MISERY LOVES COMPANY: EQUILIBRIUM PORTFOLIOS WITH HETEROGENEOUS CONSUMPTION EXTERNALITIES*

The present article extends the Arrow–Debreu portfolio model to consumption externalities. It is assumed that each investor has a von Neumann–Morgenstern utility that is a function of her own consumption and of the average consumption in the group to which she belongs. Individual degrees of risk aversion and conformism are heterogeneous within each group and between the different groups in the economy. We show that, under some conditions on the degree of conformism in the economy, the optimal portfolio and consumption choices observed at equilibrium in each group with consumption externalities are equivalent to those that are optimal without any externality, but with an adjusted degree of risk aversion. If these conditions are not fulfilled, groups have no representative agent and the demand for pure zero‐mean lotteries may be positive, thereby showing that not all diversifiable risks are washed away at equilibrium. We characterize the relationship between the distribution of conformism in the economy to the competitive allocation of risk and to the equity premium. We provide conditions for the two‐fund separation property to hold.     

A note on additive utility and bargaining

Necessary and sufficient conditions are given under which a decision maker's von Neumann-Morgenstern utility function on the Cartesian product of two prospect spaces can be expressed as a sum of coordinate utility functions, assuming that all preferences are given. A main motivation for this result is an application in axiomatic bargaining theory.     

Expected utility theory without the completeness axiom

We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a well-defined sense.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

ON NON‐MONETARY MEASURES IN THE FACE OF RISKS AND THE SIGNS OF THE DERIVATIVES

Vulnerability of both prudence and temperance towards a sure loss and towards a zero‐mean background risk seems to be a very realistic assumption on individual preference. This paper shows that when the concepts of prudence and temperance are defined in non‐monetary terms, the above assumption is equivalent to the usual signs of the successive derivatives of the utility function.     

Comparing risk aversion in a probability triangle

An agent's acceptance set consists of the probability distributions preferred to the status quo. One agent is more risk averse than another if the more risk averse agent's acceptance set is a proper subset of the less risk averse agent's acceptance set. An agent's odds premium expresses the odds in favor of winning the largest cash prize in a lottery over the best and worst alternatives that is indifferent to the the agent's initial wealth. Comparisons of two agents odds premia completely characterizes the risk aversion relations between them when facing lotteries in a probability triangle. The result applies to expected utility and some non-expected utility theories. Received: December 30, 1998; revised version: February 10, 1999     

The demand for a risky asset in the presence of a background risk

We examine the demand for a risky asset in the presence of two risks: a financial risk and a background risk which need not be financial. First, we compute the necessary and sufficient condition for a positive demand for a risky asset, showing that it depends on two terms capturing respectively the direct effect of risk premium and the dependence between the two risks. Second, we develop higher order expectation dependence concept and show that the more information about the sign of higher cross derivatives of the utility function we have, the weaker dependence conditions on distribution we achieve.     

Risk and decentralization in the theory of economic policy

In this note, uncertainty is incorporated into Tinbergen's model of economic policy. If the Central Planning Authority's preferences amomg subjective probability distributions can be described by a von Neumann-Morgenstern utility function, and if certain restrictions are imposed on these probability distributions, then the Authority can delegate control over partitions of the instrument vector to its Agencies and so decentralize its decision process.     

A theory of anticipated utility

A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.     

Life with habit and expectation: A new explanation of equity premium puzzle

Previous writers have attempted to resolve the equity premium puzzle by employing a utility function that depends on current consumption minus (or relative to) past habit consumption. This paper points out that an individual's current utility may also depend upon how well off in the recent past he or she had expected to be today. Hence we add the concept “expectation formation” to the utility modification term in a model with a habit‐formation utility function. We apply the model to the equity premium puzzle and find that it is able to fit the data with a relatively low coefficient of relative risk aversion. Furthermore, we introduce an updated data sample and apply different values of discounting factors, and find that in all circumstances, the model is able to generate coefficients of risk aversion that are consistent with theory. Hence we conclude that the model is able to resolve the equity premium puzzle.     

Strategy-proofness versus efficiency for small domains of preferences over public goods

Summary. It has long been known that when agents have von Neumann-Morgenstern preferences over lotteries, there is an incompatibility between strategy-proofness and efficiency (Gibbard, [9]; Hylland, [12]) – a solution satisfying those properties must be dictatorial. We strengthen this result by showing that it follows from the same incompatibility on a series of much smaller domains of preferences. Specifically, we first show the incompatibility to hold on our smallest domain, in which two agents are restricted to have linear preferences over one private good and one public good produced from the private good (Kolm triangle economies). This result then implies the same incompatibility on increasingly larger domains of preferences, ending finally with the class of von Neumann-Morgenstern preferences over lotteries. Received: February 6, 1997; revised version: January 29, 1998     

Another proposition on the recoverability of cardinal utility

A sufficient condition on the asset return structure, other than the existence of a riskless asset, is derived which permits recoverability of the von Neumann-Morgenstern cardinal utility index from asset demands.     

Transitive measurable utility

This paper continues a study of theories of preferences under risk that do not use the independence axiom of the von Neumann-Morgenstern theory. Unlike its predecessor, it assumes that preferences are transitive. The effects of transitivity are noted in two representations of preferences. The first, which also uses continuity and dominance axioms, involves a function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. Although u might be nonlinear, it has other features of a von Neumann-Morgenstern linear utility function. The second representation has linear functions u and w on P, with w strictly positive except perhaps at preference-extreme measures—where it might vanish, such that u(p) w(q) > u(q) w(p) if and only if p is preferred to q. A symmetry axiom along with the axioms for the first representation are necessary and sufficient for the second representation.     

A theorem on the identifiability of the von Neumann-Morgenstern utility function from asset demands

If the demand for risky assets is determined by the maximization of an analytical von Neumann-Morgenstern utility function, and if these demands are known as a function of the assets' prices, then this utility function can be constructed without ambiguity.     

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.     

Translation homotheticity

Summary. The concept of translation homotheticity is introduced and defined. It is demonstrated that translation homotheticity is necessary and sufficient for: disposable surplus to be independent of the reference utility, Luenberger's compensating and equivalent benefits to be independent of the reference utility and always equal to one another, the risk premium to be independent of reference-level utility, absolute indexes of income inequality to be reference free, and social-welfare functionals to satisfy invariance with respect to the choice of a common origin. Translation homotheticity is also sufficient for Hicks' many-market consumer surplus measure to be a second-order approximation to disposable surplus, compensating benefit, and equivalent benefit. If preferences are translation homothetic and appropriately quadratic, Hicks, many-market consumer surplus measure is exact for these welfare measures. Received: October 24, 1996; revised version: March 3, 1997     

An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom

The independence axiom used to derive the expected utility representation of preferences over lotteries is replaced by requiring only convexity, in terms of probability mixtures, of indifference sets. Two axiomatic characterizations are proven, one for simple measures and the other continuous and for all probability measures. The representations are structurally similar to expected utility, and are unique up to a generalization of affine transformations. First-order stochastic dominance and risk aversion are discussed using a method which finds an expected utility approximation to these preferences without requiring differentiability of the preference functional.     

A cardinal approach to straightforward probabilistic mechanisms

In every probabilistic mechanism, society selects an alternative, through a random device, out of a subset of indifferent alternatives. Consequently, in this context individuals face uncertainty and value the different lotteries on alternatives by their expected utility, so that they make use of a Von Neumann-Morgenstern cardinal utility function. Surprisingly, the social choice approach to probabilistic mechanisms assumes the use of ballots which preclude the complete expression of behaviour towards risk: individuals can only announce their ordinal preferences, or an approximation of their cardinal preferences, since in any case only a finite number of representations of preferences is available. This paper attempts to study voting systems in which individuals can express the cardinality of their preferences by assigning weights to the alternatives. It is shown that by voting with ballots which reflect weighting a new class of straightforward probabilistic mechanisms is defined, and that this class strictly contains the class of probabilistic straightforward mechanism designed by Gibbard.     

Expected utility theory from the frequentist perspective

We present an axiomatization of expected utility from the frequentist perspective. It starts with a preference relation on the set of infinite sequences with limit relative frequencies. We consider three axioms parallel to the ones for the von Neumann–Morgenstern (vN–M) expected utility theory. Limit relative frequencies correspond to probability values in lotteries in the vN–M theory. This correspondence is used to show that each of our axioms is equivalent to the corresponding vN–M axiom in the sense that the former is an exact translation of the latter. As a result, a representation theorem is established: The preference relation is represented by an average of utilities with weights given by the relative frequencies.