MORAL HAZARD AND FIRST-ORDER RISK ...

MORAL HAZARD AND FIRST-ORDER RISK AVERSION by Ulrich Schmidt Christian{Albrechts{Universitat zu Kiel Institut fur Finanzwissenschaft und Sozialpolitik Olshausenstr. 40 D{24098 Kiel Germany

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January 1999



I thank an anonymous referee for helpful comments

Abstract In this paper we consider the standard hidden action model of agency theory and assume that the preferences of principal and agent are represented by dual expected utility and not by expected utility. Our main result shows that dual expected utility has quite dierent implications for the optimal contract than expected utility: First, risk aversion of the agent is neither a necessary nor a sucient condition for a second{best optimum. Secondly, there exist degenerate equilibria, in which the agent does not bear any risk and, thus, chooses the minimal eort level.

1

Introduction

Since the work of von Neumann and Morgenstern (1944), expected utility theory has been the dominant framework for analysing decision making under uncertainty. In their classical publication von Neumann and Morgenstern show that if the preferences of a decision maker obey certain axioms, they can be represented by a functional, which is linear on a choice set of probability measures. From later work [e.g. Herstein and Milnor (1953), Jensen (1967)] it became clear that the axiomatic of von Neumann and Morgenstern can be reduced to three axioms: ordering, continuity, and independence. Because of the normatively appealing implication of these axioms, expected utility (EU) became widely acknowledged as the theory of rational choice under uncertainty. While the ordering and continuity axioms are standard assumptions even in consumer theory, it is the independence axiom which forces utility to be linear in the probabilities. Since the famous \Allais Paradox" [cf. Allais (1953)], however, a large body of empirical evidence has been gathered which shows that individuals tend to systematically violate the independence axiom. This empirical evidence has motivated researchers to develop alternative theories of choice under uncertainty [cf. Schmidt (1998b) for a survey] which rest on weakened variants of the independence axiom. These non{expected utility models are { apart from Machina's (1982) local expected utility analysis { usually divided into two classes of models: utility theories with the betweenness property [cf. Chew (1989), Gul (1991)] and rank{dependent utility [cf. Green and Jullien (1988), Segal (1989), (1993), and Quiggin and Wakker (1994)]. An important dierence between expected utility and some of the non{expected utility models can be characterized by the concept of rst{ order risk aversion as de ned by Segal and Spivak (1990). Note that the risk premium for a small risk is in expected utility theory proportional to the variance of the risk, at least if the utility function is twice dierentiable. This fact has been termed second{order risk aversion. In contrast, rst{order risk aversion implies that this risk premium is proportional to the standard deviation 1

rather than the variance. There are several theories of risk preference, which exhibit rst{order risk aversion. Apart from rank{dependent utility, also the theory of disappointment aversion [cf. Gul (1991)] and semi{weighted utility [cf. Chew (1989)] belong to this class. Applications of non{expected utility models have shown that rst{order risk aversion generates results, which dier signi cantly from those of expected utility theory, and lead in many cases to a better accommodation of real world data. For instance, rst{order risk aversion can resolve the equity premium puzzle [cf. Epstein and Zin (1990)] and explain that individuals buy full insurance even at unfair odds [cf. Schmidt (1996)]. Further applications of rst{order risk aversion appeared in Demers and Demers (1990), Epstein and Zin (1991), Segal and Spivak (1992), Konrad and Skaperdas (1993), Doherty and Eeckhoudt (1995), Karni (1995), and Schlee (1995). Additionally, rst{ order risk aversion is also an empirically viable hypothesis as shown by the experiment of Loomes and Segal (1994). As far as I know, rst{order risk aversion has not yet been applied to the problem of moral hazard. The moral hazard problem can arise in many dierent economic situations. In this work we will consider the well{known \hidden action" problem in a principal{agent relationship. Therefore, we will employ a modi ed variant of Holmstrom's (1979) central model of agency theory. However, in contrast to Holmstrom's model, it is assumed that the preferences of both, principal and agent, are not represented by EU but by dual expected utility, which is a special variant of rank{dependent utility and, therefore, exhibits rst{order risk aversion. Dual expected utility and the concepts of rst{order and second{order risk aversion are introduced in the next section. In section 3 we present our model of agency theory. The analysis reveals that, also for moral hazard, rst{order risk aversion has quite dierent implications than EU. 2

2

Dual Expected Utility

The two characteristic features of rank{dependent utility (RDU) are, rst, that outcomes are ordered in an increasing sequence of preference (i.e.:      n for a preference ordering   ) and, second, cumulative probabilities are distorted by a transformation function. The most general form of RDU was developed by Green and Jullien (1988) and Segal (1989), (1993). Dual expected utility (DEU) is a special case of RDU and was axiomatized by Yaari (1979). The axiomatization of DEU involves, besides the standard ordering and continuity axioms, two additional axioms: First, a dominance axiom, which forces preferences to be consistent with rst{ order stochastic dominance. Secondly, as a substitute for the independence axiom, the dual independence axiom, which, roughly speaking, demands independence with respect to \consequence mixtures" of lotteries rather than with respect to \probability mixtures" as the standard independence axiom. DEU has some interesting features: On the one hand, it is consistent with the experimental evidence, which motivated the devolepment of non{expected utility theories. On the other hand, DEU allows for a separation of the concepts of risk aversion and decreasing marginal utility of money since the representation can accommodate risk aversion although marginal utility of money is constant. This becomes clear by considering the representation of preferences , which is given by x1

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