no adverse selection or moral hazard, where both A’s type s
and action e are contractible upon. Second, we consider the case of pure moral hazard, i.e., with
no adverse selection, where A’s type is observable ex-post when an audit is performed and thus
also contractible upon. In both models we maintain our assumption of an ex-ante participation
constraint for A.11 To simplify the exposition, when studying both benchmarks we focus on the
case where it is pro…table for P to induce all types of A to exert e¤ort. We then consider the
general case where this assumption is dropped when studying the full-‡edged model with moral
hazard and adverse selection.
3.1
The Full-Information Benchmark
When P observes ex-post both A’s type and the e¤ort he exerted, monitoring plays no role. P thus
o¤ers a contract fe0 (s); w0 (s)gs2[s;s] , where (i ) e0 (s) is the e¤ort required from type s, and (ii ) w0 (s)
is the wage promised to type s in exchange.12 The only constraint that P faces is A’s participation
11
Since random auditing plays a role only under moral hazard, we forgo discussing the less interesting benchmark
with adverse selection but no moral hazard.
12
It is implicitly assumed here and in the rest of the paper that such a contract is binding for A after he learns his
type s, and thus we do not need to account for A’s participation constraint at an interim stage.
9
constraint, so the optimal contract under full information is the solution to the problem
max
fe(s) 0;w(s) 0gs2[s;s]
s.t.
Z
s
Z
[u (w(s))
s
[y (e(s))
w(s)] f (s)ds
(1)
s
c(s; e(s))] f (s)ds
u
(2)
s
The next proposition, whose proof is straightforward and thus omitted, elicits the conditions
de…ning the corresponding optimal contract when A is risk averse.13
Proposition 1 Assume u00 (w) < 0 for all w. Also, assume that it is optimal for P to induce all
types of A to exert e¤ ort. The optimal contract under full information is determined by w0 (s) = w0
for all s 2 [s; s] and some w0 2 R+ , (2) satis…ed with equality, and
1
ce (s; e0 (s))
u0 (w0 )
= y 0 (e0 (s))
(3)
Since providing A with incentives to exert e¤ort or reveal information is unnecessary, P insures
A and o¤ers a constant wage across all states. The condition in (3) equates the marginal cost and
marginal bene…t for P of implementing e¤ort. An additional amount of e¤ort
s’s cost by ce (s; e(s)) e; to compensate it, P has to increase A’s wage by
e increases type
1
u0 (w) ce (s; e(s))
e. Since
the return for P from A’s additional e¤ort is y 0 (e(s)) e, P sets e0 (s) so as to satisfy (3).
Note that since e0 (s) varies across types (it decreases in s), the utility delivered ex-post to each
type, u(w0 )
c(s; e0 (s)), varies as well (the e¤ect of s on this utility is ambiguous), implying that
some types of A enjoy ex-post more than their reservation utility u, while others less.
Finally, it is straightforward to see that when A is risk neutral, the optimal e¤ort pro…le is deterRs
mined by ce (s; e0 (s)) = y 0 (e0 (s)), and that any wage pro…le fw0 (s)gs2[s;s] satisfying s w0 (s)f (s)ds =
13
By the boundedness of y( ) and the continuity of the relevant functions, an optimal contract exists. Moreover, it
is unique up to a set of zero measure.
10
u+
3.2
Rs
s
c(s; e0 (s))f (s)ds and w0 (s)
0 for all s 2 [s; s] is optimal.
The Pure Moral Hazard Benchmark
When an audit reveals both the e¤ort level exerted by A and his type s,14 P o¤ers a contract
n
o
w1n ; fe1 (s); w1 (s)gs2[s;s] where for each type s, (i ) w1n is the salary, paid if no audit is performed,
(ii ) e1 (s) is the e¤ort required, and (iii ) w1 (s) is the wage paid if an audit reveals that A exerted at
least e¤ort e1 (s). Since ce > 0, type s of A exerts either e¤ort e1 (s) or no e¤ort. To implement e1 (s),
the contract must thus satisfy the incentive compatibility condition ru(w1 (s)) + (1
c (s; e1 (s))
(1
r) u(w1n )
r) u(w1n ), for any s 2 [s; s]. The optimal contract then solves the problem
max
wn 0;fe(s) 0;w(s) 0gs2[s;s]
s.t. ru(w(s))
Z
Z
s
[y (e(s))
rw(s)] f (s)ds
r) wn
(4)
s
c(s; e(s))
0 for all s 2 [s; s]
s
[ru (w(s))
(1
c (s; e (s))] f (s)ds + (1
r) u (wn )
(5)
u
(6)
s
The next proposition, whose proof is in appendix A1, elicits the conditions that determine the
optimal contract when A is risk averse, and the e¤ect of r on the value of this contract.15 To focus
the exposition, we assumed here that the constraint w1n
0 does not bind and then consider the
generic case when this constraint may bind for the model with moral hazard and adverse selection.
Proposition 2 Assume that u00 (w) < 0, for all w. Also, assume that it is optimal for P to induce
all types of A to exert e¤ ort. The optimal contract under pure moral hazard is determined by (5),
14
An alternative speci…cation of a model with pure moral hazard is one where the type s is observable ex-post
even without an audit, i.e., with probability 1. In line with the discussion from section 2, we focus in this paper on
modeling situations where it is infeasible for P to acquire on a regular basis information about all employees, be that
their e¤ort level or their cost type. We therefore chose the modeling speci…cation as de…ned above.
15
Given proposition 2, the optimal contract can be computed in principle as follows. First, (8) and a binding (5)
determine the pairs (w1 (s); e1 (s)) for any s in the set on which w1 (s) > w1n ; clearly, this set depends on w1n . On the
other hand, (8) determines e1 (s) for s with w1 (s) = w1n also as a function of w1n . Substituting these into the binding
constraint from (6) determines w1n , and then the rest of the contract.
11
(6) satis…ed with equality, and for all s 2 [s; s],
w1 (s)
w1n
0, and = 0 whenever ru(w1 (s))
c(s; e1 (s)) > 0
1
ce (s; e1 (s)) = y 0 (e1 (s))
u0 (w1 (s))
(7)
(8)
The value of the optimal contract is increasing in r.
The participation constraint in (6) is satis…ed with equality; otherwise P can reduce the salary
w1n without a¤ecting (5). (8) equates again, for each type s, the marginal bene…t and marginal
cost for P of implementing additional e¤ort. To understand (7), note that since A is risk averse,
P aims to minimize the wage risk imposed on A, subject to providing the right incentives. If,
contrary to (7), w1 (s) < w1n , then
1
u0 (w1 (s))
<
1
u0 (w1n )
and thus the cost for P of delivering additional
utility to A is lower when done by means of increasing w1 (s) than by that of w1n ; therefore, P can
increase w1 (s) and decrease w1n so that A’s participation constraint in (6) continues to be satis…ed
but with a lower expected wage paid. On the other hand, when w1 (s) is set higher than w1n , the
corresponding risk is imposed on A so as to create incentives to exert e¤ort; this is again done with
a minimum variance in wages, and therefore the incentive constraint in (5) binds, as stated by (7).
As proposition 2 suggests, the optimal contract under pure moral hazard essentially speci…es a
salary w1n to be paid to all types s as long as A is not caught shirking, and a reward w1 (s)
w1n > 0
o¤ered to some types when they pass an audit. As we show in appendix A1, the e¤ort pro…le e1 (s) is
decreasing, the wage pro…le is generically non-monotonic, while the revenue generated by di¤erent
types, y (e1 (s))
rw1 (s)
(1
r) w1n , is decreasing in s. In terms of ex-post experienced utility,
types s with w1 (s) > w1n enjoy less than their reservation utility, while some of the remaining types
enjoy more.16
16
To see this, note that the expected utility delivered to type s, i.e., ru (w1 (s)) + (1 r) u (w1n ) c (s; e1 (s)) equals
(1 r) u (w1n ) for s with w1 (s) > w1n and (5) binding, and is higher than (1 r) u (w1n ) for the rest. Since on average
the utility experienced by A is u, it must be that (1 r) u (w1n ) < u.
12
The following proposition considers the case when A is risk neutral. The result follows imn
o
n
mediately from the fact that there exists a wage pro…le w1 ; fw1 (s)gs2[s;s] that implements the
full-information e¤ort pro…le fe0 (s)gs2[s;s] , while delivering A the same ex-ante expected wage as
under full information.17 Therefore, if A is risk neutral, P can attain the same payo¤ as under full
information. This payo¤ is independent of the intensity of monitoring.
Proposition 3 Assume that u(w) = w, for all w. Then e1 (s) = e0 (s), for all s 2 [s; s]. Moreover,
the value of the optimal contract equals that under full information.
3.3
The Model with Moral Hazard and Adverse Selection
It is straightforward to see that whenever it is optimal for P to induce some particular type to exert
e¤ort, it must be optimal to also do so for all lower cost types. We consider therefore contracts
where P induces types s 2 [s; sb] to exert e¤ort, with the threshold sb 2 [s; s] optimally chosen by
P. To simplify the exposition, we implicitly assume in most of the ensuing analysis a set of model
parameters such that in the corresponding optimal contract, the salary wn is nonnegative. We
then present in proposition 15 the conditions de…ning the optimal contract when we allow for the
nonegativity constraint on wn to potentially bind.18
By the Revelation Principle, one can think of P’s problem as that of selectn
o
ing an optimal contract sb 2 [s; s]; wn ; fe (s); w (s)gs2[s;bs] that extracts A’s private information
Principal’s Problem
from types in the set [s; sb], induces each type s 2 [s; sb] to exert e¤ort e (s), and the types in (b
s; s]
17
For instance, the wage schedule de…ned by w1 (s) = r1 c(s; e0 (s)) for all s, and w1n = 1 u r satis…es (5) and (6),
Rs
Rs
and thus implements fe0 (s)gs2[s;s] . Moreover, s [rw1 (s) + (1 r) w1n ] f (s)ds = u + s [c(s; e0 (s))] f (s)ds. Thus A’s
expected wage equals that under full information implying that this contract is optimal since its value attains the
theoretical upper bound, the value under full information.
18
Note here that (11) from P’s problem ensures that in any incentive compatible contract, w(s) > 0 for all s 2 [s; sb].
13
to shirk. The optimal contract is thus the solution to the problem
max
sb2[s;s];wn
0;fe(s) 0;w(s) 0gs2[s;b
s]
Z
s
sb
[y (e(s))
s.t. s 2 arg max [ru (w(e
s))
se2[s;b
s]
ru (w(s))
c (s; e (s))
max [ru (w(e
s))
Z
s
se2[s;b
s]
sb
[ru (w(s))
rw(s)] f (s)ds
r) wn
(1
c (s; e (e
s))] , for all s 2 [s; sb]
0, for all s 2 [s; sb]
c (s; e (e
s))]
0, for all s 2 (b
s; s]
c (s; e (s))] f (s)ds + (1
r) u (wn )
(9)
(10)
(11)
(12)
u.
(13)
where (10) is the incentive compatibility condition that induces types in [s; sb] to truthfully reveal
themselves, while (12) induces types in (b
s; s] to shirk rather than exert an e¤ort level speci…ed for
one of the types in [s; sb]. While (10) is a somewhat standard incentive compatibility condition under
adverse selection, the speci…c forms of (11) and (12) di¤er from other types of incentive compatibility
constraints under moral hazard from the literature and are determined by the particular type of
monitoring technology (random perfect monitoring) that we examine here.
The following lemma implies that we can replace (11) and (12) with the weaker condition from
(14) in the above problem.
Lemma 4 Any contract that satis…es (10), will satisfy (11) and (12) if and only if
ru (w(b
s))
c (b
s; e (b
s))
0, and = 0 whenever sb < s.
(14)
Proof. Consider …rst the case when sb < s. We assume throughout that (10) is satis…ed and start by
showing that then (14) implies (11) and (12). Note …rst that ru (w(s))
c (s; e (b
s))
ru (w(b
s))
the second by s
c (s; e (s))
ru (w(b
s))
c (b
s; e (b
s)) for all s 2 [s; sb], where the …rst inequality is implied by (10) and
sb. Therefore, (14) implies (11). Next, since whenever s
14
sb, we have ru (w(e
s))
c (s; e (e
s))
ru (w(e
s))
s))
max ru (w(e
s]
se2[s;b
s))
c (b
s; e (e
s)) for any se 2 [s; sb], it follows that max ru (w(e
c (b
s; e (e
s)) = ru (w(b
s))
c (s; e (e
s))
s]
se2[s;b
c (b
s; e (b
s)), where the equality follows from (10). Thus,
(14) implies (12). For the converse, note that (11) immediately implies ru (w(b
s))
Assuming by contradiction that ru (w(b
s))
c (b
s; e (b
s))
0.
c (b
s; e (b
s)) > 0, by the continuity of c ( ) in s, there
exists " > 0 such that for all s 2 (b
s; sb + "), we have ru (w(b
s))
c (s; e (b
s)) > 0, which contradicts
(12). When sb = s, then (12) is automatically satis…ed, while from the above argument, it is clear
that (11) is satis…ed if and only if ru (w(b
s))
c (b
s; e (b
s))
0.
To solve for the optimal contract, we employ the standard First-Order Approach. Lemma 5
validates this approach in the current framework by showing the equivalence between the incentive
compatibility of a contract with respect to truthful type revelation, on the one hand, and the …rst
order condition of A’s problem in (10), plus the monotonicity of the e¤ort pro…le e(s), on the other.
Its proof, which builds on a standard strategy in the literature, is presented in appendix A2.
Lemma 5 A contract induces truthful type revelation for all s 2 [s; sb] if and only if
e0 (s)
0 a.e. s 2 [s; sb]
ru0 (w(s)) w0 (s) = ce (s; e(s))e0 (s) a.e. s 2 [s; sb]
(15)
(16)
To keep the exposition in the main text simple we will ignore the monotonicity constraint in
(15) and solve the relaxed problem, as de…ned by (9), (13), (14) and (16). We present the analysis
of the optimal contract problem for the case when the monotonicity constraint from (15) binds in
appendix A6. Note also at this point that (15) and (16) imply that in any incentive compatible
contract w0 (s)
0 (and also that w0 (s) < 0 whenever e0 (s) < 0).
Finally, given our assumption that wn
0 does not bind, the participation constraint in (13)
must bind at optimum since otherwise wn can be reduced to increase the value of the contract. We
consider therefore in the following that (13) is satis…ed with equality.
15
Optimal Control Approach We consider …rst the case where A is risk averse. To solve P’s
problem we employ methods from optimal control theory. We …rst recast the problem in terms
of induced utilities un
u (wn ) and u(s)
u (w (s)), for s 2 [s; sb]; these utilities will replace
the respective contingent wages as P’s choice variables. By denoting the inverse utility function
h
u
1,
de…ned on the range of the function u, we have that wn = h(un ) and w (s) = h (u (s)).19
s)
Under these transformations, (10) becomes s 2 arg max [ru (e
c (s; e (e
s))], and so, under the First
s]
se2[s;b
Order Approach, the incentive compatibility condition in (16) is ru0 (s) = ce (s; e (s)) e0 (s). Given
this, the control variable in the optimal control problem is x(s)
e0 (s), while the state variables
are e(s) and u(s). In addition, to account for the participation condition in (13), we introduce a
new state variable
v (s)
Z
s
[ru ( )
c ( ; e ( ))] f ( )d
(17)
s
and rewrite the binding constraint in (13) as the transversality condition v (b
s) = un
u (1
r) un .
The other transversality condition on v is v (s) = 0. There are no transversality conditions on the
two remaining state variables, e and u.
We solve for the optimal contract in two steps. First, for any …xed value of un , we solve an
optimal control problem where the decision variables are sb and fx(s)gs2[s;bs] . In the second step, we
maximize the resulting optimal value function with respect to un , as a standard static optimization
19
This transformation requires the additional assumption that for every e¤ort level there exists a wage such that
the participation constraint of the agent is satis…ed (see Bolton and Dewatripont (2005) pp. 154.)
16
problem. The optimal control problem in the …rst step is
max
sb2[s;s];fx(s)gs2[s;b
s]
Z
s
sb
[y (e(s))
rh (u(s))] f (s)ds
s.t. e0 (s) = x(s)
(19)
1
u0 (s) = ce (s; e (s)) x (s)
r
(20)
v 0 (s) = [ru (s)
(21)
c (s; e (s))] f (s)
v (s) = 0; v (b
s) = un
ru (b
s)
(18)
c (b
s; e (b
s))
0 and = 0 if sb < s
(22)
(23)
Current existence theorems for solutions of optimal control problems do not yield the complete
set of properties of the solution to the above problem required in the ensuing analysis. We therefore
make assumption 7 presented below in the following. Part (i ) of the assumption can alternatively be
derived as an implication of some su¢ cient boundness conditions on cee and ces .20 Part (ii ) ensures
that the solution for the optimal cuto¤ sb is determined by the standard in the literature (equality)
condition from (33). Part (iii ) ensures that un is determined optimally by a standard …rst-order
condition and that the Dynamic Envelope Theorem has the standard form. The superscript sb elicits
the fact that the respective trajectory is the solution corresponding to a given cuto¤ sb.
(1)
De…nition 6 We say a function is Cp if it is continuous and piecewise continuously di¤ erentiable.
Assumption 7 (Existence and Smoothness) (i) For any …xed sb, there exists a solution to
20
The two Filipov-Cesari type existence theorems that could potentially be applied to a situation where the Hamiltonian is linear in the control variable and the control has an unbounded support are presented in section 11.C on
page 392 in Cesari (1983). Theorem 11.4.vii does not apply as stated since none of the growth conditions are satis…ed.
However, these growth conditions are employed in the proof of the theorem to conclude that the value function of the
corresponding optimization problem is bounded. In our case, for any …xed value of sb, the boundedness follows from
that fact that the value is lower than that of the relaxed problem where conditions (19), (20) and (23) are dropped
and r = 1, i.e., by the value of the contract under full information, which is …nite. The theorem can then be applied
under the additional assumptions that cee and ces are bounded, which are used to infer the required properties on
what the theorem in Cesari (1983) denotes by A0 (t; x) and B(t; x).
17
(18)-(23) with the corresponding state variables esb(s); usb(s)
(1)
s2[s;b
s]
being Cp
functions of s. (ii)
(1)
The functions sb ! esb(s) and sb ! usb(s) are Cp for all s 2 [s; s]. (iii) The optimal value of problem
(18)-(23) is continuously di¤ erentiable as a function of un and of r.
The Hamiltonian associated with the problem (18)-(23) is
H (e; u; v; x;
1;
2;
3 ; s)
[y (e)
rh (u)] f (s) +
1x +
2
1
ce (s; e) x +
r
3 [ru
c (s; e)] f (s) (24)
Since this Hamiltonian is linear in the control variable x, while the domain of x is unbounded,21 a
solution to this problem necessarily involves a so-called singular control, i.e., it must satisfy
(1)
for all s.22 By the Pontryagin’s Maximum Principle,23 there exist Cp
3 (s),
functions
1 (s),
@H
@x
=0
2 (s)
and
and a scalar , such that the following conditions are necessarily satis…ed at the optimum
21
Recall that we are solving the relaxed problem where we drop the monotonicity condition in (15) and thus the
domain is R. If instead we incorporate that condition, the solution may involve a so-called bang-singular-bang control,
= 0 when x(s) < 0, and @H
> 0 when x(s) = 0. See appendix A5 for the details.
with @H
@x
@x
22
See, for instance, page 247 in Bryson and Ho (1975) for a discussion of singular controls on unbounded domains.
In regards to that discussion, note that since in our problem there are no initial or terminal conditions on the state
variables e and u, which are those a¤ected by the control x, the optimal control will not require Dirac function
impulses at s or s meant to generate jumps to the singular solution.
23
See Theorem 4.2 on page 81 in Caputo (2005) for a more standard version of this result, or Theorem 1 on page
178 in Seierstad and Sydsaeter (1987) for a version that accounts for the state constraint at sb in (23).
18
solution of problem (18)-(23).24
@H
@x
=
1 (s)
+
1
ce (s; e (s)) = 0
r
1
y 0 (e) f (s)
2 (s) cee (s; e (s)) x (s) +
r
2 (s)
@H
=
@e
@H
= rh0 (u (s))f (s)
@u
@H
=0
@v
@
0; 1 (b
s) =
[ru (b
s)
@e (b
s)
@
0; 2 (b
s) =
[ru (b
s)
@u (b
s)
0
1 (s)
=
0
2 (s)
=
0
3 (s)
=
1 (s)
=
2 (s)
=
3 (s)
2 R;
s)
3 (b
0, with
(25)
3 (s) ce (s; e (s)) f (s)
3 (s) rf (s)
(26)
(27)
(28)
c (b
s; e (b
s))] =
ce (b
s; e (b
s))
(29)
(30)
c (b
s; e (b
s))] = r
2R
(31)
= 0 and sb = s if ru (b
s)
c (b
s; e (b
s)) > 0
(32)
In addition, since sb is a choice variable, we have the condition
H (e (b
s) ; u (b
s) ; v (b
s) ; x (b
s) ;
s) ;
1 (b
s) ;
2 (b
s) ; sb)
3 (b
0, and = 0 if sb < s
(33)
which is the standard necessary condition for free end-time optimal control problems.25 Condition
(25) equates the marginal cost,
1 (s),
and marginal bene…t,
1
2 (s) r ce (s; e (s)),
for P of decreasing
the level of e¤ort required from type s.26
There also exists a second-order necessary condition, which in the case of a singular control
takes the form of the so-called generalized Legendre-Clebsch condition.27 As we show in appendix
A4, in our problem, this condition is satis…ed if, for instance, cees
24
0 along the trajectory of the
Note that (29) is redundant given (25) and (30), which is why it is not used when deriving the optimal contract.
See, for instance, Theorem 10.2 on page 266 in Caputo (2005). The interpretation of (33) follows from the fact
that the value of the Hamiltonian at s captures the total value to P (or virtual surplus) generated by type s.
26
See page 89 Caputo (2005) for an interpretation of the costate variables in dynamic optimization problems. Note
that in our case, 1 (s) < 0, as that costate variable captures the bene…t of decreasing the state variable e(s), rather
than increasing it, since e0 (s) < 0. On the other hand, 2 (s) > 0, as it captures the bene…t of decreasing u(s).
27
Also refered to as the Kelley condition; see, for instance, page 246 in Bryson and Ho (1975).
25
19
solution to (25)-(33).28 This additional assumption on c ( ; ) is only su¢ cient, not necessary for
the generalized Legendre-Clebsch condition to be satis…ed.
Lemma 8 states the su¢ ciency of conditions in (25)-(33) for the problem (18)-(23), and the
uniqueness of the corresponding solution.
o
n
Lemma 8 (Su¢ ciency and Uniqueness) If sb ; fe (s); u (s); v (s); x (s)gs2[s;bs] satis…es (25)(33) with costate variables f
Proof of lemma 8.
1
(s);
2
(s);
3
(s)gs2[s;bs] , then it is the unique solution to (18)-(23).
We employ the Arrow Su¢ ciency Theorem (see, Theorem 3.4 on page 60
in Caputo (2005)) assuming …rst that sb is not a choice variable, but …xed. In our case, the
o
n
maximized Hamiltonian evaluated at the costate functions f 1 (s); 2 (s); 3 (s)gs2[s;bs] equals
[y (e)
rh (u)] f (s) +
3
(s) [ru
c (s; e)] f (s) by (25). Since, as we show later,
3
(s) > 0, this
maximized Hamiltonian is concave in (e; u; v) and strictly concave in (e; u) by the assumptions
imposed on y ( ) and c ( ; ) in section 2. The Arrow Su¢ ciency Theorem implies that the necessary
conditions are su¢ cient and the uniqueness of the state variables in the solution.29 The theorem
does not state the uniqueness of the control, but since x (s) = e0 (s), the uniqueness of the control
follows immediately as well. To account for the fact that sb is in fact a choice variable, one can then
employ a result from Seierstad (1984) to conclude the claim of the lemma. Since the corresponding
details are slightly more technical, they are deferred to appendix A6.
To complete the derivation of the necessary conditions for the problem in (9)-(13), we denote
by V(un ) the value function of the optimal control problem in (18)-(23), as a function of un . The
28
This condition implies that the marginal cost of e¤ort ce ( ) increases faster in e¤ort for higher s. It is immediately
satis…ed if, for instance, c(s; e) = c1 (s) c2 (e), with c1 increasing and c2 increasing and convex.
29
The Arrow Su¢ ciency Theorem, as stated in Caputo (2005), requires strict concavity of the maximized Hamiltonian in (e; u; v) for the uniqueness of the solution. However, by following its proof, it is evident that if the maximized
Hamiltonian is strictly concave in (e; u) and constant in v, as in our case, then fe (s); u (s)gs2[s;bs] must be unique.
The uniqueness of the remaining state variable fv (s)gs2[s;bs] follows then from its de…nition in (17).
20
necessary …rst order condition for the choice variable un is then
d
[V(un )
dun
Properties of the Optimal Contract
through un , i.e.,
and thus
3 (s)
dV(un )
dun
dV(un )
dun
= (1
= h0 (un ) =
=
r)
1
u0 (wn )
@V @un
@un @un .
s).
3 (b
(1
r)h(un )] = 0
(34)
Note …rst that un a¤ects the value of the contract only
By the Dynamic Envelope Theorem30 we have
Since (28) implies that
3(
@V
@un
=
s),
3 (b
) is constant, it follows from (34) that
> 0, for all s 2 [s; sb]. Employing this result and (25) into the de…nition
of the Hamiltonian H , we combine (33) with the requirement that sb = s if ru (b
s)
c (b
s; e (b
s)) > 0
from (32) to conclude the following result.
Lemma 9 An optimal contract must satisfy
y (e (b
s))
rw (b
s)
1
u0 (wn )
[ru (w (b
s))
c (b
s; e (b
s))] , and = 0 if sb < s
(35)
Intuitively, there are two e¤ects of P implementing positive e¤ort for a type s. First, it generates
an additional marginal revenue to P, y (e (s)) rw (s)). Second, it delivers an additional net utility
to A from an ex-ante point of view, ru (w (s)))
c (s; e (s)), and allows reducing the salary wn .
While the …rst e¤ect can be negative in an optimal contract, condition (35) requires that the sum
of these two e¤ects at sb be always non-negative. On the other hand, when sb < s, since the utility
ru (w (b
s))
c (b
s; e (b
s)) must be zero by (14) for incentive compatibility reasons, the marginal
revenue generated by type sb must be zero as well (otherwise sb would be increased).
The following lemma, whose proof is in appendix A3, identi…es a relationship between wn and
the wage pro…le fw (s)gs2[s;bs] in an optimal contract, representing the counterpart of (7) here. The
R sb
1
lemma follows from (32) and the fact proved in appendix that = s u0 (w1 (s)) f (s)ds u0 (w
n) .
30
See, for instance, Theorem 9.1 on page 232 in Caputo (2005).
21
Lemma 10 An optimal contract must satisfy
Z
s
sb
1
f (s)ds
u0 (w (s))
When ru (w(b
s))
1
u0 (wn )
0, and = 0 if ru (w(b
s))
c (b
s; e (b
s)) > 0
(36)
c (b
s; e (b
s)) > 0 (which by (14) can occur only when sb = s), at optimum, P can
equalize the marginal utility delivered to A through an increase of wn by a small amount with the
expected increase in utility that could be delivered to A by increasing each value w (s), for s 2 [s; sb],
R sb
1
s)) c (b
s; e (b
s)) = 0
by the same amount. Therefore s u0 (w1 (s)) f (s)ds u0 (w
n ) = 0. When ru (w(b
(which generically occurs when sb < s), P may not be able to perfectly equalize these inverse
marginal utilities. More precisely, he may not be able to set the wage pro…le fw (s)gs2[s;bs] low
enough without inducing A to shirk for at least some types in [s; sb]. To preempt shirking, P
keeps the wages w (s) high enough and lowers wn below the level that would equalize the inverse
R sb
1
marginal utilities; therefore s u0 (w1 (s)) f (s)ds u0 (w
n ) , as elicited by (36), with the inequality being
generically strict.
An implication of lemma 10 is that unlike the case from section 3.2, where the audit also revealed
A’s type, in the case with adverse selection studied here, the wage w (s) may be lower than the
salary wn for some types, i.e., A may be penalized when evaluated even if he exerted the level of
e¤ort required for his type. This is necessary as if P were to increase w (s) whenever w (s) < wn
(and simultaneously reduce wn to keep A’s participation constraint binding), aiming to reduce the
risk to A, in order to preserve the incentives for truthful type revelation, he would also need to
increase the remaining contingent wages, including those higher than wn . On net, this may subject
A to more risk, thus rendering an increase of w (s) suboptimal.
Lemma 11, whose proof is in appendix A4, determines the e¤ort level e (s) for each type s as
n
o
a function of the wage pro…le wn ; fw (s)gs2[s;bs] . Remark 12 is also proved in appendix A4.
22
Lemma 11 An optimal contract must satisfy for all s 2 [s; sb]
ce (s; e (s))
f (s) + ces (s; e (s))
u0 (w (s))
Remark 12 We have
Z
s
sh
1
u0 (w
Z
s
1
( ))
s
u0 (wn )
1
u0 (w ( ))
i
1
f ( )d = y 0 (e (s))f (s)
u0 (wn )
(37)
f ( )d > 0 for all s 2 (s; sb).
While the moral hazard in the model induces a departure from the e¢ cient outcome by requiring
a wage pro…le that subjects A to risk, the adverse selection induces ine¢ ciency in the choice of
n
o
the e¤ort level. For the given wage pro…le wn ; fw (s)gs2[s;bs] , the e¤ort level e (s) elicited by
equation (37) maximizes type s’s "virtual surplus" for contracting situations with random auditing,
i
Rsh
1
1
31 Unlike the case of pure moral
y(e)f (s) u0c(s;e)
f
(s)
c
(s;
e)
0
0
n
s
s u (w ( ))
(w (s))
u (w ) f ( )d .
hazard studied in section 3.2, P cannot implement the e¤ort that maximizes the "social surplus"
y(e)
c(s;e)
u0 (w (s)) .
If he did, some of the types in [s; s), for which their own marginal cost of e¤ort
is lower than that of type s, would choose it in place of their prescribed e¤ort levels. Instead,
P implements the lower 32 level of e¤ort e (s) which solves (37) and essentially in a discrete-type
version of the model would make the type just below s indi¤erent between his prescribed e¤ort level
and e (s) while all other types in [s; s) strictly prefer their prescribed e¤ort levels. The magnitude
Z sh
i
1
1
of this downward distortion is a¤ected by the positive factor
0
0
n
u (w ( ))
u (w ) f ( )d . This
s
distorting factor, which for any s is a measure of the cumulative utility gains delivered by the wage
adjustments awarded to types in [s; s) when passing an audit, and thus of the information rent that
type s is paid, is increasing in s on [s; ), where
solves w ( ) = wn , i.e., as long as types receive a
reward when they are audited, and is decreasing on ( ; sb]. The maximum distortion is applied by
this factor to the type
whose wage when passing an audit equals the salary wn .
31
Unlike many other agency models, where both players’utilities are quasilinear in transfers and thus these transfers
vanish from the expression of the virtual surplus, what we refer to here slightly improperly as the virtual surplus also
depends on wages. The same observation is valid for the expression which we refer to as the social surplus later on.
32
This follows from ces > 0, cee > 0, y 00 < 0 and the result of remark 12.
23
Note that at s, (37) becomes
ce (s;e (s))
u0 (w (s))
= y 0 (e (s)), implying the familiar no distortion at the
top property; when setting the optimal e¤ort and wage for type s, P does not need to account
for potential deviations from lower-cost types and can implement the e¢ cient e¤ort level for s.
Moreover, when sb = s and ru (w(s))
ce (s;e (s))
u0 (w (s))
c (s; e (s)) > 0, employing (36) into (37) it follows that
= y 0 (e (s)). In this case, and unlike most other contracting situations studied in the
literature,33 the optimal contract with random perfect monitoring also exhibits no distortion at the
bottom. Note that for types on ( ; sb], the distorting factor is decreasing in s, as over this range of
types, the wage when passing an audit is lower than the salary. When sb = s, at s the distorting
factor is zero.
Proposition 13 collects our …ndings and presents the necessary and su¢ cient conditions that
elicit the optimal contract in this model when A is risk averse.34
Proposition 13 Assume that u00 (w) < 0, for all w. The solution for the optimal contract under
moral hazard and adverse selection is given by (13) satis…ed with equality, (14), (16), and (35)-(37).
It deserves mentioning here that the constraint ru (w(b
s))
c (b
s; e (b
s))
0 from (14) does not
necessarily bind in an optimal contract when sb = s. This may occur if, for instance, (i ) u is
n
o
su¢ ciently high, requiring a high wage pro…le wn ; fw (s)gs2[s;bs] , (ii ) the marginal output y 0 is
low for high levels of e¤ort implying that P optimally chooses to implement low levels of e¤ort,
and (iii ) the marginal cost of e¤ort ce is low for low levels of e¤ort, implying that a wage pro…le
33
An exception are some models with multidimensional screening; see, for instance, Rochet and Stole (2002).
Given proposition 13, the optimal contract can be computed in principle as follows. When sb < s, then ru (w (b
s))
c (b
s; e (b
s)) = 0 by (35), and n
thus (36) is satis…ed
o with inequality. Then (37) determines implicitly e (s) for each
34
s 2 [s; sb], as a function of
wn ; fw (s)gs2[s;bs] . This also gives e0 (s) as a function of the same wage pro…le.
Then, fw (s)gs2[s;bs] is the solution of the di¤erential equation de…ned by (16), with initial conditions given by (13),
ru (w (b
s)) c (b
s; e (b
s)) = 0 and y (e (b
s)) rw (b
s) (we need three conditions because there are also the two unknowns
wn and sb). When ru (w (b
s)) c (b
s; e (b
s)) > 0, then sb = s, while from the fact that (36) is satis…ed with equality,
wn is determined as a function of fw (s)gs2[s;bs] . fe (s)gs2[s;bs] and fw (s)gs2[s;bs] are derived then as above. While
this shows that the optimal contract can in principle be computed with the conditions identi…ed in proposition 13,
a practical numerical implementation would involve constructing a system of di¤erential equations in e(s) and 2 (s)
and their derivatives, with the two equations obtained from the second derivative of the equality in (25) with respect
@H
d2
to s (see the computation of ds
in appendix A4) and (27), and two initial conditions on 2 (s) given by (30).
2
@x
24
chosen so as satisfy A’s participation constraint and to minimize his wage risk35 is su¢ cient to
also incentivize A to exert e¤ort. Essentially, when P needs A for a job that requires low and
inexpensive e¤ort, he o¤ers a contract that satis…es A’s participation constraint with a smooth
wage pro…le across all contingencies, which also provides A with su¢ cient incentives to exert e¤ort.
The following corollary states the intuitive fact that the revenue generated by di¤erent types of
agents is decreasing in the value of the type. Its proof is in appendix A6.
Corollary 14 The revenue generated by type s, y (e (s))
rw (s)
(1
r) wn , is decreasing in s.
The next proposition considers the general case where the constraint wn
Proposition 15 Consider the case where the constraint wn
0 may bind.36
0 may bind. The solution for the
optimal contract under moral hazard and adverse selection is then given by (13), (14), (16), (35)(37) with
3
replacing
3
1
u0 (wn ) ,
and the following additional conditions
0, and = 0 if
Z
s
1
3
u0 (wn )
Proof. By (28),
wn
3 (s)
sb
[ru (w (s))
c (s; e (s))] f (s)ds > u and wn = 0
(38)
0, and = 0 if wn > 0
(39)
is again a constant, which we denote by
3.
When we allow for the constraint
0 to potentially bind, the participation constraint in (13) may not necessarily bind in the
un , and therefore in (31),
optimal contract. Thus, in (22), we have v (b
s)
0, and
3
3
=0
whenever v (b
s) > un . Since v (b
s) > un can optimally occur in a solution only when the constraint
un
0 binds, this immediately implies (38). Next, (34) becomes
35
d
dun
[V(un )
(1
r)h(un )]
0
This implies that the utility u is delivered to A not only through a high value of wn , but also through high values
for the wages fw (s)gs2[s;bs] so to reduce the discrepancy between the wages with an audit and the salary wn .
36
Proposition 15 suggests two additional
potential cases when determining the optimal contract besides that conR sb
sidered in proposition 13. First, if s [ru (w (s)) c (s; e (s))] f (s)ds = u and wn = 0, then 3 is undetermined, but
R sb
wn = 0 provides the additional condition for computing the contract. Second, if s [ru (w (s)) c (s; e (s))] f (s)ds >
u and wn = 0, then (13) is no longer satis…ed with equality, but 3 = 0 and wn = 0 provide the additional conditions.
25
and = 0 whenever un > 0. Since, as argued earlier,
dV(un )
dun
= (1
r)
3,
this implies (39). Following
the analyis from the appendix leading to (35), (36) and (37), one concludes that these conditions
continue to be necessary for the optimal contract, only that
3
substitutes
1
u0 (wn )
everywhere.
We close the section with proposition 16, whose proof is in appendix A7, which states that if A
is risk neutral over monetary transfers, then the full-information payo¤ is attainable by P whenever
A’s reservation utility u is su¢ ciently high. We restrict attention again to the simpler case where
in a full-information setting, it is optimal for P to implement positive e¤ort for all types.
Proposition 16 Assume that u(w) = w, for all w, and that e0 (s) > 0, for all s 2 [s; s]. Then,
there exists U > 0 such that for all u
U , we have e (s) = e0 (s), for all s 2 [s; s], and the value of
the optimal contract under moral hazard and adverse selection equals that under full information.
Essentially, with a risk neutral agent, P can set a wage pro…le that provides A with incentives
to both reveal his type and exert e¤ort without a need to compensate him for the induced wage
risk. An immediate consequence of this result is that the value of the optimal contract is again
constant in the intensity of monitoring if A is risk neutral. The requirement that u be su¢ ciently
high ensures that salary wn is nonnegative in this contract. With limited liability of the agent, as
we argue in appendix A7, if u is low and thus the constraint wn
0 binds, P chooses to implement
a lower or equal e¤ort pro…le than the e¢ cient one.
The E¤ect of r on The Value of the Optimal Contract Denote by V (r) the value of the
(1)
optimal contract as a function of the probability of monitoring r. By assumption 7, V (r) is Cp .
The following lemma, whose proof is in appendix A8, elicits the e¤ect of r on this value.
Lemma 17 For all r 2 (0; 1), we have
0
V (r) =
Z
s
sb
u(w (s))
u0 (w (s))
w (s) f (s)ds
26
u(wn )
u0 (wn )
wn
(40)
To understand this result, note …rst that since sb and the e¤ort pro…le fe (s)gs2[s;bs] are chosen
optimally, when r increases slightly, the value of the contract is impacted only through the change
in the expected payment to A. There are two e¤ects of an increase in r on this expected payment.
First, a higher r increases the likelihood of monitoring; this e¤ect is captured in (40) by the term
R sb
n
s w (s)f (s)ds + w . Second, the increase in r induces an adjustment in wages, as it relaxes A’s
incentive constraints allowing P to reduce the wage risk by lowering w (s); to compensate for this
and continue satisfying A’s participation constraint, P must increase wn . These wage adjustments
are captured by the remaining terms in (40).37
The next proposition, whose proof is in appendix A9, states that when it is optimal for P to
implement positive e¤ort for all types, the value of the contract is increasing in r.
Proposition 18 Assume that u00 (w) < 0 for all w. For any r 2 (0; 1), if sb = s, then V 0 (r) > 0.
When r increases, P is less reliant on the power of incentives and o¤ers a smoother wage pro…le
that exposes A to less risk, thus reducing the risk premium P needs to pay and increasing the value
of the contract.
As a side note here, it deserves mentioning that while we proved the result of proposition 18
under the underlying speci…cation of an ex-ante participation constraint of the agent, it can also be
shown that it holds when the agent evaluates the contract only after he learns his type, and thus
the optimal contract problem has a more standard interim participation constraint for each type.
The next proposition, which is one of the main results of the paper, states that if the optimal
contract does not induce e¤ort for all types, then the value of the contract may be decreasing in
37
Since sb and fe (s)gs2[s;bs] do not change, ru(w(s)) stays constant as r increases slightly. In particular, if r
increases by some small percentage ", u(w(s)) must
h decrease by the
i same percentage ", for each s 2 [s; sb]. Therefore,
w(s) decreases by an amount
w(s) satisfying
d
dw(s)
ln u(w(s))
w(s) = ", i.e.,
w(s) =
n
u(w(s))
".
u0 (w(s))
To continue
u(wn )
".
u0 (wn )
The
satisfying A’s binding participation constraint, it follows by the same logic that w must increase by
net impact of these wage adjustments constitutes the second e¤ect of the change in r on V (r), as elicited by (40).
27
the probability of monitoring. Its proof constructs a numerical example, under speci…c functional
forms of the fundamentals of the model, with the property that V 0 (r) < 0 for high values of r.
Proposition 19 There exist f ( ), y( ), u ( ), c( ) and u, such that sb < s and the value of the
corresponding optimal contract is decreasing in r for all high enough r.
The numerical example employed in proposition 19 builds on the simple case of a discrete type
space fsa ; sc g. With no restrictions on the continuous type density function f ( ), one can approximate su¢ ciently well the discrete distribution from this numerical example with a continuous one
for which the corresponding optimal contract maintains the key property stated in proposition 19.
Since for any set of remaining parameters of the model, there exist values of sc high enough such
that it is optimal for P to induce e¤ort only from the lower cost type sa , we considered directly
that sc takes such a high value without setting a particular value for it.38 In an environment with
this particular type distribution, P o¤ers a contract fwn ; wa ; ea g where (i ) wn is the salary paid
to A when no audit is performed, and (ii ) wa is the wage paid to A when an audit reveals that
he exerted at least e¤ort ea . The contract is designed so as to be accepted by A ex-ante and to
induce A to exert e¤ort ea when his type is sa and no e¤ort otherwise. The remaining details of
the formal analysis and numerical implementation are presented in appendix A10.
Figure 1 presents in solid lines key variables from the corresponding optimal contract as functions of the probability r, for the values of r that allow for a positive value of the optimal contract.
1
38
with s 2 fsa ; sc g and
The functional forms that we employed are y(e) = e, u(w) = w and c(s; e) = s 1 e
a
a
a
a
p
Prfs = s g. The corresponding parameters are set at = 1:6, = 1:2, s = 0:2, p = 0:85 and u = 1. For the
functional forms selected, there are multiple sets of parameters with the property that V 0 (r) < 0 for high enough r.
28
a. V alue of Contract
b. E ffort Lev el
1.6
8
1.4
7
1.2
1
6
0.8
0.6
5
0.4
4
0.2
3
0
0.2
0.4
0.6
0.8
0.2
c. W ages
12
w
0.4
0.6
0.8
d. E xpected rev enue and expected wage
7
a
10
6
Expected rev enue
↓
8
5
6
↑
2
Expected wage
4
4
w
n
w
0.2
0.4
0.6
3
c
0.8
0.2
0.4
0.6
0.8
Figure 1: Optimal Contracts as Functions of the Probability of Auditing r
When r is small, the contract speci…es a low e¤ort level ea , and in return promises a high wage wa
if A passes an audit. Intuitively, the low probability of auditing makes it hard to provide incentives
since A knows that even when exerting e¤ort, he is unlikely to be rewarded, as in the absence of an
audit, this e¤ort is not observed. Therefore, P …nds it optimal to only induce a low level of e¤ort.
Moreover, if r is very small, (r
0:1) P cannot ensure himself a non-negative expected payo¤ if he
were to implement positive e¤ort and has to shut down. As r increases, monitoring becomes a more
e¤ective monitoring instrument which allows implementing a higher e¤ort with a lower wage wa ;
A is compensated for the higher e¤ort with an increase in the salary wn . The expected revenue,
pa y (ea ), the expected wage paid, rpa wa + (1
r) wn , and the di¤erence between the two, i.e., the
value of the contract, all increase over this range.
The increase in r also impacts the risk premium paid to A through two channels. First, it
decreases it by lowering the dispersion in the set of possible wages, fwa ; wn ; 0g up to the value of r
29
at which wa and wn become equal. Above that value, the wages wa and wn are equal and constant,
and this …rst channel shuts down. Second, the increase in r alters the probabilities with which
these wages are paid. When r is small, most of the utility is delivered to A through the salary wn ,
while the probabilities of A being paid either wa or 0 are small. As r increases, the probability that
A is paid the wage wa increases, but so does the probability that A is caught shirking and paid
nothing. This second e¤ect tends to increase the wage variance and therefore the risk premium P
needs to pay. For su¢ ciently high values of r, the consequent increase in risk premium determines
a decrease in the value of the contract and can even bring this value to zero.39
Next, we argue that when P can credibly commit not to void the contract, but to make payments
even when A fails an audit, then the optimal value of the contract is again everywhere increasing
in r. The following proposition states this result. Its proof, which is presented in appendix A11,
considers the standard case adopted in the paper of a continuous type density function.
Proposition 20 Assume that u00 (w) < 0 for all w and that P can credibly commit to make a
payment even when A fails an audit. The value of the corresponding optimal contract is increasing
in r for any r 2 (0; 1).
If P can commit to make payments even when A is caught shirking, he can reduce the dispersion in the set of possible wages faced by A to essentially partially insure him against high-cost
realizations. This allows P to lower the risk premium he needs to pay. Moreover, and perhaps more
surprisingly, the increased power of incentive determined by the higher probability of monitoring
renders the value of the contract be again everywhere increasing in r.
The optimal contract corresponding to the numerical example employed in proposition 19 is
presented in Figure 1 in broken lines. The wage wc is promised to be paid to A when an audit is
39
Note the monotonicity of V (r) changes around the value of r where wn starts increasing at a faster rate. Since
the salary wn is the channel through which this risk premium is paid, this suggests that the risk premium starts
increasing signi…cantly, and therefore, that it is indeed the factor driving the decrease of V (r).
30
performed and it reveals that A exerted less e¤ort than ea , i.e., essentially this is the wage paid
when A’s type is sc . As expected, the value of the contract with commitment is at least as high as
that without commitment for all values r. When r is low, the two values are equal; in particular, P
implements the same contract as without commitment by setting wc to zero and thus not availing
himself of the possibility of commitment. For the high values, commitment becomes valuable and P
promises a wage wc > 0 to reduce the risk premium paid. It also deserves emphasizing at this point
that the optimal wage wc in this numerical example is nonnegative. This ensures that the driving
force inducing a higher value of the contract is the ability of P to make a credible commitment
to make a net payment towards A when the latter fails an audit, and not an ability to extract
payments from A in such situations.
4
Extension: The Optimal Contract with Communication
In this section we analyze the optimal contracts with random auditing when pre-play communication
is feasible. As discussed in section 2, in this situation, P can design a contract that requires A
to declare his type after he learns it, and then de…ne the wage paid to A when an audit is not
performed as a function of this type.40 To focus the exposition, we restrict again attention to
the case when it is optimal to implement positive e¤ort for all types and when the nonnegativity
constraints on wages do not bind. Finally, we consider the case where A is strictly risk averse. P’s
40
An alternative standard interpretation of the "communication" between A and P is that A chooses a contract
fw; wn g out of a menu of contracts o¤ered by P after he learns his type.
31
problem in this case is to select a contract
max
fe(s) 0;w(s) 0;wn (s) 0gs2[s;s]
Z
n
n (s); e (s); w (s)
w+
+
+
s
[y (e(s))
rw(s)
ru (w(s))
Z
s
[ru (w(s)) + (1
that solves the problem
r) wn (s)] f (s)ds
(41)
c (s; e (e
s))]
(42)
s
r) u (wn (e
s))
s)) + (1
s.t. s 2 arg max [ru (w(e
se2[s;s]
(1
s2[s;s]
o
c (s; e (s))
0, for all s 2 [s; s]
r) u (wn (s))
c (s; e (s))] f (s)ds
(43)
u.
(44)
s
.
Following the First Order Approach, we replace (42) with the corresponding …rst order condition
ru0 (w(s)) w0 (s) + (1
r) u0 (wn (s)) wn0 (s) = ce (s; e(s))e0 (s) a.e. s
(45)
and assume that e0+ (s) < 0 in the optimal contract. On the other hand, a counterpart of the lemma
4 from the case without communication does not hold here, and thus we cannot relax (43).
We solve the resulting problem using the same optimal control methods employed in section
3.3. By denoting un (s)
u(wn (s)), the …rst order condition for A’s truthful revelation problem
from (45) becomes ru0 (s) + (1
r) un0 (s) = ce (s; e(s))e0 (s) a.e. s. In addition to the variables from
problem (18)-(23), we introduce a new state variable un (s) and a new control k(s)
32
un0 (s). The
optimal control problem is then
max
fx(s);k(s)gs2[s;s]
Z
[y (e(s))
rh (u(s))
(1
r)h(un (s))] f (s)ds
(46)
s
s.t. e0 (s) = x(s)
(47)
un0 (s) = k(s)
(48)
1
u0 (s) =
r
1
k (s) + ce (s; e(s))x(s)
r
r
(49)
r)un (s)
(50)
v 0 (s) = [ru (s) + (1
v (s) = 0; v (s)
ru (s)
s
c (s; e (s))
c (s; e (s))] f (s)
u
(51)
0, for all s 2 [s; s]
(52)
This is an optimal control problem with pure state constraints determined by the speci…c form of
(52).41 The necessary conditions delivered by Pontryagin’s Maximum Principle for such problems
are presented in Theorem 4.1 in Hartl, Sethi and Vickson (1995), with the formal proof of the
theorem for our case where we have no constraints with both state and control variables, so condition
2.3 from the text of their problem does not exist, presented in the references cited therein.
Thus, to solve the control problem, we construct the Lagrangian
L+ (e; u; un ; v; x; k;
1;
+
1x
where
1;
2;
3;
4
and
2;
+
3;
2
4;
; s)
1
r
[y (e)
rh (u)
1
k + ce (s; e) x +
r
r
(1
r) h (un )] f (s) + [ru
3 [ru
+ (1
r) un
c (s; e)] f (s)
c (s; e)] f (s) +
4k
are functions de…ned on [s; s]. Then, by Theorem 4.1 in Hartl, Sethi
and Vickson (1995), there exist almost everywhere di¤erentiable functions
41
1;
2;
3;
42
4,
and an
See Chapter 6 in Caputo (2005) for a comprehensive discussion of the problems with state and control constraints,
and Chapter 5 in Seierstad and Sydsaeter (1987) for a more detailed discussion of problems with pure state constraints.
42
The theorem does not state that the costate variables 1 ; 2 ; 3 ; 4 are continuous, but it states that at all
points where the constraint (52) binds, the functions 1 and 2 may have discontinuities given by the following jump
@
@
conditions 1 (s ) = 1 (s+ ) + (s) @e(s)
[ru (s) c (s; e (s))] and 2 (s ) = 2 (s+ ) + (s) @u(s)
[ru (s) c (s; e (s))] for
some positive function (s). Since the same type of result applies to the functions 3 and 4 , but the constraint
33
almost everywhere continuous function
such that the following conditions are satis…ed almost
everywhere.43
@L+
@x
@L+
@k
0
1 (s)
=
1 (s)
1
=
2 (s)
1
ce (s; e (s)) = 0
r
+
4 (s)
(53)
(54)
=0
(55)
y 0 (e) f (s) + (s) ce (s; e (s)) f (s)
2 (s)
1
cee (s; e (s)) x (s) +
r
(s)
3 (s) ce (s; e (s)) f (56)
@L+
= rh0 (u (s))f (s) r (s)f (s) r 3 (s) f (s)
@u
@L+
=0
@v
@L+
= (1 r) h0 (un (s)) f (s)
r) f (s)
3 (s) (1
@un
0
2 (s)
=
0
3 (s)
=
0
4 (s)
=
1 (s)
= 0;
(s)
r
2 (s)
r
@L+
=
@e
=
=
+
1 (s)
= 0;
1 (s)
= 0;
1 (s)
0, with (s) = 0 if ru (s)
= 0;
3 (s)
2 R;
3 (s)
(57)
(58)
(59)
2 R;
4 (s)
= 0;
4 (s)
=0
(60)
c (s; e (s)) > 0, for any s 2 [s; s]
(61)
There exists also a variant of the generalized Legendre-Clebsch necessary condition for problems
with multiple controls and state constraints, which we show in appendix A12 that it is satis…ed if,
for instance, we assume again that cees
0 along the trajectory of the solution to (53)-(61).
Lemma 21 states the su¢ ciency of conditions in (53)-(61) for the problem in (46)-(52) and the
uniqueness of the corresponding solution.44
Lemma 21 (Su¢ ciency and Uniqueness) If e+ (s); u+ (s); un+ (s); v+ (s); x+ (s); k+ (s)
isfy the conditions in (53)-(61) with costate variables f
1+ (s);
2+ (s);
3+ (s);
s2[s;s]
4+ (s)gs2[s;s] ,
sat-
then it
in (52) is independent of the corresponding state variables v(s) and un (s), it follows that 3 and 4 are continuous
everywhere. Given (54), it follows then that 2 is also continuous everywhere, and then (53) implies the same for
1 . We conclude thus that for this problem, as in the case of the necessary conditions from section 3.3, the costate
variables are continuous everywhere.
43
Given (53) and (54), some conditions in (60) are redundant and so are not used in deriving the optimal contract.
44
The proof of this lemma is simpli…ed by the underlying assumption that it is optimal that all types exert positive
e¤ort. If that assumption was relaxed, the result to apply for proving su¢ ciency in an optimal control problem with
pure state constraints and a free end time is Theorem 7 on page 377 in Seierstad and Sydsaeter (1987).
34
is the unique solution to (46)-(52).
Proof of lemma 21.
The lemma follows from the Arrow Su¢ ciency Theorem for optimal control
problems with mixed constraints (see Theorem 6.4 on page 166 in Caputo (2005)).45 By employing
the conditions in (53) and (54), the maximized Hamiltonian evaluated at the corresponding costate
variables equals [y (e)
rh (u)
(1
r) h (un )] f (s) +
3+ (s) [ru
+ (1
r) un
c (s; e)] f (s). From
the assumed properties of y ( ) and c ( ; ) and the fact that for any solution to (53)-(61), we have
3+ (s)
0 (we prove this in appendix A12), the maximized Hamiltonian is concave in (e; u; un ; v)
and strictly concave in (e; u; un ). This implies the claims of the lemma 21.46
Proposition 22, proved in appendix A12, is the main result of this section eliciting the conditions
that determine the optimal contract with communication when A is strictly risk averse, and the
e¤ect of r on the value of this contract.47 Remark 23 is proved in the same appendix.
Proposition 22 Assume that u00 (w) < 0, for all w. Also, assume that it is optimal to induce all
types of A to exert e¤ ort. The solution for the optimal contract with communication under moral
hazard and adverse selection is given by (44) satis…ed with equality, (43), (45), and for any s 2 [s; s]
w+ (s)
n
w+
(s)
0, and = 0 whenever ru (w+ (s)) c (s; e+ (s)) > 0
(62)
Z s
Z s
ce (s; e+ (s))
1
f (t)
f
(s)
+
c
(s;
e
(s))
dt f ( )d = y 0 (e+ (s))f (s) (63)
es
+
0 (w n ( ))
0 (w n (t))
u0 (w+ (s))
u
u
s
s
+
+
The value of the optimal contract is increasing in r for any r 2 (0; 1).
45
Note that while our problem has pure state constraints, Theorem 6.4 in Caputo (2005), which deals with optimal
problems with mixed constraints (where the control also appears in the constraint), applies to it since the rank
constraint quali…cation is not required for that theorem. This rank quali…cation is not satis…ed in problems with
pure state constraints and thus we could not apply Theorem 6.1 from Caputo (2005) to conclude that (53)-(61) are
necessary conditions for a solution to (46)-(52).
46
As in the case of lemma 8, the text of Arrow’s Su¢ ciency Theorem from Caputo (2005) requires the maximized
Hamiltonian be strictly concave in all state variables, and does not claim the uniqueness of the control variables, but
the same argument employed in the proof of lemma 8 completes the proof in this case.
47
n
(63) determines e+ (s) as a function of fw+ (s); w+
(s)gs2[s;s] . Then (45) and (62), subject to the constraint in (43)
n
and with the binding constraint in (44) as initial condition, constitute a system that determine w+ (s) and w+
(s).
35
Remark 23 We have
Z
s
s
1
n ( ))
u0 (w+
Z
s
s
f (t)
n (t)) dt
u0 (w+
f ( )d > 0 for all s 2 (s; s).
As in the model with pure moral hazard studied in section 3.2, the optimal contract sets
w+ (s)
n (s). To see why, note that otherwise, w n (s) could be reduced by an amount and w (s)
w+
+
+
0 (s) with the property that ru (w (s)) + (1
increased to a value w+
+
(1
n (s) +
r) u w+
n (s) = ru w 0 (s) +
r) u w+
+
; this adjustment would not violate any of constraints in problem (41)-(44),
but would reduce the amount of risk that A is subjected to and the risk premium that needs to be
paid. Thus, unlike the case with no pre-play communication, where P needs to use the wage scheme
fw (s)gs2[s;bs] both to induce the agent to reveal his type and to exert e¤ort, if communication
is feasible, P can make use of the ‡exibility o¤ered by his ability to adjust the wage scheme
n (s)
w+
s2[s;s]
to tailor the contract so as to not penalize the agent when an audit is performed
and the agent passes it. Moreover, as (62) states, the wages with and without an audit for a given
type s are equal whenever the corresponding level is su¢ cient to provide incentives for that type
to exert e¤ort, i.e., when ru (w+ (s))
c (s; e+ (s)) > 0. On the other hand, (63) imposes the usual
equality between the marginal cost and bene…t of requiring additional e¤ort from type s, with the
second term in the left hand side being the (strictly positive, by remark 23) distorting factor that
sets type s’s induced e¤ort below its e¢ cient level. Finally, under our underlying assumption that
all types are induced to exert positive e¤ort, the value of the contract is strictly increasing in r
whenever w+ (s)
5
n (s) > 0 for s in a set of positive measure, and is constant otherwise.
w+
Conclusion
In this paper we studied optimal contracts with random auditing de…ned as a monitoring instrument where the agent’s action is observed with some non-degenerate probability, but otherwise
the principal has no informative signal of this action. We characterized and compared the optimal
contracts under several standard information structures that combine moral hazard and adverse
36
selection. We showed that a higher precision of the monitoring instrument, as measured by the
probability of auditing, always increases the value of an optimal contract when all agent types are
optimally induced to exert e¤ort or when the principal can commit to make payments towards
the agent even when the latter fails an audit, but may decrease the value of a contract otherwise.
Finally, we characterized the optimal contracts for situations where pre-play communication is possible and thus the principal can adjust the wage paid to the agent when an audit is not performed
as a function of the signal transmitted by the agent. As an avenue for future research, we are
also currently working on a repeated version of this model where a failed audit voids the dynamic
contract and leads to a loss for the agent of the promised value of future payments.
Appendix
Appendix A1. Proof of Proposition 2
First, note that in any optimal contract, the participation constraint in (6) must bind since otherwise
w1n can be decreased without violating any of the constraints. We construct the Lagrangian
L1 =
Z
s
+
s
[y (e(s))
Z
rw(s)
(1
n
r) w ] f (s)ds +
Z
s
(s) [ru(w(s))
s
s
[ru (w(s)) + (1
r) u (wn )
s
37
c (s; e (s))] f (s)ds
u
c(s; e(s))] ds
The necessary …rst order conditions are then
@L1
@e(s)
@L1
@w(s)
@L1
@wn
@L1
(s)
@ (s)
(s)
From (65), we have
= y 0 (e(s)) f (s)
(s) ce (s; e(s))
=
rf (s) + (s) ru0 (w(s)) + ru0 (w(s))f (s) = 0
=
(1
=
(s) [ru(w(s))
r) + (1
0; ru(w(s))
(s) + f (s) =
f (s)
u0 (w(s)) .
Substituting this into (67), and noting that
w1n = 0 whenever ru(w(s))
1
u0 (w)
w1n
(65)
(66)
(67)
c(s; e(s))] = 0
0;
0
(68)
Substituting this into (64), implies (8) from the
i
h
1
1
1
= u0 (w
(s) = u0 (w(s))
n ) , and thus
0
n
u (w ) f (s).
1
u0 (w(s))
1
u0 (wn )
= 0 () w(s)
c(s; e(s)) > 0. Finally,
is increasing in w imply w1 (s)
(s)
wn = 0, it follows that
0 from (68) and the fact that
0 as stated by (7).
Denote now by V1 (r) the value of the optimal contract; we will show that V10 (r)
the Envelope Theorem, and then substituting
V10 (r)
(64)
r) u0 (wn ) = 0
c(s; e(s))
text of the proposition. From (66), we have
w1 (s)
ce (s; e (s)) f (s) = 0
and
0. Employing
(s) computed above, we have
Z s
@L1
=
=
f[ w(s) + wn ] f (s) + (s) u(w(s)) + [u (w(s)) u (wn )] f (s)g ds
@r
s
Z s
1
1
1
=
[ w(s) + wn ] + 0
u(w(s)) + 0 n [u (w(s)) u (wn )] f (s)ds
0
n
u (w(s)) u (w )
u (w )
s
Z s
n
u(w(s))
u (w )
=
w(s)
wn f (s)ds
0
u (w(s))
u0 (wn )
s
Since w(s)
wn for all s, V10 (r)
0 follows from the fact that
d
dw
h
u(w)
u0 (w)
i
w =
u(w)u00 (w)
(u0 (w))2
> 0.
Next, we show that the e¤ort pro…le e1 (s) is decreasing, and argue that the wage pro…le w1 (s)
is not necessarily monotonic. Consider any interval in [s; s] on which w1 (s) = w1n ,48 and note
48
By the continuity of the objective function and constraints in P’s problem, it follows that the sets of s on which
w1 (s) > w1n and w1 (s) = w1n , respectively, are unions of intervals.
38
ces (s;e1 (s))
cee (s;e1 (s)) y 00 (e1 (s))u0 (w1n ) ,
that from (8) it follows by the Implicit Function Theorem that e01 (s) =
which is negative since ces > 0, cee > 0, y 00
0 and u0 > 0. On the other hand, on an in-
terval in [s; s] on which w1 (s) > w1n , we have from (7) that w1 (s) = u
y 0 (e1 (s)) u0 (u
stituting this in (8), we obtain ce (s; e1 (s))
that e01 (s) =
u
1
r c (s; e1 (s))
. Sub-
) = 0. It follows then
, where we substituted w1 (s) for
. Again, the properties of the functions c, y and u imply that e01 (s) < 0. Finally,
ce (s; e1 (s)) = 0 it follows that whenever w1 (s) > w1n , we have
rewriting (8) as u0 (w1 (s))y 0 (e1 (s))
w10 (s) =
1
r c (s; e1 (s))
u00 (w (s))
ces (s;e1 (s)) y 0 (e1 (s)) u0 (w 1(s)) r1 cs (s;e1 (s))
1
u00 (w (s))
cee (s;e1 (s)) y 00 (e1 (s))u0 (w1 (s)) y 0 (e1 (s)) u0 (w 1(s)) r1 ce (s;e1 (s))
1
1
r c (s; e1 (s))
1
1
1
u0 (w1 (s))y 00 (e1 (s))e01 (s) ces (s;e1 (s)) cee (s;e1 (s))e01 (s)
.
u00 (w1 (s))y 0 (e1 (s))
Since ces (s; e1 (s)) > 0, the numerator is
generically unsigned, and thus w1 (s) is not necessarily monotonic. Moreover, the set of values of s
for which w1 (s) = w1n may be a union of disjoint intervals.
We close by arguing that the surplus generated by the di¤erent agent types, y (e(s))
r) wn , is decreasing in s. Consider …rst any interval in [s; s] on which w1 (s) > w1n , and thus,
(1
from (5), ru(w1 (s)) c(s; e1 (s)) = 0, implying w1 (s) = u
d
ds
rw(s)
[y (e1 (s))
u0 (u
rw1 (s)] =
cs (s;e1 (s))
,
1 1 c(s;e (s))
(r
))
1
d
ds
y (e1 (s))
ru
1
1
1
r c (s; e1 (s))
1
r c (s; e1 (s))
. On this interval, we have
= y 0 (e1 (s))
u0 (u
ce (s;e1 (s))
1 1 c(s;e (s))
(r
))
1
e01 (s)
which is negative since (8) implies that the …rst term is zero. On the other hand,
on intervals in [s; s] on which w1 (s) = w1n , we have
e01 (s) < 0, we conclude that
d
ds
[y (e1 (s))
d
ds
[y (e1 (s))
rw1 (s)] = y 0 (e1 (s)) e01 (s). Since
rw1 (s)] < 0 for all s.
Appendix A2. Proof of Lemma 5
We argue …rst that any incentive compatible contract must satisfy (15) and (16). Consider A’s
n
o
original problem of choosing an e¤ort level under a contract wn ; fw(e)ge 0 when his type is s,
max [ru (w(e))
e(s) 0
c (s; e)] and note that for any wage pro…le w(e), A’s objective function in (10) is
submodular in (e; s) because
@2
@e@s
[ru (w(e))
c (s; e)] =
ces (s; e) < 0. By Topkis’Monotonicity
Theorem, it follows then that the maximizer e (s) is a.e. decreasing in s. Thus, in order for a
o
n
contract wn ; fe(s); w(s)gs2[s;s] to be incentive compatible, e(s) must be a.e. decreasing in s
39
(this also implies that e(s) is a.e. di¤erentiable). The necessity of (16) follows from the …rst order
condition in A’s problem in (10).
Next, we show that a contract satisfying (15) and (16) is incentive compatible. Let
ru (w(e
s))
c (s; e (e
s)); we will argue that
to complete the proof. We have
@
@e
s
(e
s; s)
@
@e
s
(e
s; se) = 0. Thus,
@
@e
s
(e
s; s) for all se; s 2 [s; s], which will be enough
(s; s)
(e
s; s) = ru0 (w(e
s)) w0 (e
s)
(e
s; se) if and only if
e0 ( ) < 0 by (15), it follows that
@
@e
s
@
@e
s
ce (s; e (e
s)) e0 (e
s)
@
@e
s
(e
s; s)
(e
s; s)
ce (s; e (e
s)) e0 (e
s), and so note that
ce (e
s; e (e
s)) e0 (e
s). Since ces ( ) > 0 and
(e
s; se) if and only if se
(e
s; s) is increasing (decreasing) in se for se
s (e
s
s. But by (16), we have
s), implying immediately
that it is indeed maximized when se equals s.
Appendix A3. Proof of Lemma 10
Since
2 (s)
= 0 by (30), integrating equation (27) we have
2 (s) = r
Z
s
h0 (u ( ))f ( )
) f ( ) d , for all s 2 [s; sb]
3(
s
Employing
s)
2 (b
= r, as required by (30), and
=
Z
s
sb
3 (s)
h0 (u (s))
= h0 (un ), we conclude that
h0 (un ) f (s)ds
The claim of lemma 10 follows by substituting for
h0 (u(s)) = u
1 0 (u(w(s)))
=
1
u0 (u
1 (u(w(s))))
=
1
u0 (w(s)) ,
40
(69)
(70)
from (70) into (32) after observing that
and similarly that h0 (un ) =
1
u0 (wn ) .
Appendix A4.
Proof of Lemma 11.
Di¤erentiating
1 (s)
+
1
2 (s) r ce (s; e (s))
= 0 from (25) with respect to s,
we get
0
1 (s)
Plugging in
0
2 (s)
+
0
1 (s)
and
1
ce (s; e (s)) +
r
0
2 (s)
=
1
ces (s; e (s)) +
r
2 (s)
1
cee (s; e (s)) e0 (s) = 0
r
(71)
from (26) and (27), we obtain that
y 0 (e (s))f (s)
+ rh0 (u (s))f (s)
2 (s)
2 (s)
1
cee (s; e (s)) x (s) +
r
+
1
1
ces (s; e (s)) + 2 (s) cee (s; e (s)) e0 (s) =
r
r
1
y 0 (e (s))f (s) + h0 (u (s))ce (s; e (s)) f (s) + 2 (s) ces (s; e (s)) = 0
(72)
r
3 (s) rf (s)
1
ce (s; e (s)) +
r
3 (s) ce (s; e (s)) f (s)
2 (s)
2 (s)
where we used the fact that x (s) = e0 (s). Now, since (69) and 3 (s) = h0 (un ), imply
Rs 0
h0 (un )] f ( )d , it follows that the optimal contract must satisfy
s [h (u( ))
0
h (u(s))ce (s; e (s)) f (s) +
Z
s
h0 (u( ))
h0 (un ) f ( )d
r
=
ces (s; e (s)) = y 0 (e (s))f (s)
s
which can then be immediately rewritten as in (37). This completes the proof of lemma 11.
Proof of Remark 12. We will show that
for
2 (s)
2 (s)
0 for all s 2 [s; sb]. Given the expression computed
above, this will immediately imply the claim of the corollary. Now, to prove that
for all s 2 (s; sb), since
2 (s)
= 0 and
s)
2 (b
= r
0, it would be enough to show that
2
2 (s)
>0
is strictly
increasing on some interval [s; s0 ] and strictly decreasing on [s0 ; sb]. To this aim, note …rst that
d
ds
[h0 (u(s))] = h00 (u(s))u0 (s) < 0 because u(s) is decreasing while h is strictly convex as the inverse
of a concave and increasing function.49 Therefore, h0 (u(s)) is strictly decreasing in s. Since it is
also continuous in s it follows that there exists some s0 2 [s; sb] such that h0 (u(s))
49
Di¤erentiating twice each side of the equality u(h(v)) = v, we obtain h00 (v) =
41
h0 (un ) > 0 for
2
u00 (h(v)) [h0 (v)] =u0 (h(v)) > 0.
h0 (un ) < 0 for all s 2 (s0 ; sb]. Since
all s 2 [s; s0 ) and h0 (u(s))
follows that
0
2 (s)
= [h0 (u(s))
h0 (un )] f (s) it
is increasing on [s; s0 ] and decreasing on [s0 ; sb], as desired.
2 (s)
Veri…cation of the generalized Legendre-Clebsch condition. This condition requires that
( 1)n
@ d2n
@x ds2n
where 2n is the …rst higher-order derivative of
@H
@x
@H
@x
0
(73)
with respect to s in which the control x appears
(it has been proved that n 2 N+ ). We already showed above that
h0 (u (s))ce (s; e (s)) f (s) +
d2
ds2
@H
@x
1
2 (s) r ces (s; e (s)),
y 00 (e (s))x(s)f (s)
=
d
ds
@H
@x
=
y 0 (e (s))f (s) +
so di¤erentiating one more time, we have
1
y 0 (e (s))f 0 (s) + h00 (u (s)) ce (s; e (s)) x (s) ce (s; e (s)) f (s)+
r
+ h0 (u (s))ces (s; e (s)) f (s) + h0 (u (s))cee (s; e (s)) x(s)f (s) + h0 (u (s))ce (s; e (s)) f 0 (s)+
+ rh0 (u (s))f (s)
3 (s) rf (s)
1
ces (s; e (s)) +
r
2 (s)
1
cess (s; e (s)) +
r
2 (s)
1
cees (s; e (s)) x(s)
r
where employing (19), (20) and (27), we substituted x(s) for e0 (s), 1r ce (s; e (s)) x (s) for u0 (s) and
rh0 (u (s))f (s)
3 (s) rf (s)
for
0
2 (s).
Clearly, n = 1 and thus the condition in (73) requires that
1
y 00 (e (s))f (s) + h00 (u (s)) [ce (s; e (s))]2 f (s) + h0 (u (s))cee (s; e (s)) f (s) +
r
2 (s)
1
cees (s; e (s))
r
0
(74)
along the solution to (25)-(33). Since y 00 (e) < 0, h00 (u) > 0, h0 (u) > 0, cee (s; e) > 0 and
a su¢ cient condition for (74) to be satis…ed is that cees (s; e (s))
0,
0 along this solution. However,
this additional condition on the cost function c( ; ) is not necessary.
42
2 (s)
Appendix A5. The Optimal Contract when the Monotonicity Constraint Binds
To simplify exposition, we assume that it is optimal for P to induce positive e¤ort levels for all
o
n
types, i.e, that the optimal value of sb is s. Denote by fe (s); w (s)gs2[s;s] ; wn the optimal
contract when incorporating the monotonicity constraint in (15) into P’s maximization problem,
and assume that there exist intervals in [s; s] for which this constraint binds and thus e ( ) is
constant. To account formally for this additional constraint, we construct a new Hamiltonian as
H (e; u; v; x;
1;
2;
3;
4 ; s)
H (e; u; v; x;
1;
2;
3 ; s)
+
4(
x)
(75)
where H ( ) is as de…ned in (24). In addition to the necessary conditions identi…ed in (26)-(94),
(1)
which must still be satis…ed, Pontryagin’s Maximum Principle50 implies that there exists a Cp
function
4 (s)
such that (25) is replaced with
@H
=
@x
1 (s)
+
1
ce (s; e (s))
r
2 (s)
4 (s)
=0
(76)
= 0 if x(s) < 0
(77)
while the corresponding complementary slack conditions are
x(s)
Moreover,
4 (s)
0;
4 (s)
0, with
4 (s)
is continuous whenever x(s) is continuous (in this situation, since we do not have
a bang-bang solution, the optimal control is continuous everywhere). To understand (76) and (77),
note by inspecting the relaxed problem in (18)-(23), that since H ( ) is linear, we have x(s) = 0 in
the optimal solution whenever
@H
@x
(s) > 0. The newly de…ned function
captures precisely the nonnegative gradient
@H
@x
4 (s)
=
@H
@x
(s)
@H
@x
(s)
(s). In other words, to deal with the additional
monotonicity constraint in (15), rather than de…ning H and then imposing (76) and (77), we could
50
See Theorem 6.1 in Caputo (2005) on page 152 for the version with constraints on the control employed here.
43
@H
@x
have instead added a complementary slack condition
and x(s)
(s) x(s) = 0, together with
@H
@x
(s)
0
0 to the set of conditions (25)-(94). The key additional information that Pontryagin’s
Maximum Principle applied to the non-relaxed problem delivers, and which we do employ below,
is that
@H
@x
(s) is continuous when x(s) is continuous.
The only impact that these changes have on our previous analysis is on the argument and
result of proposition 11. In particular, since
@H
@x
4 (s)
is not di¤erentiable, we cannot di¤erentiate
= 0 with respect to s at all s 2 [s; s], as we did in the proof of that proposition. Consider
thus an interval [s0 ; s00 ]
[s; s] on which e (s) is constant, and thus e (s) = e (s0 ) = e (s00 ) for all
s 2 [s0 ; s00 ], which by (16) also implies that w (s) = w (s0 ) = w (s00 ) for all s 2 [s0 ; s00 ]. Note then
that we can replicate the argument from the proof of proposition 11 for all s such that x (s) < 0
in the optimal contract. Thus we can determine the e¤ort and wage at every s with x (s) < 0 by
employing the same conditions identi…ed in the case when the monotonicity constraint does not
bind; the fact that w (s) is constant on [s0 ; s00 ] and continuous everywhere implies that values for
w (s) can be imputed for all s 2 [s; s] in (36), (37) and (13) since these values equal w (s0 ) which
has been determined (also, values for e (s) can be imputed in (13)). However, the resulting wage
and e¤ort schedules will be functions of interval endpoints s0 and s00 .
To determine these values, note that since
0, which from (76) implies then that
Thus, it must be that
Z
s00
s0
Noting that
4
d
ds
4(
) is continuous, it must be that
0
0
0
0 1
1 (s ) + 2 (s ) r ce (s ; e (s ))
1 (s)
+
2 (s)
=
0
4 (s )
=
00
4 (s )
=
00
00 1
00
00
1 (s ) + 2 (s ) r ce (s ; e (s )).
1
ce (s; e (s)) ds = 0
r
(78)
does not appear in the expressions implied by Pontryagin’s Maximum Principle for
44
either
d
ds
0
1 (s)
or
1 (s)
+
=
0
2 (s),
2 (s)
by the same argument as in the proof of proposition 11, we have that
1
ce (s; e (s)) =
r
ce (s; e (s))
f (s) + ces (s; e (s))
u0 (w (s))
Z
s
1
u0 (w
s
1
( ))
u0 (wn )
y 0 (e (s))f (s)
f ( )d
It follows then from (78) that we must have
Z
s00
s0
ce (s; e (s))
f (s) + ces (s; e (s))
u0 (w (s))
Z
s
s
1
u0 (w
1
( ))
f ( )d
u0 (wn )
y 0 (e (s))f (s) ds = 0
(79)
Equation (79) together with e (s0 ) = e (s00 ) constitute a system of two equations in two unknowns
that allows determining the values for s0 and s00 . This system may have several solutions, corresponding to a case where there are multiple intervals on which e (s) is constant. It is also worth
noticing here that w (s0 ) = w (s00 ) and e (s0 ) = e (s00 ) would not constitute an alternative system
of equations to determine s0 and s00 . Given (16), these two conditions always hold simultaneously
by construction when eliciting the wage and e¤ort schedules as functions of s0 and s00 . Moreover,
the condition in (79) is necessary to be satis…ed by s0 and s00 and thus needs to be imposed.
Appendix A6.
Proof of Corollary 14. We have
d
[y (e (s))
ds
rw (s)] =
d
[y (e (s))
ds
= y 0 (e (s)) e0 (s)
rh(u (s))]
rh0 (u (s))u0 (s) = e0 (s) y 0 (e (s))
ce (s; e (s))
u0 (w (s))
(s))
where we employed (20) for the last equality. Note that from (37), we have y 0 (e (s)) cue0(s;e
(w (s)) =
Z sh
i
ces (s;e (s))
1
1
f (s)
u0 (w ( ))
u0 (wn ) f ( )d . Since the integral was shown to be strictly positive on
s
(s; sb) in remark 12, while e0 (s) < 0, it follows, as required, that
45
d
ds
[y (e (s))
rw (s)] < 0.
Proof of Lemma 8.
We present here the details left out of the proof presented in the main text.
To show the su¢ ciency of the necessary conditions, we appeal to the main theorem in Seierstad
(1984). To apply it, we need to show several facts. (i ) The functions sb ! e (b
s), sb ! u (b
s) and
sb ! v (b
s), obtained by solving the necessary conditions in (25)-(32) for each …xed value of sb, must
(1)
be Cp . The …rst two requirements are part of assumption 7, while the last follows from them by
the de…nition of the state variable v. (ii ) The costate variables should uniquely satisfy (25)-(28) for
given trajectories of the state variables, but then the ensuing Note 1 states that this requirement,
which is not necessarily obviously satis…ed here, can be dropped from the text of the theorem.
(iii ) The function (b
s)
H (e (b
s) ; u (b
s) ; v (b
s) ; x (b
s) ;
(b
s) ;
1
the property that there exists some sb0 2 [s; s] such that (b
s)
2
(b
s) ;
3
(b
s) ; sb) should have
0 for sb < sb0 and (b
s)
sb > sb0 . In our case, given the de…nition of H and (25), we have (b
s) = [y (e(b
s))
s)
3 [ru(b
0 for
rh (u(b
s))] f (b
s) +
c (b
s; e(b
s))] f (b
s). Note that
d
(b
s)
db
s f (b
s)
=
=
d
[y (e (b
s))
db
s
d
[y (e (b
s))
db
s
ru0 (b
s)
rh(u (b
s))] +
3
rh(u (b
s))]
s; e(b
s))
3 cs (b
cs (b
s; e(b
s))
ce (b
s; e(b
s)) e0 (b
s)
<0
where for the second equality we used (19)-(20) to cancel out the two terms, while for the inequality
we used the result obtained in the proof of corollary 14, and the facts
the monotonicity of
(b
s)
f (b
s) ,
there exists sb0 2 [s; s] such that
(b
s)
f (b
s)
3
> 0 and cs > 0. Thus, by
0 for sb < sb0 and
(b
s)
f (b
s)
0. Since
f (b
s) > 0 for all sb, the third condition of the theorem in Seierstad (1984) is immediately satis…ed. (iv )
The last requirement of the theorem in Seierstad (1984) is that the control variable has a bounded
support, which is clearly not satis…ed here. However, this assumption is essentially employed
R sb
to conclude that the following supremum is …nite
sup
rh (u(s))] f (s)ds
s [y (e(s))
where S
ru (b
s)
{fe(s); u(s); v(s)gs2[s;bs] :
u0 (s)
=
fe(s);u(s);v(s)gs2[s;b
s] 2S
1
0
0
r ce (s; e (s)) e (s), v (s)
= [ru (s)
c (s; e (s))] f (s),
s) = vb} for all vb in a neighbourhood of un . Since by
c (b
s; e (b
s)) = 0, v(s) = 0, and v(b
assumption 7, this supremum is …nite when vb = un , it is clearly …nite also in a neighbourhood of
46
that value by the smoothness of all functions involved. This completes the proof of lemma 8.
Appendix A7. Proof of Proposition 16
We show that there exists a wage pro…le that can implement the full-information e¤ort pro…le
fe0 (s)gs2[s;s] , i.e., that can satisfy all the incentive constraints in P’s optimal contract problem
under moral hazard and adverse selection. Moreover, when u is high enough, this contract delivers
to A the same ex-ante expected wage as in the case with full information, implying that the
corresponding contract is optimal since its value attains the upper bound, i.e., the value of the
optimal contract under full information.
Rs
Thus, let w (s) be de…ned by rw (s) c (s; e0 (s)) = 0, let w (s) = w (s) s ce ( ; e0 ( ))e00 ( )d
i
h
Rs
for s 2 [s; s), and let wn = 1 1 r u
[rw
(s)
c(s;
e
(s))]
f
(s)ds
. This wage pro…le satis…es (11)
0
s
and (13). Moreover, it also satis…es (16), which combined with e00 (s) < 0 implies that it satis…es
Rs
Rs
(10). On the other hand, s [rw (s) + (1 r) wn ] f (s)ds = u + s c(s; e0 (s))f (s)ds, which equals
the value of A’s expected wage under full information.
Note now that w (s) > 0 for all s 2 [s; s] (since w (s) = 1r c (s; e0 (s)) > 0, ce > 0 and e00 ( ) < 0),
Rs
but generically one cannot guarantee that wn 0. However, the term s [rw (s) c(s; e0 (s))] f (s)ds
from the de…nition of wn , which is strictly positive by (11), is also independent of u. To see
the latter, recall from section 3.1 that when A is risk neutral, the full-information e¤ort pro…le
fe0 (s)gs2[s;s] is de…ned by ce (s; e0 (s)) = y 0 (e0 (s)), which is independent of u, while the wage pro…le
fw (s)gs2[s;s] is de…ned above as a function of fe0 (s)gs2[s;s] . Thus, wn
0 if and only if u is high
enough. When wn , as de…ned above, is negative, by corollary 15, the optimal e¤ort level is given
by ce (s; e (s)) f (s) + ces (s; e (s)) (1
3 ) F (s)
= y 0 (e (s))f (s). Since by (39), we have
3
1, it
follows that this e¤ort level is lower or equal to e0 (s), which satis…es ce (s; e0 (s)) = y 0 (e0 (s)). This
completes the proof of proposition 16.
47
Appendix A8. Proof of Lemma 17
By the Dynamic Envelope Theorem it follows that
V 0 (r) =
Z
s
sb
@
H (e(s); u(s); v(s); x(s); s) f (s)ds
@r
@ n
u
@r
@
[ru(s)
@r
o
s 0
2 (s) 2 c (e(s))x(s) + 3 (s) u(s)f (s) ds
r
s)
3 (b
=
=
Z
Z
s
s
sb
sb n
h(u(s))f (s)
@
f(1
@r
h(u(s)) + h0 (un )u(s) f (s)
r) h(un )g +
2 (s)
2 (s)
r
3 (s)
Z
Z
+
sb
s
sb
[h0 (un )
s
u (s)
s
Z
s
h0 (un )
+ h(un ) + u(s)
h0 (un )un + u(s)
h0 (u( ))] f ( )d , while by (20),
h(u(s)) + h0 (un )u(s) f (s)ds + h(un )
0
n
= h0 (un ).
Rs
=
we have 1r ce (s; e(s)) x(s) = u0 (s), it follows that
V 0 (r) =
s) u
3 (b
1
ce (s; e(s))x(s) ds + h(un )
r2
where for the last equality we used the fact that
Now, since as elicited in appendix A4,
c(s; e(s))]
h0 (u( )) f ( )d
h0 (un )un
ds + u(s)
(80)
s
Integrating by parts the second integral in (80), we have
Z
s
=
sb
u0 (s)
u(s)
Z
Z
s
h0 (un )
h0 (u( )) f ( )d
s
s
0
n
h (u )
0
h (u( )) f ( )d
s
Z
=
s
because u(b
s)
R sb
s
sb
s
u(s) h0 (un )
[h0 (un )
sb
h0 (u(s)) f (s)ds
h0 (u( ))] f ( )d =
ds
Z
s
sb
u(s) h0 (un )
h0 (u(s)) f (s)ds
u(s)
u(b
s) since the integral equals
48
by (70).
Substituting this result into (80), we obtain that
0
V (r) =
Z
s
=
Z
s
sb
h(u(s)) + h0 (un )u(s) f (s)ds + h(un )
Z
s
sb
u(s) h0 (un )
sb
h0 (u(s)) f (s)ds
h0 (un )un + u(s)
u(s)
h(u(s)) + u(s)h0 (u(s)) f (s)ds + h(un )
h0 (un )un
which by using the facts that u(s) = u(w(s)), un = u(wn ), h0 (u(s)) =
1
u0 (w(s))
and h0 (un ) =
1
u0 (wn )
imply the result of lemma 17.
Appendix A9. Proof of Proposition 18
Denote by z n
1
u0 (wn )
1
u0 (w (s))
and z(s)
zn =
and note that (36) with sb = s implies
Z
s
z (s) f (s)ds
(81)
s
1
On the other hand, solving z = u0 (w)
for w to get w = (u0 ) 1
h
i
zu (u0 ) 1 z1
(u0 ) 1 z1 . Thus, (40) for sb = s becomes
V 0 (r) =
Z
s
z (s) u
u0
1
z (s)
1
s
znu
1
u0
1
zn
1
z
, it follows that we have
u0
1
1
z (s)
1
u0
f (s)ds
1
zn
Denoting by
'(z)
u0
zu
1
z
1
u0
1
1
z
and using (81) for the second equality, we have that
0
V (r) =
Z
s
s
'(z (s))f (s)ds
n
' (z ) =
Z
s
'(z (s))f (s)ds
s
'
Z
s
49
s
z (s) f (s)ds
u(w)
u0 (w)
w=
Rs
Rs
Rs
We will show next that (i ) ' s z (s) f (s)ds ' s z (s) f (s)ds
0, and (ii ) s '(z (s))f (s)ds
Rs
' s z (s) f (s)ds > 0, which would be enough to conclude that V 0 (r) > 0. To show (i ), since
0, it is su¢ cient to prove that ' is increasing. To show (ii ), we employ Jensen’s Inequality, for
which we need to prove that ' is strictly convex.
We have
'0 (z) = u
u0
1
= u
u0
1
1
z
+ zu0
1
z
1
u0
u00 (u0 )
1
z
h
where we used the fact that u0 (u0 )
1
1
z
1
h
i
1
z.
=
1
1
z
1
z2
i
1
h
u00 (u0 )
Since u (w)
0, for w
1
z2
1
1
h
3
z u00 (u0 )
1
1
z
i
1
z2
0, it follows that
'0 (z) > 0, as required for (i ). On the other hand,
'00 (z) = u0
u0
1
1
z
h
1
u00 (u0 )
1
1
z
i
=
1
1
z
i
Thus, '00 (z) > 0 when u is strictly concave. Therefore, (ii ) is also satis…ed. This completes the
proof of proposition 18.
Appendix A10. Proof of Proposition 19
Since ce > 0, if A accepts the contract fwn ; wa ; ea g, then after learning his type, he exerts either
e¤ort ea or 0. By making the standard transformations un
u(wn ), ua
u(wa ) and employing
the inverse utility function h ( ) de…ned earlier, P’s problem is to choose e; ua ; un so as to maximize
pa y (e) rpa h (ua ) (1
r) h (un ) subject to rua c(sa ; e)
0 and pa [rua
c(sa ; e)]+(1
r) un = u,
where we employed the fact that the participation constraint bind at optimum. The Lagrangian
for this problem is L(e; ua ; un ;
1;
2)
fpa y (e)
rpa h (ua )
50
(1
r) h (un )g +
1 [ru
a
c(sa ; e)] +
2 fp
a [rua
c(sa ; e)] + (1
r) un
@L
@ua
@L
@un
@L
e
@e
@L
1
@ 1
ug. The necessary …rst order equality conditions are
=
rpa h0 (ua ) +
=
(1
r)h0 (un ) +
= e pa y 0 (e)
=
1 [ru
a
2 rp
2 (1
a
=0
(82)
r) = 0
(83)
a
1 ce (s ; e)
c(sa ; e)] + (1
while the necessary inequality conditions are
2
+
2p
a
ce (s; e) = 0
(84)
c(sa ; e)] = 0
@L
= pa [rua
@ 2
that (83) implies
1r
@L
@e
(85)
r) un
0; rua
= h0 (un ), and then that (82) implies
u=0
c(sa ; e)
1
(86)
0;
= pa [h0 (ua )
0 and
1
2
> 0. Note
h0 (un )].51
We consider the following functional forms for the fundamentals of the model: y(e) = e, u(w) =
w
1
pa
with
> 1 (thus, implying h(u) = u ), and c(s; e) = s 1 e
with
> 1, where s 2 fsa ; sc g and
Prfs = sa g. With these functional forms, we set the model parameters as follows
= 1:6,
= 1:2, sa = 0:2, pa = 0:85 and u = 1. The values for r are chosen in the interval [0:11; 0:99] where
the value of the resulting optimal contract is positive if P implements positive e¤ort.
To solve for the optimal contract for a particular set of parameters ( ; ; sa ; ps ; u; r), we substituted for
1
and
2,
as derived above from (82)-(83), into (84)-(86) and obtained a system of
3 non-linear equations in 3 unknowns (ua ; un ; ea ) amenable to be solved numerically. We veri…ed
that the solution to this system corresponds to positive values for ua , ea ,
rua
c(sa ; e)
0 and pa y (e)
rpa h (ua )
(1
1
and
2,
and that
r) h (un ) > 0, i.e., that P has a weakly positive
expected payo¤ from the resulting contract. Since (i ) P’s expected payo¤ is strictly positive at the
unique critical point of the Lagrangian that we obtained, (ii ) the set of P’s feasible expected payo¤s
is bounded from above by the …nite expected payo¤ from the …rst-best case of full information,
(iii ) P’s expected payo¤ is continuous as a function of the contract variables, we conclude that the
51
It can be shown that V 0 (r) = pa [h0 (ua ) ua
h (ua )]
[h0 (un ) un
51
h (un )], which cannot be signed generically.
critical point corresponds to a global maximum.
Appendix A11. Proof of Proposition 20
If P can credibly promise to make a payment, w3c , when an audit is performed and it reveals that
n
o
A exerted no e¤ort, P’s problem is to select a contract sb 2 [s; s]; w3n ; w3c ; fe3 (s); w3 (s)gs2[s;bs] to
solve the problem
max
sb;wn ;wc ;fe(s);w(s)gs2[s;b
s]
Z
s
sb
[y (e(s))
rw(s)] f (s)ds
s.t. s 2 arg max [ru (w(e
s))
r [u (w(s))
wc ]
max [r [u (w(e
s))
Z
s
sb
se2[s;b
s]
[ru (w(s))
se2[s;b
s]
c (s; e (s))
wc ]
r [1
F (b
s)] wc
c (s; e (e
s))]
(89)
0, for all s 2 (b
s; s]
r) u (wn ) + r [1
(87)
(88)
0, for all s 2 [s; sb]
c (s; e (e
s))]
c (s; e (s))] f (s)ds + (1
r) wn
(1
(90)
F (b
s)] u (wc )
u.
(91)
Making the same transformations as in the solution for the optimal contract without commitment,
and denoting by uc
u (wc ), the optimal control problem for this case is the same as the one de…ned
by (18)-(23), only that un is replaced in (22) by unc (b
s)
is replaced by r [u (b
s)
uc ]
c (b
s; e (b
s))
u
(1
r)un
r [1
F (b
s)] uc , and (23)
0 and = 0 if sb < s. The Hamiltonian H3 associated with
this problem takes the same form as H de…ned in (24). The corresponding necessary conditions
elicited by the Pontryagin’s Maximum Principle are the same as those in (25)-(32), only that (32)
is replaced by
0, with
= 0 and sb = s if r [u (b
s)
uc ]
c (b
s; e (b
s)) > 0
(92)
Next, condition (33) is replaced by
H3 (e (b
s) ; u (b
s) ; v (b
s) ; x (b
s) ;
s) ;
1 (b
s) ;
2 (b
s) ; sb)
3 (b
52
3 ru
c
f (b
s)
0, and = 0 if sb < s
(93)
Finally, denoting by V3 (un ; uc ) the optimal value function of this optimal control problem, as a function of the variables un and uc , the following …rst order conditions
0 and
d
duc
[V3 (un ; uc )
r [1
d
dun
[V3 (un ; uc )
(1
r)h(un )] =
F (b
s)] h(uc )] = 0 are necessary. The …rst of these two conditions im-
plies by the same argument as in section 3.3 that
hand, the second condition implies
[1
3 (s)
nc
s) @u
3 (b
@uc
r
F (b
s)] h0 (uc ) = [1
= h0 (un ), for all s 2 [s; sb]. On the other
F (b
s)] h0 (uc ) = 0, i.e.,
r [1
F (b
s)] h0 (un )
(94)
By the same argument as in the proof of Lemma 10, it follows that (70) must hold in this
situation as well. Substituting in (70) the value of
from (94), we conclude that the following
condition, which is the counterpart of (36), must be satis…ed by the optimal contract
Z
s
sb
F (b
s)] h0 (uc )
h0 (u (s))f (s)ds + [1
h0 (un ) = 0
(95)
Denoting by V3 (r) the value of the optimal contract as a function of r, by the Dynamic Envelope
Theorem and employing the fact
V30 (r)
=
Z
s
sb
3 (s)
= h0 (un ), we have that
@
H3 (e(s); u(s); v(s); x(s); s) f (s)ds
@r
s)
3 (b
@ nc
u
@r
@
@
f(1 r) h(un ) + r [1 F (b
s)] h (uc )g +
fr [u(b
s) uc ] c(b
s; e(b
s))g
@r
@r
Z sb n
o
s 0
0 n
=
h(u(s))f (s)
2 (s) 2 c (e(s))x(s) + h (u )u(s)f (s) ds
r
s
=
Z
h0 (un ) fun
s
sb
F (b
s)] h (uc ) + [u(b
s) uc ]
Z sb
Z s
0 n
0
h(u(s)) + h (u )u(s) f (s)ds +
u (s)
h0 (un ) h0 (u( )) f ( )d
+h(un )
[1
F (b
s)] uc g + h(un )
[1
s
[1
F (b
s)] h (uc )
where for the last equality, we substituted
h0 (un ) fun
2 (s)
r
=
53
Rs
s
ds
s
[1
[h0 (un )
F (b
s)] uc g + [u(b
s)
uc ]
h0 (u( ))] f ( )d and 1r ce (s; e(s)) x(s) =
u0 (s). Integrating by parts, as in the proof of lemma 17, we have that
Z
s
sb
0
u (s)
Z
s
0
n
0
h (u )
h (u( )) f ( )d
Z
ds =
s
s
sb
u(s) h0 (un )
h0 (u(s)) f (s)ds
Z
h0 (u(s)) f (s)ds
+u(b
s)
s
Since by (95), we have
V30 (r)
=
Z
s
sb
R sb
s
[h0 (un )
Z
h(u(s)) + h (u )u(s) f (s)ds
s
n
s) [1
+h(u ) + u(b
Z sb
=
h0 (u(s))u(s)
0
c
sb
u(s) h0 (un )
h (un )
0
F (b
s)] h (u )
h(u(s)) f (s) ds + u(b
s) [1
h0 (un )
F (b
s)] [h0 (uc )
h0 (u( ))] f ( )d = [1
n
0
sb
h0 (un )], it follows that
h0 (u(s)) f (s)ds
h0 (un ) fun
F (b
s)] h (uc )
[1
F (b
s)] uc g + [u(b
s)
[1
F (b
s)] h0 (uc )
h0 (un )
uc ]
F (b
s)] h (uc )
[1
s
+h(un ) h0 (un ) fun [1 F (b
s)] uc g + [1 F (b
s)] h0 (un )
Z sb
=
h0 (u(s))u(s) h(u(s)) f (s) ds + [1 F (b
s)] h0 (uc )uc
h0 (uc ) [u(b
s)
uc ]
h (uc ) + h(un )
n
h0 (un )u(96)
s
where for the second equality, we substituted the value of
derived from (94). Given then the
results in (95) and (96), by the same argument as in the proof of proposition 18, it follows that
V30 (r) > 0. This completes the proof of proposition 20.
Appendix A12. Proof of Proposition 22
Di¤erentiating the equality in (54) with respect to s, we have
(57) and (59), it follows that (s) = h0 (u (s))
equals a nonnegative constant
0
2 (s)
= rh0 (un (s))f (s)
r
3
1 r 0
2 (s)
r
+
0
4 (s)
= 0. Employing
h0 (un (s)). Next, (58) and (60) imply that
3 (s)
for all s. Employing these results into (57), we conclude that
3 f (s).
Integrating this equality between s and any arbitrary s 2 [s; s],
Rs
and accounting for 2 (s) = 0 from (60), we have then 2 (s) = r s [h0 (un (t))
3 ] f (t)dt. Applying
Rs 0 n
this result at s, where 2 (s) = 0, it follows that 3 (s) = 3
s h (u (t))f (t)dt for all s 2 [s; s].
Moreover, since h0 > 0 it follows that
3
> 0 and thus that (44) is satis…ed with equality at
54
optimum. Collecting these …ndings into (61), we obtain condition (62) from the text of proposition
22 once we account for the fact that h is strictly convex and thus h0 is strictly increasing.
Di¤erentiating now the equality in (53) with respect to s, as in (71), and then employing (55)
and (57) to substitute for
0
1 (s)
0
2 (s),
and
y 0 (e) f (s) + (s) ce (s; e (s)) f (s)
+ rh0 (u (s))f (s)
2 (s)
1
cee (s; e (s)) x (s) +
r
3 (s) rf (s)
Substituting into (97) the expression for
0
h (u(s))ce (s; e (s)) f (s) +
Z
s
0
n
2 (s)
h (u ( ))
s
1
ce (s; e (s)) +
r
3 (s) ce (s; e (s)) f (s)
2 (s)
derived above, we obtain
Z
s
h0 (un (t))f (t)dt f ( )d
ces (s; e (s)) = y 0 (e)f (s)
s
which can be rewritten as in (63) in the text of proposition 22. Note also here that since
r [h0 (un (s))
+
1
1
ces (s; e (s)) + 2 (s) cee (s; e (s)) e0 (s) =
r
r
1
y 0 (e)f (s) + h0 (u (s))f (s)ce (s; e (s)) + 2 (s) ces (s; e (s)) = 0
(97)
r
(s)rf (s)
=
we have
3 ] f (s),
2 (s)
= 0,
2 (s)
= 0, and
the proof of remark 12, we can conclude that
d
ds
2 (s)
0
2 (s)
=
[h0 (un (s))] < 0, by the same argument as in
> 0 for all s 2 [s; s] to prove remark 23.
To evaluate the e¤ect of r on the optimal value of the contract with communication, which we
denote here by V+ (r), employing the Dynamic Envelope Theorem, we have
V+0 (r)
Z s
@
L+ (e(s); u(s); un (s); v(s); x(s); k(s); 1 (s); 2 (s); 3 (s); 4 (s); (s); s) ds
=
@r s
)
Z s(
n
[ h(u(s)) + h(un (s)) + (s)u(s) + 3 (s) u(s)
3 (s) u (s)] f (s)+
=
ds
+ 2 (s) r12 [k(s) ce (s; e(s))x(s)]
s
#
Z s ("
h(u(s)) + h(un (s)) + u(s) [h0 (u (s)) h0 (un (s))]
1
n0
=
f (s) +
2 (s) u (s)
n
r
+
[u(s)
u
(s)]
s
3
where for the third equality we used (49) to substitute for k(s)
k(s) = un0 (s) as well as the result derived above for (s).
55
u0 (s)
ce (s; e(s))x(s) and then the fact
)
ds
Integrating by parts the second term in the expression for V+0 (r) derived above, we have
Z
s
s
=
1
r
n0
2 (s) u (s)
n
[u (s)
u(s)]
u0 (s) ds =
Z
Z
s
un0 (s)
u0 (s)
s
s
0
n
h (u (t))
Z
s
h0 (un (t))
3
f (t)dtds
s
Z
s
3
f (t)dt
s
s
[un (s)
u(s)] h0 (un (s))
f (s)ds
3
s
s
Z
=
s
[un (s)
u(s)] h0 (un (s))
3
f (s)ds
s
where for the last equality we used the fact that
3
=
term in the previous expression equals zero.
Rs
s
h0 (un (t))f (t)dt to conclude that the …rst
Substituting this result into the expression for V+0 (r) we obtain
V+0 (r) =
=
Z
s
s
Z
s
"
h(u(s)) + h(un (s)) + u(s) [h0 (u (s))
+
3 [u(s)
u(s)h0 (u (s))
un (s)]
[un (s)
h0 (un (s))]
u(s)] [h0 (un (s))
3]
#
f (s)ds
h(u(s))
un (s)h0 (un (s)) + h(un (s)) f (s)ds
uh0 (u)
h(u) we have
s
Now, note that for the function
(u)
0 (u)
= h00 (u), which is strictly
positive, as argued above. Therefore, using the fact that u(s) un (s), we conclude that V+0 (r) =
Rs
(un (s))] f (s)ds 0. This completes the proof of proposition 22.
s [ (u(s))
The generalized Legendre-Clebsch condition for this problem is a combination of the corresponding condition for problems with multiple controls (see Theorem 6.2 in Krener (1977)) and the
condition for problems with state constraints (see conditions (84) or (85) Seywald and Cli¤ (1993)).
Thus, on intervals in [s; s] on which the constraint in (52) does not bind, the condition stated in
Krener (1977) applied to our problem requires that the following matrix52
0
@
@
@x
@
@k
h
d
2
h ds2
d
ds2
@L+
@x
@L+
@x
i
i
@
@x
@
@k
h
d2
2
ds
h 2
d
ds2
@L+
@k
@L+
@k
i 1
i A
(98)
52
We use here the fact that we have a pure state constraint and thus the …rst order derivatives with respect to the
control variables of the Lagrangian L+ and Hamiltonian H+ L+
[ru c(s; e)] are identical.
56
be symmetric and negative semide…nite when evaluated at the candidate solution, where we accounted for the fact that x and k appear in the second order derivatives with respect to s of
and
@H+
@k .
@H+
@x
On the other hand, on intervals in [s; s] where the constraint in (52) binds, applying
the result from Seywald and Cli¤ (1993) and other standard results from multivariate constrained
optimization (for instance, Theorem 19.7 on page 461 in Simon and Blume (1994)), it follows that
it is necessary that the following determinant be nonnegative at the candidate solution53
0
@ d
@x ds
@ d
@k ds
[ru
c(s; e)]
[ru
c(s; e)]
We have already showed that
d
ds
@L+
@x
@ d
[ru
@x ds h
d
@
@x h ds2
@
d2
@k ds2
=
c(s; e)]
i
@L+
@x
@L+
@x
i
@ d
[ru
@k ds h
d2
@
@x h ds2
@
d2
@k ds2
r) h0 (u(s))f (s) + (1
r) (s)f (s) + (1
@L+
@k
@L+
@k
(99)
i
y 0 (e)f (s)+h0 (u (s))f (s)ce (s; e (s))+
in (97). On the other hand, employing (57) and (59), we have
(1
c(s; e)]
i
d
ds
@L+
@k
=
1
2 (s) r ces (s; e (s))
1 r 0
2 (s)
r
+
0
4 (s)
=
r) h0 (un (s))f (s). Then, suppressing arguments of the
various functions when there is no risk of confusion, we have
d2
ds2
@L+
@x
1
cees x + K1
r
r
1
1
1 r
k + ce x f ce + h0 (u)cee x + 2 cees x + K2
r
r
r
=
y 00 xf + h00 (u)u0 f ce + h0 (u)cee x +
=
y 00 xf + h00 (u)
0 1
2 c
+
2
where for the second equality, we used (49) and (57). K1 and K2 are terms that do not depend on
either x or k. On the other hand, (K3 and K4 again do not depend on the controls) we have
d2
ds2
53
@L+
@k
=
(1
r) h00 (u)u0 f + (1
=
(1
r) h00 (u)
1
r
r) h00 (un )un0 f + K3
1
k + ce x f + (1
r
r
r) h00 (un )kf + K4
The function g from the result in Seywald and Cli¤ (1993) (de…ned in equation (37)) is here
57
d
ds
[ru (s)
c(s; e)].
where for the second equality, we used (49) and (48). It follows then that
@ d2
@x ds2
@ d2
@k ds2
@L+
@x
@L+
@x
@ d2
@k ds2
@L+
@k
Finally, we have
d
ds
[ru
=
r)2
(1
=
c] = ru0
@ d
[ru
@x ds
1
y 00 f + h00 (u) (ce )2 f + h0 (u)f cee +
r
2
@ d
@L+
1 r 00
=
h (u)f ce
2
@x ds
@k
r
=
r
cs
c(s; e)] = 0 and
(1
r) k
@ d
[ru
@k ds
1
cees
r
r) h00 (un )f
h00 (u)f + (1
ce x =
2
cs using (49). Therefore,
c(s; e)] =
(1
r)
2
@L+
@ d
< 0, whereas the determinant
Now, assuming again that cees 0, it follows that @x
@x
ds2
h
i
ih
2
of the matrix in (98) is y 00 f + h00 (u) 1r (ce )2 f + h0 (u)f cee + 2 1r cees (1 rr) h00 (u)f + (1 r) h00 (un )f
2
1 r 00
r h (u)f ce
> 0. Therefore, the matrix in (98) is indeed negative semide…nite. On the other
h
i
hand, the determinant in (99) equals (1 r)2 y 00 f + h00 (u) 1r (ce )2 f + h0 (u)f cee + 2 1r cees > 0.
We conclude thus that the generalized Legendre-Clebsch condition is indeed satis…ed.
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