Risk and risk aversion when states of nature matter

When uncertainty is associated with some intrinsically relevant states of nature, there is no reason for an agent to base his or her preferences only on probability distribution of claims. We propose a new concept of risk for state-contingent claims that, unlike the standard concept of Rothschild–Stiglitz, does not identify state-contingent claims with their probability distribution. This concept is called mean-independent risk, and we provide a simple characterization in terms of marginal utilities of (non-expected) utility functions that exhibit aversion to mean-independent risk. We study implications of aversion to mean-independent risk on agents’ choices under uncertainty. This research has been supported by the NSF under Grant SES-0099206. I have benefited from numerous conversations with Rose-Anne Dana and illuminating discussions with Tadeusz Miłosz about the theory of subgradients.     

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.     

Regulation of a Risk Averse Firm

We extend the Laffont–Tirole regulation model to the case of risk-averse firms. Our main results are:

• The impact of risk aversion is to shift the optimal contract toward a cost-plus contract.

• As compared with the risk-neutral case, distortions are greater and informational rents are smaller.

• For high levels of risk aversion, the optimal contract involves cost ceilings and the less efficient firms are bunched together.

Journal of Economic LiteratureClassification Numbers: D8, L5.     

Real estate and relative risk aversion with generalized recursive preferences

This paper investigates how real estate wealth affects the household’s attitude toward risk, and derives the closed-form expressions for risk aversion with generalized recursive preferences. We find three channels through which real estate wealth affects risk aversion, and these channels are absent in the traditional measure of relative risk aversion as in Arrow (1965) and Pratt (1964). First, illiquidity and fluctuations in real estate value increase consumption risk, thereby increasing risk aversion. Second, real estate as an asset provides a cushion for absorbing negative shocks to households, reducing risk aversion. Third, an increase in real estate prices lowers the profit of the firm that uses real estate as a factor of production, induces a decline in the real wage, and causes a rise in consumption risk. This channel increases risk aversion. We study how these channels as a whole determine relative risk aversion using a basic real business cycle model with generalized recursive preferences and compare the results with the case of expected utility preferences. Finally, we explore the implications of the firm’s and the household’s real estate holdings and illiquidity of real estate on the risk premiums for equity and real estate.     

Some recent developments in capital market theory: A survey

This paper surveys some recent developments in the theory of capital markets. Particular emphasis is given to two strands of the literature. The first covers some recent and fundamental extensions to the theory of risk aversion and the demand for risky assets. These papers are concerned with the effect of non-hedgeable background risk on risk attitudes. The important implications for finance are for the size of the risk premium (the equity premium puzzle) and for the demand for and pricing of contingent claims. For example, background risk may help to explain the apparent over-pricing of options on equity indices. The second topic is interest rate term structure models. Stochastic term structure models try to capture the possible future shapes of the term structure of interest rates. This is relevant for the pricing of contingent claims, in particular for the pricing of interest rate derivatives such as American-style swaptions. The paper will survey the most important recent models in the literature, each of which satisfies the fundamental no-arbitrage property. It will discuss the implications of the models for the pricing of both European-style and American-style options.     

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.     

Another Look at the Equity Risk Premium Puzzle

The model of Mehra and Prescott (1985, J. Econometrics, 22, 145–161) implies that reasonable coefficients of risk‐aversion of economic agents cannot explain the equity risk premium generated by financial markets. This discrepancy is hitherto regarded as a major financial puzzle. We propose an alternative model to explain the equity premium. For normally distributed returns and for returns far away from normality (but still light tailed), realistic equity risk premia do not imply puzzlingly high risk aversions. Following our approach, the ‘equity premium puzzle’ does not exist. We also consider fat‐tailed return distributions and show that Pareto tails are incompatible with constant relative risk aversion.     

Background risk in generalized expected utility theory

Summary. In this paper, it is shown that, for a wide range of risk-averse generalized expected utility preferences, independent risks are complementary, contrary to the results for expected utility preferences satisfying conditions such as proper and standard risk aversion. Received: August 10, 2001; revised version: June 18, 2002 RID="*" ID="*"I thank Simon Grant and an anonymous referee for helpful comments and criticism. This research was supported by an Australian Research Council Senior Fellowship and Australian Research Council Large Grant A79800678.     

Revealed preference and differentiable demand

Summary. We prove that, for finitely many demand observations, the Strong Axiom of Revealed Preference tests not only the existence of a strictly concave, strictly monotone and continuous utility generator, but also one that generates an infinitely differentiable demand function. Our results extend those of previous related results (Matzkin and Richter, 1991; Chiappori and Rochet, 1987), yielding differentiable demand functions but without requiring differentiable utility functions.Received: 1 November 2001, Revised: 5 February 2004, JEL Classification Numbers: D11, D12. Correspondence to: Kam-Chau WongThis is a much revised version of Lee and Wong (2001). We are grateful to the Referee for valuable suggestions. We also thank Professor Marcel K. Richter for his comments.     

The role of corruption and risk aversion in entrepreneurial intentions

The literature on entrepreneurship has suggested that an individual’s entrepreneurial intention depends on three types of factor: personal characteristics, the individual’s expertise and professional background, and external factors. Our study investigates how corruption, an external factor, and risk aversion, a personal characteristic, may simultaneously affect individuals’ entrepreneurial intentions. With data on 76 203 individuals in 53 countries, our estimation results indicate that risk aversion decreases the individual’s probability of having an entrepreneurial intention by 6.67 percentage points. In addition, an increase in 1 SD in the perceived level of corruption in a country decreases the individual’s probability of having an entrepreneurial intention by 0.96 percentage points.     

From sure to strong diversification

This paper presents a characterization of weak risk aversion in terms of preference for sure diversification. Similarly, we show that strong risk aversion can be characterized by weakening preference for diversification,as introduced by Dekel (Econometrica 57:163,1989), in what we call preference for strong diversification. We are grateful to Jean-Yves Jaffray, Peter Wakker and anonymous reference for very helpful suggestions and comments.     

The Risk and Incentives Trade-off in the Presence of Heterogeneous Managers

Agency theory predicts a negative relationship between risk and incentives, yet recent empirical evidence has not consistently found such a relationship. In fact, some researchers have found a positive relationship. By introducing competition for heterogeneous managers, who differ in their degrees of risk aversion, into a standard agency model, this paper demonstrates that a negative or positive relationship is theoretically possible. Which arises depends on the relative risk aversion parameters of the managers and the absolute and relative riskiness of the environments.Acknowledgement I thank two anonymous referees for helpful comments.     

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.     

Intertemporal elasticity of substitution and risk aversion: are they related empirically?

This article examines the relationship between two types of preference: preference of intertemporal choices and preference towards risk. In the simplest form of the constant relative risk aversion utility function, the intertemporal elasticity of substitution (IES) and risk aversion have an inverse relationship. However, there is no empirical evidence that suggests this inverse relationship holds. We examine the relationship between risk aversion and IES using household consumption data from the Consumer Expenditure Survey during 1996–2010. Multiple risk domains are selected to represent risk preference, and for each domain, we consider some households to be more risk averse than others. We separately estimate IES for the more risk-averse and less risk-averse households. We find that the IES estimates are generally smaller for the more risk-averse households than for the less risk-averse households and that the difference is statistically significant in the majority of the financial domains. This finding supports the inverse relationship between the two parameters, although considerable heterogeneity is found across domains.     

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.     

实际汇率与就业--基于内生劳动力供给的跨期均衡分析

文章在个体跨期最优模型中引入内生劳动力供给,并同时假定对资本流动存在限制,以此来分析实际汇率变动对就业的影响。结果发现:只有当消费者对商品消费的相对风险规避程度比较大,即消费的跨期替代弹性比较小时,本币实际贬值才会促进就业的增加,反之贬值会使就业减少。考虑到中国当前的具体情况,消费者商品消费的相对风险规避程度会比较大,因而人民币实际贬值会有利于就业。对中国的实证分析支持了这一结论,且该实证分析通过了实际汇率的超外生性检验。鉴于从长期来看中国消费者的相对风险规避程度有下降的趋势,过分的依赖于低币值的汇率政策来解决失业问题是不可行的。     

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.     

The impact of prudence on optimal prevention

Summary. What are the determinants of the optimal level of effort to reduce the probability of a loss to occur? Whereas most of the literature on this question focused on risk aversion, we show that the concept of prudence (i.e., a positive third derivative of the utility function) is essential to answer this question. We explain in this paper that prudence and prevention tend to be opponents rather than allies contrary to the intuition attached to everyday language.Received: 7 November 2003, Revised: 3 August 2004, JEL Classification Numbers: D61, D81. Correspondence to: Christian Gollier     

Left-side Relatively Weak Increases in Risk and Their Comparative Statics

This paper proposes a new concept, a left-side relatively weak increase in risk (L-RWIR) order, that extends the definition of a relatively weak increase in risk (RWIR) order. We show that, for the class of linear payoffs, one can obtain an appealing comparative statics result for L-RWIR shifts imposing additional restrictions on risk preferences of a risk-averse decision maker.JEL classification: D81.revised version received October 10, 2003Acknowledgements The authors would like to thank an anonymous referee for insightful comments and useful suggestions.     

Estimating Willingness to Pay for Improved Drinking Water Quality Using Averting Behavior Method with Perception Measure

This paper suggests an application of an averting behaviormethod to reconcile inconsistent public activity with objectiveenvironmental risk. In Korea, an historically polluted watersupply has created citizen resistance to using tap water. Koreansperceive low quality levels for tap water, although objectivelymeasured data show that the pollution levels are lower than theacceptable risk. We hypothesized that the irrational aversion tousing tap water is a consequence of inconsistencies between theobjectively measured and perceived pollution levels. We introducethe perception averting behavior method, in which we add aperception measure unit to the conventional averting behaviormethod. We found that the perception measure provided a validexplanation for citizens' aversion to using tap water in Korea.