Academic journal article Journal of Risk and Insurance

Efficient Risk-Sharing and the Dual Theory of Choice under Risk

Academic journal article Journal of Risk and Insurance

Efficient Risk-Sharing and the Dual Theory of Choice under Risk

Article excerpt


Since the work of von Neumann and Morgenstern (1944), expected utility theory has been the dominant framework for analyzing decision making under uncertainty. Consequently, Borch (1960) also assumed in his pioneering paper on efficient risksharing that all individuals are expected utility maximizers. Since the work of Borch (1960), efficient risk-sharing has been extensively heated in the expected utility framework (see Gollier, 1992, and Eeckhoudt and Gollier, 1995, for reviews).

Since the famous "Allais Paradox" (see Allais, 1953), however, a large body of empirical evidence has been gathered that shows that individuals tend to systematically violate the independence axiom of expected utility theory.

This empirical evidence has motivated researchers to develop alternative theories of choice under uncertainty (see Schmidt, 1999, for a survey) that rest on weakened variants of the independence axiom. A well-known class of these non-expected utility theories is the rank-dependent model. Variants of rank-dependent utility are anticipated utility (see Quiggin, 1982) and the dual theory of choice under risk (see Yaari, 1987), henceforth referred to as dual expected utility. An important difference between expected utility and dual expected utility can be characterized by the concept of first-order risk aversion as defined 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 differentiable. This fact has been termed second-order risk aversion. In contrast, first-order risk aversion implies that this risk premium is proportional to the standard deviation rather than the variance. Th ere are several theories of risk preference, which exhibit first-order risk aversion. Apart from dual expected utility, also all other rank-dependent models, as well as the theory of disappointment aversion (see Gul, 1991) and semi-weighted utility (see Chew, 1989) belong to this class.

Applications of non-expected utility models have shown that first-order risk aversion generates results that differ significantly from those of expected utility theory and lead in many cases to a better accommodation of real world data. For instance, first-order risk aversion can resolve the equity premium puzzle (see Epstein and Zin, 1990) and explain that individuals buy full insurance even at unfair odds (see Schlesinger, 1997). Further applications of first-order risk aversion appeared, among others, in Demers and Demers (1990), Epstein and Zin (1991), Segal and Spivak (1992), Konrad and Skaperdas (1993), Schlee (1995), and Doherty and Eeckhoudt (1995). Additionally, first-order risk aversion is also an empirically viable hypothesis as shown by the experiment of Loomes and Segal (1994).

The goal of this article is to provide a complete characterization of efficient risk-sharing with dual expected utility (DEU). Apart from first-order risk aversion, a second characteristic property of DEU is the linearity in payments. Therefore, DEU seems to be particularly appropriate to represent the behavior of firms. This has been explained by Yaari (1987, p. 96) as follows:

In studying the behavior of firms, linearity in payments may in fact be an appealing feature. Under the dual theory, maximization of a linear function of profits can be entertained simultaneously with risk aversion. How often has the desire to retain profit maximization led to contrived arguments about firms' risk neutrality?

Moreover, as shown by Guriev (1998), even the representation of the behavior of a risk-neutral firm facing a bid-ask spread on the credit market is given by DEU.

Note that efficient risk-sharing has already been analyzed by Machina (1995) in a non-expected utility framework. Machina shows that all important results derived with expected utility carry over to Frechet-differentiable preference functionals, which generalize expected utility by replacing the independence axiom with a much weaker differentiability assumption. …

Search by... Author
Show... All Results Primary Sources Peer-reviewed


An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.