Inequality aversion and risk aversion

This note shows that for two social welfare functions which are inequality averse with respect to certainty equivalents, if one is more inequality averse for certainty equivalents than the other, the household preference induced by optimally allocating aggregate bundles according to this social welfare function is more risk averse than the other. We present examples showing that this comparative static can be reversed if absolute inequality aversion is dropped. We show that the utilitarian rule always induces the least risk averse household preference among all social welfare functions (this corresponds to the sum of certainty equivalents).     

Intertemporal substitution, risk aversion and ambiguity aversion

Summary. This paper axiomatizes a form of recursive utility on consumption processes that permits a role for ambiguity as well as risk. The model has two prominent special cases: (i) the recursive model of risk preference due to Kreps and Porteus [18]; and (ii) an intertemporal version of multiple-priors utility due to Epstein and Schneider [8]. The generalization presented here permits a three-way separation of intertemporal substitution, risk aversion and ambiguity aversion.Received: 5 August 2003, Revised: 12 March 2004, JEL Classification Numbers: D80, D81, D90.I am grateful to Larry Epstein for his guidance and invaluable advice, and to a referee for helpful comments and suggestions.     

Monotone risk aversion

Summary. This paper defines decreasing absolute risk aversion in purely behavioral terms without any assumption of differentiability and shows that a strictly increasing and risk averse utility function with decreasing absolute risk aversion is necessarily differentiable with an absolutely continuous derivative. A risk averse utility function has decreasing absolute risk aversion if and only if it has a decreasing absolute risk aversion density, and if and only if the cumulative absolute risk aversion function is increasing and concave. This leads to a characterization of all such utility functions. Analogues of these results also hold for increasing absolute and for increasing and decreasing relative risk aversion.Received: 31 January 2003, Revised: 15 January 2004, JEL Classification Numbers: D81.The views, thoughts and opinions expressed in this paper are those of the author in his individual capacity and should not in any way be attributed to Morgan Stanley or to Lars Tyge Nielsen as a representative, officer, or employee of Morgan Stanley.     

Repetitive risk aversion

We propose a reasonable condition, which we call repetitive risk aversion (RRA), to be imposed on any utility function to account for the observed data on the relationship between the degree of absolute risk aversion and wealth. We deduce this condition from the concept of the fear of ruin (Aumann and Kurz 1977) and show it to be equivalent to the behaviorally meaningful condition that the risk premium is increasing at a non-increasing rate with the size of the bet. We drive mixed risk aversion, which is known to be stronger than standard and thus proper risk aversion, from RRA. We present several economic applications of RRA to demonstrate that it delivers better comparative static results.I am thankful to Jacques Drèze and Louis Eeckhoudt for their comments. I greatly appreciate the comments of an anonymous reviewer of this journal which have resulted in substantial improvement to both the content and presentation of the paper. An earlier version of this paper was presented at seminars at Brown, CORE, Hopkins, HKUST, Yale, and IMS     

On the correspondence between multivariate risk aversion and risk aversion with state-dependent preferences

This paper establishes the correspondence between multivariate risk aversion and risk aversion with state-dependent preferences. It shows that the prerequisite for comparability of risk aversion in the multivariate case, namely, identical ordinal preferences on the commodity space, corresponds to identical, properly defined, reference sets in the case of state-dependent preferences. For comparable decision makers the condition that the utility function of one is a concave transformation of that of the other on the commodity space corresponds to the condition that the expected utility of one is a concave transformation of that of the other on the reference set.     

First-order risk aversion and non-differentiability

Summary First-order risk aversion happens when the risk premium a decision maker is willing to pay to avoid the lottery , is proportional, for smallt, tot. Equivalently, 0$$ " align="middle" border="0"> . We show that first-order risk aversion is equivalent to a certain non-differentiability of some of the local utility functions (Machina [7]).We are grateful to the Social Sciences and Humanities Research Council of Canada for financial support and to Kim Border, Larry Epstein, Mark Machina and Joe Ostroy for helpful discussions and suggestions.     

Housing and relative risk aversion

This paper derives closed-form and numerical solutions for relative risk aversion in a standard consumption-based model enriched with housing. The presence of housing enables the household to hedge against unexpected shocks and may decrease relative risk aversion. In addition, housing may generate state-dependent, time-varying risk aversion.     

Risk breeds risk aversion

We examine whether exposure to a more or less risky environment affects people’s subsequent risk-taking behavior. In a laboratory setting, all subjects went through twelve rounds of multiple-price-list decisions between a risky alternative and a safe alternative. In the first six rounds, subjects were randomly assigned to a high-, moderate-, or low-risk environment, which differed in the variances of the lotteries they were exposed to. In the last six rounds, subjects in all treatments made decisions on an identical set of lotteries. We found that subjects who had experienced a riskier environment exhibited a higher degree of risk aversion. Our experimental design allows us to conclude that this effect is driven by the risk environment per se, rather than the realized outcomes of the risk. This finding has important theoretical and policy implications.     

A note on surplus labour and risk aversion

The necessary and sufficient conditions for the existence of surplus labour, as derived by A.K. Sen, have been found to be less plausible in the usual framework of the theory of economic planning and project evaluation than seems to be commonly realized. Under a very reasonable assumption concerning strict risk aversion on the part of peasant family members, surplus labour is an impossibility in such a framework.     

Incomplete insurance and absolute risk aversion

Suppose that a subset of states of nature are not verifiable individually. Given an optimal feasible insurance scheme, the expected utility across a group of unverifiable states is greater (less) than that of a verifiable state, if the degree of absolute risk aversion is decreasing (increasing).     

A note on comparative downside risk aversion

We provide comparative global conditions for downside risk aversion, which are similar to the ones studied by Ross for risk aversion. We define a coefficient of downside risk aversion, and study its local properties.     

Subjective performance appraisal and inequality aversion

Making use of a subjective performance appraisal system, it is a well-established fact that many supervisors tend to assess the employees too good (leniency bias) and that the appraisals hardly vary across employees of a certain supervisor (centrality bias). We explain these two biases in a simple theoretical model and discuss determinants of the size of the biases.     

More on parametric characterizations of risk aversion and prudence

Summary. This note provides an alternative proof for the equivalence of decreasing absolute prudence (DAP) in the expected utility framework and in a two-parametric approach where utility is a function of the mean and the standard deviation. In addition, we elucidate that the equivalence of DAP and the concavity of utility as a function of mean and variance, which was shown to hold for normally distributed stochastics in Lajeri and Nielsen [4], cannot be generalized. Received: November 27, 2000; revised version: November 26, 2001 Correspondence to: T. Eichner     

Survey measures of risk aversion and prudence

This paper utilizes a thought experiment conducted by the Bank of Italy to estimate absolute and relative risk aversion along with absolute and relative prudence for a broad cross-section of Italian households. Upper and lower bounds are calculated for each parameter, and comparisons are made across socio-demographic groups. Evidence is found of decreasing absolute risk aversion, decreasing absolute prudence, increasing relative risk aversion, and increasing relative prudence.     

Endogenous time-varying risk aversion and asset returns

Stylized facts about statistical properties for short horizon returns in financial markets have been identified in the literature, but a satisfactory understanding for their manifestation is yet to be achieved. In this work, we show that a simple asset pricing model with representative agent is able to generate time series of returns that replicate such stylized facts if the risk aversion coefficient is allowed to change endogenously over time in response to unexpected excess returns under evolutionary forces. The same model, under constant risk aversion, would instead generate returns that are essentially Gaussian. We conclude that an endogenous time-varying risk aversion represents a very parsimonious way to make the model match real data on key statistical properties, and therefore deserves careful consideration from economists and practitioners alike.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Information,endogenous uncertainty and risk aversion

The importance of incomplete information and risk aversion for the allocation of economic resources is shown to depend critically on whether uncertainty is exogenously imposed or endogenously related to the ability of the price system to aggregate and disseminate the information possessed by the agents in a decentralized market economy. The specific example analysed in this paper is a two-period exchange model with competitive markets and a homogeneous product.Comments and suggestions by anonymous referees, Jacques Drèze, Jan Rose Sørensen, and financial support from the Danish Social Science Research Council are gratefully acknowledged.