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Working Paper Series

Repeated Moral Hazard with Effort
Persistence

WP 08-04

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Arantxa Jarque
Federal Reserve Bank of Richmond

Repeated Moral Hazard with Effort Persistence∗
Arantxa Jarque†
FRB— Richmond
and U. Carlos III de Madrid
July 2008
Federal Reserve Bank of Richmond Working Paper 08-04

Abstract
I study a problem of repeated moral hazard in which the effect of effort is persistent over
time: each period’s outcome distribution is a function of a geometrically distributed lag of past
efforts. I show that when the utility of the agent is linear in effort, a simple rearrangement
of terms in his lifetime utility translates this problem into a related standard repeated moral
hazard. The solutions for consumption in the two problems are observationally equivalent,
implying that the main properties of the optimal contract remain unchanged with persistence.
To illustrate, I present the computed solution of an example.
Journal of Economic Literature Classification Numbers: D30, D31, D80, D82.
Key Words: mechanism design; repeated agency.

∗

I would like to thank Harold Cole, Huberto Ennis, Juan Carlos Hatchondo, Nicola Pavoni and especially
Hugo Hopenhayn and Iván Werning for very useful suggestions and discussions. I would also like to thank
an anonymous referee for feedback on an earlier version of the article. The paper benefited from comments
by seminar audiences at CEMFI, the University of Iowa, the Federal Reserve Bank of Richmond, and the
Meetings of the SED 2005. I thank the Ministry of Education, Culture and Sports of Spain for its support
through the Research Project SEJ2004 - 08011ECON. This article is based on part of my dissertation; I
gratefully acknowledge financial support from the Banco de España during my Ph.D.
†
Email: Arantxa.Jarque@rich.frb.org.

1

1

Introduction

The moral hazard literature has pointed out repeatedly the importance of generalizing the current
models of asymmetric information to setups in which either the hidden endowment of the agent
or the effect of the agent’s effort are correlated in time, or “persistent.” The difficulty of this
generalization lies on the fact that, with persistence, incentives for deviation in a given period
depend on the history of private information of the agent (what is sometimes referred to as the
“joint deviations” problem). As with the case in which the agent has access to hidden savings
(Abraham and Pavoni, 2006, and Werning, 2001), persistence implies that there is no common
knowledge of preferences at the beginning of each period. Hence, the standard recursive formulation
with continuation utility as a state variable (Spear and Srivastava, 1987) is not valid in the presence
of persistence.
In this paper, I characterize a class of agency problems of repeated moral hazard (hidden effort)
with persistence for which a simple solution exists. I model persistence by assuming that the effort of
the agent affects not only the current output distribution but also the distribution of output in every
future period. The class of problems that I characterize satisfies two key assumptions: the agent is
risk averse with linear disutility of effort, and output depends on the sum of depreciated past efforts.
I show that the (constrained) optimal contract can be found by solving an auxiliary problem — a
related repeated moral hazard problem without persistence. The two problems are observationally
equivalent. Hence, the intertemporal properties of consumption under persistence do not differ from
those of the optimal contract in problems without persistence. The (unobservable) effort sequences
do typically exhibit different properties when effort is persistent. I present a numerical example
that illustrates the main conclusions.

1.1

Related Literature

The existing literature on repeated moral hazard with persistence has not yet generally characterized
the effect of persistence on the properties of effort and consumption in the optimal contract. There
are, however, some interesting examples for which a solution can be found. Fernandes and Phelan
(2000) provide the first recursive treatment of agency problems with effort persistence. In their
paper, the current effort of the agent affects output in the same period and in the following one.
Their setup is characterized by three parameters: the number of periods for which the effect of
effort lasts, the number of possible effort levels and the number of possible outcome realizations. All
three parameters are set to two, and this makes their formulation and their computational approach
feasible. The curse of dimensionality applies whenever any of the three parameters is increased.
Moreover, no results are given in their paper on how the properties of the optimal contract differ
from the case without persistence. In my formulation, I allow for a continuum of efforts, infinite
persistence and multiple outcomes, but under my assumptions the recursive formulation of the
problem turns out to be particularly simple. Mukoyama and Sahin (2005) and Kwon (2006) show
in a discrete two—effort model that, if persistence is high, it may be optimal for the principal to
perfectly insure the agent in the initial periods. The results that I present here apply to a different
class of problems in which past efforts influence output less than current effort; in this context,

2

perfect insurance is never optimal.

2

Model

I study a T —period relationship between a risk neutral principal and a risk averse agent, where
T may be infinite. I consider the repeated moral hazard (RMH) problem that arises because the
effort of the agent is unobservable. I generalize the standard RMH problem by considering the case
of “persistent” effort : effort carried out by the agent each period affects current as well as future
output distributions.
I assume that both parties commit to staying in the contract and that the principal has perfect
control over the savings of the agent. They both discount the future at a rate β. I assume that the
agent has additively separable utility that is linear in effort.
A1 The agent’s utility is given by U (ct , et ) = u (ct )−et , where u is twice continuously differentiable
and strictly concave and ct and et denote consumption and effort at time t, respectively.
There is a finite set of possible outcomes each period, Y = {yi }N
i=1 , and the set of histories of
t
outcome realizations up to time t is denoted by Y , with typical element y t = (y1 , ..., yt ) . Histories
of outcomes are assumed to be common knowledge. I assume both consumption and effort lie in a
compact set: ct ∈ [0, yt ] and et ∈ E = [0, emax ] for all t.
To capture the persistence of effort, I model the probability distribution of current output as a
¢
¡
function of the whole history of past efforts, and I denote it by π yt |et , where et is the history of
¢
¡
effort choices up to time t. I assume the distribution has full support: in every period t, π yt |et > 0
for all yt ∈ Y and for all et ∈ E t .
The strategy of the principal consists of a sequence of consumption transfers to the agent
© ¡ ¢ª
contingent on the history of outcomes, c = c y t , to which he commits when offering the contract
at the beginning of time. The agent’s strategy is a sequence of period best—response effort choices
© ¡ ¢ª
that maximize his expected utility from t onward, given the past history of output: e = e y t .
At the end of each period, output yt is realized according to the distribution determined by the
¡ ¢
effort choices up to time t, and the corresponding amount of consumption c y t is given to the
agent.
© ¡ ¢ª
© ¡ ¢ª
and e∗ = e∗ yt
An optimal contract is a pair of contingent sequences c∗ = c∗ yt
that maximize the expected discounted difference between output and the promised contingent
payments, subject to two constraints: the Participation Constraint (PC), which states that the
initial expected utility of the agent in the contract should be at least as large as his outside utility,
w0 , and the Incentive Constraint (IC), which states that the sequence e∗ should be a solution to the
maximization problem faced by the agent, given the contingent consumption transfers established
by c∗ :
max

T X
X

e∈E T ,
t=1 yt
c: c(yt )∈[0,yt ] ∀t

t
³ © ¡
¡ ¢¤ Y
£
¢ªτ ´
β t−1 yt − c yt
π yτ | ej y j−1 j=1
τ =1

s.t.

3

∞ X
X
t=1 y t

t
³ © ¡
£ ¡ ¡ ¢¢
¡
¢¤ Y
¢ªτ ´
β t−1 u c y t − et yt−1
π yτ | ej yj−1 j=1 ≥ w0

e ∈ arg max
b
e∈E T

(PC)

τ =1

T X
X
t=1 yt

t
³ © ¡
£ ¡ ¡ ¢¢
¡
¢¤ Y
¢ªτ ´
β t−1 u c yt − ebt y t−1
π yτ | ej y j−1 j=1

(IC)

τ =1

Following the influential work of Spear and Srivastava (1987), the usual procedure in the literature to solve for the optimal contract in the standard RMH problem without persistence is to
write the problem of the principal recursively using the continuation utility of the agent as a state
variable. In this recursive formulation, the IC is a per—period constraint on the level of effort. If
the problem of the agent stated in the IC is concave in his effort choice, the corresponding first
order condition is both necessary and sufficient. In such case, the IC can be replaced by this first
order condition and a Lagrangian can be constructed for the problem of the principal.1
In general, it is not possible to follow a similar approach to find the solution to the problem with
persistence . When effort is persistent, incentives for deviation may also depend on the particular
sequence of past and future efforts chosen by the agent.2 Therefore, one needs to check for the
possibility of joint deviations involving effort choices in more than one period; this implies that
the standard recursive formulation is no longer valid, complicating the computation of the optimal
contract.3
In what follows, I characterize a class of problems where the specification of persistence is such
that the optimal contract can be found in a simple way. Within this class, the problem above can
be translated into a standard RMH problem, where the usual recursive tools can be used to derive
the optimal contract. I make the following simplifying assumption about how the effect of effort
persists in time:
A2 The distribution of output depends on a “productive state,” denoted s, determined by the
history of effort choices in the following way:
st =

t
X

ρt−τ eτ ,

(1)

τ =1

were ρ ∈ (0, 1) measures the persistence of effort through its effect on future productive states,
and s0 = 0.
Under A2, st is a sufficient statistic for the history of effort up to time t, and the distribution
of outcomes at time t can now be simply written as π (yt |st ) .4 The limit case of ρ = 0 corresponds
1

This solution procedure is often called the First Order Approach. See Rogerson (1985b) and Jewitt (1988).
For a given continuation contract, the first order condition of the IC problem with respect to et includes the term
∂π (yt+j |{eτ }t+j
τ =t )
for all j periods between t and T. Persistence implies that this derivative depends on {ek }t−1
k=1 and
∂et
that it is non—zero for some or all future j’s.
3
A modified recursive formulation including three state variables may be possible. See Fernandes and Phelan
(2000) for an example of this approach.
4
Under A2 but without A1, any recursive formulation would in general need three state variables (Jarque, 2002).
It is the combination of the two assumptions that further simplifies the problem.
2

4

to the standard RMH problem in which the probability distribution of current outcomes depends
only on current effort. The set of feasible productive states at time t, St , is derived recursively from
the set of feasible efforts and equation (1):
St = [ρst−1 , ρst−1 + emax ] .

(2)

I can express the strategy of the agent using s as a function of the history of outputs, by substituting
¡ ¢
in (1) the corresponding e yt for each t. This allows me to write the expected utility of the agent
© ¡ ¢ªT
in terms of the sequence s = s yt t=1 :
T X
X
t=1 yt

t
£ ¡ ¡ ¢¢ ¡ ¡
¢
¡
¢¢¤ Y
¢¢
¡
¡
β t−1 u c yt − s yt−1 − ρs y t−2
π yτ |s yτ −1 .
τ =1

After a simple rearrangement of terms in the above expression, the problem of the principal can
be written as follows:
T X
X

max

s∈S T ,
t=1 y t
c: c(yt )∈[0,yt ] ∀t

t
¡ ¢¤ Y
¡
¡
£
¢¢
β t−1 yt − c y t
π yτ |s yτ −1 ,

(OP)

τ =1

s.t.

w0 =

T X
X
t=1 yt

s ∈ arg max
b
s∈S T

t
£ ¡ ¡ ¢¢
¡
¢¤ Y
¢¢
¡
¡
β t−1 u c yt − (1 − βρ) st y t−1
π yτ |s yτ −1
τ =1

T X
X

β

t=1 y t

t−1

t
¡
¡
£ ¡ ¡ t ¢¢
¡ t−1 ¢¤ Y
¢¢
π yτ |b
s yτ −1 .
u c y − (1 − βρ) sbt y
τ =1

Simple inspection of this formulation of the problem shows that the solution for the optimal
contract, c∗ , and the sequence of productive states that it implements, s∗ , are also a solution for
the optimal consumption and effort sequences in an RMH problem without persistence in which
the utility of the agent is given by

and the distribution function is

ct ) − (1 − βρ) eet ,
U (e
ct , eet ) = u (e
et ) = π (yt |e
et ) ,
π
e (yt |e

where eet and e
ct denote the choice variables in this new problem. I refer to this related moral hazard
e∗ . The solution for effort in the
problem as the auxiliary problem (AP), with solution e
c∗ and e
original problem (OP) can be recovered from e
e∗ according to
e∗1 = s∗1 = ẽ∗1 ,
e∗t

=

s∗t

− ρs∗t−1

(3)

=

ẽ∗t

− ρẽ∗t−1

t > 1.

Some additional technical conditions are needed for the solutions of problems OP and AP to
coincide:
5

A3 The expected utility of the agent for a given consumption scheme c is concave in each st .
A4 The distribution function π (yt |st ) satisfies
∂E [y]
= +∞ and
s→0 ∂s
lim

lim

s→emax

∂E [y]
= 0.
∂s

These two assumptions ensure that, in problem AP, the solution for effort is always interior
and the expected utility of the agent is a concave function of his effort choice. Hence, the IC can be
substituted by the implied first order conditions and problem AP can be solved with the standard
recursive techniques.5
In any RMH problem with linear disutility of effort, effort choices must lie in a closed set; this
ensures that the domain of the problem is not empty by putting a limit on the utility that the
agent can get by choosing a very low effort.6 In problem OP, I assume a closed set E = [0, emax ] ,
which translates into a closed St . Ideally, in problem AP we would like to impose eet ∈ St . However,
the set St is endogenously determined according to Eq. (2), given the effort solution for problem
OP. In practice, the auxiliary problem must be solved using an exogenously determined domain
for effort, such as
ft = [0, emax ] ∀t.
E
ft shifted to the right by
At any t after the first period, the true set St corresponds to the set E
f
ρst . To establish that Et is a good alternative set, one needs to check two things: first, that the
ft ∩ St = [ρst , emax ] , so that s∗ is in fact a feasible sequence in problem AP;
solution e
e∗ belongs to E
and second, that the different domain does not make the solutions differ across the two problems.
ft , both conditions are satisfied under a restriction on the
I now argue that, for the proposed set E
values of ρ.
First, consider the feasibility of s∗ . That eet < emax for all t is implied by the second condition
e∗ to problem AP to determine
in A4. That eet > ρst−1 for all t can be checked using the solution e
if ρ satisfies the following condition:7
¡
¢
¡ ¢
¡
¢
∀ y t , yt+1 and ∀t.
s∗ yt , yt+1 ≥ ρs∗ yt
Note that this condition rules out combinations of parameters that would imply negative effort
recommendations, which would be necessary if the drop in the required s from one period to the
next was too big to be achieved just by letting the current s depreciate. In particular, I denote as
ρ∗max the maximum persistence that guarantees that effort is positive at all times:
à ¡
¢!
∗ yt , y
e
e
i
min
ρ∗max =
,
ee∗ (yt )
yt ∈Y T ,yi ∈Y
5

Kocherlakota (2004) studies a related repeated moral hazard problem in which the agent has access to hidden
savings. He demonstrates that the problem of the agent may fail to be globaly concave in effort and savings when the
disutility of the agent is linear. Effort persistence could, in general, also imply that the problem in the IC of problem
OP fails to be globally concave in effort across periods. Here, however, I only require the IC of the auxiliary problem
AP to be concave in the current period’s effort; since this problem is a standard repeated moral hazard, the usual
conditions on the probability function suffice to guarrantee A4 (see Rogerson (1985b) and Jewitt, 1988.)
6

I thank an anonymous referee for raising this issue.

7

Numerical simulations suggest that this condition is fulfilled by a wide range of parameter values.

6

and study only problems with ρ < ρmax .
Second, consider the effect of the difference in the domains in the solution. Under the concavity
and interiority implied by A3 and A4, the deviations added and those eliminated when using
et are always dominated by deviations involving effort choices closer to the optimum,
domain E
et and St ; hence, the solutions to the two problems are in fact the same
which is included both in E
et as the domain for eet .
when one uses E
I summarize the above discussion in the following proposition:
Proposition 1 Consider a repeated moral hazard problem between a risk neutral principal and a
risk averse agent with utility u (c) − e in which the distribution of output is given by π (yt |st ) , where
P
st = tτ =1 ρτ −1 eτ and et ∈ [0, emax ] ∀t. If the expected utility of the agent for a given contract is
concave in st for all t and the optimal choice for st is always interior, there is an auxiliary repeated
moral hazard problem where the agent’s utility is given by u (e
c) − (1 − βρ) ee and the distribution of
et ) = π (yt |e
et ) which can be solved recursively. Furthermore, if ρ ≤ ρ∗max ,
output is given by π
e (yt |e
we have that (i) the optimal consumption coincides in both problems and (ii) the optimal sequence
of effort in the original problem, e∗ , can be obtained from the solution to the auxiliary problem, ee∗ ,
using the following system of equations:
e∗1 = s∗1 = ẽ∗1 ,

e∗t = s∗t − ρs∗t−1 = ẽ∗t − ρẽ∗t−1 t > 1.
In the presence of persistence, when the agent increases effort today — incurring some disutility
today — he can achieve the same productive state tomorrow with less effort, therefore saving some
disutility tomorrow. This intertemporal trade—off implies, in general, that persistence introduces
history dependence. The assumption of the productive state being a geometrically discounted sum
of past efforts, combined with linearity in the disutility function, make the marginal cost and the
marginal benefit of effort depend only on s, simplifying the history dependence.8
The intuition for the equivalence between problem OP and AP relies on the fact that with
linear disutility, the actual period in which the agent experiences the cost of effort is not important.
This means that the principal can solve the problem by choosing directly the optimal level of s in
every period, modifying accordingly the utility function: persistence of effort can be understood as
a lower marginal cost of acumulating s.
An important implication of the relationship I established between the problem with persistence
and the auxiliary problem is that the sequence of contingent consumption will be exactly the same
in both problems. This makes the two problems observationally equivalent. The results found in
the moral hazard literature on the long run distribution of utilities and the individual consumption
paths will also hold in the environment studied here with persistence and linear disutility.9
8

Farhi (2006) independently proposes a similar set of simplifying assumptions in a capital taxation problem.
See Rogerson (1985a) for a seminal contribution to the study of the properties of consumption dynamics in the
optimal contract. See Spear and Srivastava (1987), Thomas and Worral (1990), Phelan and Townsend (1991), Phelan
(1994, 1995), and Atkeson and Lucas (1995) for long run results and applications.
9

7

3

An example

Consider a problem in which there are only two possible outcomes each period, yH and yL . The
utility of the agent is
c1−σ
− e.
U (ct , et ) =
1−σ
The probability of the high outcome depends on s in the following way:
√
π (yH |st ) = 1 − exp {−r st } ∀t.
Under these specifications, A1—A4 are satisfied.
I compute the numerical solution to the optimal contract for ρ = 0.4, and compare it to the
solution under the same parameters but with no persistence of effort, i.e. ρ = 0.10 As seen in Figure
1, for a given w, the levels of consumption and continuation utilities are lower with persistence.
Also, the set of feasible and incentive compatible values of w is bigger with persistence (higher
values are included). As seen in the last plot of Fig. 1, for a given w, the value to the principal is
higher with persistence. Also, higher levels of s (w) are implemented (Fig. 2, bottom): consistently
with what we expect from our knowledge of RMH problems, a lower marginal disutility means the
solution ee (w) in the AP problem is higher, which translates into higher s (w) in problem OP.
The solution for effort is plotted in Fig. 2 (top). When ρ = 0, we have e (w) = s (w) ∀w.
When ρ = 0.4, however, effort depends on both the current and the last period’s promised utility
(see Eq. 3). In the two—output case studied in this numerical example, there are (at most) two
values of last period’s w that are compatible with a given w today, each corresponding to a different
output realization yesterday. This allows me to label the effort solution as eL (w) and eH (w) and
to plot them as a function of w only.11 The figure shows that, for a given w, effort is generally
slightly lower with persistence than without. Note, however, that in the first period e1 = s1 (w0 )
since s0 = 0, which creates a front—loading effect on effort. Fig. 3 plots the average time path of
both s and e over 7,000 realizations. The persistence of effort makes it efficient to build up the
optimal level of s from the first period. If the disutility of effort was strictly convex, I conjecture
that this front—loading force would still be present, although the build up of s would presumably
take place over several periods. With convex disutility, however, the contract may exhibit quite
different properties because of the joint deviations problem. The case characterized here can be a
useful benchmark for future research in this subject.
10

The rest of the parameters are: σ = 1/2, yH = 25, yL = 8, r = 0.8, β = 0.85, T = ∞. Note that the assumption
ct ≤ yt puts a lower bound on s for high values of continuation utility, and hence the condition ρ ≤ ρmax is satisfied
in the parameterization of the example.
11
Generally, effort is a function of two states, that is, e (w, w−1 ) , where w−1 denotes the previous period continuation
utility. Hence, ei (w) , for i = L, H, is the effort recommendation today if the current promised utility is w, and it
is the case that w = wi (w−1 ) , where wi (·) is the continuation value that the contract prescribes for the previous
period contingent on output i. Some values of w are never chosen as a continuation utility (corresponding to the
discontinuities in ei (w)) because the numerical solution uses a discretized state space. Fitting a polynomial through
the solutions, for example, produces a smooth and continuous effort function.

8

Contingent Consumption
25
cH, ro = 0
cH, ro = 0.4
cL, ro = 0
cL, ro = 0.4

ch(w),cl(w)

20
15
10
5
0

0

10

20

30

40

50

40

50

w
Contingent Continuation Values
50
wH, ro = 0
wH, ro = 0.4
wL, ro = 0
wL, ro = 0.4

wH(w),wL(w)

40
30
20
10
0

0

10

20

30
w

Value function
120
ro = 0
ro = 0.4

100

V(w)

80
60
40
20
0

0

10

20

30

40

50

w

Figure 1: Computed solution for consumption, continuation utility and value function.

9

Effort
1.4
e(w), ro = 0
eH(w), ro = 0.4
eL(w), ro = 0.4

1.2

e(w)

1
0.8
0.6
0.4
0.2
0

0

10

20

30

40

50

w

Accumulated effort
2

ro = 0
ro = 0.4

s(w)

1.5

1

0.5

0

0

5

10

15

20

25
w

30

35

40

45

Figure 2: Computed solution for effort and accumulated effort.

10

Average Effort and s path, w0 = 7.0352
e(t) = s(t), ro = 0
s(t), ro = 0.4
e(t), ro = 0.4

2
1.8
1.6
1.4

e(t)

1.2
1
0.8
0.6
0.4
0.2
0

0

50

100
time

150

200

Figure 3: Example of an average 200 period long path for effort and accumulated effort, with 7,000
different path draws.

11

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12