Abstract
In several problems of decision-making under uncertainty, it is necessary to study the sign of the covariance between marginal utilities. All of the works that study the covariance signs are based on Chebyschev’s integral inequality. However, this inequality requires that both functions be monotonic. There are many cases, originated basically by new alternative theories, which assume that the marginal utilities of interest are non-monotonic. Thus, we cannot use Chebyschev’s result as it relies on monotonic functions. In this article, I derive some new covariance inequalities for utility functions which have non-monotonic marginal utilities. I also apply the theoretical results to two problems in economics: First, I study some properties of the indiference curve in the mean-variance space for Prospect Theory and for Markowitz utility functions. Second, I analyze the asset hedging policies of a bank that behaves as predicted by Prospect Theory.
Highlights
Many problems of choice under uncertainty involve studying the sign of a covariance
Broll et al (2010) extend this result considering S-shaped utility function. They show that the mean has an important role to determine the covariance sign for a particular type of S-shaped utility functions, as we can see in the following theorem
I derive new results to study the sign of Cov[X,u (X)], when the marginal utility is non-monotonic
Summary
Many problems of choice under uncertainty involve studying the sign of a covariance. In particular, many times it is necessary to determine the sign of the covariance of two real functions and of a random variable X: Cov[ (X), (X)]. (1)The sign of (1) is deduced with the following argument: If these two functions are increasing (or both are decreasing), its sign is non-negative, while if one function is increasing and the other is decreasing the sign is non-positive (cf. e.g. Gurland, 1967; Lehmann, 1966; McEntire, 1984; Schmidt, 2014). Note that for power utility functions as presented in Definition 2 condition (4), reversed loss aversion implies that λ < 1. Egozcue, Fuentes Garcıa, & Wong (2009), Egozcue et al (2011) derive some new covariance inequalities relaxing the assumption of monotonicity, but their results work only for symmetric random variables.
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