A Continuous-Time Optimal Insurance Design with Costly Monitoring
Hisashi Nakamura∗
Hitotsubashi University
hisashi.nakamura@r.hit-u.ac.jp

Koichiro Takaoka
Hitotsubashi University
k.takaoka@r.hit-u.ac.jp

October 4, 2012

Abstract

We provide a theoretical and numerical framework to study optimal insurance properties forplayers’ general utility forms. We consider a continuous-time model where neither the eﬀorts nor
the outcome of an insured ﬁrm are observable to an insurer. The insured may then cause twointerconnected information problems: moral hazard and fraudulent claims. We show that, whencostly monitoring is available, an optimal insurance contract distinguishes the one problem fromthe other. Furthermore, if the insured’s downward-risk aversion is weak and if the participation
constraint is not too tight, then a higher level of the monitoring technology can mitigate bothproblems.
Keywords: Insurance, Costly monitoring, Moral hazard, Fraudulent claims.
JEL Classiﬁcation: D82, D86, G22, G32.

1

Introduction

As it is well known, insurers are often exposed to information problems in corporate insurance practices;
see e.g. MacMinn and Garven (2000). Speciﬁcally, neither the eﬀorts nor the outcome of insured ﬁrms areobservable to insurers directly without a cost. The insureds may then cause two information problems:(1) causing losses purposely and (2) ﬁling fraudulent claims. Call the former the problem of moral hazard
and the latter the problem of ex-post informational asymmetry.
The ex-post informational asymmetry problem distorts the insured’s eﬀort incentives in the moral hazard
problem. In standard moral hazard models, it is often assumed that the insurer can observe the insured’s
outcome ex post; e.g., see Rogerson (1985). The insurer then faces only the incentive problem of inducing theinsured to make desired eﬀorts. However, when there exists the ex-post informational asymmetry problem
as well, the insured may also exaggerate a claim. The insurer then needs to provide additionally the insured
with an incentive to tell the truth ex post. Thus the insurer cannot ﬁnd directly whether the ﬁled claim isdue to the insured’s laziness, to the fraud, or neither. Much worse, when the insured has a chance to tella lie about the ex-post outcome, he may have an incentive to be lazy in order to minimize his eﬀort cost.
Since the two information problems are interconnected in such ways, it is very diﬃcult to distinguish the one
problem from the other directly without a cost. In practice, to overcome this diﬃculty, the insurer routinely
investigates claims via a costly monitoring technology after they are ﬁled (Harrington and Niehaus (2004)).
The purpose of this paper is to provide a theoretical and numerical framework to study optimal insurance
properties when the costly monitoring is available under the problems of moral hazard and ex-post informational asymmetry in an optimal contracting model with two players: an insurer (i.e., insurance company)∗ Corresponding author: Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi,
Tokyo 186-8601, Japan. Phone: +81-42-580-8826. Fax: +81-42-580-8747. Email: hisashi.nakamura@r.hit-u.ac.jp. We would
like to thank Takau Yoneyama for his constant encouragement and advice throughout the preparation of this work. We
are also thankful to Ricard Gil, Chiaki Hara, Hongjoo Jung, Jean Kwon, Soichiro Moridaira, Mahito Okura, Min-Teh Yu,and all seminar participants at 2011 Nakanoshima Workshop (Osaka University), Asia-Paciﬁc Risk and Insurance Association
16th Annual Conference (Sungkyunkwan University, Seoul), Japanese Association of Risk, Insurance and Pensions Annual
Conference (Meiji University, Tokyo), 12th SAET Conference on Current Trends in Economics (University of Queensland,Brisbane), Working Group on Globalization of Insurance, and Yonsei University for valuable comments. Financial supportfrom the Grants-in-Aid for Scientiﬁc Research (C) and the Dai-ichi Life Insurance Company, Limited and the Dai-ichi Life
Research Institute Inc. is gratefully acknowledged.

1