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.     

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.     

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.     

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.     

Prudence, Intertemporal Substitution, and Savings under Uncertainty

This paper derives the relations between the coefficient of absolute prudence, the equivalent precautionary premium, risk aversion to concentration, and the normality and shift of current consumption under uncertainty, without the time-separable utility assumption. Examples show that Kimball's coefficient of absolute prudence does not fully characterize precautionary saving or saving behavior under uncertainty. It is proved that, whereas a higher rate of intertemporal substitution and a larger coefficient of absolute prudence imply more savings when current consumption is normal, a larger coefficient of intertemporal substitution and a smaller coefficient of absolute prudence imply more savings when current consumption is inferior. Received July 17, 2001; revised version received November 20, 2001     

Optimal pricing and capacity choice for a public service under risk of interruption

We develop rules for pricing and capacity choice for an interruptible service that recognize the interdependence between consumers’ perceptions of system reliability and their market behavior. Consumers post ex ante demands, based on their expectations on aggregate demand. Posted demands are met if ex post supply capacity is sufficient. However, if supply is inadequate all ex ante demands are proportionally interrupted. Consumers’ expectations of aggregate demand are assumed to be rational. Under reasonable values for the consumer’s degrees of relative risk aversion and prudence, demand is decreasing in supply reliability. We derive operational expressions for the optimal pricing rule and the capacity expansion rule. We show that the optimal price under uncertainty consists of the optimal price under certainty plus a markup that positively depends on the degrees of relative risk aversion, relative prudence and system reliability. We also show that any reliability enhancing investment—though lowering the operating surplus of the public utility—is socially desirable as long as it covers the cost of investment.     

Relative risk aversion, relative prudence and comparative statics under uncertainty: The case of (μ, σ)‐preferences

From the expected‐utility approach, relative risk aversion being smaller than one and relative prudence being smaller than two emerge as preference restrictions that fully determine the optimal responses of decisions under uncertainty to certain shifts in probability distributions. We characterize the magnitudes of relative risk aversion and relative prudence in terms of the two‐parameter, mean‐standard deviation approach. We demonstrate that this characterization is instrumental in obtaining comparative static results in the two‐parameter setting. We further relate our findings to the results in the expected‐utility framework.     

Time Horizon and the Discount Rate

We consider an economy à la Lucas (1978, Econometrica46 , 1429-1446) with a risk-averse representative agent. The exogenous growth rate of the economy follows a random walk. We characterize the set of utility functions for which it is efficient to discount more distant cash flows at a lower rate. The benchmark result is that, when the growth rate is almost surely nonnegative, the yield curve is decreasing if and only if relative risk aversion is decreasing with wealth. Relaxing the assumption on the absence of recession requires more restrictions on preferences, such as increasing relative prudence. Journal of Economic Literature Classification Numbers: D81, D91, Q25, Q28.     

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.     

Decreasing absolute prudence: Characterization and applications to second-best risk sharing

In this paper, we provide two basic properties of utility functions u which exhibit decreasing absolute prudence i.e. (− u/u″)′ ≤ 0. These properties are used to examine the allocation of risks in an economy when some agents bear non-transferable risks. We show that it is fair Pareto-efficient to let those with a non-transferable risk bear relatively less of the transferable risk in the economy if and only if absolute prudence is decreasing. In another model, there is a complete set of contingent markets, but some agents have no direct access to them. We examine the fair efficient allocation of risk in a pool gathering a trader and a non-trader. Decreasing absolute prudence provides an upper bound to the share of the pool's risk that should be borne by the trader.     

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.     

Does the Optimal Size of a Fish Stock Increase with Environmental Uncertainties?

We analyze the effect of environmental uncertainties on optimal fishery management in a bio-economic fishery model. Unlike most of the literature on resource economics, but in line with ecological models, we allow the different biological processes of survival and recruitment to be affected differently by environmental uncertainties. We show that the overall effect of uncertainty on the optimal size of a fish stock is ambiguous, depending on the prudence of the value function. For the case of a risk-neutral fishery manager, the overall effect depends on the relative magnitude of two opposing effects, the ‘convex-cost effect’ and the ‘gambling effect’. We apply the analysis to the Baltic cod and the North Sea herring fisheries, concluding that for risk neutral agents the net effect of environmental uncertainties on the optimal size of these fish stocks is negative, albeit small in absolute value. Under risk aversion, the effect on optimal stock size is positive for sufficiently high coefficients of constant relative risk aversion.     

Risk aversion,prudence and temperance: An experiment in gain and loss

We characterize the individual's attitude towards risk, prudence and temperance in the gain and loss domains. We analyze the links between the three features of preferences for a given domain and between domains for each feature of preferences. Consequently, the reflection effect, the mixed risk aversion and the risk apportionment, are key concepts of our study. We also display some determinants for risk aversion, prudence and temperance in each domain. To do this, we conducted a lab experiment with students eliciting risk aversion, prudence and temperance in the two domains, and collected information about each subject's characteristics.     

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.     

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).     

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.     

Wealth Inequality and Asset Pricing

In an Arrow–Debreu exchange economy with identical agents except for their initial endowment, we examine how wealth inequality affects the equilibrium level of the equity premium and the risk-free rate. We first show that wealth inequality raises the equity premium if and only if the inverse of absolute risk aversion is concave in wealth. We then show that the equilibrium risk-free rate is reduced by wealth inequality if the inverse of the coefficient of absolute prudence is concave. We also prove that the combination of a small uninsurable background risk with wealth inequality biases asset pricing towards a larger equity premium and a smaller risk-free rate.     

The impact of prudence on optimal prevention revisited

We re-examine the link between absolute prudence and self-protection activities. We show that the level of effort chosen by an agent with decreasing absolute prudence is larger than the optimal effort chosen by a risk-neutral agent if the degree of absolute prudence is less than a threshold that is utility-independent and empirically verifiable.     

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.