A Continuous-Time Optimal Insurance Design with Costly
Monitoring under Moral Hazard and Adverse Selection
Hisashi Nakamura∗
Hitotsubashi University, Graduate School of Commerce and Management
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
September 25, 2011

Abstract
This paper provides a theoretical framework to study optimal insurance properties with monitoring under two information problems: moral hazard and adverse selection. It explores continuous-time optimal
insurance arrangements for general utility forms in an environment in which an insurer cannot observe
directly either the eﬀort or the outcome of an insured ﬁrm but can verify the outcome via a costly
monitoring technology. It shows that, due to the technology, the insurer can write explicitly the optimal
insurance contract that diﬀerentiates the two information problems. Particularly, when the ﬁrm’s outcome is low, adverse selection is removed whereas moral hazard is reduced.
Keywords: Insurance, Costly monitoring, Moral hazard, Adverse selection.
JEL Classiﬁcation: D82, D86, G22, G32.

1Introduction

In practice, most ﬁrms purchase corporate insurance to hedge against various business risks. On the otherhand, in theory, we know that, under perfect markets, there would be no rationale for such corporate
insurance. Thus the actual demand for corporate insurance should be due to market imperfections suchas default costs, agency problems, information problems, and tax asymmetries (for example, see MacMinnand Garven [11]). Among the imperfections, this paper looks at how equilibrium insurance properties areinﬂuenced by two information problems: moral hazard and adverse selection.
The two information problems coexist in actual insurance practices. Meanwhile, in contract theory, theconclusion in the basic moral hazard problem is essentially close to the one in the adverse selection problem,
except for an inverted causality, from a static (i.e., cross-sectional) viewpoint (Chiappori [4]). Speciﬁcally, inboth cases, controlling for observables, the choice of a contract will be correlated with the probability of bad
corporate outcome – more comprehensive coverage being associated with high risk. This implies that, whenobserving ﬁrms’ poor outcomes in static data, the insurer is not sure if it would be due to their low abilities
or to their laziness. Accordingly, the insurer faces diﬃculty with writing appropriate insurance contracts todistinguish adverse selection and moral hazard, although they are essentially diﬀerent issues.
Against such diﬃculty, much of the insurance literature has diﬀerentiated moral hazard and adverse
selection by making use of some diﬀerent dynamic properties of the incentive structures between the twoinformation problems in insurance contracts for various exogenous cases (typically, a reform of regulatoryframework), without using dynamic optimal (i.e., endogenous) insurance properties (e.g., Abbring et al. [1]).
Furthermore, most of the literature assumes only costless reporting as a communication method, as usual
in contract theory.1 On the other hand, however, in insurance practices, insurance companies verify the∗ I would like to thank Takau Yoneyama for his constant encouragement and advice throughout the preparation of this work.
I am also thankful to Jean Kwon and all seminar participants at APRIA 15th Annual Conference in Tokyo (Meiji University).
All errors are my own.
1

In the previous contract-theory literature, communication games with costless reporting have been studied a lot in ﬁnite-horizon (typically, two or three period) discrete-time models. There are a few exceptions
in a literature on insurance fraud (e.g., Dionne, Giuliano and Picard [6], Picard [13]).

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