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Economic Theory 19, 311–331 (2002)
Information and experimentation
in short-term contracting
Thomas D. Jeitschko1and Leonard J. Mirman2
1Department of Economics, Michigan State University, East Lansing, MI 48824, USA
(e-mail: jeitschk@msu.edu)
2Department of Economics, University of Virginia, Charlottesville, VA 22903, USA
Received: February 15, 2000; revised version: August 29, 2000
Summary. The impact of information dissemination and experimentation on
dynamic adverse selection in noisy agency relationships is examined. Significant
deviations in terms of equilibrium actions and payments occur, when compared to
deterministic environments. Information dissipates slowly, so payments to agents
who stand to lose informational rents over time are lower than compared to
deterministic settings. Moreover, the principal manipulates the agent’s actions to
affect the informativeness of the signal. Thus, the principal trades-off lower initial
payments with higher informational rents later. Simultaneously, the principal
manipulates the signal distribution to enhance his ability to learn about the agent’s
type.
Keywords and Phrases: Bayesian learning, Experimentation, Agency.
JEL Classification Numbers: D8, C73.
1 Introduction
Ever since the pioneering works of Holmstr¨
om (see Hart and Holmstr¨
om (1987),
and the literature cited therein) and Mirrlees (1976), contracts between parties
when there is hidden information and hidden actions have been extensively stud-
ied. Stemming from this initial work, two extensions of agency relationships have
been given particular attention, bringing this theory closer to many real-world
problems. First, contract design when adverse selection and moral hazard are
simultaneously present; and, second, the dynamics of incentive contracts when
We thank Fritz Laux, Farshid Vahid, and an anonymous referee for helpful comments and
discussion.
Correspondence to: T.D. Jeitschko
312 T.D. Jeitschko and L.J. Mirman
relationships are of longer duration. Indeed many, if not most, instances of agency
have elements of both noisy observations and on-going interactions as charac-
teristic features. In this paper we analyze an environment in which both of these
aspects are present. That is, we study a model of dynamic adverse selection in a
stochastic environment.
Specifically, the purpose of this paper is to shed light on the impact of noisy
environments when the principal and the agent interact over time and the agent’s
“type” remains the same over the course of the interaction.1An example illustrat-
ing the significance of these issues is given in Jeitschko, Mirman, and Salgueiro
(JMS) (2002) for the context of the ratchet effect in procurement settings. Three
central issues arise in such environments. The first issue is the contract between
the principal and the agent. In particular, when the signal generated by the agent
is stochastic, and the signal is the sole basis for the contract, the principal may
not learn the agent’s type. Thus, the principal cannot base a payment to the agent
on knowledge of the agent’s type, even in equilibrium. The other issues stem
from the degree to which the principal learns about the agent’s type in the course
of the interaction. Indeed, the principal learns only slowly about the agent’s type.
Since the principal designs the contracts, however, the principal can affect the
flow of information and thus his ability to learn about the agent’s type. The prin-
cipal uses this ability (a) to reduce up-front payments that induce the equilibrium
actions, and (b) to experiment and reduce future informational rents of the agent.
Instances of the simultaneous occurrence of adverse selection and moral haz-
ard have been studied in static settings by Laffont and Tirole (1986), Picard
(1987), and Caillaud, Guesnerie, and Rey (CGR) (1992). The former two pa-
pers incorporate observations or messages in addition to the stochastic signal, in
designing the contract. In the latter paper the case in which only the stochastic
signal is contractible is also considered and there it is shown under which con-
ditions incentive compatible mechanisms can be either fully or approximately
implemented.
The issue of learning in stochastic settings has also been studied in the con-
text of individuals learning about their environments. Examples of this literature
include Prescott (1972), Grossman, Kihlstrom, and Mirman (1977), Fusselman
and Mirman (1993), Mirman, Samuelson, and Urbano (MSU) (1993), and, in the
context of games, Mirman, Samuelson, and Schlee (1994). These papers study
optimal behavior in dynamic stochastic environments. Through observing market
outcomes, agents update their beliefs about parameters affecting their payoffs,
that is, they are able to learn. Moreover, since agents’ actions affect market out-
comes, and thus the signals generated in the market, they can manipulate the
flow of information in the market to enhance their learning, that is, they deviate
from myopic behavior to experiment.
The impact of learning on the rewards paid to the agent in a dynamic
principal-agent context has been previously examined in deterministic environ-
ments. Most notable are the major contributions by Freixas, Guesnerie, and Tirole
1In somewhat related work Baron and Besanko (1984) consider dynamic agency interactions.
There, however, the agent’s type changes stochastically in the course of the interaction.
Information and experimentation in short-term contracting 313
(1985), and Laffont and Tirole (1987,1993). These works have shed light on dy-
namic agency relationships that are governed by short term contracts. They study
two period adverse selection models in a deterministic environment in the context
of regulation. The regulated firm can be one of two possible types – high cost,
or low cost. The principal does not know which type the firm is and can only
contract on an observed cost realization, but not the agent’s actions. Since the
relationship between the firm’s action and the signal is deterministic, any pure
strategy separating incentive contract yields distinct cost realizations of the two
types, and the regulated firm reveals its type. Consequently, the low-cost firm
loses all informational rents in the course of the interaction. In order to yet induce
the low-cost firm to reveal its type, this firm must be paid an additional up-front
bonus. If this bonus is very large, the incentive compatibility constraint of the
high-cost firm may become binding, so that only a semi-separating or pooling
contract is feasible (an implication of the take-the-money-and-run strategy).
In what follows we consider a general agency interaction of two-period du-
ration. There are two possible types of agent and the agent maintains his char-
acteristics over the course of the interaction. In each period the agent generates
a signal through unobservable actions. We suppose that only the observable sig-
nal is contractible. After the first period of the interaction, the principal uses
the signal generated in the first period to learn about the agent’s type, and then
constructs the second period contract on the basis of the updated beliefs.
The assumption that contracts are based solely on the observable signal is
critical because any contract that incorporates additional messages from the agent
to the principal – e.g., a menu-based contract – immediately reveals the agent’s
type. Due to risk-neutrality of both the agent and the principal, the larger message
space thus simply yields the well-understood outcomes of the deterministic case
and does not yield additional insights into short-term contracting in long-term,
stochastic environments.
The reason for studying this environment of short-term contracting in long-
term stochastic settings in general, and in particular in environments in which
contracts are based solely on the observable signal, is three-fold. First, the ubiq-
uitousness of this particular type of contractual arrangement in the real world
warrants a closer scrutiny of the subtle informational issues that arise in this
setting.
Second – and this partly explains why contracts based exclusively on observ-
able signals are so common – in many instances the principal may not be able to
contract on the larger message space. This may be the case, for instance, if the
principal is contracting with many agents in a similar (or identical) environment
and all agents must work under the same contract. Or, if contracting on additional
signals prevents a pure strategy equilibrium so that only a mixed strategy can be
used, but the exact mixing cannot be verified and hence not contracted on.
The third reason for studying the case in which only the signal is contracted
on is also a factor in explaining the prevalence of these types of contracts.
Namely, the principal may not want to contract on additional messages even if
this is, in principle, possible. Melumad and Reichelstein (1989) study the issue
314 T.D. Jeitschko and L.J. Mirman
of eliciting an agent’s type in static stochastic settings. Our environment differs
from theirs, however, due to the dynamic nature of the problem. Specifically, in
our model due to non-commitment to longer term contracts, a ‘good’ agent must
be compensated for the decrease in future information rents that result from the
principal’s increased information in the future. This compensation takes the form
of an up-front payment and, since it depends on the amount of information the
principal expects to obtain about the agent’s type in the course of the interaction,
this up-front payment is endogenous. In particular, these payments can be pro-
hibitively high if information assimilation occurs very quickly, as is the case in
menu-based contracts that are designed to lead to immediate and full information
(see, e.g., JMS).
An implication of the endogeneity of the up-front payments in stochastic set-
tings in which contracts are based exclusively on observable signals is, of course,
that the principal structures the contract in order to reduce these payments. This
is done by designing the contract to reduce information transmission and thus
preserve the good agent’s future informational rents. In addition to this, the prin-
cipal’s ability to manipulate the flow of information and learn from the observed
signal leads him to experiment. Unlike other work on Bayesian experimentation,
where the Bayesian learner chooses actions directly to influence the signal in
order to enhance learning, in the context of agency, the principal must construct
a contract that induces the agent to take actions that enhance the principal’s
ability to learn about the environment (the agent’s type). In sum, in dynamic
environments in which contracts are based on observable stochastic signals, the
principal uses the pure strategy actions of the agent to manipulate the degree of
information that the signal carries. The principal uses this ability both to affect
payments made to the agent as well as to experiment and learn about the agent’s
type.
The remainder of the paper is structured as follows. In Section 2 the model
and the equilibrium notion are specified. In Section 3 the final period of the
two-period interaction is discussed. This section states the well-known results
from static contract theory that apply to the interaction, and serves as a basis
for Section 4, in which the first period contract is studied. Section 5 contains a
discussion of the types of contracts that support the equilibrium actions derived
in the previous sections. We conclude with an outlook on issues that remain
unresolved.
2 The model
2.1 The environment
The principal and the agent contract over a one-dimensional observable and
verifiable stochastic signal, e.g., the profit of a firm, the outcome of a process,
or the cost of a project, whose realization in period tis denoted by Xt. The
interaction between the principal and the agent has a two-period duration. For
each period t=1,2, the contract between the principal and the agent, denoted
Information and experimentation in short-term contracting 315
by Pt(Xt), specifies a payment made to the agent upon observing the signal Xt.
For simplicity, assume that the payment is in the same units as the signal.
The signal Xtconsists of a nonnegative deterministic component, x, and a
stochastic element, . The agent affects the distribution of the signal through
the effect of unobservable actions on the deterministic part of the signal. Let
at∈Rndenote all the agent’s relevant activities in period t, and suppose that
xt(at,·) is continuous and monotone in its first narguments. In addition to the
agent’s actions at, the signal Xtis affected by time-invariant parameters, χ, that
characterize the technology that transforms the agent’s actions into the signal.
Thus, Xtcan be represented as the sum of a function x(at,χ) and zero-mean
homoskedastic noise, t; i.e.,
Xt=Xt(at,
t)=x(at,χ)+t,
where thas distribution Fwith density f, positive on some connected set Ω⊆R,
such that the distribution satisfies the monotone likelihood ratio property (MLRP),
i.e., log(f) is concave. For expositional ease, assume that the MLRP is strict and
fis differentiable except possibly at the endpoints of Ω. Since Et=0,x(at,χ)
denotes the expectation of Xtconditioned on at, therefore we write EX =x.
Since xis continuous in a, the inverse mapping of the deterministic part of the
signal generating mechanism with respect to ais a continuous compact-valued
correspondence, denoted by x−1(x).
2.2 The agent
The agent is assumed to be risk neutral with respect to the payments received.
His actions yield a disutility that depend on parameters ν. Suppose the util-
ity of the agent in period tis represented by the additively separable function
Ut(Pt,a)=Pt(Xt)−w(a,ν), where Ptdenotes the payment made to the agent by
the principal based on the signal Xtand wis a continuous real-valued function
that measures the agent’s disutility from the choice of aand the time-invariant
disutility parameters ν. Let the agent’s reservation utility be 0. With the inverse
mapping of xwith respect to abeing a continuous compact-valued correspon-
dence, the relation ω(x)≡min{a}w(a,ν) s.t.a∈x−1(x) is a continuous function.
For ease of exposition, assume that ω(x) is twice continuously differentiable.
Risk-neutrality of the agent allows the following representation of the agent’s
expected utility in period t, as a function of the expected signal in that period,
given the contract specified by the principal:
ut(xt)=pt(xt)−ω(xt),
where pt(xt)≡EPt(x(at,χ)+t).
There are two types of agent, who differ in the marginal disutility they ex-
perience from taking the actions necessary to increase the expected signal x.We
refer to the type with the smaller marginal disutility from affecting the expected
signal as the high type (e.g. high productivity, high efficiency, high tolerance
316 T.D. Jeitschko and L.J. Mirman
for effort exertion) and the other as the low type and distinguish their respective
disutility levels of the expected signal by ωand ω, so that ω<ω
,∀x≥0.
One of two things explains the difference in types. First, the difference can be
due to the signal responding more strongly to one type’s actions than the other’s.
That is, the signal generating mechanism may be inherently different for the two
types, so that there are differences between the two types with respect to the
time-invariant parameters χ. Second, one type’s disutility incurred due to actions
may be less sensitive than the other’s. That is, there are idiosyncratic differences
between the two types, reflected in differences in the disutility parameters ν.
Lastly, a combination of the two may occur. Assume that the agent knows his
type.
2.3 The principal
Having standardized the payments made to the agent in the same units as the
signal, the principal maximizes the sum of the per-period excesses of the signal
over the payment. That is, the principal’s utility in period tis given by Vt=
Xt−Pt(Xt). One can think of the principal’s problem in two steps. First, in light
of the principal’s uncertainty regarding the agent’s type, the principal chooses
for each type of agent actions that yield expected signals (xt,xt) and expected
payments (p(xt),p(xt)) that together maximize the total expected per-period gains
of the signals over the payments made. In a second step, the principal designs a
reward schedule, Pt(Xt), mapping the observed signals into rewards for the agent
such that the proper expected rewards of both types of agent are obtained and
the equilibrium actions are implemented. We refer to this reward schedule as the
equilibrium contract.
Thus, letting ρtdenote the principal’s beliefs in period tthat the agent is a
high type, the principal finds the contract Pt(Xt), that maximizes the principal’s
expected utility in period t=1,2, given by
Ev(2)
1=ρ1(x1−p1(x1))+(1−ρ1)x1−p1(x1)+E1v2,(1)
Ev2=ρ2(x2−p2(x2))+(1−ρ2)x2−p2(x2).(2)
2.4 The equilibrium concept and solution method
The equilibrium notion utilized in this paper is that of a perfect Bayesian equi-
librium. In the context of this model this means that in each period the princi-
pal maximizes his current and future expected utility by designing a payment
schedule that takes into account the principal’s beliefs about the agent’s type.
Specifically, in the first period, the principal takes account of how the first period
contract affects the principal’s second period expected payoffs. In the second pe-
riod, the principal uses updated beliefs about the agent’s type in designing the
second period contract. The principal updates his beliefs according to Bayes’ rule
Information and experimentation in short-term contracting 317
after observing the first period signal, given the expected actions of the two types
of agent. The principal cannot commit himself to ignore the information gleaned
about the agent’s type in the course of the interaction with the agent. Thus, con-
tracts are short term and newly designed for each period, on the basis of the
principal’s updated beliefs. Given a contract, each type of agent decides whether
to accept the contract or to disengage the principal and receive the reservation
utility of 0. An agent who accepts the contract chooses actions that maximize
the agent’s expected utility in light of the expected signal that results from the
actions taken, given the contract that specifies a payment to the agent on the
basis of the actually observed signal.
The strategy we pursue in the remainder of the paper is as follows. Due to
learning in the course of the interaction and the fact that first period payments
to the high type agent depend on the second period contract, we focus on the
second period first (Section 3), before considering the initial interaction between
the principal and the agent (Section 4). In these Sections we establish sufficient
conditions for the principal to consider the two different types of agent separately.
In particular, we establish the conditions that assure that (a) all second order
conditions hold and (b) that the indifference surfaces of the two types of agent
intersect only once, that is, that the single-crossing property of the indifference
curves in the expected signal space holds.
Throughout Sections 3 and 4 we focus exclusively on the equilibrium ex-
pected signals and rewards – not the actual reward function that implements the
proper actions of the agent. This separation of reward function and expected re-
wards is possible due to the risk neutrality of the agent. In particular, any current
decision is affected only by expected future outcomes, not the distribution of
possible outcomes. Similarly, the high type’s up-front payment to compensate
for possible decreases in future informational rents also depends only on second
period expected rents, not the actual distribution of second period rents. Issues
concerning the reward functions that yield implementation of the expected signals
and expected rewards of the two periods are addressed in Section 5.
3 The second period (static) contract
At the outset of the second period the principal knows what contract was in
force in the first period, P1, as well as the first period signal X1. This information
is used to update beliefs via Bayes’ Rule. The conditions under which a fully
separating contract exist in the first period are analyzed in the next section. For
now, assume that the first period contract has the two types of agent targeting
distinct expected signals. The principal’s posterior is given by
ρ2=ρ1f
ρ1f+(1−ρ1)f,(3)
where f=f(X1−x1), and f=f(X1−x1). Recall that X1=x(a1,χ)+1,
so, if the agent is a high type, the expected signal is x1and 1=X1−x1,
318 T.D. Jeitschko and L.J. Mirman
and if the agent is a low type, the expected signal is x1and 1=X1−x1.
Notice that if Ωis sufficiently large and fhas little weight in its tails, then in
equilibrium with high probability ρ2∈(0,1), and the principal remains less than
fully informed about the agent’s type. In fact, if Ω=R, it is impossible for the
principal to become fully informed about the agent’s private information, even
if the interaction between the principal and the agent were to last infinitely long.
For completeness, however, assume that out-of-equilibrium observations lead to
the beliefs ρ2=1.
Given the posterior beliefs, the principal designs the second period contract.
This section is devoted to determining the equilibrium expected signals for the
two types of agents. The following two assumptions shall hold for both types of
agent:
Assumption 1 ω(0)=0,ω
(0) <1; and
Assumption 2 ω>0,ω
>0.
Assumption 1 states that an expected signal of zero is obtained at no cost to
the agent, and that at low levels of the expected signal the marginal disutility from
the expected signal is less than the marginal signal strength. Given the principal’s
objective function, with a reservation payoff of zero and marginal gain in the
signal being unity, this assures that the principal will design the contract in such
a way that both types of agent enter the contract.
The first part of Assumption 2 asserts that increases in the expected signal
are associated with increases in the agent’s disutility from the activities that
increase the expected signal. The second part of Assumption 2 says that the
agent experiences increasing marginal disutility from increasing the expected
signal. These conditions are met if, for instance, the agent applies effort to a
decreasing returns to scale production process, and has non-decreasing marginal
utility from effort.
Together Assumptions 1 and 2 assure that both types of agent are considered
separately when designing the contract. That is, the equilibrium contract has
the property that the expected signals are distinct for the two types of agent.
Moreover, both types of agent expend activity so that the expected signal is
greater than zero. In particular, in equilibrium, the only binding constraints are
the high type’s incentive compatibility constraint and the low type’s individual
rationality constraint.
With x2and x2denoting the expected signals resulting from the specified
activity level of the two types respectively, the second period binding incentive
compatibility constraint of the high type is
p2(x2)−ω(x2)=p2(x2)−ω(x2).(4)
Where the left side of Equation (4) measures the difference between the expected
payment and the disutility from the action taken when targeting the expected
signal x2, and the right side is the difference between the expected payment and
the disutility from taking actions that yield the low type’s expected signal, x2.
Information and experimentation in short-term contracting 319
Given the reservation level of utility of 0, the low type’s binding individual
rationality constraint is given by
p2(x2)−ω(x2)=0.(5)
Hence, applying the binding constraints (4) and (5) to the principal’s second
period payoff, given in Equation (2), the principal’s second period problem can
be viewed as one of choosing the two targeted expected signals x2and x2in
order to maximize the expectation of
ρ2x2−ω(x2)+ω(x2)−ω(x2)+(1−ρ2)x2−ω(x2).(6)
Given that ω<ω
, Assumption 2 implies that ω > ω. Hence the princi-
pal’s first order conditions are sufficient for the maximization problem. Regarding
the expected signals generated in equilibrium, the situation is well understood
in the literature. Therefore we restrict attention to the main findings as they are
needed for deriving the first period contract.
The high type generates the first best expected signal, determined by having
his marginal disutility from generating the expected signal equal to the principal’s
marginal utility from observing the signal, i.e., the principal’s first order sufficient
conditions yield ω(x2) = 1. An implication of which is that (d/dρ2)x2= 0. Thus,
the targeted expected signal of the high type in the second period is independent
of the principal’s beliefs, and hence independent of the information generated in
the first period.
The low type generates an expected signal below his first best level. That is,
from the principal’s first order sufficient conditions one obtains ω(x2)<1. For
the low type, the degree of distortion is a function of the principal’s beliefs. In
particular, the less likely the principal thinks the agent is a low type, the greater
the downward distortion of the low type’s expected signal from the first best,
i.e., (d/dρ2)x2<0.
Regarding the expected second period payoffs the usual results also hold.
Through the binding individual rationality constraint, the low type is kept at his
reservation utility level, regardless of the principal’s beliefs. That is, u2=0,∀ρ2.
The high type’s individual rationality constraint is not binding and thus the high
type earns informational rents. These informational rents depend on the beliefs
of the principal. The informational rent of the high type can be represented as
his second period utility as a function of the principal’s posterior beliefs, i.e.,
u2(ρ2)=ω(x2(ρ2)) −ω(x2(ρ2)).
Thus, the high type obtains an expected payoff equal to the difference between
the disutility incurred by the two types when choosing actions that target the low
type’s expected signal. Since ω> ωand (d/dρ2)x2<0,
(d/dρ2)u2=(ω−ω)(d/dρ2)x2<0.(7)
That is, the less likely the principal thinks that the agent is a high type, the greater
is the target expected signal of the low type. This yields increased informational
rents for the high type in the second period.
320 T.D. Jeitschko and L.J. Mirman
Finally, the principal’s second period expected payoff is given by,
v2(ρ2)=ρ2x2−ω(x2(ρ2)) + ω(x2(ρ2)) −ω(x2)
+(1 −ρ2)x2(ρ2)−ω(x2(ρ2)).
4 The first period contract
4.1 The first period constraints
The principal’s first period objective is to determine expected signal levels for the
two types of agent that maximize his total utility from the two-period interaction.
Thus, the principal’s objective function in the first period is given by,
V(2)
1=ρ1(x1−p1(x1))+(1−ρ1)x1−p1(x1)+E1v2(ρ2(x1,x1)),
where ρ2is a function of x1, and x1, due to the updating after observing the first
period signal.
Not only does the principal consider the two period problem, but the agent
also considers implications of first period actions on learning and hence the
second period contract. Recall that the principal’s second period beliefs affect
the low type’s second period targeted expected signal. However, the second
period contract keeps the low type agent at his reservation utility regardless of
the principal’s posterior. Therefore the low type’s current actions do not affect
future payoffs, and hence, from the perspective of the low type, the first period
problem is essentially static. In the context of the principal’s problem this means
that the binding individual rationality constraint of the low type is as before and
given by, p1(x1)−ω(x1)=0.(8)
This is different for the high type agent. Although the high type’s second
period targeted expected signal is independent of the principal’s beliefs, the high
type’s second period payoff depends on the principal’s posterior. In particular,
the more likely the principal thinks that the agent is a low type, the higher the
high type’s informational rents in the second period are (see Equation (7)). This
is because the more likely the principal thinks that the agent is the low type, the
closer he wants to keep the low type agent at the first best expected signal. How-
ever, through the high type’s binding incentive compatibility constraint, keeping
the low type closer to the first best expected signal is obtained only by increasing
the high type’s expected informational rent.
Thus, the high type has an incentive to manipulate the principal’s beliefs
through his actions in the first period – in particular the agent has an incentive
to choose the low type’s expected signal. This affects the high type’s first period
incentive compatibility constraint. Specifically, if a high type mimics the low
type by choosing the low type’s expected signal, the high type’s second period
expected payoff is given by Ru2(ρ2)f dX1.2The high type choosing the expected
2We integrate on the entire real line, letting f= 0 off the range of the signal X1.
Information and experimentation in short-term contracting 321
signal intended for him has a second period expected payoff of Ru2(ρ2)fdX
1.
Thus the high type’s first period binding incentive compatibility constraint be-
comes:
p1(x1)−ω(x1)+R
u2(ρ2)fdX
1=p1(x1)−ω(x1)+R
u2(ρ2)f dX1,
or, solving for the high type’s equilibrium first period expected payment,
p1(x1)=p1(x1)−ω(x1)+ω(x1)+R
u2(ρ2)f−fdX1.(9)
That is, compared to a static setting, for given levels x1and x1, the high type’s
first period payment differs by Ru2(ρ2)f−fdX1– the value to the high type
of mimicking the low type’s actions in the first period.
Theorem 1 For given targets, x1>x1, in a dynamic stochastic environment the
first period expected payment made to the high type is greater than in the static
environment and less than in the dynamic deterministic setting, i.e.,
0<R
u2(ρ2)f−fdX1<u2(0).
Proof. By the MLRP dρ2/dX1>0, so
du2/dX1=(du2/dρ2)(dρ2/dX1)<0 (10)
Thus, the high type’s second period expected payoff is decreasing in the first
period observed signal. Hence, for all x1>x1,Ru2(ρ2)f dX1>Ru2(ρ2)fdX
1,
and R
u2(ρ2)f−fdX1>0.
For the same reason, u2(0) >u2(ρ2), so
R
u2(ρ2)
u2(0) f−fdX1<1
The first part of Theorem 1 is standard in dynamic agency relationships, it
simply states that if a principal can reduce an agent’s informational rents by
learning about the agent’s private information, then the agent must be compen-
sated for these losses in order to induce the agent to take actions that allow the
principal to learn.
The second part of Theorem 1 is explained by the fact that, unlike the de-
terministic environment in which the fully separating contract leads to full in-
formation, there is incomplete learning in a stochastic environment. Thus, noise
has two critical functions in determining the size of the additional payment for
given expected signal targets. First, a high type who chooses the expected signal
intended for his type is able to preserve some informational rents in the second
period, because the signal generated in the first period is stochastic and hence the
322 T.D. Jeitschko and L.J. Mirman
principal cannot fully infer the agent’s type. Second, in the deterministic envi-
ronment if a high type agent takes actions that generate the low type’s signal, the
principal is completely deceived about the agent’s type. However, in the stochas-
tic environment this deception is not complete, since the principal continues to
believe that with some probability the agent is the high type.
Theorem 1 illustrates a subtle yet important difference between the stochastic
and the deterministic environments. In a deterministic setting learning is often
immediate and complete. Should complete learning not be possible immediately,
then the only way that the principal can affect the the flow of information, is
if the principal includes as part of the first period contract specific instructions
on how agents should employ mixed strategies. In the stochastic setting regard-
less of the principal’s instruction regarding the agent’s actions, full information
dissemination is rarely possible, so that learning is almost always incomplete.
Nevertheless, as is demonstrated in the next section, the principal has an incen-
tive to manipulate the actions of the agents and hence the flow of information,
in order to affect current and future payments.
4.2 The first period equilibrium expected signals
Consider now the derivation of the first period targets. To obtain distinct targets,
neither type of agent can have an incentive to deviate from the equilibrium
actions. A sufficient condition for this to be the case is that the high type’s
marginal disutility of the first period expected signal must be smaller than that
of the low type for all expected signals. That is, in the first period, the single-
crossing property holds with regard to the agents’ indifference curves with respect
to actions and payments. Formally, letting f=f(X1−x),we state,
Assumption 3 ω(x)≥ω(x)+Ru2(ρ2)fdX1,∀x.
The left side of the inequality in Assumption 3 measures the low type’s
marginal disutility from the first period expected signal, caused by the activities
undertaken to generate this expected signal. The right side measures the high
type’s marginal disutility from the first period expected signal. This consists of
two parts. First, just as in the static case, the disutility that the agent incurs through
the activity level taken to generate the expected signal. The second part is the
marginal impact that the first period expected signal, x1, has on the distribution
of the first period realized signal, X1, and hence the expected value of the second
period contract for the high type.
With respect to the informational element of the right side, notice that
it is increasing in the agents effort. Specifically, since the argument of fis
decreasing in x1, the distribution of X1shifts to the right with increases in
x1. And since u2(ρ2) is positive and decreasing in X1(see Equation (10)),
−(d/dx1)Ru2(ρ2)f−fdX1=Ru2(ρ2)fdX1>0. Loosely speaking the in-
tuition is as follows. The higher the high type’s expected signal in the first period,
the higher the probability of a high first period observed signal. The higher the
Information and experimentation in short-term contracting 323
first period observed signal, the more likely the principal thinks that the agent is
the high type (this is the MLRP). The more likely the principal thinks the agent
is the high type, the smaller the informational rent of the high type. Hence, the
high type’s expected second period utility is decreasing in the expected signal of
the first period.
4.2.1 The impact of payments on the equilibrium expected signals
Using the binding constraints, (8) and (9), one can examine the principal’s first
period problem and hence the principal’s incentives in choosing the two expected
signals. Specifically, the principal’s first period objective function is given by
ρ1x1−ω(x1)+ω(x1)−R
u2(ρ2)f−fdX1
+(1 −ρ1)x1−ω(x1)+E1v2(ρ2),
where ρ2and fare functions of x1, and ρ2and fare functions of x1.
In the previous subsection it is demonstrated that the magnitude of the impact
that noise has on the second-period incentive compatibility constraint of the high
type depends critically on the informativeness of the distribution of the first period
signal, given equilibrium actions in the first period. In particular, the closer the
two expected signals are, the less informative the distribution of the first period
signal is, since fand fmove closer together. Notice that it is the principal
who chooses the first period targeted expected signals and hence, the equilibrium
distribution of the first period signal. Thus, the principal affects the magnitude
of the reward needed to induce the high type to target a distinct expected signal.
The following Theorem demonstrates that the principal uses his ability to affect
the reward of the high type by manipulating the distribution of the first period
signal, and how this manipulation is accomplished.
Theorem 2 Compared to a deterministic environment, the principal reduces the
high type’s first period reward by the choice of the first period expected signals,
i.e.,
(d/dx1)R
u2(ρ2)f−fdX1>0,
(d/dx1)R
u2(ρ2)f−fdX1<0.
Proof. Consider the impact of the principal’s choice of x1on the reward paid to
the high type.
(d/dx1)R
u2(ρ2)f−fdX1=Ru
2dρ2
dx1f−f+u2fdX1
Integrating the last term under the integral by parts yields
324 T.D. Jeitschko and L.J. Mirman
Ru
2dρ2
dx1f−f−u
2dρ2
dX1fdX1
=R
u
2dρ2
dx1f−dρ2
dX1+dρ2
dx1fdX1.
Recall f=f(X1−x1), f=f(X1−x1), and ρ2from Equation (3). Then, defining
D≡ρ1f+(1−ρ1)f,
dρ2
dX1=ρ1(1 −ρ1)ff−ff
D2,(11)
dρ2
dx1=−ρ1(1 −ρ1)ff
D2.(12)
Therefore, letting c=−ρ1(1−ρ1)
D2, the integral is
R
u
2−ρ1(1 −ρ1)
D2ff2+ρ1(1 −ρ1)
D2ff2dX1
=R
u
2cff2−ff2dX1.
Since both u
2and care negative, their product is positive and it is sufficient to
show that ff2−ff2>0 for the integral to be positive. Dividing ff2−ff2by
f, does not change its sign, and one obtains H(X1)≡fff/f−ff, which is
to be shown to be positive.
Let g(X1)=ff−ff, then ff=ff+g(X1), and hence
H(X1)=ff+g(X1)f/f−ff
=f/f−1ff+f/fg(X1).
Define ˆx1implicitly by f/f= 1, and notice that, by the MLRP, x1<ˆx1<x1.
H(X1) is positive for all X1<x1because, by the MLRP, g(X1) is positive
for all X1; and f/f>1, and f>0 for X1<x1.
Also, H(X1) is positive for all X1>ˆx1because, by the MLRP, f/f<1, and
f<0 for X1>ˆx1.
Now consider H(X1)on[x1,ˆx1]. Notice that
H(X1)=f/f−1ff+f/fg(X1)
=f/f−1ff+f/fff−ff
=−ff+f/fff,
which is positive, since f<0, for X1>x1and f>0, since X1<ˆx1(<x1).
Thus,
(d/dx1)R
u2(ρ2)f−fdX1>0.
Information and experimentation in short-term contracting 325
Similarly, one can show that
(d/dx1)R
u2(ρ2)f−fdX1=−(d/dx1)R
u2(ρ2)f−fdX1<0
Hence, although the principal cannot commit to a long term contract, that is,
although the principal uses all information gleaned in the first period to design
the second period contract, by choice of the first period targets, the principal can
commit to a less informative first period signal, and thus reduce the amount by
which the high type’s rents will be diminished in the second period. This is done
by choosing the target expected signals closer together, thus leading the principal
to obtain less information.
This highlights an added feature of the contract in a stochastic environment
when compared to the deterministic setting. In the deterministic setting the prin-
cipal is either immediately fully informed, or he can only influence the amount
of learning (and hence the amount of the first period reward paid to the high
type) by determining the degree of mixing on part of the high type in a so-called
“semi-separating” equilibrium contract. In the stochastic environment, however,
the principal uses the ability to affect the informativeness of the distribution of
the first period signal as a means of influencing the first period rewards even
when the equilibrium contract yields distinct pure strategy expected signals.
4.2.2 The impact of experimentation on the equilibrium expected signals
Despite the incentive to preserve the agent’s future informational rents, the prin-
cipal clearly also has an incentive to become better informed. That is, obtaining
information about the true type of the agent is to the principal’s benefit in the
second period as it reduces the deviations from the first best action of the low
type and reduces informational rents of the high type in the latter interaction.
This is captured in the following Lemma.
Lemma 1 Information is valuable to the principal. Formally this is the case
whenever the principal’s second period payoff function (the value function) is
convex in the posterior beliefs, ρ2, i.e.,
d2v2
dρ2
2
≡v
2≥0.
Proof. The proof is a variation of the Envelope Theorem. Specifically, notice
that v2(ρ2) = max{x2,x2}Ev2(x2,x2,ρ
2), so v2(ρ2) is convex if Ev2(x2,x2,ρ
2)is
convex in ρ2. As seen in Equation (2), Ev2(x2,x2,ρ
2) is linear in ρ2, and hence
v2(ρ2) is convex.
Since obtaining better information is beneficial to the principal, he has an
incentive to manipulate the distribution of the first period signal to enhance
learning, by making the signal more informative. Obviously this is exactly the
326 T.D. Jeitschko and L.J. Mirman
opposite incentive to that captured in Theorem 2, above. In other words, the
principal has an incentive to chose a first period signal distribution that reveals
much information.
Theorem 3 In determining the expected signals, the principal experiments.
Specifically, in order to enhance the flow of information the principal increases
the distance between the expected signals, i.e.,
(d/dx1)E1v2(ρ2(x1,x1)) >0,
(d/dx1)E1v2(ρ2(x1,x1)) <0.
Proof. Notice that E1v2=ρ1Rv2fdX
1+(1−ρ1)Rv2f dX1,so
(d/dx1)E1v2=ρ1Rv
2dρ2
dx1f−v2fdX1+(1−ρ1)R
v
2dρ2
dx1f dX1.
Integrating the first integral by parts yields
ρ1R
v
2dρ2
dx1+dρ2
dX1fdX
1+(1−ρ1)R
v
2dρ2
dx1f dX1.
Applying (11) and (12) as before, the derivative is
−ρ1R
v
2ρ1(1 −ρ1)ff
D2fdX
1−(1 −ρ1)R
v
2ρ1(1 −ρ1)ff
D2f dX1
=−R
v
2(1 −ρ1)ρ2
2f+ρ1(1 −ρ2)2fdX1.
This integral is analogous to Equation (22) in (MSU), where it is shown to reduce
to
(d/dx1)E1v2=ρ1R
v
2(1 −ρ2)dρ2
dX1fdX
1>0,
since information is valuable, v
2≥0, and by the MLRP, dρ2/dX1>0.
Similarly one can show that
(d/dx1)E1v2(ρ2(x1,x1)) = −(d/dx1)E1v2(ρ2(x1,x1)) <0.
Theorems 2 and 3, demonstrate that the first period optimal contract involves
deviations from the first best expected signals for both types of agent. However,
which of the two effects dominates depends on the parameters of the model. In
particular, with regard to the high type, the case in which x1<x2has been
discussed previously in the literature. This is the classic ratchet effect, in which
high types have their targets increased in the course of the interaction (see, e.g.,
Weitzman, 1980; Holmstr¨
om, 1982; Jeitschko, Mirman, and Salgueiro, 2002).
The opposite case, in which x1>x2, implies an activity level above the first best,
a phenomenon known in the signaling literature (see, e.g., Akerlof, 1976; Spence,
1974). This can reflect instances in which new members to an organization have
to “earn their spurs” before becoming integrated, at which point expected targets
may very well be decreased. That is, instances in which a high type has the
incentive to initially take actions above the efficient level, and then in the course
of the interaction reduces the activity level to the first best.
Information and experimentation in short-term contracting 327
5 The equilibrium payment schedules
Having derived all equilibrium actions and expected payments, we consider the
question of implementability. That is, we consider issues in constructing the
equilibrium reward schedules Pt(Xt),t=1,2, under which each type of agent
has the incentive to choose the actions (or, equivalently, target the expected
signal) intended for him by the principal.
To achieve full implementation of the targeted expected signals derived in
the previous section, two conditions must be fulfilled. First, it must be assured
that the expected reward schedule lies (weakly) below the lower envelope of the
equilibrium utility indifference curves of both types of agent. Second, each type
of agent must obtain the exact equilibrium expected payment when choosing
equilibrium actions. In this way, for each type of agent, any actions that do
not target the equilibrium expected signal, given the agent’s type, will not be
chosen, since the average utility associated with any such actions is less than
his utility from taking the equilibrium actions. Moreover, it is assured that the
targeted expected signal for each type of agent will be chosen, since the difference
between the expected reward and the disutility of taking the corresponding actions
leaves each type of agent with his equilibrium expect utility.
Since single crossing of the indifference curves of the two types with respect
to the agent’s actions and (present and future) expected payments is assured
in both periods, the problem is structurally the same in both periods. In order
to reduce extraneous notation, we discuss only the contract of the first period
and suppress the period subscripts where this does not lead to ambiguities. The
second period contract follows by omitting the dynamic aspects involved in the
first period.
As in the previous sections, due to the risk-neutrality of both the agent and
the principal, one can restrict attention to expected signals when formalizing the
two conditions regarding the specifications of the reward function made at the
outset of the section. In order to determine the lower envelope of the equilib-
rium indifference curves of both types of agent, consider the two equilibrium
indifference curves in turn.
First, as noted in Section 4.1, since the low type is kept at his reservation
level of utility in the second period, his first period indifference curves depend
solely on his first period actions and the first period payments made to him.
Thus, the low type’s first period utility given a payment of pand an action that
produces the expected signal x, is given by u(x,p)=p−ω(x). In equilibrium,
from Equation (8), u= 0, so the low type’s equilibrium indifference curve over
payments and expected signals is given by,
p=ω(x).
Due to the principal’s slow learning and the implied second period informa-
tional rents that accrue to the high type, the high type’s first period actions
affect his second period expected payoff in that the actions affect the prin-
cipal’s posterior beliefs. Consequently, the high type’s expected utility from
328 T.D. Jeitschko and L.J. Mirman
the two-period interaction, given the second period equilibrium, a first period
payment of p, and first period actions that yield the expected signal x,is
given by p−ω(x)+Ru(ρ2)fdX1, where again f=f(X1−x). In equilibrium,
from Equation (9), the expected utility of the high type in the first period is
ω(x)−ω(x)+Ru2(ρ2)f dX1, so the high type’s equilibrium indifference curve
is given by,
p=ω(x)−ω(x)+ω(x)+R
u2(ρ2)f−fdX1.
Thus, the condition that the expected reward function lie everywhere (weakly)
below the lower envelop of the equilibrium indifference curves is given by,
p(x)=R
P(X)f(X−x)dX
≤minω(x),ω(x)−ω(x)+ω(x)+R
u2(ρ2)f−fdX1.(13)
Of course, the single-crossing occurs at the point x, with ω(x)<ω(x)−ω(x)+
ω(x)+Ru2(ρ2)f−fdX1, for x<xand ω(x)>ω(x)−ω(x)+ω(x)+
Ru2(ρ2)f−fdX1, for x>x.
The second condition that the expected reward function must fulfill is again
implied by the equilibrium expected payoffs given in Equations (8) and (9),
namely
p(x)=ω(x) and
p(x)=ω(x)−ω(x)+ω(x)+R
u2(ρ2)f−fdX1.(14)
Notice that there is considerable freedom in selecting the actual reward sched-
ule P(X), since the two Conditions (13) and (14) are sufficient conditions and are
not particularly restrictive. Indeed, it is well-known that under general conditions
when the agent is characterized by a continuous distribution of types, there exist
equilibrium reward schedules P(X) that induce an expected reward function p(x)
satisfying the above conditions for all types of agent.3In that case, the lower
envelope corresponding to Condition (13) is implied by the equilibrium payments
corresponding to Condition (14), and yields a smooth expected reward function
p(x).
The current two-type setting is less restrictive than the continuous type case
in that Condition (14) fixes only two points of the expected reward schedule, as
opposed to an entire interval – thus yielding additional freedom in the choice of
P(X). However, the two-type setting is in some sense more restrictive in that there
is a point of non-differentiability in the lower envelope implied by Condition (13).
This non-differentiability stems from the fact that the lower envelope is made up
of only two indifference curves, which have the single-crossing property.
3See, e.g., CGR and the literature cited therein.
Information and experimentation in short-term contracting 329
Single-crossing in itself does not imply the non-differentiability, but single-
crossing in conjunction with the (strict) differences between the types’ marginal
disutility of the signal strength and the informational impact of the high type’s
first period actions on his second period expected utility is sufficient to estab-
lish non-differentiability. Specifically, given that p(x)=ω(x) from Condition
(14), the envelope in Condition (13) implies that for p(x) differentiable at x,
limh→0,h>0p(x−h)≡p(x−)≥ω(x) and, using analogous notation, p(x+)≤
ω(x)+Ru2(ρ2)fdX1. Yet, by Assumption 3, ω(x)<ω
(x)+Ru2(ρ2)fdX1,∀x,
so that p(x) cannot be differentiable at the point x.
In general, one cannot expect to find an explicit reward schedule. This leaves
two possibilities. The first is to make appropriate restrictions on the model. The
second is to find an approximate reward schedule. Consequently, the first question
that needs to be resolved is under what conditions can a non-differentiability on
p(x) be obtained. The following are sufficient conditions, any one of which allows
this non-differentiability to occur.
First, suppose that Ωis bounded below and define i= infΩ. Then if f(i)>0
any discontinuity in P(·)atx+iyields a non-differentiability of p(·)atx, and
a sufficiently large discontinuous increase of the reward function at x+ican be
used to meet the required Conditions.
Similarly, if Ωis bounded above with s= supΩ, then if f(s)>0 a non-
differentiability in p(·)atxcan be achieved by a discrete step down in the
payment function P(·)atx+s.
Lastly, suppose that Ωis unbounded. In this case fis positive on the entire
real line (an implication of the MLRP). If fis non-differentiable at some point
y,4then a discrete step in the payment function yields a non-differentiability in
the expected payment as in the second case, with ytaking the place of s.
If none of the above three conditions hold, that is, if f(i)=0,f(s)=0,
or Ωis unbounded, a non-diferentiability in p(x) cannot occur, since if fde-
fined on the entire real is positive and differentiable on the entire real line, then
p(x)=−RP(X)f(X−x)dX,∀x. In this case it is only possible to come within
an arbitrarily small deviation of the equilibrium targeted expected signals, by in-
creasing either type of agent’s equilibrium expected reward by an arbitrarily
small amount and then smoothing out the expected reward function below the
envelope around x.
6 Conclusion
This paper demonstrates that in short term contracting of multi-period agency re-
lationships, significant issues regarding equilibrium actions and payments emerge
when the signals generated by the different types of an agent are stochastic. No-
tably, the principal’s ability to learn about the agent’s type is impeded in the
course of the interaction. Stemming from this are three significant differences
4We have assumed that fis differentiable for simplicity of the proofs in the main body of the
text. However, the Theorems hold also when fis continuous, but not differentiable.
330 T.D. Jeitschko and L.J. Mirman
between the optimal incentive compatible contracts, when compared to static, or
deterministic environments. First, since the dissemination of information is slow,
additional payments made to induce information revealing actions are lower, as
informational rents are protected over time and do not immediately dissipate.
Although a reduced flow of information may occur in dynamic deterministic
settings, this only occurs when the principal instructs the agent to mix across
actions.
Second, in specifying equilibrium actions, the principal can influence the
informativeness of the observed signal. The principal uses this ability to trade-
off up-front payments to the agent to reduce future informational rents with
becoming less informed and thus preserving these rents in latter interaction.
Finally, given that information about the agent’s type is valuable to the prin-
cipal, the informativeness of the observed signal will also be manipulated to
enhance the principal’s ability to learn. That is, the principal uses his ability to
manipulate the distribution of the signal in order to experiment.
Given the importance of these insights, it seems worthwhile to explore how
these insights are affected by changes in the model specification. There are three
immediate extensions worth studying. First, is to examine the scenario with a
richer type space, in particular, a continuum of types. The question then arises to
what degree the support of equilibrium actions taken by the agent is impacted by
the principal’s desires to keep payments low, yet enhance the flow of information.
Moreover, it is then no longer clear that there need be “much pooling,” as is the
case in the deterministic setting (see Laffont and Tirole, 1988). Second, it should
be analyzed how the principal trades-off different incentives when more tools
are available, such as costly observation of some of the agent’s actions, but
possibly not all. Finally, since the optimal reward scheme depends heavily on
the distribution of the noise, one might want to consider how noise impacts the
principal’s objectives when contracts are restricted to adhere to specific structures.
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