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.     

Insurance and the behavior of competitive firms under revenue risks: a note

This paper re-examines the model of Ford, Mpuku, and Pattanaik [“Revenue Risks, Insurance, and the Behavior of Competitive Firms”.Journal of Economics 64 (1996): 233–246] wherein a risk-averse competitive firm faces insurable revenue risk. The optimal output and insurance cover of the firm are shown to be deterministically related in that the marginal costs of self-insurance and market insurance are equated. In response to increasing risk aversion, the firm always takes a higher insurance cover. Increasing fixed costs generate an income effect which induces the firm to take a higher insurance cover should the preference of the firm satisfy decreasing absolute risk aversion. Market insurance and self-insurance can be either substitutes or complements, depending on the shape of the variable insurance-premium schedule.     

Wealth inequality and dynamic stability

We explore the link between wealth inequality and stability in a two-sector neoclassical growth model with heterogeneous agents. We show that when the inverse of absolute risk aversion (or risk tolerance) is a strictly convex function, wealth inequality is a factor that favors instability. In the opposite case, inequality favors stability. Our characterization also shows that whenever absolute risk tolerance is linear, as when preferences exhibit hyperbolic absolute risk aversion (HARA), wealth heterogeneity is neutral.     

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.     

Invariant risk attitudes

Concepts of constant absolute risk aversion and constant relative risk aversion have proved useful in the analysis of choice under uncertainty, but are quite restrictive, particularly when they are imposed jointly. A generalization of constant risk aversion, referred to as invariant risk aversion is developed. Invariant risk aversion is closely related to the possibility of representing preferences over state-contingent income vectors in terms of two parameters, the mean and a linearly homogeneous, translation-invariant index of riskiness. The best-known index with such properties is the standard deviation. The properties of the capital asset pricing model, usually expressed in terms of the mean and standard deviation, may be extended to the case of general invariant preferences.     

BACKGROUND RISK AND THE THEORY OF THE COMPETITIVE FIRM UNDER UNCERTAINTY*

This paper examines the optimal production decision of a firm under output price risk á la Sandmo when the firm also faces a dependent background risk. It is shown that standard risk aversion plus a non-negative association between the output price risk and the background risk are sufficient to ensure a reduction in the firms optimal output upon introduction of the background risk. The paper investigates the impact of a deterministic transformation of the background risk on the firms optimal production decision. It is shown that decreasing absolute risk aversion in Ross' sense is among the sufficient conditions that generate an unambiguous negative comparative static result.     

Economic indices of absolute and relative riskiness

Following Aumann and Serrano (J Polit Econ 116:810–836, 2008) who characterize by axioms an index of riskiness defined on absolute returns, we characterize a new index of riskiness defined on relative returns. Both indices are characterized by a similar principle of duality between risk and risk aversion, but while the index of absolute riskiness refers to absolute risk aversion, the index of relative riskiness refers to relative risk aversion. The similarities and differences between the two indices are studied.     

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.     

On the measurement of risk aversion from experimental data

Attitudes towards risk are measured for households in Northern Zambia using an experimental gambling approach with real payoffs that at maximum were equal to 30% of average total annual income per capita. The results of the experiment show decreasing absolute risk aversion and increasing partial risk aversion. Determinants of risk aversion are investigated using random effects interval regression model exploiting the panel data structure of the repeated experiments. Wealth indicator variables are found to be significant, and partial relative risk aversion decreases as wealth increases. Females are found to be more risk averse than males.     

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     

Production and savings under uncertainty

This paper studies an integrated model of production and savings under uncertainty in which production inputs and the amount of savings are jointly chosen. The analysis shows that if the agent's risk preferences exhibit constant absolute risk aversion, then all results from nonintegrated or separate models of savings and production extend to the integrated framework. Under decreasing absolute risk aversion, the comparative static properties of optimal production decisions with respect to mean preserving spread and spread preserving mean parameters extend from the non-integrated to the integrated framework. However, extension of the savings model results for the same parameters requires a restriction on production technology.     

Further results on comparative statics under uncertainty. A comment on Machnes

This paper re-examines the results in Machnes (1993). When a risk-averse firm faces both uncertain demand and uncertain fixed costs, the Arrow-Pratt theory of decreasing absolute risk aversion may be too weak to yield unambiguous comparative statics. Herein, it is shown that the stronger notions of risk behavior proposed by Ross (1981) and Kimball (1993) are useful in providing intuitive results in this context.     

Background risk and the demand for state-contingent claims

Summary. We consider the demand for state-contingent claims, in the presence of an independent zero-mean, non-hedgeable background risk. An agent is defined to be generalized risk averse if he/she chooses a demand function for contingent claims with a smaller slope everywhere, given a simple increase in background risk. We show that the conditions for standard risk aversion, that is positive, declining absolute risk aversion and prudence, are necessary and sufficient for generalized risk aversion.Received: 13 February 2002, Revised: 10 February 2003, JEL Classification Numbers: D52, D81, G11. Correspondence to: Guenter FrankeWe are grateful to Louis Eeckhoudt, Christian Gollier, Harris Schlesinger and an unknown referee for valuable comments.     

Estimating risk aversion coefficients for dry land wheat,irrigated corn and dairy producers in Kansas

The risk attitudes of dry land wheat, irrigated corn, and dairy producers in Kansas are examined using the nonlinear mean–standard deviation approach. Results of analyses indicated that dryland wheat and dairy producers are characterized by increasing absolute and increasing relative risk aversion while irrigated corn producers are characterized by constant absolute and increasing relative risk aversion. Both crop enterprises exhibited constant returns to scale technology while the dairy enterprise exhibited decreasing returns to scale. Gross farm income was significant and positively related to relative risk aversion.     

Marginal risk aversion and preferences in a betting market

An individual's behavioural attitudes toward variance and non-symmetry in the payoff distributions of pari-mutuel gambles are empirically examined using the von Neumann - Morgenstern expected utility of wealth paradigm. Preferences over payoff distributions for a representative bettor are estimated from observed payoffs at a greyhound racetrack. The results indicate that the representative bettor exhibits increasing absolute risk aversion and, given that the representative bettor is locally non-satiated with regard to wealth, exhibits preference for variance and aversion to positive skewness in the payoff distributions of the gambles examined.     

Risk and regret aversions on optimal bank interest margin under capital regulation

This paper examines the bank's optimal loan rate (and thus the bank's interest margin) under more stringent capital regulation when the bank is not only risk-averse but also regret-averse. Risk-averse preferences are characterized by an option-based utility function that includes disutility from the dislike of bank equity risk. Regret-averse preferences feature an option-based utility function that includes disutility from having chosen ex-post suboptimal alternatives. We show that an increase in bank capital requirement results in an increased margin under risk aversion dominating regret aversion, whereas it results in a reduced margin under regret aversion dominating risk aversion. The former holds when risk aversion domination stems from increasing risk-averse preference, but not from decreasing regret-averse preference, while the latter holds when regret aversion domination results from either decreasing risk-averse or increasing regret-averse preference. Risk aversion, as such, makes the bank more prudent and less prone to risk-taking, while regret aversion, as such, makes the bank less prudent and more prone to risk-taking.     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

Convex and decreasing absolute risk aversion is proper

Proper risk aversion, a pivotal concept in the study of behavioral conditions on utility functions, states that an undesirable risk can never be made desirable by the presence of an independent risk. It is well known that standard risk aversion is sufficient for this concept. We show in this short article that convex and decreasing absolute risk aversion is an alternative sufficient condition.     

Risk aversion and prudence in the large

The familiar measures of absolute and relative risk aversion constructed by Pratt and Arrow, along with the measures of absolute and relative prudence inspired by Leland and later developed by Kimball, are local instruments based on the first and second derivatives of utility at a specific level of wealth. As such, they are applicable only to infinitesimal risks—those for which differential calculus is a suitable analytical tool. Consequently, they may not accurately gauge preferences regarding the larger risks typically encountered in practice. To address this problem, the present paper develops more general, closed-form index measures of risk aversion and prudence that are applicable to either large or small risks. The new measures are exact in that they do not rely on approximations, they can be implemented empirically without knowledge of the functional form of utility, and they do not require information regarding pre-existing wealth.     

Growth and financial liberalization under capital collateral constraints: The striking case of the stochastic AK model with CARA preferences

We consider a small-open, collateral-constrained AK economy. We show that the combination of CARA preferences and uncertainty on capital inflows generates long-term growth while the deterministic counterpart does not: long-term growth is entirely driven by precautionary savings, and the asymptotic growth rate of the expected capital stock is increasing in both the risk magnitude and the Arrow–Pratt absolute risk aversion parameters.