Utility Functions whose Parameters depend on Initial Wealth

Conventional one-period utility functions in Economics assume that initial wealth only enters preferences through the definition of final wealth. Consequently, those utility functions most utilized (i.e., exponential and quadratic) have implausible risk characteristics. The authors characterize a new class of utility function whose risk parameters depend upon initial wealth and obtain several desirable results. In particular, investors with quadratic and exponential utility functions can have decreasing risk aversion, and risky assets in a quadratic utility multi-asset environment do not have to be inferior as implied by the traditional framework.     

Modeling non-monotone risk aversion using SAHARA utility functions

We develop a new class of utility functions, SAHARA utility, with the distinguishing feature that it allows absolute risk aversion to be non-monotone and implements the assumption that agents may become less risk averse for very low values of wealth. The class contains the well-known exponential and power utility functions as limiting cases. We investigate the optimal investment problem under SAHARA utility and derive the optimal strategies in an explicit form using dual optimization methods. We also show how SAHARA utility functions extend the class of contingent claims that can be valued using indifference pricing in incomplete markets.     

Optimal financial investments for non-concave utility functions

We prove a formula for the computation of optimal financial investments in an expected utility framework with arbitrary (not necessarily concave) utility functions. This extends classical results on optimal financial investments for strictly concave utility functions and is of importance particularly for applications of prospect theory where the utility function has a convex-concave shape.     

Systems of Benevolent Utility Functions

This paper studies systems of utility functions in which each person's utility depends on his or her own consumption as well as on the utilities of others. We consider the question of when a system of interdependent utility functions induces unique utility functions over allocations and identifies the class of transformations on interdependent utility functions that are equivalent in the sense of inducing the same preferences over allocations. We show that well-behaved systems of this kind can be studied by means of the theory of dominant-diagonal matrices and that the theory of dominant-diagonal matrices with finitely many elements extends in a satisfactory way to denumerable matrices. The theory of denumerable dominant diagonal matrices allows an elegant analysis of systems of intergenerational benevolence. We also revisit and extend the theory of two-sided altruism as formulated by Kimball and by Hori and Kanaya.     

Parametric weighting functions

This paper provides preference foundations for parametric weighting functions under rank-dependent utility. This is achieved by decomposing the independence axiom of expected utility into separate meaningful properties. These conditions allow us to characterize rank-dependent utility with power and exponential weighting functions. Moreover, by allowing probabilistic risk attitudes to vary within the probability interval, a preference foundation for rank-dependent utility with parametric inverse-S shaped weighting function is obtained.     

Dual interpersonal comparisons of utility and the welfare economics of income distribution

Interpersonal comparisons can be of utility levels and/or of utility differences. Comparisons of levels can be used to define equity in distributing income. Comparisons of differences can be used to construct an additive Bergson social welfare function over income distributions. When both utility levels and utility differences are compared, one can require the constructed additive Bergson social welfare function to indicate a preference for more equitable income distributions. This restricts the form of both the individual utility functions and the optimal distribution of income. The form of these restrictions depends on whether the levels and differences of the same utility functions are being compared.     

Non-paternalistic Altruism and Utility Interdependence

This paper clarifies the notion of non-paternalistic altruism through the use of utility aggregators. It presents conditions for the existence of non-paternalistically altruistic utility functions and provides a complete characterization of such utility functions. The results are used to generalize the Second Theorem of Welfare Economics and to prove the existence of an equilibrium in a game of voluntary gift-giving.
JEL Classification Nos.: D11, D64.     

Distortion of preferences and the Nash theory of bargaining

It is shown that in Nash bargaining over division of a single good, when agents are allowed to distort their von Neumann-Morgenstern utility functions into any (weakly) concave form, reporting linear utility functions constitutes a unique dominant-strategy Nash equilibrium.     

Adaptive behavior and the existence of Weizsäcker's long-run indifference curves

Suppose that past consumptions of the first r commodities (r < n) influence present consumption. Then, the long-run demand function to which demand converges maximizes the equilibrium short-run utility function only under very restrictive conditions. The long-run demand functions can be rationalized by a utility function, different from the equilibrium short-run utility function, if and only if the short-run utility function is such that past consumptions of any good that experience, learning, or taste changes is separable from all other goods. The class of such utility functions has been found.     

Correlated equilibria in continuous games: Characterization and computation

We present several new characterizations of correlated equilibria in games with continuous utility functions. These have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions. We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions.     

The effect on optimal consumption of increased uncertainty in labor income in the multiperiod case

We consider a multiperiod, additive utility, optimal consumption model with a riskless investment and a stochastic labor income. The main result is that for utility functions belonging to the set F, consumption decreases when we go from any sequence of distribution functions representing labor income to a more risky sequence. A concave utility function belongs to F if its first derivative exists everywhere and is convex.     

Minimal and maximal product differentiation in Hotelling's duopoly

Hotelling's (1929) model of duopolistic competition is re-examined. A family of utility functions is used which has as a special case Hotelling's original utility function. In a two-stage location-price game it is shown that an equilibrium exists when the curvature of the utility functions in the space of characteristics is sufficiently high. The (subgame-perfect) equilibrium never exhibits minimum product differentiation. On the other hand, not all equilibria are at maximal product differentiation.     

Extrinsic uncertainty revisited

This paper is devoted to extending the concept of extrinsic uncertainty introduced by Cass and Shell for von Neumann-Morgenstern utility functions to a class of ordinal utility functions. It is shown that the important result that extrinsic uncertainty does not matter in a pure exchange economy extends to the more general framework.     

Risk Behavior Under Linear Utility Function

An uncertainly model based on money-market imperfections and on linear utility functions (of money) is reformulated so as to be directly with the traditional non-linear utility approach. We postulate that the utility is a linear functions of “augmented” income. But the augmented income is a non-linear function of the nominal income, due to differential interest rates. This new formulation is shown to generate all the essential implications of the traditional approach as in Friedman and Savage, Markowtiz, Arrow, and Pratt. [020]     

Saving and retirement behavior under quasi-hyperbolic discounting

This paper investigates saving and retirement behavior using a quasi-hyperbolic discounting model with endogenous labor supply. The behavior of quasi-hyperbolic-discounting consumers is compared with optimal behavior, which is obtained under exponential discounting. The quasi-hyperbolic discounters, whether naïve or sophisticated, under-save and retire early compared with an exponential discounter, if and only if the present-biased marginal utility of future consumption decreases with stronger present bias. Logarithmic utility functions and constant-absolute-risk-aversion utility functions can both exhibit this property. In other words, quasi-hyperbolic discounting explains why, consistent with previous empirical studies, under-savers might also be early retirers. Under logarithmic utility, a wage tax and an interest subsidy can counteract the under-saving and early retirement and improve consumer welfare.     

A note on duality in consumer theory

Summary. In this paper we will show that upper semicontinuity of the indirect utility function implies the upper semicontinuity of the direct utility function. By strengthening the assumptions, one can also deduce the continuity of the utility function. Based on indirect utility functions a model of consumer behavior will be established. Received: December 26, 1996; revised version: August 25, 1997     

Preservation of “more risk averse” under expectations

The preservation of risk aversion properties of utility functions under expectation operations is of interest in the study of derived utility functions in sequential decision problems and in problems with multiple (contemporaneous) sources of uncertainty. The purpose of this note is to present sufficient conditions for the Pratt relation “more risk averse” to be preserved under expectations.     

Quasi-stationary cardinal utility and present bias

This note gives an axiomatic foundation for utility exhibiting quasi-geometric discounting. In addition, it introduces a wider class of utility functions satisfying weakened stationarity, called quasi-stationary utility. Both are established as von Neumann-Morgenstern utility indices in a model of risk preference.     

A nested CES-type utility function and its demand and price-index functions

The nested CES-type utility function is introduced in this paper: it consists of an N-level utility tree of slightly modified CES-type functions. Attractive price- and quantity-indices are developed and an elegant system of demand functions, specified in terms of price-indices, results. In addition, some suggestions are made with respect to the estimation of this system.