Optimal Contracts for Lenient Supervisorsa
Thomas Giebeb
Humboldt University at Berlin
Oliver Gürtlerc
University of Cologne
January 13, 2011
a We
would like to thank Patrick Bolton, Florian Englmaier, Guido Friebel,
Veronika Grimm, Karl Ludwig Keiber, Nicolas Klein, Johannes Koenen, Matthias
Kräkel, Sebastian Kranz, Tymofiy Mylovanov, Judith Przemeck, Andreas Roider,
Ferdinand von Siemens, Dirk Sliwka, Roland Strausz, Achim Wambach and Elmar
Wolfstetter for helpful comments. Financial support by the Deutsche Forschungsgemeinschaft (DFG), grants SFB/TR 15 (”Governance and the Efficiency of Economic Systems”), is gratefully acknowledged.
b (Corresponding author) Thomas Giebe, Dept. of Economics, Economic Theory I, Humboldt University at Berlin, Spandauer Str. 1, D-10099 Berlin, Germany,
e-mail: [email protected], phone: +49-30-20935773, fax: +49-3020935619.
c Oliver Gürtler, University of Cologne, Dept. of Economics, Albertus-MagnusPlatz, D-50923 Cologne, Germany, email: [email protected], phone:
+49-221-4701450, fax: +49-221-4701816.
Abstract
We analyze optimal contracts in a hierarchy consisting of a principal, a supervisor and an agent. The supervisor is either neutral or altruistic towards the
agent, but his preferences are private information. In a model with two supervisor types, we find that the optimal contract may be very simple, paying
the supervisor a flat wage independent of his type and his evaluation of the
agent’s effort. Such a contract induces one type of supervisor to report the
agent’s performance truthfully, while the other reports favorably independent
of performance. Accordingly, overstated performance (leniency bias) may be
the outcome of an optimal contract under informational asymmetries.
Keywords: Subjective performance evaluation, leniency, supervisor, private information
JEL classification: D82, D86, J33, M52
1
1
Introduction
Many firms use subjective measures to assess and reward their employees’
performance. This gives discretion to those who are assigned the task of
performance evaluation. Several studies indicate that supervisors seem to
exploit the power given to them.1 In particular, leniency bias — a practice of
systematically overstating employee performance — is a regular phenomenon
in subjective performance evaluation.
We study the implications of subjective performance evaluation in a three–
tiered hierarchy consisting of a principal, a supervisor, and an agent. The
procuction of output requires the agent’s effort. In order to motivate the
agent, the principal hires a supervisor whose task it is to monitor and report
the agent’s performance. The supervisor is assumed to be either altruistic or
neutral towards the agent and, hence, may be lenient.
We distinguish different settings where the supervisor’s preference for the
agent (his type) might be private or public information and where the parties
might be wealth–constrained.2
The aim of the paper is to derive optimal contracts for these settings and to
find out whether leniency prevails if contracts are structured optimally.
We find that, if the supervisor’s type is public information, the optimal con1
2
See e.g. the excellent survey by Prendergast (1999).
Most of the literature on alternative preferences assumes common knowledge of the
parties’ preferences and so does not consider the problem we address. See, however, von
Siemens (2010a,b) on the screening of inequity–averse agents.
2
tract eliminates leniency by punishing the supervisor for good ratings of the
agent. The first-best solution can be implemented regardless of wealth constraints. Under private information, it is feasible to eliminate leniency by
offering an incentive–compatible menu of contracts inducing correct evaluation of the agent. Due to excessive rents, however, these contracts are not
generally optimal. Instead, the optimal contract can be very simple, paying the supervisor a flat wage independent of his type and how he evaluates
the agent.3 This induces a neutral supervisor to always report the agent’s
performance truthfully, while an altruistic supervisor reports favorably independent of performance. This implies that leniency is consistent with optimal
contractual arrangements under informational asymmetries.4 Under unlimited liability, the first–best solution can be implemented even though leniency
occurs.
The paper is organized as follows: In the next section we relate the paper to
the literature. In Section 3 we present the model and the first–best solution.
Section 4 looks at the public information case, while Section 5 deals with
private information. Section 6 gives a discussion and Section 7 concludes.
All proofs are in the Appendix.
3
In our model, the supervisor’s only task is to monitor the agent. The term ”flat wage”
indicates that the supervisor’s wage does not depend on how he rates the agent. Of course,
a supervisor might have other tasks with variable pay.
4
For other explanations see MacLeod (2003) or Grund and Przemeck (2008).
3
2
Related Literature
The paper is closely related to the literature on subjective performance evaluation. Particularly relevant is the work on biased ratings, e.g., Baker et al.
(1988), Grund and Przemeck (2008), Murphy (1992), Harris (1994), Prendergast and Topel (1996), or Prendergast (2002). In contrast to the present
text, none of those papers deals with eliciting the supervisor’s preferences.
The paper is clearly related to the literature on optimal contracts in threetiered hierarchies consisting of a principal, a supervisor and an agent, see, e.g.,
Tirole (1986), Strausz (1997) or Vafai (2004). These papers differ from ours in
two respects. First, they typically assume hard information, i.e., a supervisor
may conceal but not misrepresent information. In our model, information is
soft.5 Second, in those models the supervisor has known standard (egoistic)
preferences. Eliciting the supervisor’s preferences is, however, at the core of
the present paper.
Finally, the paper is related to literature combining adverse selection and
moral hazard, such as Laffont and Tirole (1986), McAfee and McMillan
(1987) or Lewis and Sappington (1997). In those papers, the player who
chooses an unobservable action has superior information. In contrast, our
supervisor has private information (his type) and a hidden action (to report
effort truthfully or not) while the agent’s action is hidden from the principal
but not from the supervisor.
5
See Laffont and Martimort (1997) and Faure-Grimaud et al. (2003) for models with
soft information.
4
Summarizing, there are two important features that distinguish our paper
from the rest of the literature. We consider a model where the supervisor’s
preference is private information and where performance information is soft.
While we know of no other model, where these two assumptions are imposed, we think it is worthwhile to analyze this combination: First, taking
for granted that biased ratings can be due to supervisors’ preferences, it goes
without saying that private information about preferences is more realistic
than public information. Second, if the agent’s action is not observable and
his evaluation by the supervisor is subjective, then, we believe, it is realistic
to assume that performance information can be misrepresented, i.e., that it
is soft. Accordingly, we think that it is important to analyze contracts in
this setting.
3
The model
A principal, P, a supervisor, S, and an agent, A, play the following basic
game. At stage 1, P offers a menu of contracts. A contract obliges A to exert
effort and S’s task is to monitor and report A’s effort. The contract specifies
an effort level and wages for A and S that (may) depend on reported performance. At stage 2, either both A and S simultaneously sign the contract
or the game ends. At stage 3, A chooses effort while S observes and reports
A’s effort to P. Then payments are made according to the contract and the
game ends.
5
The cost of A’s effort, e ≥ 0, is c(e). It is twice differentiable and satisfies
c (0) = 0, c0 (0) = 0, c0 (e) > 0 for e > 0, c0 (e) = ∞ for e → ∞, and c00 (e) > 0.
Effort produces an output, f (e), that accrues to P, where f (0) = 0, f 0 (e) > 0
and f 00 (e) ≤ 0.
Effort and output are unobservable for P and not verifiable by third parties.
Moreover, there is no other objective measure of e. Hence, P cannot write an
explicit performance contract to motivate A. Thus, it is S’s task to monitor
A and to observe and report the agent’s effort choice.6
The contract obliges S to send a report m ∈ {y, n} to P where m = y
indicates that A has chosen (at least) the requested effort while m = n
means underperformance. Reports are verifiable and we also assume that S
observes A’s effort perfectly and costlessly.7
All parties are risk-neutral and have zero reservation value. A central assumption is that S may care for A’s well-being and thus has an incentive to
distort the effort report. We model S’s preferences as8
US = wS + λ̂UA ,
(1)
where wS denotes the income of S and UA is A’s utility. The parameter
λ̂ ∈ [0, 1) measures how strongly S cares for A’s well-being. We assume that
6
7
Of course, S may have other tasks as well.
As an example, consider a supervisor and an agent who share an office, possibly
working on the same project. Here, S might observe A’s effort without additional effort
or cost.
8
See Rotemberg and Saloner (1993), Prendergast and Topel (1996), Prendergast (2002),
Rotemberg (2004, 2007) or Sliwka (2007) for similar specifications.
6
S reports honestly if he is indifferent (i.e., he might be lying–averse9 or there
might be a risk of detection and punishment).10
A’s utility is given by
UA = wA − c (e) ,
(2)
with wA as A’s income.
Except for Section 4, we assume that S’s preference, or type, is private information, i.e., a supervisor of type i, denoted by Si , knows that his preference
parameter is λ̂ = λi , while for P and A, λ̂ is a random variable.
Throughout the paper we assume that there are two S–types. In Section 6 we
briefly address the case of more types. Supervisor S1 has standard (egoistic)
preferences, λ1 = 0. We call him neutral. The other type, S2 , is altruistic,
with λ2 ∈ (0, 1). S1 (resp. S2 ) occurs with probability q (resp. 1 − q).
We confine analysis to pure strategies and (menus of) incentive–compatible
contracts where S self–selects into the contract designed for his type and
A makes the demanded effort. A contract designed for Si is of the form
{ai , bi , ei , wyi , wni }. It specifies the requested (minimum) effort level, ei , Si ’s
wages {wyi , wni } conditional on report m ∈ {y, n}. Moreover, A receives a
base wage ai , and, if S reports m = y, a bonus bi .
P offers a menu of contracts and, at stage 2, A and S simultaneously sign
9
10
Gneezy (2005) gives experimental evidence for lying aversion.
We do not formally incorporate this into the model. Also note that this assumption
is not needed for the existence of our equilibria but it ensures that they are unique.
Otherwise, e.g., an indifferent S might be dishonest with some probability which need not
affect his payoff but would reduce effort.
7
this menu. Then P may choose to conceal from A which particular contract
from the menu is selected by S. In that case, A cannot infer S’s type but he
knows each type’s incentives. Note that, in principle, A’s wage might depend
on S’s type also when the true type is concealed from A but then, obviously,
the demanded effort must be equal for both types. For that case, we assume
that it is ex-post verifiable, which particular contract has been selected by S.
We call S lenient if he reports favorably, m = y, independent of A’s performance.
As a benchmark, consider the first-best solution assuming that effort is contractible and no contractual frictions arise. Here, P does not need S.11 She
simply imposes the effort level that solves
max f (e) − wA
e,wA
s.t. wA − c (e) ≥ 0.
(3)
Hence, the first-best effort, eF B , is implicitly given by
f 0 (eF B ) = c0 (eF B ).
(4)
We denote fF B := f (eF B ) and cF B := c(eF B ).
11
Note that P cannot do better by leaving a rent r to A and hiring a supervisor at a
negative wage wS = −λr. This is due to λ < 1.
8
4
Complete Information
As another benchmark case, suppose S’s type λ̂ is common knowledge.12 The
optimal contracts solve the following program:13
max
a,b,wy ,wn ,e
f (e) − a − b − wy
s.t. (ICA ) a + b − c (e) ≥ a
(ICSn ) wn + λ̂ (a − c (ẽ)) ≥ wy + λ̂ (a + b − c (ẽ)) , ∀ ẽ < e
(5)
(ICSy ) wy + λ̂ (a + b − c (ē)) ≥ wn + λ̂(a − c (ē)), ∀ ē ≥ e
(IRA ) a + b − c (e) ≥ 0
(IRS ) wy + λ̂ (a + b − c (e)) ≥ 0
The constraint (ICA ) induces the requested effort. (ICSy ) and (ICSn ) make
S report A’s effort truthfully. (IRA ) and (IRS ) ensure participation.
Proposition 1. The optimal contract under complete information is a∗ = 0,
b∗ = cF B , e∗ = eF B , wy∗ = 0, wn∗ = λ̂cF B . It implements the first-best solution
and eliminates leniency.
If λ̂ is strictly positive, the optimal contract has wn∗ > wy∗ = 0, i.e., it
rewards an altruistic S for giving a negative performance report. A positive
report implies that A gets the bonus b∗ in which S participates with utility
λ̂b∗ = λ̂cF B . Thus, a truthful negative report requires that S is compensated
for his ”loss“ due to the unpaid bonus to A. Truthful reporting, in turn, makes
12
13
We drop the subscript i.
The program maximizes P’s profit on the equilibrium path, i.e., where A exerts positive
effort, S reports truthfully and A earns a bonus.
9
effort basically verifiable and allows us to implement the first–best solution.
Wages are nonnegative which makes the contract feasible even under wealth
constraints.
5
Incomplete Information
Suppose the distribution of S–types is common knowledge while the realization λi is Si ’s private information. The optimal contracts depend on whether
the players are wealth–constrained. We analyze unlimited liability in 5.1 and
limited liability in 5.2.
5.1
Unlimited liability
Proposition 2. If the players are unlimitedly liable, the optimal contract is
unique, type–independent, and satisfies a∗ =
q−1
c
q FB
< 0, b∗ =
cF B
,
q
e∗ =
eF B , wy∗ = wn∗ = 0. It implements the first-best solution and the altruistic
supervisor is lenient.
The optimal contract pays the same flat wage to both S–types. A flat wage
induces S1 to report truthfully, since his own wage makes him indifferent
between reports and he does not care about A’s wages. S2 is lenient, i.e.,
reports favorably independent of A’s effort. This is because his wage is not
affected by his report, but the bonus payment to A increases his utility. A
does not know which S–type he is facing. S2 ’s leniency does not provide
effort incentives, since if S2 occurs, A gets the bonus anyway. Thus, positive
10
effort can only be induced via the truthfully reporting S1 . Since S1 occurs
with probability q, the expected bonus, qb, must cover the effort cost. Thus,
efficient effort is achieved by setting qb∗ = cF B . Finally, A’s base wage is
negative in order to extract all rents from him. This makes the contract
infeasible if A is wealth-constrained. Limited liability of S, in contrast, is
unproblematic since S’s wages are nonnegative.
An implication of Proposition 2 is that leniency is not necessarily detrimental
to performance. In this respect, the result complements a recent study by
Bol (2009) that analyzes compensation data from a Dutch financial service
provider and finds that leniency is positively associated with agents’ future
performance. Unlike the current model, she explains this result by agents’
concerns for a fair evaluation.
5.2
Limited liability
Limited liability is defined as
ai ≥ 0, ai + bi ≥ 0, wyi ≥ 0, wni ≥ 0,
i ∈ {1, 2}.
(6)
Recall that we only consider incentive–compatible (menus of) contracts, i.e.,
supervisors self–select according to their types. While we assume that A
knows the menu of contracts, P may choose to conceal from A which particular contract was chosen by S. Then A only knows each type’s incentives.
By (5), (ICSy ) and (ICSn ) hold simultaneously for Si if and only if wyi +λi bi =
wni . Thus, truthful reporting of effort requires that Si is indifferent between
11
reports m ∈ {y, n}. Truthful reporting, in turn, is necessary in order to
induce positive effort. Thus, positive equilibrium effort requires that at least
one S–type reports truthfully.
Accordingly, we partition the potential (menus of) contracts into the following cases, depending on which S-type reports effort truthfully and whether
the type is revealed to A.
truthful report by
A does not learn S’s type A learns S’s type
S1 only
(A)
(D)
S2 only
(B)
(E)
S1 and S2
(C)
(F)
The following lemma gives the optimal wages for each of the cases (A)–(F)
for given requested effort levels, while for the moment ignoring the issue
of optimal effort levels. However, the lemma states that optimal requested
efforts are strictly positive with one exception: In cases (D) and (E), A learns
S’s type and, by design, one type does not report effort truthfully. If A faces
this type then he will not exert positive effort. For that S–type, the only
feasible effort level is zero.
In order to simplify notation in Lemma 1 and its proof, we denote λ := λ2 ,
f := f (e), c := c(e), fi := f (ei ), ci := c(ei ), i ∈ {1, 2}.
Lemma 1. Consider the contract cases (A) – (F). Under contracts where A
learns S’s type and can infer that S reports independent of effort, the only
implementable effort is zero if that type occurs. Otherwise, it is optimal for
P to request (and implement) strictly positive effort. For given effort levels,
12
the sets of optimal wages and the corresponding P’s profits are given by
(A) :
(B) :
wy1 = wn1 = wy2 = wn2 = 0, a1 = 0, b1 =
c
or a2 =
a2 ≥ 0, b2 > 0, a2 + b2 =
q
c
π (A) = f − ,
q
c
,
q
c
c
, b2 ∈ − , 0
,
q
q
λc
c
wy2 = 0, wn2 =
, a2 = 0, b2 =
,
1−q
1−q
λc
λc
λc
a1 = t ≥ 0, b1 = −t, wy1 =
, wn1 ∈ 0,
, wn1 + λt ≤
1−q
1−q
1−q
λc
λc
λc
, wy1 + λb1 ≤
, wn1 =
,
or a1 = 0, b1 ≥ 0, wy1 ∈ 0,
1−q
1−q
1−q
q
π (B) = f − c 1 +
λ ,
1−q
(C) :
wy1 = wn1 = wn2 = λb2 , wy2 = 0, a1 = 0, a2 = b1 ,

2


qλ
c


0, 1−q
, f − c 1 + 1−q
if λ < 1−q

q



2
1−q
c
c
,
b1 , b2 , π (C) =
,
0,
f
−
if
λ
>
q
q
q




h
i
2


1−q

c
c
 t, c−qt
,
f
−
,
t
∈
0,
if
λ
=
1−q
q
q
q
13
(7)
(8)
(9)
(D) :
a1 = 0, b1 = c1 ,
(wn2 = 0, a2 = 0, wy2 = wy1 = wn1 = ŵ ,




)


, ˜ ∈ [0, ], if λ = 1 − q 
˜, (−˜

λ






(ŵ, b2 ) =

0,
if
λ
>
1
−
q
λ










(, 0)
if λ < 1 − q
or
(wy2 = 0, wn2 = wy1 = wn1 = ŵ ,



)
(−˜
)


˜, (−˜
,
−
, ˜ ∈ [0, ],

λ
λ



(ŵ, a2 , b2 ) =
0, λ , − λ







(, 0, 0)



q(f1 − c1 ) − if λ ≤ 1 − q
(D)
π
=
,


q(f1 − c1 ) − 1−q if λ > 1 − q,
(10)

if λ = 1 − q 



,
if λ > 1 − q 



if λ < 1 − q
λ
(where > 0 is arbitrarily small),
(E) :
a2 = b2 = c2 , wy2 = 0, wn2 = λc2 ,
(wy1 = λc2 , wn1 ∈ [0, λc2 ) , a1 = t ≥ 0, b1 = −t, wn1 + λa1 ≤ λc2 )
(11)
or
(wy1 ∈ [0, λc2 ) , wn1 = λc2 , a1 = b1 = 0) ,
π (E) = (1 − q)(f2 − c2 ) − c2 (1 − q + qλ),
(F ) :
a1 = 0, b1 = c1 , a2 = b2 = c2 , wy1 = wn1 = wn2 = λc2 , wy2 = 0,
(12)
π (F ) = q(f1 − c1 ) + (1 − q)(f2 − c2 ) − c2 (1 − q + qλ).
Evaluating the results of Lemma 1, we find
14
Proposition 3. The principal’s maximum expected profit is
(C)
(F )
max max π (e), max π (e1 , e2 )
e
where e, e1 , e2 > 0 and




π (A) (e)
(C)
π (e) =



π (B) (e)
e1 ,e2
if λ ≥
if λ <
1−q
q
1−q
q
(13)
2
(A)
(B)
=
max
π
(e),
π
(e)
.
2
(14)
Thus, the global maximum profit is either equal to π (F ) or π (C) (after maximizing them over efforts) where π (C) is equal to the larger of π (A) and π (B) .
While there is only one optimal contract for (F), there are several optimal
wage combinations for (C) (and (A), (B)).
We now demonstrate by example that there are model parameters such that
either case, (A), (B), (C), and (F) can contain the (globally) optimal contract.
Suppose f (e) = e and c(e) = e10 . Figures 1 and 2 plot the maximum profits
for (A), (B), (C) and (F). Recall that π (C) = max{π (A) , π (B) }. Figure 1 is
drawn for λ = 2/3 and it also shows the first–best profit (FB). In Figure 2,
the profits for (A), (B), (F) are given by the dark grey, light grey and white
areas, respectively.14
The example illustrates properties of π (A) , π (B) , π (C) , and π (F ) that we discuss
in the following.
In (A) and (B), only one S–type reports truthfully. A knows that effort does
not pay if the ”wrong“ type occurs, but A does not learn which type he is
14
The figures look very similar for, e.g., f (e) =
15
√
e and c(e) = e2 /2.
maximum profits for Λ=23
0.70
FB
0.69
C, B
C, A
0.68
F
0.67
F
0.66
F
0.65
A
B
q
0.0
0.2
0.4
0.6
0.8
1.0
Figure 1: Optimal contracts for f (e) = e, c(e) = e10 and λ = 2/3.
Figure 2: Optimal contracts for f (e) = e and c(e) = e10 .
16
facing. Thus, the optimal (implementable) effort increases in the probability
of facing the truthtelling supervisor. In particular, (A) and (B) approach
first–best if q → 1 for (A) and q → 0 for (B). Intuitively, this makes sense
since we approach the case of complete information. However, the more
likely the ”wrong“ S–type is, the worse is the performance of the (A), (B)
contracts.
Under (C), both types report truthfully but A doesn’t learn which type he
faces. Thus, his effort only depends on the expected bonus payment, but A
does not care for which S–type the bonus is larger. But for P, generically,
one of the two types is always more cost–effective in incentivizing A than the
other: As can be seen from (9), either b1 or b2 is zero, depending on q and
λ. Thus, (C) replicates the better one of (A) and (B). Since both (A) and
(B) become worse for ”more uniform“ distributions15 , contracts (C) perform
worst where π (A) = π (B) , see (9).
Under (F ), both types report truthfully. This, however, requires two kinds
of rents: Both S–types have the same utility from wages, but are differently
affected by A’s utility. Thus, the only way to induce self–selection is to treat
A differently for each S–type. In fact, P has to leave A a rent if S chooses the
contract designed for type 2. By limited liability, S2 then has positive utility.
This, in turn, requires wn2 > 0 in order to induce him to report truthfully.
In contrast to S2 , S1 does not benefit from A’s rent. For him, this contract
is only interesting because of wn2 > 0. In order to prevent S1 from imitating
15
To be precise, the worst point is where π (A) = π (B) , or, λ = (1 − q)2 /q 2 , see (9).
17
S2 (and reporting m = n), S1 receives an informational rent.
In π (F ) , see (12), the first term refers to S1 , the neutral type. Note that the
optimal e1 is the first–best effort, eF B . Thus, with probability q, we get the
first–best profit. The remaining terms refer to S2 , where, with probability
1 − q the profit is below first–best, going to π (F ) = f2 − 2c2 for q → 0. Thus,
(F) cannot perform as badly as (A) and (B) do if the ”wrong“ type is very
likely. By the above, (F) becomes better with respect to (C) when both (A)
and (B) perform worst.
Finally, consider cases (D) and (E), that can never contain optimal contracts,
by Proposition 3. In each case, A learns S’s type but only one type reports
truthfully. A does not exert effort whenever this is not rewarded. Thus,
either with probability q or 1 − q P’s profit is zero. As is clear from (10),
(11), and (12), π (F ) > π (D) + π (E) (recall that q ∈ (0, 1)).
It is hard to pin down the general relation between the maximum profits
under (C) and (F). But it seems plausible that either (A) or (B) (and thus
(C)) perform well if one S–type is very likely. This is because (A) and (B)
are geared to one of the S–types and with a more extreme distribution we
approach the complete information case.
Next, we derive one of the main observations of the paper: It can be optimal
to have a very simple contract that does not eliminate leniency. Consider
case (A), see (7). We have wy1 = wy2 = wn1 = wn2 = 0 and, for A, we choose
a1 = a2 = 0 and b1 = b2 = c/q. With these wages, S2 is lenient and always
reports favorably. Obviously, this menu of contracts can be replaced by a
18
single contract since neither P nor A need to learn S’s type. This contract is
optimal whenever (A) contains an optimal contract. By our example, there
are parameters for which this holds. By the above argument, this contract’s
profit increases in the probability of the neutral type, i.e., the ”closer“ we
are to the standard preferences setting. We summarize:
Proposition 4. Suppose P offers a single contract (rather than a menu of
contracts) where S is paid a flat wage regardless of type and effort report.
Moreover, A is paid independent of S’s type. Such a contract can be optimal.
It does not eliminate leniency.
Note that this contract is similar to the optimal one under unlimited liability
(see Proposition 2). The difference is that here we cannot have a negative
base wage for A. Thus, P cannot extract A’s rents, which lowers the optimal
effort and therefore the bonus.
6
Discussion
Leniency We have shown that optimal contracts do not necessarily eliminate leniency. In particular, Proposition 2 states that under incomplete
information and unlimited liability the altruistic supervisor is always lenient.
Under limited liability (and incomplete information), the altruistic supervisor is lenient under contracts (A), and, in particular, the simple contracts of
Proposition 4. Under all other optimal contracts, leniency does not occur.
19
More than two types Some of our results easily generalize to the case of
arbitrarily many supervisor types or continuous type space. These are the
complete information case and the incomplete information case with unlimited liability (Propositions 1 and 2).
As regards incomplete information with limited liability, the number of contract cases becomes very large with even a few more supervisor types. It
should, however, be straightforward to determine the optimal wages for any
particular case (in the fashion of Lemma 1). For instance, contracts of type
(F), that eliminate leniency, are always feasible. However, satisfying the
many incentive constraints (e.g., preventing imitation of other types) leads
to excessive rents such that the required efforts are (near) zero for many supervisor types. Thus, these contracts perform much worse with more types
unless one (or some) types have high probability.16 Case (B) generalizes to
pooling contracts where some altruistic type reports truthfully. Optimality
of these contracts requires that this type is more likely than others (and, in
particular, the neutral type). Case (A) generalizes to pooling contracts where
only the neutral supervisor reports truthfully while all others are lenient.
Recall that we assumed lying aversion, i.e., S prefers reporting effort truthfully if his disadvantage in terms of payoff is not too large. Then, depending
on the degree of lying aversion, a given contract can induce several types to
report truthfully which would reduce information rents and, thus, increase
P’s profit.
16
A solution for the general case (F) is available from the authors.
20
If A knows S’s type Suppose A is limitedly liable and knows S’s type
when he chooses effort. Then contracts (A), (B), and (C) are not feasible
because they require concealing the type. Then the unique optimal contract
is (F). Obviously, it is never in P’s interest that A knows S’s type.17
Practical Implications We have shown that, in principle, the cases (A),
(B), (C), and (F) can contain optimal contracts. While there is only one
feasible contract in (F), there are many optimal wage combinations in the
other cases, for given profitability.
From a practical perspective, some contracts are simpler or seem to be more
realistic than others. In particular, we can use the following two criteria to
evaluate contracts. First, consider contracts with type–independent wages
(for S and A). There, P can offer a single contract rather than a menu of
contracts, uncertainty for A is reduced since A’s wages are independent of
S’s private information, and, neither P nor A need to infer S’s type. Second,
consider flat wages for S, that do not depend on how A is evaluated. These
wages are the simplest and seem to be the most prevalent in reality.
The contract from Proposition 4, contained in (A), fits both criteria and it is
the unique optimal contract in the model with standard preferences where S
is neutral for sure. We have seen that it can remain optimal if the probability
17
A practical implication might be that P should try to prevent A from learning S’s
preferences, e.g., by introducing job rotation. See e.g. Tirole (1986) and Prendergast and
Topel (1993) for different motives of job rotation in a principal-supervisor-agent relationship.
21
of this neutral type is sufficiently large.
The first criterion is satisfied by (B) if we set a = 0, b = c/(1 − q), wy = 0,
and wn = λc/(1 − q). This contract has wn > wy and the neutral supervisor
always reports negatively. In practice, we might observe this if supervisors
are rewarded on the basis of cost savings. Contracts (C) and (F) satisfy
neither criterion. Recall, however, that, in terms of profit, (C) is equal to
either (A) or (B).
Collusion A full analysis of collusion is beyond the scope of this paper.
However, even without a detailed collusion model, we can evaluate the potential gains from collusion under the optimal contracts. The most obvious
way of collusion is that the agent exerts zero effort, saving the effort cost,
and pays the supervisor a bribe in order to make him report favorably such
that the agent gets the bonus anyway.
At first glance, collusion should be more profitable in the presence of an
altruistic supervisor as compared to the model with standard (neutral) preferences because any utility gain of the agent implies an additional indirect
gain for the supervisor.
However, the potential gains from collusion are actually lower than under
the assumption of standard preferences, for several reasons.18 First, by the
altruistic preferences, for any given bribe paid by the agent, the agent’s loss
18
A detailed analysis of the collusion problem under all optimal contracts is available
on request.
22
is larger than the supervisor’s gain.19 Second, the optimal contracts imply
various rents to S or A which lead to efforts below the first–best level and
thus reduce the potential gains from collusion. Third, the optimal contracts
already take into account the supervisor’s tendency for favorable reports.
They induce the altruistic supervisor to tell the truth by punishing good
reports through lower wages. Thus, the altruistic supervisor’s only gain from
collusion may be the bribe, as in the setting with neutral supervisor. Fourth,
under the simple contract from Proposition 4, the altruistic supervisor is
lenient anyway and thus the agent only wants to bribe the neutral supervisor.
But since he does not know which type he is facing, the bribe is wasted with
positive probability.
All in all, the potential rents that can be shared between the colluding parties
are lower than those under standard preferences. These gains have to be
weighed up against the cost of collusion, like, e.g., the risk of detection and
punishment.
7
Conclusion
We looked at the optimal design of contracts in a principal–supervisor–agent
relationship where the supervisor may be lenient because he cares for the
agent’s well–being.
19
A bribe payment of t to the supervisor reduces the agent’s utility by t. The bribe has
two effects on the supervisor. His utility increases directly by t but it also decreases by λt
through the agent’s utility loss. The supervisor’s net gain is (1 − λ)t < t.
23
The main insight is that, under incomplete information and limited liability,
it can be optimal to have a very simple contract that does not eliminate
leniency although this is feasible. In particular, this contract pays the supervisor a flat fee, independent of his performance reports, and does not
reveal his type. Under this contract the supervisor who cares for the agent
is lenient. This kind of contract is always optimal under unlimited liability.
8
Appendix
Proof of Proposition 1. It is easily verified that the proposed contract satisfies (5). The principal’s profit simplifies to π = fF B − cF B .
Proof of Proposition 2. The proof is done in two steps. First, we show that
the proposed contract yields the first-best solution and is therefore optimal.
Second, we show that it is the only contract with this property.
(1) If wy∗ = wn∗ = 0, S1 reports A’s effort truthfully, while the other type
reports m = y. Thus, A chooses effort e, if
a + b − c(e) ≥ a + (1 − q) b
⇔
b≥
c(e)
.
q1
(15)
Thus, b∗ induces e = eF B . S and A participate since their payoffs are zero.
P gets the first–best profit: π = fF B − b∗ − a∗ = fF B − cF B .
(2) P’s profit can be written as π = f (e) − wS − wA , where wS and wA
are the wage payments to S and A. From S’s participation constraints it
follows that wS ≥ −λUA . Hence, P’s profit is not above f (e) + λUA − wA or
24
f (e) + λUA − (UA + c (e)). As λ < 1, first-best requires UA = 0, i.e., A must
not receive a rent.
If the contract is such that no supervisor type reports A’s effort truthfully,
then positive effort cannot be induced, i.e. A chooses e = 0. Moreover, a
contract can only be optimal if S accepts it and no type of supervisor receives
a positive rent. The latter requirement implies max {wy , wn } ≤ 0. Otherwise,
S1 receives a strictly positive rent. As UA = 0, the only contract satisfying
max {wy , wn } ≤ 0, inducing at least one type of S to report y truthfully and
ensuring participation of S has wy = wn = 0.
Proof of Lemma 1. In all cases below we will see that the expected bonus
payment covers the expected effort cost. This, together with (6) takes care of
A’s and S’s participation constraints. In cases (D) and (E), A learns S’s type.
Whenever he faces a type who does not have incentives to report truthfully,
then the only implementable effort is zero. In all other cases, i.e., when A
knows (or expects with positive probability) that S reports effort truthfully,
then strictly positive effort is feasible and optimal. Throughout the proof we
assume strictly positive requested effort (and thus strictly positive cost and
output) unless indicated otherwise. At the end of the proof, we show why this
is optimal. Throughout the proof, we repeatedly refer to ”relevant“ wages
and ”optimal“ wages. The first term refers to those wage components that
are payoff–relevant on the equilibrium path. The second term refers to wages
that minimize the expected value of total relevant wages in equilibrium (to S
25
and A) for given effort. The proof proceeds by minimizing the expected total
relevant wages for each case observing all relevant constraints. We explicitly
give those constraints but the easy task of verifying that the proposed wages
satisfy the constraints is left to the reader. We also characterize the set(s) of
non–relevant wages that support the equilibrium in each case.
First, we state the set of conditions that have to be satisfied by the optimal
contract(s). Limited liability is defined by (6). Note that negative boni,
bi < 0, are generally feasible. Si ’s incentive constraints for truthful reporting
of A’s effort, given that P demands that effort be at least e = ê, are
wni + λi (ai − ci (ẽ)) ≥ wyi + λi (ai + bi − ci (ẽ)) ∀ ẽ < ê
(16)
wyi + λi (ai + bi − ci (ē)) ≥ wni + λi (ai − ci (ē)) ∀ ē ≥ ê,
(17)
which is equivalent to wyi + λi bi = wni . Thus, truthful reporting implies
S1 : wy1 = wn1
(18)
S2 : wy2 + λb2 = wn2 .
(19)
Moreover, wages wyi + λi bi > wni (resp. < wni ) induce Si to always report
m = y (resp. m = n) regardless of effort. In such a case, A will not exert
effort if he learns S’s type (as under contract cases (D), (E)).
Under any contract, Si (where Sj is the other type) selects the right contract
only if his corresponding payoff, πSi , is at least as large as the payoff from
imitating the other type,
πSi ≥ max {wyj + λi (aj + bj − cj ), wnj + λi (aj − cj )} .
26
(20)
For S1 (where λ1 = 0) we have πS1 ∈ {wy1 , wn1 } and thus, depending on the
relation of wy1 and wn1 , (20) simplifies to
wy1 = wn1 ⇒ (20) : wy1 = wn1 ≥ max {wy2 , wn2 } ,
(21)
wy1 > wn1 ⇒ (20) : wy1 ≥ max {wy2 , wn2 } ,
(22)
wy1 < wn1 ⇒ (20) : wn1 ≥ max {wy2 , wn2 } .
(23)
Cases (A), (B), (C) Here, P conceals S’s type from A. Therefore, obviously, the requested (and equilibrium) efforts are the same for both types,
e1 = e2 =: e > 0 (and c, f without subscript).
Case (A) Here, only S1 reports effort truthfully, i.e., (18) holds while (19)
does not. We divide (A) into subcases (A1) where S2 always reports m = y,
and (A2) where the report is always m = n. Since only S1 ’s report depends
on effort, A’s incentive constraints for (A1) and (A2) are the same:
(A1) : q(a1 + b1 ) + (1 − q)(a2 + b2 ) − c ≥ qa1 + (1 − q)(a2 + b2 ) ⇐⇒ b1 ≥
(A2) : q(a1 + b1 ) + (1 − q)a2 − c ≥ qa1 + (1 − q)a2 ⇐⇒ b1 ≥
c
.
q
c
q
(24)
In addition to the above and (6), we have to satisfy constraints (18), (20),
27
and (21), while (19) is violated. Collecting and simplifying, we get
b1 ≥
c
q
(25)
wy1 = wn1
(26)
wy1 ≥ max{wy2 , wn2 }
(27)
(A1) : wy2 + λb2 > wn2
(28)
(A1) : wy2 + λ(a2 + b2 ) ≥ max{wy1 + λ(a1 + b1 ), wn1 + λa1 }
(29)
(A2) : wy2 + λb2 < wn2
(30)
(A2) : wn2 + λa2 ≥ max{wy1 + λ(a1 + b1 ), wn1 + λa1 }.
(31)
(A1) By (25), b1 > 0. Together with (26), the RHS of (29) evaluates to
wy1 + λ(a1 + b1 ). By (27), wy1 − wy2 ≥ 0. Thus, (29) implies
wy1 − wy2 ≤ λ(a2 + b2 − (a1 + b1 ))
⇒ a2 + b 2 ≥ a1 + b 1 .
(32)
(33)
The relevant wages are ai , bi , wyi , while wni are not relevant, i ∈ {1, 2}.
We minimize total relevant wages. First, (25) holds with equality, b1 = qc .
Second, the cheapest way to satisfy (33) is with a1 = 0 (limited liability)
and a2 + b2 = qc . Third, the cost-minimizing payoff-relevant supervisor wages
are wy1 = wy2 = 0. This requires wn1 = wn2 = 0, by (26) and (27). In
equilibrium, only the sum a2 + b2 is relevant. Thus, there is a degree of
freedom in choosing a2 and b2 as long as b2 > 0 (in order to satisfy (28)). It
28
follows that P’s maximum profit is
c
π (A1) = f − q(a1 + b1 + wy1 ) − (1 − q)(a2 + b2 + wy2 ) = f − .
q
(34)
(A2) By (25), b1 > 0. Together with (26), the RHS of (31) evaluates to
wy1 + λ(a1 + b1 ). By (27), wy1 − wn2 ≥ 0. Thus, (31) implies
wy1 − wn2 ≤ λ(a2 − (a1 + b1 ))
⇒ a2 ≥ a1 + b 1 .
(35)
(36)
The relevant wages are a1 , a2 , b1 , wy1 , and wn2 , while b2 , wn1 , and wy2 are
not. First, (25) holds with equality, b1 = qc . Second, the cheapest way to
satisfy (36) is with a1 = 0 and a2 = qc . Third, the cheapest relevant S–wages
are wy1 = wn2 = 0. By (26) and (27), this requires wn1 = wy2 = 0. By (30),
we must set b2 < 0 and, by (6), a2 + b2 ≥ 0. Since b2 is not relevant, we get
b2 ∈ [−a2 , 0) = [−c/q, 0). Then P’s profit is the same as under (A1):
c
π (A2) = f − q(a1 + b1 + wy1 ) − (1 − q)(a2 + wn2 ) = f − .
q
(37)
Case (B) Here, only S2 reports effort truthfully, i.e., (19) holds, while we
have either subcase (B1), where (22) holds, or subcase (B2) where (23) is
satisfied. Positive effort requires b2 > 0 since S1 ’s report is independent of
effort. Similar to (24) we get b2 ≥ c/(1 − q). Collecting and simplifying all
29
relevant constraints, we need to satisfy
b2 ≥
c
(1 − q)
(38)
(B1) : wy1 > wn1
(39)
(B1) : wy1 ≥ max{wy2 , wn2 }
(40)
(B2) : wy1 < wn1
(41)
(B2) : wn1 ≥ max{wy2 , wn2 }
(42)
wy2 + λb2 = wn2
(43)
wy2 + λ(a2 + b2 ) ≥ max{wy1 + λ(a1 + b1 ), wn1 + λa1 }.
(44)
(B1) The relevant wages are ai , bi , wyi , while wni are not relevant, i ∈ {1, 2}.
Choose b2 and wy2 as low as possible, i.e., b2 = c/(1 − q), by (38), and
wy2 = 0. By (43) this leads to wn2 = λc/(1 − q). By (40), it follows
that wy1 = λc/(1 − q). By (39) and (6), wn1 ∈ [0, λc/(1 − q)). For the
remaining relevant wages, it is feasible to have the minimum values a2 = 0
and a1 + b1 = 0. Note that only the sum a1 + b1 but not its composition is
payoff–relevant. Inserting the above into (44), we get
λc
≥ max
1−q
λc
, wn1 + λa1 .
1−q
(45)
Since wn1 is not relevant and a1 only matters indirectly, as part of the sum
a1 + b1 , there is some degree of freedom as long as the value of the RHS is
not affected. Precisely, we need wn1 + λa1 ≤ λc/(1 − q), where a1 = t ≥ 0,
30
b1 = −a1 , and wn1 ∈ [0, λc/(1 − q)), by (39). P’s maximum profit is
π (B1) = f − q(a1 + b1 + wy1 ) − (1 − q)(a2 + b2 + wy2 )
q
=f −c 1+
λ .
1−q
(46)
(B2) Here, the relevant wages are a1 , a2 , b2 , wn1 , and wy2 , while b1 , wy1 and
wn2 are not relevant. By (38), b2 = c/(1 − q). We set wy2 = 0 and, by (43),
wn2 = λc/(1 − q). By (42), we cannot pay less than wn1 = λc/(1 − q). For
wy1 , which is not relevant, we get, by (41), wy1 ∈ [0, λc/(1 − q)). For the
remaining relevant wages, we claim that a1 = a2 = 0 is feasible. Inserting
the above into (44), we get
λc
λc
≥ max wy1 + λb1 ,
,
1−q
1−q
(47)
which is satisfied, provided that wy1 and b1 (which are not relevant) are set
such that they don’t increase the RHS of (47), i.e., wy1 + λb1 ≤ λc/(1 −
q). Together with b1 ≥ 0 (by limited liability and a1 = 0) and wy1 ∈
[0, λc/(1 − q)) (see above), we have determined all wages. P’s profit becomes
π (B2) = f − q(a1 + wn1 ) − (1 − q)(a2 + b2 + wy2 )
q
=f −c 1+
λ .
1−q
(48)
Case (C) Here, S1 and S2 report effort truthfully, i.e., (18) and (19) hold.
A doesn’t learn the supervisor’s type and thus exerts positive effort if
q(a1 + b1 ) + (1 − q)(a2 + b2 ) − c ≥ qa1 + (1 − q)a2
(49)
⇒ qb1 + (1 − q)b2 ≥ c,
(50)
31
By limited liability, (6), a negative bonus, bi < 0, is feasible as long as
ai + bi ≥ 0. For the moment, suppose that bi ≥ 0. We derive the optimal
contract(s) for that case and afterwards prove that bi < 0 is never optimal.
Inserting (19) into the RHS of inequality (21), we get
wy1 ≥ wy2 + λb2 .
(51)
wy2 + λ(a2 + b2 ) ≥ max{wy1 + λ(a1 + b1 ), wn1 + λa1 }.
(52)
Under (C), (20) simplifies to
By (18) and bi ≥ 0, the RHS of (52) is equal to wy1 + λ(a1 + b1 ). After
replacing wy1 by the RHS of (51) and simplifying, we get
a2 ≥ a1 + b 1 .
(53)
First, a1 and a2 are relevant. From (53) we get a1 = 0 (limited liability) and
a2 = b1 . Second, (50) is binding. Third, wy1 and wy2 are relevant. From (51)
and bi ≥ 0 we get wy2 = 0 and wy1 = λb2 . By (18) and (19), it follows that
wn1 = wn2 = λb2 . Thus, P ’s expected profit is, for bi ≥ 0,
π (C) = f − q(a1 + b1 + wy1 ) − (1 − q)(a2 + b2 + wy2 )
= f − q (b1 + λb2 ) − (1 − q) (b1 + b2 )
= f − b1 − b2 (1 − q + qλ).
(54)
P’s maximization problem becomes
max f (e) − b1 − b2 (1 − q + qλ)
b1 ,b2 ,e
s.t. qb1 + (1 − q) b2 = c (e) .
32
(55)
(56)
By (56), P can achieve a certain fixed increase in effort by increasing either
b1 by k units or b2 by
qk
1−q
units. By (55), the corresponding cost of this
qk
, respectively. Hence, if k >
additional effort are either k or (1 − q + qλ) 1−q
2
qk
(1 − q + qλ) 1−q
, or, equivalently, if λ < 1−q
, P sets b1 = 0 and b2 =
q
c/(1 − q) and expected profit becomes
π
Similarly, if λ >
1−q
q
2
(C)
=f −c 1+
q
λ .
1−q
, b2 = 0 and b1 = c/q are optimal and
c
π (C) = f − .
q
Finally, if λ =
1−q
q
2
(57)
(58)
, both, b1 and b2 are equally effective (and costly). In
that case, (57) and (58) are equal and there is a degree of freedom in setting
b1 and b2 , i.e., they may both be strictly positive (as long as (56) holds).
Now we rule out the optimality of negative boni, bi < 0. Consider the set of
optimal wages derived above where b1 = 0. Now suppose b1 is reduced by any
amount t > 0 to b1 = −t. Then, by (50), b2 increases, b2 ≥ tq/(1−q)+c/(1−
q). By (6), the negative b1 must be compensated by a1 ≥ t. By (19), wn2
increases and, thus, by (21), wy1 and wn1 increase as well. As before, we need
wn2 , wy1 , wn1 ≥ λb2 but b2 is larger now. Consider (52). Since wy1 = wn1 and
a1 ≥ t now, the RHS of (52) has increased (as compared to the case bi ≥ 0).
Thus, the LHS must also increase. This would be achieved by setting a2 ≥ t,
since raising wy2 or b2 would again increase wn2 , wy1 and wn1 . Applying the
33
cost minimization argument, we get an expected profit of
π
(C),b1 <0
=f −c 1+
q
q2
λ −t 1+
λ ,
1−q
1−q
(59)
which is less than (57). By the previous analysis, P can get the maximum
of (57) and (58) with bi ≥ 0. Thus, b1 < 0 is never desirable.
A similar argument can be made for b2 < 0. Consider the set of optimal
wages derived above where b2 = 0. Now suppose b2 is reduced by some
amount t > 0 to b2 = −t. We get b1 = c/q + t(1 − q)/q. Then a1 = 0 and, by
(19), wn2 = 0 and wy2 = λt. It follows that wy1 = wn1 = λt. Inserting into
(52), we get
λt + λ(a2 − t) ≥ λt + λ
c
1−q
+t
q
q
⇐⇒ λa2 ≥ λ
c+t
.
q
(60)
Note that, for any given t, in order to satisfy (60), raising wy2 on the LHS
of (52) does not help: it would in turn increase wy1 and thus the RHS. Thus
we need a2 = (c + t)/q. We get a profit of
π
(C),b2 <0
c
1−q
=f − −t λ+
,
q
q
(61)
which is less than (58), and thus never desirable (see the argument above).
Cases (D), (E), (F) Here, P reveals the type report to A before A chooses
effort. Then A chooses type-dependent efforts, e1 and e2 .
Case (D) Here, only S1 tells the truth, i.e., (18) holds while (19) does not.
We divide the case into (D1) where S2 always reports m = y, and (D2) where
he reports m = n. Since S2 reports independent of effort, A chooses e2 = 0
(and thus c2 = f2 = 0). Therefore, A’s incentive constraint is a1 +b1 −c1 ≥ a1 ,
34
or, b1 ≥ c1 . Collecting (and simplifying) all incentive constraints (see (18),
(20), and (21)), we need to satisfy (6) and
b1 ≥ c1
(62)
wy1 = wn1
(63)
wy1 ≥ max{wy2 , wn2 }
(64)
(D1) : wy2 + λb2 > wn2
(65)
(D1) : wy2 + λ(a2 + b2 ) ≥ max {wy1 + λ(a1 + b1 − c1 ), wn1 + λ(a1 − c1 )}
(66)
(D2) : wy2 + λb2 < wn2
(67)
(D2) : wn2 + λa2 ≥ max {wy1 + λ(a1 + b1 − c1 ), wn1 + λ(a1 − c1 )} .
(68)
(D1) By (62) and (63), the RHS of (66) simplifies to wy1 + λ(a1 + b1 − c1 ).
Since wy1 ≥ wy2 , by (64), and λ > 0, inequality (66) gives us
wy1 − wy2 ≤ λ(a2 + b2 − (a1 + b1 − c1 ))
⇒ a2 + b2 ≥ a1 + b1 − c1 .
(69)
(70)
The relevant wages are ai , bi , and wyi (i ∈ {1, 2}), while wni are not relevant.
We cannot pay less than b1 = c1 and a1 = 0. Then (70) would be satisfied
for any a2 + b2 ≥ 0 (which must hold by (6)). Now consider (65). Suppose
we want to achieve a difference of some > 0 between the LHS and the RHS
of (65) and do so in a cost–minimizing way. On the LHS of (65), both wy2
and b2 are relevant. Thus, we set the RHS (which is not relevant) as low
as possible, wn2 = 0. Although wy2 and b2 need to be as small as possible,
in order to satisfy (65) we must set one of them (or both) strictly positive.
Raising the LHS of (65) to > 0 requires b2 = /λ and costs (1 − q)/λ in
35
expectation while wy2 = leads to wy1 = wn1 = (by (64) and (63)) and
costs q + (1 − q) = . Thus, if λ > 1 − q, a positive b2 is cheaper and we
set b2 = /λ, wy2 = wy1 = wn1 = 0 while if λ < 1 − q, we set b2 = 0 and
wy2 = wy1 = wn1 = . Finally, suppose λ = 1 − q. Then it is optimal to have
wy2 = wy1 = wn1 = ˜ ≤ and b2 = ( − ˜)/λ.
Finally, a2 = 0. Note that b2 < 0 does not make sense since, by (6), this
must be offset with a2 > 0 while, by (65) and (64), it requires even higher
wy2 and wy1 . The principal’s expected profit becomes
π (D1) = q(f1 − a1 − b1 − wy1 ) + (1 − q)(−a2 − b2 − wy2 )




q(f1 − c1 ) − if λ ≤ 1 − q
=



q(f1 − c1 ) − 1−q if λ > 1 − q
(71)
λ
(D2) By (62) and (63), the RHS of (68) simplifies to wy1 + λ(a1 + b1 − c1 ).
Since wy1 ≥ wn2 , by (64), and λ > 0, inequality (68) gives us
wy1 − wn2 ≤ λ(a2 − (a1 + b1 − c1 ))
⇒ a2 ≥ a1 + b1 − c1 .
(72)
(73)
The relevant wages are a1 , b1 , wy1 , a2 , and wn2 , while wn1 , b2 , and wy2 are not
relevant. We cannot pay less than a1 = 0 and b1 = c1 . Consider (67) and for
the moment suppose that we want to achieve a difference of > 0 between
the RHS and LHS of (67). The wages on the LHS of (67) are not relevant,
but the RHS is, thus, wy2 = 0. The only way to make the LHS negative is
with b2 < 0, i.e., b2 = −/λ. However, by (6), this requires at least a2 = /λ
36
which is relevant and costs (1 − q)/λ in expectation. The only alternative is
wn2 = . This, however, is relevant and, by (64), it requires wy1 = which is
also relevant, and, by (63), wn1 = which is not relevant. The total cost of
this is q + (1 − q) = . Thus, if λ < (1 − q), we set wn2 = wy1 = wn1 = and
a2 = b2 = 0 while if λ > (1 − q) we set wn2 = wy1 = wn1 = 0 and b2 = −/λ
and a2 = /λ. Finally, if λ = (1 − q), we get wn2 = wy1 = wn1 = ˜ ≤ ,
b2 = −( − ˜)/λ and a2 = ( − ˜)/λ. P’s expected profit becomes
π (D2) = q(f1 − a1 − b1 − wy1 ) + (1 − q)(−a2 − wn2 )




q(f1 − c1 ) − if λ ≤ 1 − q
=



q(f1 − c1 ) − 1−q if λ > 1 − q,
(74)
λ
which is equal to π (D1) .
Case (E) Only S2 reports effort truthfully, i.e., (19) holds while (18) does
not. We divide (E) into (E1) where S1 always reports m = y and (E2) where
m = n. Since S1 reports independent of A’s effort, A chooses e1 = 0 (and,
thus, c1 = f1 = 0) while A’s incentive constraint for e2 is a2 + b2 − c2 ≥ a2 , or,
b2 ≥ c2 . Collecting (and simplifying) all relevant constraints (see (6), (19),
37
(20), (22) for (E1), and (23) for (E2)), we need to satisfy
b2 ≥ c2
(75)
(E1) : wy1 > wn1
(76)
(E1) : wy1 ≥ max{wy2 , wn2 }
(77)
(E2) : wy1 < wn1
(78)
(E2) : wn1 ≥ max{wy2 , wn2 }
(79)
wy2 + λb2 = wn2
(80)
wy2 + λ(a2 + b2 − c2 ) ≥ max {wy1 + λ(a1 + b1 ), wn1 + λa1 } .
(81)
(E1) The relevant wages are ai , bi , wyi , for i ∈ {1, 2}, while wni are not
relevant. Set b2 to its minimum value, b2 = c2 , see (75). We cannot pay less
than wy2 = 0. It follows, by (80), that wn2 = λc2 . By (77), we get wy1 = λc2 .
Inserting these values in (81), we get
λa2 ≥ max {λc2 + λ(a1 + b1 ), wn1 + λa1 } .
(82)
The LHS of (81) is S2 ’s equilibrium payoff, so every component is relevant.
By the above and (6), the RHS of (82) cannot be less than λc2 . It is feasible
to have both sides equal to λc2 : Consider the LHS of (81). Raising wy2 or
b2 is not an option: both increase wn2 , by (80), and therefore wy1 , by (77),
which would increase the RHS of (82) to a value above λc2 . Thus, we set
a2 = c2 which makes the LHS of (82) equal to λc2 . Then, in order to keep
the RHS equal to λc2 , we need a1 + b1 = 0 and wn1 + λa1 ≤ λc2 . There is
some degree of freedom in setting the remaining wages, as long as we observe
these two conditions, as well as (6) and (76). Thus, we get a1 = t, b1 = −t,
38
for t ≥ 0, and wn1 ∈ [0, λc2 ). The principal’s profit is therefore
π (E1) = q(−a1 − b1 − wy1 ) + (1 − q)(f2 − a2 − b2 − wy2 )
(83)
= (1 − q)(f2 − c2 ) − c2 (1 − q + qλ).
(E2) The relevant wages are a1 , wn1 , a2 , b2 , and wy2 , while b1 , wy1 , and wn2
are not relevant. Set b2 to its minimum value, b2 = c2 , see (75). We cannot
pay less than wy2 = 0. It follows, by (80), that wn2 = λc2 . By (79), we get
wn1 = λc2 . Inserting these values in (81), we get
λa2 ≥ max {wy1 + λ(a1 + b1 ), λc2 + λa1 } .
(84)
The LHS of (81) is S2 ’s equilibrium payoff, so every component is relevant.
By the above and (6), the RHS of (84) cannot be less than λc2 . It is feasible
to have both sides equal to λc2 : Consider the LHS of (81). Raising wy2 or
b2 is not an option: both increase wn2 , by (80), and therefore wn1 , by (79),
which would increase the RHS of (84). Thus, we set a2 = c2 which makes
the LHS equal to λc2 . Then, in order to keep the RHS equal to λc2 , we need
a1 = 0 which, by (6), implies b1 = 0. Finally, wy1 ∈ [0, λc2 ) in order to satisfy
(78) and (84). The principal’s profit is equal to π (E1) :
π (E2) = q(−a1 − wn1 ) + (1 − q)(f2 − a2 − b2 − wy2 )
(85)
= (1 − q)(f2 − c2 ) − c2 (1 − q + qλ).
Case (F) Here, both S–types report truthfully, see (18) and (19). Since A
learns S’s type, his incentive constraints are ai + bi − ci ≥ ai , or, bi ≥ ci
for i ∈ {1, 2}. A chooses positive but (potentially) different efforts, e1 , e2 .
39
Collecting (and simplifying) all relevant constraints (see (6), (18), (19), (20),
and (21)), we need to satisfy
b1 ≥ c1 , b2 ≥ c2
(86)
wy1 = wn1
(87)
wy1 ≥ max{wy2 , wn2 }
(88)
wy2 + λb2 = wn2
(89)
wy2 + λ(a2 + b2 − c2 ) ≥ max {wy1 + λ(a1 + b1 − c1 ), wn1 + λ(a1 − c1 )}
(90)
The relevant wages are ai , bi , wyi for i ∈ {1, 2}, while wni are not relevant.
By (86), the minimum boni are b1 = c1 and b2 = c2 . By (87) and b1 > 0,
the RHS of (90) is equal to wy1 + λ(a1 + b1 − c1 ). Set wy1 = wy2 + λb2 since
combining (88) and (89) implies that it cannot be lower than that. Then
(90) delivers
a2 − c 2 ≥ a1 + b 1 − c 1 .
(91)
By b1 = c1 and (6), the RHS of (91) is nonnegative, which implies that we
cannot have less than a2 = c2 . We set the remaining wages equal to their
minimum feasible values: wy2 = 0, then, by (89), wn2 = λc2 . As shown
above, wy1 = wy2 + λb2 = λc2 , and, by (87), wn1 = λc2 . Finally, a1 = 0.
P’s profit becomes
π (F ) = q(f1 − a1 − b1 − wy1 ) + (1 − q)(f2 − a2 − b2 − wy2 )
(92)
= q(f1 − c1 ) + (1 − q)(f2 − c2 ) − c2 (1 − q + qλ).
Collecting results, we find that in all cases P’s expected profits, π (A) –π (F ) ,
are weighted sums of output and effort cost (with constant weights). Thus,
40
by our model assumptions we can confirm that the optimal requested efforts
are positive. An exception is π (D) : Here, a small amount is subtracted that
is independent of c and f . Although the optimal (smallest) amount does
not exist, it can always be chosen such that positive effort strictly dominates
zero effort.
Proof of Proposition 3. First, by Lemma 1, π (F ) > π (D) +π (E) : The first term
of π (F ) is larger than π (D) and the two remaining terms in π (F ) are equal to
π (E) . Since q ∈ (0, 1), the assertion already holds if one inserts the optimal
efforts for (D) and (E) in π (F ) . Thus, contracts as in (D) and (E) are strictly
dominated by (F). Thus, the remaining cases are (A)-(C) and (F).
Second, the first equality in (14) obviously holds by Lemma 1.
Third, given any positive effort e, it is straightforward to show that π (A) R
π (B) iff λ R (1 − q)2 /q 2 which gives us the second equality in (14).
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