OSIPP Discussion Paper: DP-2022-E-005

The Provision of High-powered Incentives under
Multitasking
November 25, 2022
Kohei Daido*
School of Economics, Kwansei Gakuin University

Takeshi Murooka†
Osaka School of International Public Policy (OSIPP), Osaka University

Abstract: We study multitasking problems where an agent engages in both a contractible task and a
non-contractible task, which are substitutes. The agent has private information on the value of the
non-contractible task, and there are followers who can also contribute to this task. We highlight a newmechanism by incorporating leading-by-example (Hermalin, 1998) in a multitasking model. To prevent
excessive effort by the agent with low value on the non-contractible task, the principal provides
high-powered incentives for the contractible task. We discuss its organizational implications to pay for
performance, incentives to help colleagues, and prevention of overwork.

JEL Codes: D82, D86, J33, M52
Keywords: Multitasking, Signaling, Leadership, Pay for Performance, Help, Overwork*

Email: daido@kwansei.ac.jp†

Email: murooka@osipp.osaka-u.ac.jp

The Provision of High-powered Incentives under Multitasking∗Kohei Daido†

Takeshi Murooka‡

November 25, 2022
Abstract
We study multitasking problems where an agent engages in both a contractible task and a
non-contractible task, which are substitutes. The agent has private information on the valueof the non-contractible task, and there are followers who can also contribute to this task. We
highlight a new mechanism by incorporating leading-by-example (Hermalin, 1998) in a multitasking model. To prevent excessive effort by the agent with low value on the non-contractible
task, the principal provides high-powered incentives for the contractible task. We discuss its
organizational implications to pay for performance, incentives to help colleagues, and preventionof overwork.
JEL Codes: D82, D86, J33, M52.
Keywords: Multitasking, Signaling, Leadership, Pay for Performance, Help, Overwork∗

We thank Matthias Fahn, Daisuke Hirata, Akifumi Ishihara, Hajime Kobayashi, Klaus Schmidt, Peter Schwardmann, Joel Sobel, seminar and conference participants, and especially Shintaro Miura for their helpful discussions
and suggestions. Kohei thanks JSPS KAKENHI (JP15K03529, JP18H03640, JP19K01568), Murata Science Foundation, Nomura Foundation, and the hospitality of the Department of Economics at UCSD. Takeshi is grateful for thefinancial support from JSPS KAKENHI (JP15K03529, JP18H03640, JP19K01568, JP20K13451, and JP22K13365).
†
School of Economics, Kwansei Gakuin University, 1-155 Ichiban-cho Uegahara Nishinomiya, Hyogo, JAPAN
662-8501 (Email: daido@kwansei.ac.jp).
‡
Osaka School of International Public Policy, Osaka University, Machikaneyama 1-31, Toyonaka, Osaka, JAPAN
560-0043 (Email: murooka@osipp.osaka-u.ac.jp).

11

Introduction

After the seminal work by Holmstrom and Milgrom (1991), multitasking problems have been one of
the most important topics in contract theory and organizational economics, as well as many otherfields.1 In a standard multitasking model, an agent is responsible for multiple tasks, some of which
are not easy to monitor and reward (called “non-contractible tasks”). One of the main results is thatwhen the tasks are substitutes or effort costs are complements, a principal prefers to provide a lowerpowered incentive for a task which outcome is verifiable (called a “contractible task”) compared
to a case in which the agent works only on the contractible task. However, even when some tasks
are non-contractible, the effort spent on the task can be observed by other agents (e.g., helping
colleagues or working while on vacation). In addition, the value of such a non-contractible task isoften uncertain, and some agents know more about it than other agents (e.g., senior employees know
more about the importance of non-contractible work than junior employees do, or a salesperson
who directly meets customers knows more about their demand than administrative officers).
This study builds a multitasking model in which an agent’s effort can signal the non-contractible
task’s value to others. We highlight a new rationale that, for an agent to work harder on a noncontractible task, a principal may prefer to provide higher-powered incentives to a contractible task.
Because the signaling game between agents (leading-by-example by Hermalin, 1998) is embeddedin the multitasking model, the principal can indirectly control the incentives for agents to workon the non-contractible task by designing a payment scheme for the contractible task. We derive
the condition in which the principal offers higher-powered incentives to a contractible task, incontrast to the result of the standard multitasking model (with no signaling). The results have
implications to pay for performance, incentives to help other agents, and the prevention of overwork
in multitasking situations.
In this model, the agent engages in two substitute tasks simultaneously. The principal canverify the performance of one task, while she cannot write any contract with regard to the othertask. A key element in our model is that the agent has private information on the value of the
non-contractible task (either high or low), and there is a follower (e.g., another agent, third party,
1

For example, 2016 Scientific Background on the Prize in Economic Sciences in Memory of Alfred Nobelfeatures three extensions of the basic moral-hazard model as highly influential and important: the multitasking
model, incentives in teams, and career concerns. https://www.nobelprize.org/prizes/economic-sciences/2016/press-release/ (accessed October 19, 2022).

2

or principal) who can also contribute to the non-verifiable task after observing the agent’s action. In this setting, the agent and follower play a signaling game concerning the value of the
non-contractible task. The signaling role of the agent’s effort can be interpreted as “leading-byexample” studied by Hermalin (1998). The signaling effect has significant implications on the
optimal incentive contracts the principal offers: the optimal contract may specify a high-powered
incentive to the contractible task to suppress the agent’s excessive effort on the non-contractible
task. Intuitively, by providing a sufficiently high bonus to the contractible task, the agent does
not work hard on the non-contractible task if its value is low. Hence, under the high-powered
incentive to the contractible task, working hard on the non-contractible task becomes a credible
signal with regard to the value of the non-contractible task. If this effect is sufficiently important
for the principal, she provides a high-powered incentive to the contractible task (with a low fixedwage).2 The difference between the optimal contract in our model and the one in the standard
multitasking model can be substantial: The bonus upon the success of the contractible task may
exceed the highest possible revenue from the contractible task.
These results have implications for managerial incentives in organizations. For example, supposethat a salesperson engages in both the main sales activity and other informal activities (e.g.,
participating in events in the community of a sales territory). Unlike the main sales activity,
the informal activities do not directly contribute to sales performance, but they might affect thecompany’s overall performance. Only the salesperson knows how each informal activity contributes
to overall performance. Even if an informal activity is worthwhile, the company’s support isnecessary to succeed. Hence, if the salesperson knows that the informal activity contributes tooverall performance, the company should support it. On the other hand, if the salesperson indulges
in the informal activity just for fun (e.g., to host parties), the company should not support it. To
alleviate inefficiencies from information asymmetry, it may be optimal for the company to provide
a high bonus for the main sales activity (with low fixed payment), which suppresses excessive effort
in the informal activity by the salesperson who knows that the informal activity is not worthwhile.
Related literature. This paper relates mainly to two literatures: multitasking and leadership.3
In the literature on multitasking initiated by Holmstrom and Milgrom (1991), our study is mainly
2
3

We use female pronouns to refer to a principal and male pronouns to refer to an agent.
We will introduce other related studies in section 5 which provides some possible applications.

3related to those in which some agents have private information. Bénabou and Tirole (2016) investigate the impact of labor-market competition on contracts in a Hotelling model, in which both
multitasking and screening problems matter. They show that competition for unobservable talents
causes high-powered incentives for the task easily measured, which leads to distorted decisions
and efficiency losses. Although they mainly focus on private information about the quality of acontractible task, they also highlight private information about the quality of a non-contractible
task (called “ethical motivation”) as an alternative setting. However, in this setting, they focus on
screening and show that the first-best outcome could be achieved under perfect labor-market competition. Kosfeld and von Siemens (2011) study multitasking problems with worker heterogeneity
regarding social preferences. They show that firms employing “conditionally cooperative” workers
moght make strictly positive profits even in a competitive labor market due to adverse selectionof the workers and discuss its implications for different corporate cultures. While these studies
focus on adverse selection and labor market competition, we shed light on the signaling role inthe standard multitasking setting. We show that the principal may offer a high-powered incentive
with a lower fixed wage relative to the symmetric information situation because the high-powered
incentive improves the credibility of the signal for the value of the non-contractible task. By thismechanism, the bonus on the contractible task may exceed the highest possible revenue from thecontractible task.4
Our signaling effect is closely related to the literature on the economics of leadership. Specifically, Hermalin (1998) and the following literature on leading-by-example investigate how an action
taken by a leader agent can credibly convey the agent’s private information about the value of atask to the following agents. Hermalin (1998) studies a team-production problem and shows thatthe sequence of moves may improve team performance and enhance efficiency because the leader’saction serves as a signal of the value of the task.5 While Hermalin (1998) assumes that an agent’s
action fully conveys the true value, Komai et al. (2007) extend Hermalin (1998) to state that theleader’s action partially informs the value. They show that efficiency can be improved under partial
4

Sinclair-Desgagné (1999) shows how selective auditing can lead to high-powered incentives under multitasking.
Our model features a different mechanism and also provides additional insights, e.g., when the bonus upon the successon the contractible task exceeds the highest benefit of the contractible task.
5
Kobayashi and Suehiro (2005) and Huck and Rey-Biel (2006) study how leadership can arise endogenously inleading-by-example models. Costa (2014) provides historical evidence on leading-by-example. Vesterlund (2003)andAndreoni (2006) investigate dynamic public-good provision problems with a focus on leading-by-example. For thereview on the economics of leadership, see Hermalin (2012).

4

information. Komai and Stegeman (2010) analyze the above team-production model by incorporating coordination problems, and highlight a new trade-off between shirking and coordination.
Komai and Stegeman (2010) also study the problem of choosing an optimal leader by assuming
heterogeneous agents. Zhou (2016) studies leadership in a hierarchical structure and optimal organizational structure endogenously. To the best of our knowledge, however, previous studies on
leading-by-example have not investigated the implications of performance-based payment scheme.
We analyze optimal performance-based payment schemes in the presence of leading-by-example byfocusing on multitasking situations, highlighting the new mechanism and implications discussed
above.
Organization of the paper. This remainder of this article is organized as follows. Section 2
sets up our model. Section 3 illustrates our result with a binary action space. Section 4 analyzes
the optimal contract and discusses its implications with providing an example. Section 5 covers
applications of our model to CEO payments, incentives to help other agents, and overwork. Section
6 concludes. All proofs are in Appendix.

2

Setup

There are three parties: a principal (P ), an agent (A), and a follower (F ).6 When the principal
hires the agent, the agent can make non-verifiable efforts on two tasks (n = 1, 2). The outcome oftask 1 is verifiable and solely depends on the agent’s effort. Following the standard multitasking
framework, we focus on a situation in which the principal cannot write any contract with regard
to the other task (task 2). The outcome of task 2 depends both on the agent’s effort and on thedecision made by the follower after observing the agent’s effort in task 2.
At the beginning, the principal offers the agent a contract (w, b), where w is an unconditional
fixed payment and b is a bonus upon success in task 1. If the agent rejects the contract, then hereceives the outside option ū ∈ R. If the agent accepts the contract, he exerts effort en ∈ R+ for
n = 1, 2. Let e = (e1 , e2 ) denote the agent’s effort profile.7 Task 1 succeeds with probability q(e1 )and fails with probability 1 − q(e1 ) where q(e1 ) ∈ [0, 1) for all e1 ∈ R+ . q(·) is strictly increasing,6
Throughout the analysis, we assume that there are three different parties. However, the follower may notnecessary be a third party. For example, the principal can also serve the role of the follower.
7
Section 3 analyzes the illustrative case in which the agent’s effort choice is binary.

5twice continuously differentiable, and lime1 →0 q 0 (e1 ) = +∞. When task 1 succeeds, the principal
receives V > 0; otherwise, she receives 0 from task 1.
The agent has private information on the value of task 2, denoted by θ ∈ {θ, θ} with 0 < θ < θand ∆θ ≡ θ − θ. There is a common prior belief that θ = θ with probability µ ∈ (0, 1). Denote byE(θ) = µθ + (1 − µ)θ and ∆θ = θ − θ. When the agent takes up the contract, he privately observes
the true value of θ. Because the follower cannot observe θ, the follower infers the value of θ byobserving the agent’s effort for task 2. The decision of the follower is denoted by d ∈ R+ .
The payoff from task 2 for each player i (i = P, A, F ) is (xi e2 + yi d)θ, where xi , yi ≥ 0 and
xA , yA > 0. The agent’s effort cost for (e1 , e2 ) is cA (e1 , e2 ), where cA (e1 , e2 ) is strictly increasing,2cA (e1 ,e2 )twice continuously differentiable, strictly convex in (e1 , e2 ), and ∂ ∂e
> 0. The follower’s
1 ∂e2effort cost is cF (d), where cF (0) = c0F (0) = 0, cF (·) is strictly increasing, strictly convex, twice
continuously differentiable, and limd→∞ c0F (d) = ∞. We assume that for any fixed d ≥ 0 andb > 0, the agent’s optimal effort choice (e1 , e2 ) is unique, interior, and satisfies the second-ordercondition.8
The timing is summarized as follows:
1. The principal offers a contract (w, b) to the agent.
2. The agent decides whether or not to accept the contract.
3. The agent receives a signal about θ.
4. The agent chooses e = (e1 , e2 ).
5. The follower observes e2 and then makes a decision d.
6. The outputs are realized and payments are executed.
Note that the agent’s action depends on θ and the follower’s action depends on e2 . Let eθ = (eθ1 , eθ2 )be the action profile for the agent with type θ. Let Ie2 be the information set in which the follower
who observed e2 makes a decision. Let µe2 = P rob{θ = θ|e2 } be the follower’s posterior belief
8


 2
 2
2
2
2
(e1 ,e2 )(e1 ,e2 )∂ cA (e1 ,e2 )∂ cA (e1 ,e2 )Specifically, we assume q 00 (e1 )b − ∂ cA∂e
< 0 and − q 00 (e1 )b − ∂ cA∂e
−
>0
2
2
2
∂e1 ∂e2∂e1

12

for the second-order condition. A sufficient condition for the unique and interior effort choice is that xA > 0, q 00 (·) ≤ 0,(e1 ,e2 )(e1 ,e2 )and ∂cA∂e|e1 =0 = ∂cA∂e|e2 =0 = 0.
126

on θ after observing e2 .9 Let d(µe2 ) be the decision of the follower who holds the posterior beliefµe2 . For continuation games played between the agent and follower, we adopt perfect-Bayesianequilibria (PBE).

3

Illustration with Binary Effort

This section illustrates a simple case in which each party’s effort choice is binary: e ∈ {(1, 0), (0, 1)},d ∈ {0, 1}, task 1 succeeds with probability q(1) = q ∈ (0, 1) and q(0) = 0, the agent’s effort cost iscA = 0, and the follower’s effort cost is cF · d with cF > 0.10 To focus on the most interesting case,
we make the following assumptions throughout this section:
Assumption 1. (i) qV > u, (ii) yF E(θ) < cF < yF θ, (iii) xA θ < (xA + yA )θ.
Assumption 1 (i) states that working on task 1 yields a positive social surplus. Assumption 1 (ii)
implies that the follower does not prefer to work on task 2 if there is no information. Assumption 1(iii) is to focus on our most interesting case in which the agent’s payoff on task 2 crucially depends
on the follower’s decision. In the rest of this section, we analyze the case where the principal sets
the lowest bonus by which her intended equilibrium is played.11 For continuation games played
between the agent and the follower, we focus on a pure-strategy PBE whenever it exists.
We first illustrate optimal contracts in the benchmark case where there is no asymmetric information; all parties observe θ when the agent observes it. Let (w∗ , b∗ ) denote the optimal contract
for this benchmark case. In this case, the follower chooses d = 1 if and only if θ = θ regardless of
the agent’s effort. Also, the type-θ agent chooses (1, 0) if and only if qb ≥ xA θ. Given the above
actions, the optimal bonus level is either b∗ = xAq θ (when the principal prefers both types of agent
to work on task 1), b∗ = xAq θ (when the principal prefers only the type-θ agent to work on task 1),
or b∗ = 0 (when the principal prefers both types of agent to work on task 2). Hence, the highest
possible bonus level in this benchmark case is b∗ = xAq θ .
9For simplicity, we assume that the follower observes e2 but not e1 . All our results remain qualitatively the sameif the follower can observe both e1 and e2 .
10
In this setup, none of the following results changes when e = (0, 0) is included in the agent’s effort choice. Thisis because e = (0, 0) never constitutes the optimal contract under cA = 0.
11
Under binary effort and linear utility, multiple optimal contracts exist under certain parameters. As we show insection 4, the optimal contract becomes unique under continuous effort. Alternatively, if we incorporate the agent’s
risk aversion into the binary model, the optimal contract will be uniquely pinned down to the specified one.

7

In the following, we demonstrate that the principal may set a bonus strictly higher than xAq θ
when θ is the private information of the agent. After observing the principal’s offer (w, b), there
are three pure-strategy PBE played by the agent and follower, which depend on the values of b.12
First, there exists a separating equilibrium in which the type-θ agent works on task 2 and the
A )θ
A )θ
type-θ agent works on task 1 if (xA +y≤ b ≤ (xA +y. In this equilibrium, the follower worksqq

on task 2 if and only if the agent exerts effort on task 2. Second, a pooling equilibrium exists inwhich both types of the agent work on task 1 if b ≥ xAq θ .13 Third, there is a pooling equilibrium inwhich both types of the agent work on task 2 if b ≤ xAq θ . In both pooling equilibria, the follower
never works on task 2 on the equilibrium path by Assumption 1 (ii).
We now derive the conditions under which the principal offers higher-powered incentives than thebenchmark cases. Let (w∗∗ , b∗∗ ) denote the optimal contract in the case of asymmetric information.
By Assumption 1 (i), the principal prefers the agent to accept a contract. Under the optimal
contract, the agent’s individual rationality constraint is binding. There are three possible cases.
First, given that both types of the agent work on task 1, the optimal contract is (w∗∗ , b∗∗ ) =u − xA θ, xAq θ and the principal’s payoff is πpool−1 = qV − u. Second, given that both types of
the agent work on task 2, the optimal contract is (w∗∗ , b∗∗ ) = (u − xA E(θ), 0) and the principal’s
payoff is πpool−2 = (xP + xA )E(θ) − u. Third, given that the type-θ agent works on task 2 and theA )θ
type-θ agent works on task 1, the optimal contract is (w∗∗ , b∗∗ ) = u − (xA + yA )E(θ), (xA +yq
and the principal’s payoff is πsep = µ(xP + xA + yP + yA )θ + (1 − µ)qV − u.
We derive the following proposition by comparing among the principal’s payoffs in these threecases.
Proposition 1. Suppose Assumption 1 holds. Then, the optimal contract isA )θ
(w∗∗ , b∗∗ ) = u − (xA + yA )E(θ), (xA +y
if and only if the following condition holds:q(xP + xA )θ −µ(yP + yA )θ < qV < (xP + xA + yP + yA )θ.
1−µ

(1)

12

For some values of b, a pure-strategy PBE does not exist; hence a mixed-strategy PBE is played. In the proof of
Proposition 1, we show that such mixed-strategy PBE is never implemented in the optimal contract. Furthermore, in
the Supplementary Material, we derive all the PBE (including mixed strategies) and show that the optimal contractstated in Proposition 1 remains the same.
13
If we impose the Intuitive Criterion (Cho and Kreps, 1987), then a pooling equilibrium in which both types
A )θ
A )θ
of the agent working on task 1 fails to survive for (xA +y< b < (xA +y. However, note that the lowest bonusqqin this pooling equilibrium, which survives the Intuitive Criterion, is still b = xAq θ by Assumption 1 (iii). A fullcharacterization by imposing the Intuitive Criterion is presented in the Supplementary Material.

8

Proposition 1 characterizes the condition in which the optimal contract is supported by the separatA )θ
ing equilibrium. From Assumption 1 (iii), b∗∗ = (xA +yis strictly higher than xAq θ ; the principalqoffers a strictly higher bonus level than the benchmark case. Intuitively, by providing a sufficiently
high bonus to task 1 so that the agent never works on task 2 if the value of task 2 is low, theprincipal can credibly infer the value of the task 2. The following comparative statics holds. First,
Condition (1) is more likely to be satisfied as θ is higher. Second, Condition (1) is more likely to
be satisfied as the benefit from the follower’s support for task 2 (i.e., yP or yA ) increases. Third,
Condition (1) is more likely to be satisfied as the likelihood of θ (i.e., µ) increases.
The signaling game between the agent and the follower can be viewed as a model of “leadership”
by Hermalin (1998) and the following literature. In our model, the principal facing a multitasking
problem attempts to take advantage of the agent’s leadership. The type-θ agent increases his effort
level, which is a credible signal to make the follower support task 2. Our multitasking model— which embeds a signaling game — shows that the principal needs to provide a high-powered
incentive for task 1 to prevent the type-θ agent from pretending to be the type-θ agent. As a result,the principal can save the fixed wage payment and increase her expected payoff. We examine thismechanism in detail in the next continuous-effort model.

4

Analysis

This section analyzes the model with continuous effort choices and derives its implications. Section
4.1 presents a benchmark case with symmetric information. Section 4.2 characterizes the PBE
played between the agent and follower. Section 4.3 analyzes the optimal contract and provide an
example.

4.1

Benchmark Result

In this subsection, we analyze a benchmark case with symmetric information. Suppose that all
parties observe θ when the agent observes it.
We first characterize the actions of the follower and agent. Suppose that contract (w, b) is
accepted. The follower maximizes yF dθ − cF (d) and hence chooses d∗ such that yF θ = c0F (d∗ ).
For consistency of notation with the analysis in the main model with private information, let d∗ (1)denote the follower’s action when θ = θ and d∗ (0) denote the follower’s action when θ = θ.
9Anticipating the follower’s action, an agent with type θ maximizes q(e1 )b + [xA e2 + yA d∗ ]θ −
θ∗cA (e1 , e2 ). In this benchmark case, the agent with type θ chooses (eθ∗1 , e2 ) such thatθ∗∂cA (eθ∗1 , e2 )
= 0,
∂e1
θ∗
∂cA (eθ∗1 , e2 )xA θ −= 0.
∂e2

q 0 (eθ∗1 )b −In the proof of Proposition 2, we show that

(2)
(3)

∂eθ∗
∂eθ∗
1
2
∂b > 0 and ∂b < 0.

Next, we derive the optimal contract. Let (w∗ , b∗ ) denote the optimal contract for this benchmark case. In the optimal contract, the agent’s individual rationality constraint binds (otherwise,
the principal can reduce w∗ ):
θ∗

θ∗

θ∗

θ∗

θ∗
θ∗ θ∗
w = ū−µ[q(eθ∗1 )b+(xA e2 +yA d(1))θ−cA (e1 , e2 )]−(1−µ)[q(e1 )b+(xA e2 +yA d(0))θ−cA (e1 , e2 )].

Then, the principal’s profits are
n
o
n
o
θ∗
θ∗
θ∗
Π =µ q(eθ∗
+
(1
−
µ)
q(e
)(V
−
b)
+
[x
e
+
y
d(0)]θ
)(V
−
b)
+
[x
e
+
y
d(1)]θ
−w
P 2
P
P 2
P
1
1
n
o
θ∗
θ∗ θ∗
=µ q(eθ∗1 )V + [(xP + xA )e2 + (yP + yA )d(1)]θ − cA (e1 , e2 )
n
o
θ∗
θ∗
θ∗ θ∗
+ (1 − µ) q(e1 )V + [(xP + xA )e2 + (yP + yA )d(0)]θ − cA (e1 , e2 ) − ū,(4)
Proposition 2 provides the optimal level of bonus in the benchmark case:
Proposition 2 (Symmetric Information). Suppose all parties observe θ. Then, the optimal contract (w∗ , b∗ ) specifies
<0

z"
xP µ
∗

b =V +

∂eθ∗
2
∂b

}|
θ + (1 − µ)

θ∗
∂e2

∂b
θ∗

#{
θ

∂eθ∗
∂e1 0 θ∗µ 1 q 0 (eθ∗q (e1 )
1 ) + (1 − µ)
∂b
∂b
|
{z
}

.

(5)

>0

In particular, b∗ ≤ V holds strict inequality when xP > 0.
The intuition behind Proposition 2 is akin to Holmstrom and Milgrom (1991): when effort costs
are complements, the principal sets a low incentive for a contractible task.14
14

As another benchmark case, if there is no benefit from the follower’s decision (i.e., yi = 0 for i = P, A), then the
optimal contract is the same as that specified in Proposition 2.

10

4.2

Perfect-Bayesian Equilibria played by the Agent and the Follower

We now analyze the main model where θ is the agent’s private information. This subsection analyzes
the continuation games played by the agent and follower after observing the principal’s offer (w, b).
We adopt perfect-Bayesian equilibria that satisfy the Intuitive Criterion (Cho and Kreps, 1987).
Follower’s Best Responses

Given the follower’s belief µe2 ∈ [0, 1], the follower maximizes[µe2 θ + (1 − µe2 )θ]yF d − cF (d). Hence, the follower chooses d(µe2 ) such that [µe2 θ + (1 − µe2 )θ]yF =c0F (d(µe2 )). By the assumption of cF (·), d(µe2 ) is unique and strictly increasing in µe2 .
Perfect-Bayesian equilibria

Given µe2 and b, an agent with type θ maximizes q(e1 )b + [xA e2 +yA d(µe2 )]θ − cA (e1 , e2 ). Let eθ∗∗denote the equilibrium action of the type-θ agent in task i. In
iwhat follows, we derive a separating equilibrium by restricting the off-path belief that the follower
always thinks that a deviating agent must be the low type: µe2 = 0 for any e2 6= eθ∗∗2 .
Then, the type-θ agent’s action profile must satisfy the following conditions:θ∗∗ θ∗∗∂cA (e1 , e2 )θ∗∗= 0,
q (e1 )b −∂e1θ∗∗ θ∗∗∂cA (e1 , e2 )xA θ −
= 0,
∂e2
0

(6)

(7)
o
n
θ∗∗
θ∗∗
θ∗∗ θ∗∗
θ∗∗q(e1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 ) ≥ max q(e1 )b + [xA eθ∗∗+yd(1)]θ
)
.
−
c
(e
,
e
A
A 1 2
2
e1

(8)
(6) and (7) are conditions in which the type-θ agent takes his best responses for e2 6= eθ∗∗2 . (8)means that the type-θ agent does not mimic the type-θ agent by choosing eθ∗∗2 .
The type-θ agent’s action profile must satisfy the following conditions:q 0 (eθ∗∗1 )b −∂cA (e1θ∗∗ , eθ∗∗
2 )
= 0,
∂e1

(9)


θ∗∗
θ∗∗ θ∗∗q(eθ∗∗
1 )b + [xA e2 + yA d(1)]θ − cA (e1 , e2 ) ≥ max q(e1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 ) .
e1 ,e2(10)(9) implies that the type-θ agent optimally chooses e1 given e2 = eθ∗∗2 . (10) represents the conditionin which the type-θ agent does not have the incentive to choose e2 6= eθ∗∗2 . By the assumption of
the second-order condition, the set of separating equilibria given d(µe2 ) is characterized as a tupleof actions that satisfies (6)-(10).
11Note that eθ∗ , which is characterized by (2) and (3) in the benchmark case, maximizes thetype-θ agent’s payoff when µe2 = 1, i.e.,eθ∗ = argmax(e1 ,e2 ) q(e1 )b + [xA e2 + yA d(1)]θ − cA (e1 , e2 ) .
Given eθ∗∗ and eθ∗∗ , let ẽ2 denote the effort level of task 2 where ẽ2 is the highest value among
which (8) binds. Let ẽ1 denote the associated effort level of task 1.
In the proof of Proposition 3, we show that if the type-θ agent’s action profile in a separating
equilibrium survives the Intuitive Criterion, then it is either eθ∗∗ = eθ∗ or eθ∗∗ = ẽ2 with ẽ2 > eθ∗
2 .
In either case, the type-θ agent’s equilibrium action profile is the same as in the benchmark case:eθ∗∗ = eθ∗ . Note that each action profile of the type-θ agent is uniquely determined, implying thatthe separating equilibrium that survives the Intuitive Criterion is unique. We also show that all
pooling equilibria (i.e., both types of the agent choose the same task-2 effort level) fail to survive
the Intuitive Criterion. Proposition 3 summarizes the results.
Proposition 3. In continuation games played by the agent and follower, a perfect-Bayesian equilibrium that survives the Intuitive Criterion is unique. It is a separating equilibrium: different
types of the agent choose different levels of e2 .

4.3

Optimal Contract

We now characterize the optimal contract given the above equilibrium actions. Note that theequilibrium effort levels of the type-θ agent are eθ∗∗ = eθ∗ , and the equilibrium effort level of taskθ∗
2 of the type-θ agent is given by eθ∗∗2 = max{e2 , ẽ2 }.

In the optimal contract, the individual rationality constraint binds; otherwise, the principal can
reduce w. Hence, the principal sets:θ∗∗θ∗∗ θ∗∗
w = ū−µ[q(eθ∗∗1 )b + (xA e2 + yA d(1))θ − cA (e1 , e2 )]
θ∗∗

θ∗∗

−(1 − µ)[q(e1 )b + (xA e2

θ∗∗

θ∗∗

+ yA d(0))θ − cA (e1 , e2 )].

The principal’s profits are
n
o
n
o
θ∗∗
θ∗∗
θ∗∗
Π =µ q(eθ∗∗1 )(V − b) + [xP e2 + yP d(1)]θ + (1 − µ) q(e1 )(V − b) + [xP e2 + yP d(0)]θ − wonθ∗∗θ∗∗ θ∗∗
)V
+
[(x
+
x
)e
+
(y
+
y
)d(1)]θ
−
c
(e
,
e
)
=µ q(eθ∗∗
P
A
P
A
A
1
2
1
2
n
o
θ∗∗
θ∗∗
θ∗∗ θ∗∗+ (1 − µ) q(e1 )V + [(xP + xA )e2 + (yP + yA )d(0)]θ − cA (e1 , e2 ) − ū,(11)12where eθ∗∗= max{e∗2 , ẽ2 }. By solving this with respect to b in each case of eθ∗∗2
2 , Proposition 4
provides the optimal level of bonus in each case:∗∗ ∗∗Proposition 4. (i) Consider the case where eθ∗2 ≥ ẽ2 . Then, the optimal contract (w , b ) is

given by (5). In particular, b∗∗ ≤ V holds strict inequality if xP > 0.
∗∗ ∗∗(ii) Consider the case where eθ∗2 < ẽ2 . Then, the optimal contract (w , b ) specifies<0b∗∗ = V +

<0
z"
}|
#{
z }| {
θ∗∗i∂ẽ2 h∂e∂ẽ2(ẽ1 ,ẽ2 )µθ + (1 − µ) 2 θ
xA θ − ∂cA∂e+ xP µ
2
∂b
∂b
∂b
θ∗∗

.

(12)

∂ẽ1 0∂eθ∗∗µq (ẽ1 ) + (1 − µ) 1 q 0 (e1 )
∂b
∂b
|
{z
}
>0

(ẽ1 ,ẽ2 ).
Specifically, when xP = 0, b∗∗ > V if and only if xA θ < ∂cA∂e
2

Proposition 4 (i) covers the case where the signaling effect is not a relevant concern. In this case,the optimal contract is derived using the same logic as in the benchmark case under symmetric
information. The intuition behind Proposition 4 (ii) is as follows. In this case, the type-θ agent
exerts excessive effort on task 2 because of a signaling motive. Then, setting a higher bonus ontask 1 reduces the type-θ agent’s effort level on task 2, which improves efficiency under excessive
signaling. By doing so, the principal can lower the fixed wage and increase expected profits.
(ẽ1 ,ẽ2 )Note that if xA θ ≥ ∂cA∂e(i.e., the marginal return on task 2 for the type-θ agent is non2

negative), the optimal contract specifies b∗∗ ≤ V . When xP = 0 so that b∗∗ = V in the benchmark(ẽ1 ,ẽ2 ), then the optimal contract specifies b∗∗ > V . Whencase, the inverse is also true: if xA θ < ∂cA∂e
2

xP > 0 so that b∗∗ < V in the benchmark case, another effect arises: the principal directly receives
payoffs from task 2, which puts downward pressure on the optimal bonus level. Still, b∗∗ > V is
optimal if this effect is outweighed by the effect of mitigating the excessive signaling by the type-θ
agent.
Example2

We describe an example for Proposition 4 in which q(e1 ) = e1 (< 1), cF (d) = d2 ,22cA (e) = (e12) + (e22) + se1 e2 where s ∈ (0, 1), xA = xF = yA = yF = yP = 1, and xP ≥ 0.
We first analyze the benchmark case under symmetric information. Because the follower also
observes θ, d∗ = θ for θ ∈ {θ, θ}. The type-θ agent maximizes his expected payoff w + eθ1 b + (eθ2 +13θ∗d∗ )θ − cA (eθ ) given (w, b); hence, the optimal action profile is (eθ∗1 , e2 ) =b−sθ θ−sb,
1−s2 1−s2. Given theabove and binding individual rationality constraint, the principal solves the following problem:
i
h
i
h
θ∗
θ∗
θ∗
θ∗
θ∗
V
+
{(x
+
1)e
+
2θ}θ
−
c
(e
)
+
(1
−
µ)
e
V
+
{(x
+
1)e
+
2θ}θ
−
c
(e
)
− ū.
max µ eθ∗
P
A
P
A
1
2
1
2
b

By solving it, b∗ = V − sxP E(θ).
Next, we analyze our main case. Let eθ∗∗denote the equilibrium action of a type-θ agentiθ∗∗ θ∗∗θ∗ θ∗b−sθ θ−sb
in task i. For the type-θ agent, (e1 , e2 ) = (e1 , e2 ) = 1−s,
2 1−s2 . For the type-θ agent,θ∗eθ∗∗= max{eθ∗22 , ẽ2 }, where e2 is determined at the type-θ agent’s payoff and ẽ2 is determined byholding the type-θ agent’s incentive compatibility constraint with equality. When max{eθ∗
2 , ẽ2 } =eθ∗
2 , the optimal action profile and the optimal contract are the same as those in the benchmarkpb−sθ−sD θ−sb+D
θ∗
θ∗∗
θ∗∗
case. When max{e2 , ẽ2 } = ẽ2 , (e1 , e2 ) =
where
D
=
,
2(1 − s2 )θ∆θ . The
2
2
1−s
1−s
optimal bonus is b∗∗ = V − sxP E(θ) + µs(D − ∆θ ) and w∗∗ is determined as holding the individual
rationality constraint with equality. Consequently, when D > ∆θ or 2(1 − s2 )θ > ∆θ , the optimal
bonus for task 1 when the agent has private information on task 2 is higher than when all parties
have symmetric information (i.e., b∗∗ > b∗ ). This example illustrates how the bonus upon the
success of the contractible task may exceed the revenue from contractible task V .

5

Applications

This section presents three applications of our model: pay for performance, incentives to help otheragents, and prevention of overwork.

5.1

Pay for Performance

Murphy and Zábojnik (2004) discuss that there are two components for CEOs: general management
skills (i.e., the skills transferable across companies) that are priced in a labor market and firmspecific managerial skills (i.e., information about its products, suppliers, and clients) that are
unpriced.
In our model, unpriced managerial skills are private information. Investing in such an unpriced
skill by closely communicating with suppliers and clients can be beneficial to the company. On theother hand, a CEO might enjoy having parties, wining and dining, or empire-building. As mentioned in Hermalin (2016), J.P. Morgan estimated that US corporate expenditure on entertainment
14

was about $7.85 billion in 2011.15 Harford (1999) reports that CEOs may use abundant internal
resources for empire-building: making bad acquisitions and destroying shareholder value.
For optimal signaling (to stakeholders, creditors, or employees), our results imply that the pricefor the general management skills, which is discussed in Murphy and Zábojnik (2004), is evenhigher in the multitasking setting than in the corresponding single-task setting. We highlight thathigh payments for general management skills can improve the efficiency of unpriced firm-specificmanagement skills, because it alleviates excessive unpriced activities, such as wining and dining or
empire-building.

5.2

Help

Our results have implications for an agent’s incentives to help other agents (Itoh, 1991; Ishihara,
2017; Tymula, 2017). An agent may allocate his effort to a non-contractible activity that helps
other agents (i.e., task 2 in our model). We show that a lower (indirect) incentive to help otheragents (i.e., increasing b) may increase the credibility of the signal and incentives to help otheragents. In this sense, our results highlight a new rationale of the enhancement of cooperation
among agents by providing high-powered incentives to individual performance.
For example, consider a sales department that provides high-powered incentives for individual
sales performance. While each agent can work on individual sales, he can also help his colleagues(e.g., by sharing market information among agents), though it crowds out his sales performance.
While such helpful activities can be beneficial, an agent may indulge in them even when it does
not contribute much to organizational performance. The problem is that it is difficult for the
principal to evaluate the extent to which how each helpful activity contributes to organizational
performance. Hence, the principal may prefer to indirectly control the agents’ informal activities
through the individual sales activities, which is objectively easier to measure its performance. We
show that when the value of the helpful activity is private information of the agent, the principal
may provide high-powered incentives to the contractible activity. This leads to a separation between
beneficial and non-beneficial activities.
Although we are unaware of any empirical study that jointly investigates incentives to help andagents’ signaling motives, Drago and Garvey (1998) and Danilov et al. (2019) provide evidence of
15

Hermalin (2016) analyzes how such wining and dining can facilitate inter-firm cooperation, even when firms canuse their own incentive systems.

15incentives to help other work members. From the perspective of the classical multitasking model,
in which tasks are substitutes, an increased incentive would be detrimental as it reduces the effort
on the non-contractible task. However, given our signaling model, the reduced effort may beefficiency-improving, as it alleviates excessive signaling in the non-contractible task.

5.3

Overwork

Our mechanism could also have implications for overworking. As an example of overwork, theMinistry of Health, Labour and Welfare in Japan reported that more than half of full-time employees
experienced a high (or an extremely high) degree of fatigue (Mizuho Information & ResearchInstitute Inc., 2016). Galinsky et al. (2005) review overwork in the US and argue that one third of
all US employees could be viewed as chronically overworked. Importantly, Galinsky et al. (2005)
report that being overworked seems to be caused by multitasking (e.g., jumping from task to task)
and exerting effort on non-contractable work (e.g., cell-phone calls during non-work times).
In contrast to the conventional wisdom that high-powered incentives lead to excessive effort,
our multitasking model implies that the principal may provide high-powered incentives to thecontractible task to mitigate an agent’s excessive effort on the non-contractible task. In this sense,
our results offer new insights into bonus cultures and overwork.

6

Concluding Remarks

This study investigates multitasking problems in which some agents have private information on
the value of a non-contractible task. We characterize the conditions in which the principal provides
high-powered incentives for the contractible task, which contrasts with the finding by Holmstrom
and Milgrom (1991) Our results help to understand leadership in organizations, CEO pay for
performance, incentives to help other workers, and prevention of overwork.
Beyond the applications discussed, our mechanism can be applied to several important topics.
One potential application is signaling in charitable giving (e.g., Vesterlund 2003; Andreoni 2006).
Suppose a donor has private information (e.g, quality of charity) and initially contributes. This leaddonor’s contribution may signal to other potential donors who might also contribute. Designing
optimal charity schemes (e.g., tax deduction) in such a situation could be worth considering. Another potential application is the introduction of monitoring technologies, such as Sinclair-Desgagné
16

(1999) and Gil and Mondria (2011). How our signaling effect interacts with monitoring is left for
future research.

17

Appendix: Proofs
Proof of Proposition 1.
We first study a benchmark case in which all parties observe the signal on the value of task 2.
Next, we characterize PBE in the binary model with private information. Then, we derive optimal
contracts.
Benchmark: Symmetric Information Because there is no private information, the follower’s
choice does not depend on the agent’s choice of task 2. The follower chooses d = 1 when θ = θ andd = 0 when θ = θ by Assumption 1 (ii).
We first characterize the agent’s behavior. Given (w, b) and θ is realized, the agent’s payoff for
his each choice is uθ (1, 0) = w+qb+yA d(θ)θ if he chooses e = (1, 0) and uθ (0, 1) = w+(xA +yA d(θ))θ
if he chooses e = (0, 1), where d(θ) = 1 and d(θ) = 0 which do not depend on the agent’s action.
Because uθ (1, 0) ≥ uθ (0, 1) if and only if b ≥ xAq θ , the agent with type θ chooses e = (1, 0) if andonly if b ≥ xAq θ .
We next derive the optimal contract in the benchmark case (w∗ , b∗ ) depending on parameters.
In each case, w∗ is determined at holding the agent’s individual rationality constraint with equality:µ[w∗ + qb∗ + yA d(θ)θ] + (1 − µ)[w∗ + qb∗ + yA d(θ)θ] = w∗ + qb∗ + µyA θ = u

if b∗ ≥xA θ,qµ[w∗ + (xA + yA d(θ))θ] + (1 − µ)[w∗ + qb∗ + yA d(θ)θ] = w∗ + (1 − µ)qb∗ + µ(xA + yA )θ = uµ[w∗ + (xA + yA d(θ))θ] + (1 − µ)[w + (xA + yA d(θ))θ] = w∗ + µ(xA + yA )θ + (1 − µ)xA θ = u

xA θ xA θ
,
q
q
xA θ
∗
if b <.
q

if b∗ ∈

,

Then, the principal’s expected payoffs for each care are as follows:
π∗

b ≥

xA θ
q

= q(V − b∗ ) + µyP θ − w∗ = qV + µ(yA + yP )θ − u,π ∗ h xA θ xA θ  = (1 − µ)q(V − b∗ ) + µxP θ + µyP θ − w∗ = (1 − µ)qV + µ(xA + xP + yA + yP )θ − u,
b ∈

q

,

q

πb∗ < xA θ = xP E(θ) + µyP θ − w∗ = µ(xA + xP + yA + yP )θ + (1 − µ)(xA + xP )θ − u.
q

By comparing these payoffs and by the assumption that the principal sets the lowest bonus by18

which her intended equilibrium is played, the optimal contracts are derived as follows:(xA + xP )θ
xA θ∗ ∗if V ≥,(w , b ) = u − xA θ − µyA θ,
q
q




(xA + xP )θ (xA + xP )θ
xA θ∗ ∗if V ∈,(w , b ) = u − (1 − µ)xA θ − µ(xA + yA )θ,
,
q
q
q

(xA + xP )θ
(w∗ , b∗ ) = u − (1 − µ)xA θ − µ(xA + yA )θ, 0 if V <.
q
Note that in any of the three cases, the principal sets b∗ ≤ xAq θ . Hence, the optimal bonus undersymmetric information is at most b∗ = xAq θ .
Perfect-Bayesian Equilibria Now we analyze the binary model with private information.
First, we derive separating equilibria. Because of Assumption 1 (ii), the candidate of a separating equilibrium is unique: eθ = (0, 1), eθ = (1, 0), d(0) = 0, and d(1) = 1. It is indeed an equilibrium
A )θ
A )θ
≤ b ≤ (xA +y.
if and only if uθ (1, 0) ≤ uθ (0, 1) and uθ (0, 1) ≤ uθ (1, 0), or equivalently, (xA +yqq

Second, we derive pooling equilibria. Note that in any pooling equilibrium, because of Assumption 1 (ii), the follower chooses d = 0 on the equilibrium path. We focus on off-path beliefs in which
the follower’s belief is θ = θ after observing any deviation. We first consider the pooling equilibrium
in which eθ = eθ = (1, 0). In this case, uθ (1, 0) = w + qb. This kind of pooling equilibrium exists
when uθ (0, 1) ≤ uθ (1, 0) for all θ, that is, b ≥ xAq θ .16 We next consider the pooling equilibrium inwhich eθ = eθ = (0, 1). In this case, uθ (0, 1) = w + xA θ. This kind of pooling equilibrium exists
when uθ (1, 0) ≤ uθ (0, 1) for all θ, that is, b ≤ xAq θ .
Third, for parameters under a pure-strategy PBE does not exists, i.e., b ∈

xA θ xA θq , q, we derive

mixed-strategy equilibria.17 In this case, the type-θ agent randomizes his actions. Suppose that the
type-θ agent takes eθ = (0, 1) with probability l ∈ (0, 1) and that the follower takes d(1) = 1 withprobability r ∈ (0, 1) and d(0) = 0 with probability one. Then, the type-θ agent takes eθ = (0, 1)
with probability one. Under the mixed-strategy equilibrium, the follower must choose r∗∗ whichAθ
induces the type-θ to be indifferent: qb = (1 − r∗∗ )xA θ + r∗∗ (xA + yA )θ ⇐⇒ r∗∗ = qb−xyA θ . NoteA )θthat r∗∗ ∈ (0, 1) when b ∈ ( xAq θ , (xA +y). Also, the type-θ agent must choose l∗∗ which induces
q
16

A )θ
In Supplementary Material, we show that this type of equilibrium fails the Intuitive Criterion when (xA +y
<
q

A )θ
b < (xA +y.
q
17
In Supplementary Material, we derive all mixed-strategy PBE. We also show that the optimal contract neverinduces a mixed-strategy PBE to be played.

19

∗∗

(1−µ)lµ∗∗ = µ θyF −cF . Notethe follower to be indifferent: µ+(1−µ)l∗∗ yF θ + µ+(1−µ)l∗∗ yF θ − cF = 0 ⇐⇒ l
1−µ cF −θyF

that l∗∗ ∈ (0, 1) by Assumption 1 (ii).
Optimal Contracts

We now derive the conditions in which (even when the principal sets thelowest bonus by which her intended equilibrium is played) the principal offers higher-powered
incentives compared with the benchmark case. Note that the principal prefers the agent to accept
a contract and work on some task because ū < qV − cA and xP , yP ≥ 0. In the optimal contract,
the agent’s individual rationality constraint must bind:
θ∗∗

θ∗∗

θ∗∗
θ∗∗
w∗∗ = ū − µ[q(eθ∗∗1 )b + {xA e2 + yA d(e2 )}θ] − (1 − µ)[q(e1 )b + {xA e2

θ∗∗+ yA d(e2 )}θ],

where (eθ1 , eθ2 ) is the optimal action profile of the agent with θ ∈ {θ, θ} derived above. Then, theprincipal’s expected payoff is
n
o
n
o
θ∗∗
θ∗∗
θ∗∗
θ∗∗
+ yP d(eθ∗∗Π =µ q(eθ∗∗+ yP d(e2 )]θ − w∗∗
1 )(V − b) + [xP e22 )]θ + (1 − µ) q(e1 )(V − b) + [xP e2
n
o
θ∗∗
θ∗∗
)V
+
[(x
+
x
)e
+
(y
+
y
)d(e
)]θ
=µ q(eθ∗∗
P
A
P
A
1
2
2
n
o
θ∗∗
θ∗∗
θ∗∗
+ (1 − µ) q(e1 )V + [(xP + xA )e2 + (yP + yA )d(e2 )]θ − u.

There are three possible cases under pure-strategy PBE. First, given that both types of theagent work on task 1, the optimal contract with the lowest bonus is (w∗∗ , b∗∗ ) = (u − xA θ, xAq θ ) andthe principal’s payoff is πpool−1 = qV − u.
Second, given that both types of the agent work on task 2, the optimal contract with the lowest
bonus is (w∗∗ , b∗∗ ) = (u − xA E(θ), 0) and the principal’s payoff is πpool−2 = (xA + xp )E(θ) − u.
Third, given that the type-θ agent works on task 2 and the type-θ agent works on task 1,A )θ
the optimal contract with the lowest bonus is (w∗∗ , b∗∗ ) = (u − (xA + xP )E(θ), (xA +y) and theqprincipal’s payoff is πsep = µ[(xP + xA + yP + yA )θ] + (1 − µ)qV − u.
Note that
πsep > πpool−1 ⇐⇒ (xP + xA + yP + yA )θ > qV,(13)

πsep > πpool−2 ⇐⇒ µ[(yP + yA )θ] + (1 − µ)[qV − (xA + xP )θ] > 0.

(14)

From (13) and (14), we derive Condition (1) in Proposition 1.
Finally, we show that inducing a mixed strategy is not an optimal contract. Consider the mixedµ θyF −cF
strategy equilibrium in which the type-θ agent takes eθ = (0, 1) with probability l∗∗ = 1−µcF −θyF

20

Aθ
and the follower takes d(1) = 1 with probability r∗∗ = qb−xyA θ . In the optimal contract, theprincipal chooses (w∗∗ , b∗∗ ) which holds the agent’s individual rationality constraint with equality:w∗∗ + (1 − µ)(1 − l∗∗ )qb∗∗ + (1 − r∗∗ ){µθ + (1 − µ)l∗∗ θ}xA + r∗∗ {µθ + (1 − µ)l∗∗ θ}(xA + yA ) = u.
hiA )θThe principal chooses b∗∗ ∈ xAq θ , (xA +yby which she maximizes her expected payoff:qΠ =(1 − µ)(1 − l∗∗ )q(V − b) + (1 − r∗∗ ){µθ + (1 − µ)l∗∗ θ}xP + r∗∗ {µθ + (1 − µ)l∗∗ θ}(xP + yP )
− u − (1 − µ)(1 − l∗∗ )qb − (1 − r∗∗ ){µθ + (1 − µ)l∗∗ θ}xA − r∗∗ {µθ + (1 − µ)l∗∗ θ}(xA + yA )=(1 − µ)(1 − l∗∗ )qV − u + [µθ + (1 − µ)l∗∗ θ](xA + xP ) + [µθ + (1 − µ)l∗∗ θ](yA + yP )r∗∗µ θyF − cF
µ θyF − cF=(1 − µ) 1 −qV − u + µθ + (1 − µ)
θ (xA + xP )1 − µ cF − θyF
1 − µ cF − θyFµ θyF − cF
qb − xA θ.
+ µθ + (1 − µ)
θ (yA + yP )1 − µ cF − θyFyA θNote that the payoff function is linear in b and its coefficient is positive because yA > 0. Hence,A )θ
the principal should choose b∗∗ = (xA +y, which implies r∗ = 1 and the optimal contract inducesqa separating equilibrium.

Proof of Proposition 2.
We first prove

∂eθ∗
∂eθ∗
1
2
∂b > 0 and ∂b < 0. By applying the Implicit Function Theorem to (2) and (3),

we obtain:θ∗∂ 2 cA (eθ∗1 ,e2 )2∂e1θ∗
∂ 2 cA (eθ∗1 ,e2 )
− ∂e1 ∂e2

q 00 (eθ∗1 )b −

θ∗∂ 2 cA (eθ∗1 ,e2 )
∂e1 ∂e2θ∗∂ 2 cA (eθ∗1 ,e2 )
−
∂e22



∂eθ∗
1
∂b
∂eθ∗
2
∂b



θ∗
∂ 2 cA (eθ∗1 ,e2 )
−
2
∂e2

 2

−





=

−q 0 (eθ∗
1 )
0


,

Solving this system of equations,
∂eθ∗
1
=
∂b
∂eθ∗
2
=
∂b

θ∗
∂ 2 cA (eθ∗1 ,e2 )
∂e22

−



2

q 0 (eθ∗
1 )

θ∗ θ∗

∂ cA (e1 ,e2 )q 00 (eθ∗1 )b −∂e21−q 0 (eθ∗
1 )

θ∗ θ∗


∂ 2 cA (e1 ,e2 )− q 00 (eθ∗1 )b −∂e21θ∗∂ 2 cA (eθ∗1 ,e2 )
∂e1 ∂e2θ∗∂ 2 cA (eθ∗1 ,e2 )−∂e22

2 > 0,θ∗∂ cA (eθ∗1 ,e2 )
∂e1 ∂e22 < 0,θ∗∂ cA (eθ∗1 ,e2 )
∂e1 ∂e2 2

where the denominator is positive by the assumption.
We next derive the optimal level of the bonus b∗ . By taking the derivative of (4) with respect

21

to b, b∗ is characterized by:"#"#θ∗ , eθ∗ )θ∗θ∗ , eθ∗ )
∂c
(e
∂e
∂c
(e
∂eθ∗
A 1
A 1
2
2
+ µ 2 (xP + xA )θ −µ 1 q 0 (eθ∗1 )V −
∂b
∂e1
∂b
∂e2
"
"
#
#
θ∗
θ∗
θ∗ θ∗
θ∗ θ∗
∂e1
∂e2
∂cA (e1 , e2 )∂cA (e1 , e2 )0 θ∗
+ (1 − µ)q (e1 )V −+ (1 − µ)(xP + xA )θ −
= 0.
∂b
∂e1
∂b
∂e2
By substituting (2) and (3) into the above equality, we obtain:
θ∗

µ

θ∗

∂eθ∗
∂e
∂e1θ∗ 0 θ∗∂eθ∗q (e1 )(V − b∗ ) + µ 2 xP θ + (1 − µ) 1 q 0 (e1 )(V − b∗ ) + (1 − µ) 2 xP θ = 0.
∂b
∂b
∂b
∂b

By rearranging this equality, we obtain (5).

Proof of Proposition 3.
Separating Equilibria

We refine the separating equilibria derived in the main text by adoptingθ∗∗θ∗
the Intuitive Criterion. We first show that if (eθ∗∗1 , e2 ) 6= e , then (8) must hold with equality.
θ∗∗θ∗ andSuppose, toward a contradiction, that there is a separating equilibrium with (eθ∗∗1 , e2 ) 6= e
θ∗
θ∗∗
θ∗
θ∗∗ θ∗∗(8) holds with strict inequality. Note that eθ∗∗2 6= e2 ; otherwise, e1 = e1 and hence (e1 , e2 ) =eθ∗ by the right hand side of (8) and (9). Because (8) holds with strict inequality, there exists  > 0such thatθ∗∗θ∗∗q(e1 )b + [xA e2θ∗∗ θ∗∗+ yA d(0)]θ − cA (e1 , e2 ) > max q(e1 )b + [xA e02 + yA d(1)]θ − cA (e1 , e02 )
e1

θ∗∗
0
θ∗∗
θ∗∗
for any e02 ∈ (eθ∗∗2 − , e2 + ). This implies that any action e2 ∈ (e2 − , e2 + ) is equilibrium

dominated for the type-θ agent. Next, we check the case of the type-θ agent. Note that q(e1 )w +[xA e2 + yA d(1)]θ − cA (e1 , e2 ) is strictly concave by the assumption of the second-order condition;
0
θ∗∗
θ∗∗
it implies that any eθ∗∗6= eθ∗
2
2 is not a local maximum. Hence, there exists e2 ∈ (e2 − , e2 + )such thatθ∗∗θ∗∗ θ∗∗00q(eθ∗∗
1 )b + [xA e2 + yA d(1)]θ − cA (e1 , e2 ) < max q(e1 )b + [xA e2 + yA d(1)]θ − cA (e1 , e2 ) .
e1That is, such an e02 is not equilibrium dominated for the type-θ agent. By the Intuitive Criterion,
after observing e02 , the follower must conclude that the deviation comes from the type-θ agent (i.e.,θ∗∗µe02 = 1). But given this belief, the type-θ agent has an incentive to deviate from (eθ∗∗1 , e2 ) to

the above action profile — a contradiction. This result implies that in any separating equilibriumθ∗∗θ∗ or (8) binds.
surviving the Intuitive Criterion, either (eθ∗∗1 , e2 ) = e

22

θ∗∗
θ∗
θ∗∗ is the highest value among which (8) binds,
We now show that if (eθ∗∗1 , e2 ) 6= e , then e2i.e., eθ∗∗2 = ẽ2 . Suppose, toward a contradiction, that in some separating equilibrium which survivesθ∗∗ θ∗∗
θ∗ and eθ∗∗ 6= ẽ . By thethe Intuitive Criterion, there exists (e1θ∗∗ , eθ∗∗22 ) such that (e1 , e2 ) 6= e2θ∗∗ θ∗∗
definition of ẽ2 , eθ∗∗2 < ẽ2 holds. Because (8) binds at both (e1 , e2 ) and (ẽ1 , ẽ2 ),
θ∗∗

θ∗∗

q(e1 )b + [xA e2noθ∗∗ θ∗∗
θ∗∗+ yA d(0)]θ − cA (e1 , e2 ) = max q(e1 )b + [xA eθ∗∗+yd(1)]θ
−
c
(e
,
e
)
,
A
A 1 2
2
e1

(15)
θ∗∗

θ∗∗

q(e1 )b + [xA e2

θ∗∗

θ∗∗

+ yA d(0)]θ − cA (e1 , e2 ) = max {q(e1 )b + [xA ẽ2 + yA d(1)]θ − cA (e1 , ẽ2 )} .
e1

(16)
Combining (15) and (16), we havenoθ∗∗
max q(e1 )b + [xA eθ∗∗2 + yA d(1)]θ − cA (e1 , e2 ) = max {q(e1 )b + [xA ẽ2 + yA d(1)]θ − cA (e1 , ẽ2 )} .
e1e1

< ẽ2 , 0 < xA eθ∗∗Because eθ∗∗22 + yA d(1) < xA ẽ2 + yA d(1). Hence, replacing θ with θ in the above
equality leads to
n
o

θ∗∗
max q(e1 )b + [xA eθ∗∗2 + yA d(1)]θ − cA (e1 , e2 ) < max q(e1 )b + [xA ẽ2 + yA d(1)]θ − cA (e1 , ẽ2 ) .
e1

e1

(17)
Note that the left hand side of (17) is none other than the equilibrium payoff of the type-θ agent
θ∗∗by choosing (eθ∗∗1 , e2 ). Similarly to the previous paragraph, any action e2 > ẽ2 is equilibrium

dominated for the type-θ agent. (17) implies that there exists e02 = ẽ2 +  with a sufficiently
small  > 0 which is not equilibrium dominated for the type-θ agent. Hence, the type-θ agent can
profitably deviate by choosing such an e02 — a contradiction.
θ∗∗
We show that in any separating equilibrium surviving the Intuitive Criterion where (eθ∗∗1 , e2 ) 6=
θ∗
θ∗∗
θ∗
θ∗∗ = eθ∗ . Hence, towardeθ∗ , ẽ2 > eθ∗
2 must hold. Note that if ẽ2 = e2 , then e1 = e1 and hence e

a contradiction, suppose ẽ2 < eθ∗
2 . Then, as discussed in the previous paragraph, any action e2 > ẽ2is equilibrium dominated for the type-θ agent; implying that eθ∗
2 is equilibrium dominated for the
type-θ agent. Because eθ∗ is the action profile that maximizes the type-θ agent’s payoff when
µe2 = 1, eθ∗
2 is not equilibrium dominated for the type-θ agent. Hence, by the Intuitive Criterion,
after observing eθ∗
2 , the follower must conclude that the deviation comes from the type-θ agent (i.e.,µeθ∗ = 1). But then, the type-θ agent can profitably deviate by choosing eθ∗ — a contradiction.
2

23

We also check that (10) is not binding in any separating equilibrium surviving the Intuitive
Criterion. Note that the the right hand side of (10) is maximized by choosing eθ∗ . Hence, wheneθ∗∗ = eθ∗ , (10) is not binding. When eθ∗∗ = ẽ, (8) hold with equality, so by rewriting it yields:
θ∗∗

θ∗∗

q(e1 )b + [xA e2
θ∗∗

θ∗∗

θ∗∗

+ yA d(0)]θ − cA (e1 , e2 ) = max {q(e1 )b + [xA ẽ2 + yA d(1)]θ − cA (e1 , ẽ2 )} .
e1

θ∗∗Because (e1 , e2 ) maximizes the type-θ agent’s payoff given d = d(0), we have
θ∗∗

θ∗∗

q(e1 )b + [xA e2

θ∗∗

θ∗∗

θ∗
θ∗ θ∗
+ yA d(0)]θ − cA (e1 , e2 ) > q(eθ∗1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 ).

By combining it with the above equality, we obtain:θ∗θ∗ θ∗
max {q(e1 )b + [xA ẽ2 + yA d(1)]θ − cA (e1 , ẽ2 )} > q(eθ∗1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 ).
e1Because e2θ∗ < ẽ2 , 0 < xA eθ∗
2 + yA d(0) < xA ẽ2 + yA d(1). Hence, replacing θ with θ in the above
inequality leads to

θ∗
θ∗ θ∗
max q(e1 )b + [xA ẽ2 + yA d(1)]θ − cA (e1 , ẽ2 ) > q(eθ∗1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 ). (18)e1Note that the left hand side of (18) is none other than the equilibrium payoff of the type-θ agent.
Because the right hand side of (10) is maximized by choosing e∗ , we have shown that (10) is notbinding in any separating equilibrium surviving the Intuitive Criterion.
In summary, we have shown that in any separating equilibrium surviving the Intuitive Criterion,
either eθ∗∗ = eθ∗ or eθ∗∗ = ẽ where ẽ2 > eθ∗
2 . In either case, the type-θ agent’s equilibrium action
profile is given by (6) and (7) and hence eθ∗∗ = eθ∗ .
Pooling Equilibria We next analyze the set of pooling equilibria in that both types of the agent
take e∗∗2 ≥ 0. We derive it by restricting the off-path belief that the follower always thinks that adeviating agent must be the low type: µe2 = 0 for any e2 6= e∗∗
2 . Then, each type of the agent’s

24action profile must satisfy the following conditions:
θ∗∗

θ∗∗

q 0 (e1 )b −θ∗∗

∂cA (e1 , e∗∗
2 )
= 0,
∂e1

(19)

θ∗∗

∗∗
q(e1 )b + [xA e∗∗
2 + yA d(µ)]θ − cA (e1 , e2 ) ≥ max {q(e1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 )} ,(e1 ,e2 )(20)
q 0 (eθ∗∗1 )b −∂cA (e1θ∗∗ , e∗∗
2 )
= 0,
∂e1

(21)


∗∗
θ∗∗ ∗∗q(eθ∗∗
1 )b + [xA e2 + yA d(µ)]θ − cA (e1 , e2 ) ≥ max q(e1 )b + [xA e2 + yA d(0)]θ − cA (e1 , e2 ) .
(e1 ,e2 )

(22)
θ∗∗

θ∗∗ ∗∗A pooling equilibrium is characterized by a tuple of actions ((e1 , e∗∗2 ), (e1 , e2 )) that satisfies(19)-(22).
We next show that there is no pooling equilibrium that survives the Intuitive Criterion. Suppose,
toward a contradiction, that there is such an equilibrium. When (20) holds with strict inequality,
by the same argument as in the proof of separating equilibria, there exists  > 0 such that any∗∗
action e02 ∈ (e∗∗2 − , e2 + ) is equilibrium dominated for the type-θ agent and the type-θ agent

has an incentive to deviate to e02 . When (20) holds with equality, take the highest value ē2 which
satisfies the following equality:
θ∗∗

θ∗∗

∗∗
q(e1 )b + [xA e∗∗
2 + yA d(µ)]θ − cA (e1 , e2 ) = max {q(e1 )b + [xA ē2 + yA d(1)]θ − cA (e1 , ē2 )} .
e1

Note that ē2 > e∗∗
2 , as (20) holds with equality. By (19), the above equality can be written as∗∗
max {q(e1 )b + [xA e∗∗
2 + yA d(µ)]θ − cA (e1 , e2 )} = max {q(e1 )b + [xA ē2 + yA d(1)]θ − cA (e1 , ē2 )} .
e1

e1

Because θ > θ, ē2 > e∗∗
2 and d(µ) < d(1), the above equality implies∗∗
max q(e1 )b + [xA e∗∗
2 + yA d(µ)]θ − cA (e1 , e2 ) < max q(e1 )b + [xA ē2 + yA d(1)]θ − cA (e1 , ē2 ) .
e1

e1

(23)
Note that the left hand side of (23) is none other than the equilibrium payoff of the type-θ agent. As
any action ē2 +  with  > 0 is equilibrium dominated for the type-θ agent, by taking a sufficiently
small  > 0, the type-θ agent can profitably deviate by choosing ē2 +  — a contradiction.

25

Proof of Proposition 4.
= eθ∗(i) When eθ∗∗22 , neither (8) nor (10) is binding; the signaling effect is not a relevant concern
for each type of the agent. Hence, the optimal contract specifies b∗∗ ≤ V (and b∗∗ < V if xP > 0)by the same derivation with the benchmark case under symmetric information.
m(ii) Suppose eθ∗∗2 = ẽ2 . Let e1 denote the effort level of task 1 which satisfies (8) with equality(i.e., the effort level of task 1 chosen by the type-θ agent when he mimics the type-θ agent), which
is characterized as:
q 0 (em
1 )b −
θ∗∗

Because e22∂cA (em
1 , ẽ2 )
= 0.
∂e1
θ∗∗

cA (e1 ,e2 )< ẽ2 and ∂ ∂e
> 0, em
1 < e1
1 ∂e2

(24)
θ∗∗

and hence q(em1 ) < q(e1 ) holds. Note also that(8) holds with equality as follows:
o
n
θ∗∗
θ∗∗
θ∗∗ θ∗∗
m
q(em
)b
+
[x
ẽ
+
y
d(1)]θ
)b
+
[x
e
+
y
d(0)]θ
,
e
)
= 0.
−
c
(e
,
ẽ
)
−
q(e
−
c
(e
2
A 2
A
A 2
A
A 1
A 1
1
1
2
(25)
In this case, the equilibrium effort levels are characterized by (6), (7), (24), (9), and (25). By
applying the Implicit Function Theorem, we obtain:












θ∗∗

θ∗∗

∂ 2 cA (e1 ,e2∂e21θ∗∗ θ∗∗∂ 2 cA (e1 ,e2 )−∂e1 ∂e2θ∗∗q 00 (e1 )b −

)

θ∗∗

θ∗∗

∂ 2 cA (e1 ,e2 )∂e1 ∂e2θ∗∗ θ∗∗∂ 2 cA (e1 ,e2 )
−
∂e22

−

0

0

0

0

0

0

∂ 2 cA (em1 ,ẽ2 )
∂e21

0

0

0

q 00 (em
1 )b −000(ẽ1 ,ẽ2 )q 00 (ẽ1 )b − ∂ cA∂e
2

0

0

0

0

∂ 2 cA (em1 ,ẽ2 )∂e1 ∂e22cA (ẽ1 ,ẽ2 )− ∂ ∂e1 ∂e2∂cA (em
1 ,ẽ2 )xA θ −
∂e2

−

2

1







·





θ∗∗

∂e1
∂b
θ∗∗
∂e2
∂b
∂em
1
∂b
∂ ẽ1
∂b
∂ ẽ2
∂b





 
 
 
 
 
=
 
 
 
 



−q 0 (em
1 )






.





0
θ∗∗

−q 0 (e1 )θ∗∗q(e1 ) − q(em1 )Let A denote the above 5 × 5 matrix. Note that (5, 1), (5, 2), and (5, 3) elements of A are zeroby using (6), (7), and (24), respectively. By Cramer’s rule, as well as q(em1 ) < q(e1 ) and the

26












θ∗∗

−q 0 (e1 )θ∗∗assumptions, we obtain the following comparative statics:

=


−

θ∗∗

−q 0 (e1 )θ∗∗ θ∗∗

−∂ 2 cA (e1 ,e2θ∗∗
q 00 (e1 )b −∂e21

−q 0 (em1 )+∂ 2 cA (e1 ,e2
∂e22

)

θ∗∗

θ∗∗

θ∗∗

θ∗∗

∂ 2 cA (e1 ,e2
∂e22

)



∂ 2 cA (e1 ,e2∂e1 ∂e2∂ 2 cA (e1 ,e2∂e1 ∂e2

−

θ∗∗

θ∗∗

)

θ∗∗

θ∗∗

)

2 > 0,))−2 < 0,θ∗∗m∂ 2 cA (em1 ,ẽ2 ) q(e1 )−q(e1 )∂cA (em,ẽ2 )∂e1 ∂e2
1
xA θ−
∂e
2

m

2

∂ cA (e1 ,ẽ2 )q 00 (em
1 )b −
∂e21
θ∗∗

∂ẽ1
=
∂b

θ∗∗

)

θ∗∗ ∂ 2 cA (e1 ,e2∂e1 ∂e2

=

∂em
1
=
∂b

θ∗∗

)

∂ 2 cA (e1 ,e2θ∗∗
q 00 (e1 )b −
∂e21

θ∗∗

∂e2
∂b

θ∗∗

θ∗∗ ∂ 2 cA (e1 ,e2∂e22

q 0 (e1 )θ∗∗ θ∗∗

θ∗∗

∂e1
∂b

2

cA (ẽ1 ,ẽ2 )−q 0 (e1 ) + ∂ ∂e
1 ∂e2

> 0,

θ∗∗

)−q(em1 )∂cA (em
1 ,ẽ2 )
xA θ−
∂e2
q(e1

2

(ẽ1 ,ẽ2 )q 00 (ẽ1 )b − ∂ cA∂e
2

> 0,

1

θ∗∗
∂ẽ2
q(e1 ) − q(em1 )=m ,ẽ ) < 0.
∂c
(e
2
A
1
∂b
xA θ −∂e2= ẽ2 . By taking the derivative of
We next derive the optimal level of the bonus b∗∗ when eθ∗∗2(11) with respect to b, b∗∗ is characterized by:∂ẽ1 0∂cA (ẽ1 , ẽ2 )∂cA (ẽ1 , ẽ2 )∂ẽ2µq (ẽ1 )V −(xP + xA )θ −
+µ
∂b
∂e1
∂b
∂e2
"
#
"
#
θ∗∗ θ∗∗
θ∗∗θ∗∗ θ∗∗
θ∗∗∂cA (e1 , e2 )∂e2∂cA (e1 , e2 )
∂e1
0 θ∗∗
q (e1 )V −+ (1 − µ)(xP + xA )θ −= 0.
+ (1 − µ)
∂b
∂e1
∂b
∂e2
By substituting (6), (7), and (9) into the above equality, we obtain:∂ẽ1 0∂ẽ2∂cA (ẽ1 , ẽ2 )∗∗µq (ẽ1 )(V − b ) + µ(xP + xA )θ −
∂b
∂b
∂e2
θ∗∗

+ (1 − µ)θ∗∗∂e∂e1 0 θ∗∗q (e1 )(V − b∗∗ ) + (1 − µ) 2 xP θ = 0.
∂b∂b

By rearranging this equality, we obtain (12).

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29

Supplementary Material for “The Provision of High-powered Incentives under Multitasking” by Daido and Murooka (not intended
for publication)A

Illustration with Binary Effort: Full Characterization

In the binary illustrative model with private information, Proposition 1 derives the optimal contracts under the assumption that a pure-strategy PBE is played whenever it exists. Here, wecharacterize PBE including all mixed-strategy equilibria, as well as applying the Intuitive Criterion(Cho and Kreps, 1987). We then argue that the optimal contracts shown in Proposition 1 remain
the same, so focusing on pure-strategy PBE is without loss of generality when we derive the optimal
contracts.
In addition to PBE derived in Proposition 1, there exist other two types of mixed-strategy
equilibria. One kind of mixed-strategy equilibria is that the type-θ agent randomizes his actions.
In such an equilibrium, the type-θ agent must choose eθ = (1, 0) with probability one. Given
that, by Assumption 1 (ii), the follower chooses d(1) = 1 with probability one and d(0) = 0 withA )θprobability one. Then, for the type-θ agent to be indifferent, b = (xA +y. In this case, anyqmixture of the actions of the type-θ agent constitutes an equilibrium.
Another kind of mixed-strategy equilibria is that the type-θ agent randomizes his actions, while
the follower does not. Suppose that the type-θ agent takes eθ = (0, 1) with probability l ∈ [0, 1]
and that the follower takes d(1) = 1 and d(0) = 0 with probability one. Then, the type-θ agent
takes eθ = (0, 1) with probability one. Under the mixed-strategy equilibrium, the type-θ mustA )θbe indifferent: qb = (xA + yA )θ ⇐⇒ b = (xA +y. Also, the type-θ agent must choose l whichq(1−µ)lµinduces the follower to weakly prefer to play d(1) = 1: µ+(1−µ)lyF θ + µ+(1−µ)lyF θ − cF ≥ 0 ⇐⇒µ θyF −cFl ≤ 1−µcF −θyF .

We next show that the pooling equilibrium eθ = eθ = (1, 0), which is described in Section 3,
A )θ
A )θ
fails the Intuitive Criterion when (xA +y< b < (xA +y. For the type-θ agent, his equilibriumqq

payoff is w + qb, whereas his payoff upon deviation to e0 = (0, 1) is at most w + (xA + yA )θ. Hence,eθ = (1, 0) is equilibrium dominated for the type-θ agent. For the type-θ agent, his equilibrium
payoff is w + qb, whereas his payoff upon deviation to e0 = (0, 1) is at most w + (xA + yA )θ. Hence,1eθ = (1, 0) is not equilibrium dominated for the type-θ agent. Hence, the pooling equilibrium
A )θ
A )θ
eθ = eθ = (1, 0) survives the Intuitive Criterion when xAq θ ≤ b ≤ (xA +yor b ≥ (xA +y.
qq

As a result, equilibrium strategies in the perfect-Bayesian equilibria that survives the Intuitive
Criterion are summarized depending on the level of b as follows:(i) (task-2 pooling) If b ≤ xAq θ , then eθ = eθ = (0, 1) and d(1) = d(0) = 0.
(ii) (type-θ mixed) If b ∈

xA θ xA θq , q, then eθ = (0, 1), the type-θ agent takes eθ = (0, 1) with

qb−xA θµ θyF −cF
probability 1−µ∈cF −θyF ∈ (0, 1), and the follower takes d(1) = 1 with probabilityyA θ(0, 1) and d(0) = 0 with probability one.
(iii) (task-1 pooling and type-θ mixed) If b ∈hxA θ (xA +yA )θ
q ,
q



, then there exist two kinds ofequilibria: (i) eθ = eθ = (1, 0) and d(1) = d(0) = 0. (ii) eθ = (0, 1), the type-θ agentµ θyF −cF
takes eθ = (0, 1) with probability 1−µcF −θyF ∈ (0, 1), and the follower takes d(1) = 1 withAθ
probability qb−xyA θ ∈ (0, 1) and d(0) = 0 with probability one.
A )θ(iv) (task-1 pooling, separating, and type-θ mixed) If b = (xA +y, then eθ = (0, 1), the type-θhiqµ θyF −cF
agent takes eθ = (0, 1) with probability l ∈ 0, 1−µcF −θyF , d(1) = 1, and d(0) = 0.

(v) (separating) If b ∈(xA +yA )θ (xA +yA )θ
,
q
q



, then eθ = (0, 1), eθ = (1, 0), d(1) = 1, and d(0) = 0.

A )θ(vi) (task-1 pooling, separating, and type-θ mixed) If b = (xA +y, then the type-θ agent takesqeθ = (0, 1) with probability h ∈ [0, 1], eθ = (1, 0), d(1) = 1, and d(0) = 0.
A )θ, then eθ = eθ = (1, 0) and d(1) = d(0) = 0.
(vii) (task-1 pooling) If b > (xA +yq

Next, we argue that the optimal contracts shown in Proposition 1 remains the same. In the
proof of Proposition 1, we show that the principal does not induce the agent and the follower toplay a mixed strategy in (ii) and (iii) above. By the similarly derivations, it is easy to check thatthe principal does not induce the agent and the follower to play a mixed strategy in (iv) and (vi)above.

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