A class of multiattribute utility functions

A function u(z) is a utility function if u′(z) > 0. It is called risk averse if we also have u′′(z) < 0. Some authors, however, require that u ⁽ⁱ⁾(z) > 0 if i is odd and u ⁽ⁱ⁾(z) < 0 if i is even. The notion of a multiattribute utility function can be defined by requiring that it is increasing in each variable and concave as an s-variate function. A stronger condition, similar to the one in case of a univariate utility function, requires that, in addition, all partial derivatives of total order m should be positive if m is odd and negative if m is even. In this paper, we present a class of functions in analytic form such that each of them satisfies this stronger condition. We also give sharp lower and upper bounds for E[u(X ₁,... , X _s )] under moment information with respect to the joint probability distribution of the random variables X ₁,... , X _s assumed to be discrete and representing wealths. Partially supported by OTKA grants F-046309 and T-047340 in Hungary.     

On the characterization of excess demand functions

Summary. In this paper, we give the necessary and sufficient conditions that characterize the individual excess demand function when it depends smoothly on prices and endowments. A given function is an excess demand function if and only if it satisfies, in addition to Walras law and zero homogeneity in prices, a set of first order partial differential equations, its substitution matrix is symmetric and negative semidefinite. Moreover, we show that these conditions are equivalent to the symmetry and negative semidefiniteness of Slutsky matrix, Walras law and zero homogeneity of Marshallian demand functions.Received: 25 November 2002, Revised: 11 March 2004, JEL Classification Numbers: D11.Marwan Aloqeili: I would like to thank an anonymous referee for helpful comments.     

Remarks concerning concave utility functions on finite sets

Summary. A direct construction of concave utility functions representing convex preferences on finite sets is presented. An alternative construction in which at first directions of supergradients (prices) are found, and then utility levels and lengths of those supergradients are computed, is exhibited as well. The concept of a least concave utility function is problematic in this context.Received: 28 November 2002, Revised: 28 June 2004, JEL Classification Numbers: D11, C60.I am indebted to an anonymous referee, Marcel K. Richter and Kam-Chau Wong, for many valuable remarks and suggestions.     

Subjective recursive expected utility

We axiomatize a subjective version of the recursive expected utility model. This development extends the seminal results of Kreps and Porteus (Econometrica 46:185–200 (1978)) to a subjective framework and provides foundations that are easy to relate to axioms familiar from timeless models of decision making under uncertainty. Our analysis also clarifies what is needed in going from a represention that applies within a single filtration to an across filtration representation.Part of this research was conducted when Ozdenoren visited MEDS in Fall 2003. We thank Tapas Kundu, Costis Skiadas, Jean-Marc Tallon and Tan Wang for helpful discussions and also thank audiences at Koc University, Northwestern University, the CERMSEM conference “ Mathematical Models in Decision Theory” at Universite Paris I, and the FUR XI conference on foundations and applications of utility, risk and decision theory     

Subjective ambiguity, expected utility and Choquet expected utility

Summary. Using the Savage set up, this paper provides a simple axiomatization of the Choquet Expected Utility model where the capacity is an inner measure. Two attractive features of the model are its specificity and the transparency of its axioms. The key axiom states that the decision-maker uses unambiguous acts to approximate ambiguous ones. In addition, the notion of ‘ambiguity’ is subjective and derived from preferences. Received: March 23, 2000; revised version: April 24, 2001     

Option price without expected utility

We consider the meaning of the option price, commonly acknowledged as the preferred ex ante welfare measure, in the rank-dependent expected utility (RDEU) framework. The importance of this pertains to performing benefit-cost analysis when RDEU maximizers are prevalent in society.     

A note on linear utility

Summary We show that any complete, lower-semicontinuous, and translation-invariant preorder defined on a topological vector space admits a linear and continuous utility representation.Thanks are given to an anonymous referee for his/her valuable suggestions and comments.     

Strength of preference and cardinal utility

Summary. This paper provides simpler and more general preference foundations for difference representations than known before and shows how to obtain cardinal utility from those difference representations. In addition, this paper unifies all earlier derivations of cardinal utility by showing that they are implied as direct corollaries.Received: 9 November 2002, Revised: 27 December 2004, JEL Classification Numbers: C6.The author particularly thanks Peter P. Wakker and an anonymous referee for many helpful comments and suggestions. The author also thanks Han Bleichrodt and Horst Zank. The paper was partly written during a visit at the Kellogg School of Management, Northwestern University. The author thanks the Dutch Science Foundation NWO for financial support under grant 425-11-003.     

Saving under rank-dependent utility

Summary. In this note we use the rank-dependent utility (RDU) model to analyze saving decisions. The RDU model enables us to separate the effects of pessimism and optimism on saving from that of concavity of the utility function. While pessimism induces more saving, the importance of this effect is shown to depend upon properties of the utility function such as prudence and temperance.JEL Classification Numbers: D11, D81.An anonymous referee provided helpful comments. Han Bleichrodts research was made possible by a grant from the Netherlands Organisation for Scientific Research (NWO). Louis Eeckhoudt acknowledges support from the Interuniversity Attraction Poles Programme - Belgian State - Federal Office for Scientific, Technical, and Cultural Affairs.     

Utilitarianism and the role of utility in Adam Smith

Abstract

In this paper we confront attempts to bring Smith closer to utilitarianism. We show that Smith's conception of utility is not utilitarian. While the pursuit of ‘pleasure’ could lie behind human behaviour, it is not the pleasure referred to by utilitarianism. Instead, utility, in its colloquial sense, plays a greater role that suggests a type of consideration which is foreign to utilitarianism and which also introduces a rationalist element to Smith's moral analysis. Thus, utility, in the utilitarian sense, is neither a guide to action nor a means for moral evaluation.     

Fiscal policy,wealth divergence,and lifetime utility

Using a two-country dynamic optimization model where the less patient country decumulates and the more patient one accumulates wealth, we analyze spillover effects of lump-sum and consumption taxes. A lump-sum tax on a country definitely harms the other country through a change in the rate of interest. A lump-sum tax on either country always improves the less patient country's asset position. A consumption tax has no spillover effect, although it is Pareto-inferior. Applying these results into a closed-country context with heterogeneous agents, we also discuss policy implications of a discriminatory tax.     

Preferences for environmental quality under uncertainty

Although the expected effects of environmental policies and interventions are rarely known with certainty, stated preference surveys rarely elicit preferences over uncertain environmental outcomes. This article presents empirical results challenging the view that ignoring such uncertainty during preference elicitation is of no consequence so long as people only care about final environmental states. We show that measured preferences for final environmental states, water quality in this case, depend on whether people choose between final states or between lotteries over final states. In contrast to the typical finding for monetary lotteries, we find significant under-weighting of low probability events related to water quality.     

Upper and lower bounds for expected utility

Summary.   The decision-theoretic literature has developed very few techniques to bound the expected utility of a random variable when only simple statistics like its median or mode or mean are known. One reason for this lack of results is that we are missing a convenient way to link probability theory and expected utility. This paper is written to demonstrate a general (and genuinely probabilistic) technique to obtain upper and lower bounds for the expected utility of a lottery. Received: December 14, 1999; revised version: March 8, 2000     

Recursive utility with unbounded aggregators

A new framework is presented for the study of the existence and uniqueness of solutions to the Koopmans’ equation in the unbounded case, that is based on the contraction mapping approach. In the bounded below case with bounded consumption streams, uniqueness of the solution in the whole class of weak-star continuous utility functions is obtained. When the aggregator is unbounded below and/or consumption streams are unbounded, existence of a weak-star continuous solution is shown, and a simple criterium to check the sufficient conditions for existence is provided. Juan Pablo Rincón-Zapatero and Carlos Rodríguez-Palmero gratefully acknowledge financial support by the Ministerio de Educación y Ciencia and FEDER funds under Research Projects MTM2005-06534 and SEJ2005-08709/ECON, respectively, and by Consejería de Educación de la Junta de Castilla y León under Research Projects VA99/04 and VA017B05, respectively. This paper has substantially benefited from the comments of an anonymous referee. Particular and special thanks are due to Robert Becker for his very helpful comments and valuable suggestions.     

Differentiable von Neumann-Morgenstern utility

Summary. Differentiability is a convenient property of von Neumann-Morgenstern utility functions which is almost always imposed but has not been translated into behavioral terms. In applications, expected utility is usually maximized subject to a constraint, and the maximization is carried out by differentiating the utility function. This paper presents two sets of necessary and sufficient conditions for a risk averse von Neumann-Morgenstern utility function to be differentiable. The first of them is formulated in terms of the equivalent risk premia of small gambles. It says, in brief, that the equivalent risk premium is of a smaller order of magnitude than the risk itself, as measured by the expectation of the absolute value of the risk. The second set of necessary and sufficient conditions is formulated in terms of the probability premium of small lotteries. It says, essentially, that the probability premium for small binary lotteries goes to zero as the size of the lottery goes to zero. Received: May 11, 1997; revised version: May 14, 1998     

Risk preference and indirect utility in portfolio-choice problems

We consider a portfolio-choice problem with one risky and one safe asset, where the utility function exhibits decreasing absolute risk aversion (DARA). We show that the indirect utility function of the portfolio-choice problem need not exhibit DARA. However, if the (optimal) marginal propensity to invest is positive for both assets, which is true when the utility function exhibits nondecreasing relative risk aversion, then the DARA property is carried over from the direct to the indirect utility function.     

A simple axiomatization and constructive representation proof for choquet expected utility

>P>Summary. We provide a set of simple and intuitive set of axioms that allow for a direct and constructive proof of the Choquet Expected Utility representation for decision making under uncertainty. Received: October 29, 2002; revised version: November 13, 2002 RID="*" ID="*" We thank Matthew Ryan for very useful comments and suggestions on related work and for encouraging us to write this note. Correspondence to: S. Grant     

Expected utility theory and prospect theory: one wedding and a decent funeral

Choice behavior is typically evaluated by assuming that the data is generated by one latent decision-making process or another. What if there are two (or more) latent decision-making processes generating the observed choices? Some choices might then be better characterized as being generated by one process, and other choices by the other process. A finite mixture model can be used to estimate the parameters of each decision process while simultaneously estimating the probability that each process applies to the sample. We consider the canonical case of lottery choices in a laboratory experiment and assume that the data is generated by expected utility theory and prospect theory decision rules. We jointly estimate the parameters of each theory as well as the fraction of choices characterized by each. The methodology provides the wedding invitation, and the data consummates the ceremony followed by a decent funeral for the representative agent model that assumes only one type of decision process. The evidence suggests support for each theory, and goes further to identify under what demographic domains one can expect to see one theory perform better than the other. We therefore propose a reconciliation of the debate over two of the dominant theories of choice under risk, at least for the tasks and samples we consider. The methodology is broadly applicable to a range of debates over competing theories generated by experimental and non-experimental data.     

Medical insurance with rank-dependent utility

Summary. A well-known result in the medical insurance literature is that zero co-insurance is never second-best for insurance contracts subject to moral hazard. We replace the usual expected utility assumption with a version of the rank-dependent utility (RDU) model that has greater experimental support. When consumers exhibit such preferences, we show that zero co-insurance may in fact be optimal, especially for low-risk consumers. Indeed, it is even possible that the first-best and second-best contracts are identical. In this case, there is no “market failure”, despite the informational asymmetry. We argue that these RDU results are in better accord with the empirical evidence from US health insurance markets. Received: February 26, 2001; revised version: October 4, 2002 RID="*" ID="*"The authors would particularly like to thank Simon Grant, John Quiggin, Peter Wakker and an anonymous referee for valuable comments and suggestions on earlier drafts. The paper has also benefitted from the input of seminar audiences at The Australian National University, University of Auckland, University of Melbourne and University of Sydney. Ryan also gratefully acknowledges the financial support of the ARC, through Grant number A000000055. Correspondence to:R. Vaithianathan