INFORMATION, RISK SHARING, AND INCENTIVES IN AGENCY PROBLEMS

• Published date: 01 February 2017
• Author: Jia Xie
• DOI: http://doi.org/10.1111/iere.12212
• Date: 01 February 2017

INTERNATIONAL ECONOMIC REVIEW

Vol. 58, No. 1, February 2017

INFORMATION, RISK SHARING, AND INCENTIVES IN AGENCY PROBLEMS∗

BYJIA XIE1

Bank of Canada, Canada

This article studies the use of information for incentives and risk sharing in agency problems. When the principal is

risk neutral or the outcome is contractible, risk sharing is unnecessary or dealt with by a contract on the outcome, so

information systems are used for incentives only. When the outcome is noncontractible, a risk-averse principal relies on

imperfect information for both incentives and risk sharing. Under the first-order approach, this article relaxes Gjesdal’s

criterion for ranking information systems and finds conditions justifying the first-order approach when the principal is

risk averse and the outcome is noncontractible.

1. INTRODUCTION

The principal–agent problem consists of two stages: the first stage is the principal’s choice of

an information system, which generates contractible (i.e., commonly observable and verifiable)

signals or performance measures. The second stage is the optimal design of a contract based on

the information system chosen in the first stage. Much of the literature focuses on the second

stage, assuming that the information system is given. This article focuses on the first stage and

studies the choice of information systems by the principal.

Comparing information systems in agency problems was first raised by Holmstr ¨

om (1979)

and further studied by Kim (1995); Jewitt (1997, 2007); Dewatripont et al. (1999); Demougin

and Fluet (2001); Fagart and Sinclair-Desgagne (2007); and Xie (2011). These studies assume

either a risk-neutral principal or a contractible outcome. In these cases, information systems

are used to provide incentives to the agent’s action, and two main predictions are derived: (i)

An information system is valuable if it is informative about the agent’s action,2and (ii) the

statistical condition for being “informative” can be relaxed under the first-order approach, by

which the principal can predict the agent’s action using the agent’s first-order conditions alone.

In many cases, however, the principal is strictly risk averse and the outcome is noncontractible.

Gjesdal (1982) gives three main reasons for the outcome being noncontractible.3First, the

outcome may be unobservable at the time when the agent is paid. For instance, the manager of

a firm may be paid irreversibly before the outcome of his action is observed. Second, contracting

on outcome may be too costly. For instance, perfect auditing of income tax returns is expensive.

Finally, outcomes are often imperfect estimates of the “real” outcomes. For instance, the quality

measure of a project is an imperfect estimate of the “real” quality.

When the principal is strictly risk averse and the outcome is noncontractible, information

systems are used not only to provide incentives but also to allocate risk in the outcome of the

∗Manuscript received February 2013; revised April 2015.

1The analysis and conclusions set forth are those of the author and do not indicate concurrence by the Bank of

Canada. I am especially indebted to Frank Page and Michael Rauh for advice and encouragement, and I am grateful

to the editor and two referees for insightful comments that greatly improve the article. For comments and discussions,

I thank Eric Rasmusen, Michael Baye, Lars Stole, James Walker, Jason Allen, Hanna Halaburda, and Brian Peterson

as well as seminar and conference participants at Indiana University, Washington University in St. Louis, University

of Texas at Dallas, and the Midwest Economics Association. Needless to say, any mistakes are mine. Please address

correspondence to: Jia Xie, The Department of Real Estate Management, Ted Rogers School of Management, Ryerson

University, 55 Dundas St. W, Toronto, ON M5G 2C3 Canada. E-mail: jia.xie@ryerson.ca.

2See also Shavell (1979), Laffont and Martimort (2002), Tirole (2006), and Bolton and Dewatripont (2005).

3See also Mirrlees (1976), Baker (2002), and Maskin (2002).

157

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(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

158 XIE

agent’s action. Following this line of reasoning, Gjesdal (1982) develops a criterion—composed

of an incentive condition and a risk-sharing condition—that compares information systems by

how informative they are about the agent’s action and the outcome, separately. Gjesdal (1982),

however, does not assume the first-order approach. Therefore, it is natural to wonder if his

criterion can be weakened under the first-order approach and if the first-order approach can be

justified for the case with a strictly risk-averse principal and a noncontractible outcome.

The main purpose of this article is to provide the first answers in the literature to these

questions. First, we find that the incentive condition in Gjesdal (1982) can be relaxed under

the first-order approach, because only local incentive compatibility constraints matter, and

therefore one only needs to focus on a neighborhood of the action in question. The risk-sharing

condition, on the other hand, is the same because one has to consider all possible values of the

outcome, whether the first-order approach applies or not.

Second, the first-order approach can be justified by conditions ensuring that the agent’s utility

is concave in his action.4Finding these conditions for the case with a strictly risk-averse prin-

cipal and a noncontractible outcome, however, presents technical difficulties: The interaction

between the two roles of information systems may cause the agent’s utility to be nonconcave

in his action. Nevertheless, we show that, by imposing restrictions on the information structure

and both parties’ risk aversions, we can justify the first-order approach.

The rest of the article is organized as follows: I set up the model in Section 2. In Section 3,

I study the two cases where risk sharing is not a challenge, so the only concern is providing

incentives. In Section 4, I study the case where information systems are ranked for both incen-

tive and risk-sharing purposes. I justify the first-order approach in Section 5, and conclude in

Section 6.

2. THE MODEL

A principal faces a set of contractible—that is, commonly observable and verifiable—

information systems, each of which can be represented by a random vector ˜x.5After choosing

an ˜x, the principal makes a take-it-or-leave-it contract, s˜x(·)∈[s,¯s], with an agent, who has an

outside reservation utility of 0. I abuse notation by dropping the subscript in s˜xif the context

makes it clear which information system sis based on. The parameters sand ¯sare the lower

and upper bounds of the agent’s payments, respectively.6If the contract is accepted, the agent

chooses an unobservable real-valued action a∈R+and incurs private cost c(a)withc>0

and c ≥0. The action astochastically generates a real-valued outcome ˜

bon a fixed support

b∈[b,¯

b], and ˜

bis imperfectly correlated with the information system ˜x.

The principal’s utility v(b−s) is defined on her residual. To avoid negative residuals, I

assume b≥¯s. The agent derives utility from the received payment minus the private cost of

action, u(s)−c(a). I assume that v>0, v ≤0, u>0, and u <0. In particular, the agent is

strictly risk averse, whereas the principal could be risk neutral or strictly risk averse.

All distribution and density functions are denoted by Fand f, respectively, with the subscript

indicating which random variables are intended. For instance, F˜

b(b|a) is the marginal distri-

bution function of ˜

b, given a, and f(˜

b,˜x)(b,x|a) is the joint density of ˜

band ˜x, given a. I abuse

notation by dropping the subscript if the context makes things clear; in this case, the arguments

indicate which random vectors are intended. All density functions are positive and continuous

4See Rogerson (1985), Jewitt (1988), Sinclair-Desgagne (1994), and, more recently, Conlon (2009) for justification

of the first-order approach in cases where the principal is risk neutral and/or the outcome is contractible.

5I therefore exclude the cases considered by Maskin and Tirole (1999) and Maskin (2002), where information systems

are either observable by one party but not the other or are commonly observable but nonverifiable. The principal’s

problems in these cases are mechanism design problems mixed with moral hazard, which are not the focus of this article.

6As explained by Jewitt et al. (2008), contracts are always bounded due to legal and other constraints on payments.

Page (1987) also introduces the upper and lower bounds to avoid nonexistence problems. However, more recently,

Kadan et al. (2011) prove the existence of an optimal contract without relying on a compact contract space.