Job design and randomization in principal agent models
ABSTRACT We analyze task allocation and randomization in Principal Agent models. We identify a new rationale that determines the allocation of tasks and show that it can be optimal to assign tasks that are very different to one agent. Similar to randomization, the reason to assign several tasks to one agent is to mitigate the effect of the participation constraint. We show that the allocation of tasks can be used as a substitute if randomization is not feasible.
Institute for Empirical Research in Economics
University of Zurich
Working Paper Series
Working Paper No. 381
Job Design and Randomization in Principal Agent
Wolfgang R. Köhler
Job Design and Randomization in Principal Agent Models1
Wolfgang R. Köhler
IEW, Universität Zürich,
8006 Zürich, Switzerland
Summary: We analyze task allocation and randomization in Principal Agent models. We
identify a new rationale that determines the allocation of tasks and show that it can be optimal
to assign tasks that are very di¤erent to one agent. Similar to randomization, the reason to assign
several tasks to one agent is to mitigate the e¤ect of the participation constraint. We show that
the allocation of tasks can be used as a substitute if randomization is not feasible.
Keywords: and Phrases: job design; multi-task agency; ex-ante randomization; moral
JEL Classi…cation Number: D 82
1I would like to thank Christian Ewerhart, Christoph Nitzsche, and Curtis Taylor.
Two important questions for the organization of …rms are the design of incentive contracts and
the allocation of tasks. We analyze two seemingly unrelated contractual regimes in principal
agent relationships: ex-ante randomization in incentive contracts and the allocation of tasks. It
is well known that randomization over simple contracts can be optimal in the second-best (e.g.,
Fellingham et.al. 1984, Arnott and Stiglitz, 1988). However, we rarely observe contracts that
include randomization and randomization is usually regarded as a technical result with little
relevance for real-world incentive contracts. On the other hand, contracts regularly specify some
allocation of tasks. Starting with Holmstrom and Milgrom (1991), an extensive literature on multi-
task agency analyzes the optimal allocation of tasks. This literature focuses on e¤ort substitution
and argues that only one task should be allocated to an agent (or, more general, that tasks should
be homogeneous with respect to ease of performance measurement).
We identify a new rationale that determines the optimal allocation of tasks: the e¤ect of the
outside option. We analyze randomization in a simple model with one task and then develop a
multi-task model where the principal allocates working time across di¤erent tasks. Randomization
over wage schedules and the allocation of di¤erent tasks to one agent are similar in the sense that
the only reason to randomize or to allocate two tasks to an agent is to mitigate the e¤ect of the
outside option. In most contractual relationships, randomization is not feasible because parties
do not have access to a veri…able randomization device. We show that the allocation of two tasks
to one agent can serve as a substitute for randomization.
Our result that it can be optimal to assign two tasks that are very di¤erent is the opposite
of the conclusions of the multi-task literature. The reason for the di¤erent results is that most
studies of multi-task agency use the linear model of Holmstrom and Milgrom. The linear model
is special in the sense that the outside option does not a¤ect the second-best contract except for
a transfer. Hence in the linear model, e¤ort substitution is the only factor that determines the
optimal allocation of tasks.
The paper is organized as follows. In section 2 we analyze randomization if there exists only one
task. In section 3 we analyze the allocation of tasks, relate our results to the multi-task literature
and show that the allocation of tasks can serve as a substitute for randomization. Section 4
The literature distinguishes between two types of randomization. Under ex-post randomization,
the wage depends on output and on a signal that is realized after the agent chooses an action
where the signal contains no information about the action that the agent has chosen. Under
ex-ante randomization, the contract speci…es a lottery over wage functions that map output into
wages. Before the agent chooses an action, a signal determines which wage function is selected.
Of course, this signal cannot contain information about the action that the agent has not yet
chosen. We study ex-ante randomization but we also allow for contracts that include ex-post
Consider a standard principal-agent problem. The principal’s payo¤ v(?;w) = ? ? w depends
on output ? and wages w. The agent chooses an action a 2 A. Output is a stochastic function
of a. Let F(:ja) be the conditional distribution function of output withR?dF(?ja) < 18a 2 A.
The agent’s utility is u(w;a) with u0
w> 0 and u00
w< 0 8a 2 A. Let ubbe the supremum of u with
ub< 1. If the agent rejects the contract, he receives outside utility u < ub. The agent is possibly
protected by limited liability, i.e., w ? l with l ? ?1. There exists an independent random
variable e ? with generic realization ? and c.d.f. G where ? is realized after the agent chooses a.
Output and the realization of e ? are veri…able while a is unobservable. If the wage depends in a
non-trivial way on e ?, the contract includes ex-post randomization. A contract without ex-ante
randomization consists of a measurable wage function w that maps output and realizations of e ?
into R. An ex-ante random contract speci…es a set of wage functions and a probability distribution
over this set of wage functions. From now on, non-random contract refers to a contract without
Suppose ex-ante randomization is not feasible. Consider the second-best problem for arbitrary,
…nite outside utilities:
a = argmax
u(w(?;?);a)dF(?ja)dG(?) ? k (PC)
w(?;?) ? l 8?;?(L)
Let wk;ak be the solution to (1) subject to (IC), (PC), and (L) for ?1 < k < ub. There
exists an extensive literature that discusses various topological restrictions on payo¤s, actions
and the stochastic relation between action and output that guarantee the existence of a second-
best contract (see Page, 1997, and literature cited therein). Since the focus of this paper is not
on existence but on randomization and job design, we assume that the non-random second-best
problem has a unique solution in the sense that akis unique and that wkis unique except for a
set of outputs that is realized with probability zero.
Assumption 1: For all k 2 (?1;ub) exist wk;akwhere akis unique and wkis unique except
for some set P of outputs withR
PdF(?jak) = 0.
If ex-ante randomization is not feasible, the principal o¤ers the contract wu. Let rkdenote the
rent under a non-random second-best contract with rk=R Ru(wk(?;?);ak)dF(?jak;?)dG(?)?u.
Second-best rents can be positive for two reasons: limited liability and non-separability of the