Relative Risk Aversion: What Do We Know?

The relative risk aversion measure that represents the risk preferences of a decision maker depends on the outcome variable that is used as the argument of the utility function, and on the way that outcome variable is defined or measured. In addition, the relationship between any two such relative risk aversion measures is determined by the relationship between the corresponding outcome variables. These well-known facts are used to adjust several reported estimates of relative risk aversion so that those estimates can be directly compared with one another. After adjustment, the significant variation in the reported relative risk aversion measures for representative decision makers is substantially reduced. JEL Classification: D81     

Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

The Diversification Theorem Restated: Risk-pooling Without Assignment of Probabilities

Bayesian decision theory assumes that agents making choices assign subjective probabilities to outcomes, even in cases where information on probabilities is obviously absent. Here we show that agents that presume that they are equal risks can share risks mutually beneficially, even if the probabilities of losses are unpredictable or genuinely uncertain. We show also that different risk aversions among pool members do not exclude mutually beneficial loss sharing at uncertainty. Sharing when individuals’ losses differ in probabilities or in amount may still make individuals better off. Our findings are related to the theory of the insurance firm, to the management of development risks, and to the theory of justice.JEL Classification: D8, G22, L10, L30     

How Many Balance Functions Does it Take to Determine a Utility Function?

Call b your balance function at wealth W if you are indifferent between W and a 50–50 lottery with outcomes x and b(x). Given one b, u is arbitrary on one side of W but then determined on the other. Given two b‘s, u is arbitrary between the two W′ s but then determined elsewhere. Additional properties of u restrict the b’s but do not ordinarily make u unique. Contradictions can occur. Given three b′ s, an algorithm is developed using minimal domains of definition that determines the relative utility of the W’s. If it is irrational, then the set S generated by applying all combinations of b’s to W′ s is dense and u is determined. If finitely many b’s are rationally related, then S is discrete, a further algorithm determines it, the values of u on S are equally spaced, and u is arbitrary between any two adjacent points of S but then determined elsewhere. Infinitely many balance functions determine u unless they are rationally related in a uniform way.JEL Classification: D81     

工资上涨能提升劳动报酬份额吗?——基于要素替代弹性的解释

基于要素替代理论研究工资上涨与劳动报酬份额上升的关系,发现在不同经济状态和发展时期,提升劳动报酬份额不应单纯依赖增加工资.工资水平的上升会导致劳动相对资本要素价格提升,而生产要素相对价格的变动与劳动报酬份额的关系依赖于要素替代弹性水平.从实证角度出发,首先使用上市企业数据测算得到要素替代弹性的数值,检验了要素替代弹性对资本相对价格与劳动报酬份额关系的中介作用.在此基础上,从Hicks提出的要素替代弹性概念内涵中发掘了理论模型,结合中国实证数据分析了要素替代弹性的影响因素.研究发现,2001年至今中国要素替代弹性整体小于1且呈增长趋势,但部分行业存在要素替代弹性大于1的情形,在此状况下提高工资水平会导致劳动报酬份额下降.因此工资上涨并不一定带动劳动报酬份额提升,只有在劳动要素以技能工人为中心、产业结构以非资本密集型为核心的情况下,工资上涨才能拉动劳动报酬份额持续上升.     

Dopamine and risk choices in different domains: Findings among serious tournament bridge players

We explore how risk-taking in the card game contract bridge, and in a financial gamble, correlate with variation in the dopamine receptor D4 gene (DRD4) among serious tournament bridge players. In bridge risk-taking, we find significant interactions between genetic predisposition and skill. Among men with the 7-repeat allele of DRD4, namely 7R + men, those with more bridge skill take more good risks and fewer bad risks, while the opposite is found for less-expert 7R + men. Conversely, skill does not predict risk-taking among men without the 7R + allele. Consistent with some prior studies, we also find that 7R + men take more risk in the financial gamble. We find no relationship between 7R + and either risk measure among our female subjects. Our results suggest that the dopamine system plays an important role in individual differences in risk-taking among men, and is the first to distinguish between advantageous and disadvantageous risk-taking.     

Separating marginal utility and probabilistic risk aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a probabilistic risk attitude towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaari's axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (aversion) of preferences with respect to probability mixtures, and convexity (pessimism) of the probability transformation — are equivalent.     

Risk Aversion when Gains are Likely and Unlikely: Evidence from a Natural Experiment with Large Stakes

In the television show Deal or No Deal a contestant is endowed with a sealed box, which potentially contains a large monetary prize. In the course of the show the contestant learns more information about the distribution of possible monetary prizes inside her box. Consider two groups of contestants, who learned that the chances of their boxes containing a large prize are 20% and 80% correspondingly. Contestants in both groups receive qualitatively similar price offers for selling the content of their boxes. If contestants are less risk averse when facing unlikely gains, the price offer is likely to be more frequently rejected in the first group than in the second group. However, the fraction of rejections is virtually identical across two groups. Thus, contestants appear to have identical risk attitudes over (large) gains of low and high probability.      

Risk Perceptions and Value of a Statistical Life for Air Pollution and Traffic Accidents: Evidence from Bangkok,Thailand

This paper presents results of two contingent valuation surveys conducted in Bangkok measuring individuals’ willingness to pay (WTP) to reduce mortality risk arising from two risk contexts: air pollution traffic accidents Results from the risk perception survey disclose that respondents view the two risks differently. WTP to reduce air pollution risk is influenced by degrees of dread, severity, controllability and personal exposure, while WTP to reduce traffic accident risk is influenced by perceived immediate occurrence. Nevertheless, the value of a statistical life (VSL) for both air pollution and traffic accidents are comparable (US$0.74 to $1.32 million and US$0.87 to $1.48 million, respectively). This indicates that the risk perception factor alone has little impact on the VSL, a finding similar to previous studies using program choice indifferences.JEL Classification: I18, D61, J17, J28     

Background Risks and the Value of a Statistical Life

We examine the effects of background mortality and financial risks on an individual's willingness to pay to reduce his mortality risk (the value of statistical life or VSL). Under reasonable assumptions about risk aversion and prudence with respect to wealth in the event of survival and with respect to bequests in the event of death, background mortality and financial risks decrease VSL. The effects of large mortality or financial risks on VSL can be substantial, but the effects of small background risks are negligible. These results suggest that the commonplace failure to account for background risk in evaluating VSL is unlikely to produce substantial bias in most applications.     

Ross' measure of risk aversion and portfolio selection

This article shows that if Ross' definition of riskier is replaced by a more traditional definition, such as a mean-preserving spread or second-degree stochastic dominance, then the application of Ross's stronger measure of risk aversion to the portfolio problem may no longer produce the desired result. It is also shown that the stronger measure may not perform satisfactorily when applied to exponential utility functions.The authors are grateful to John Pratt for his helpful comments.     

Does financial development cause economic growth? Implication for policy in Korea

The causal relationship between financial development and economic growth is examined, utilizing the superexogeneity methodology. We use annual data for Korea during 1971–2002, during which Korea has experienced both phenomenal economic growth and a variety of financial liberalization and reforms. In our tests for superexogeneity, we find that financial development control causes economic growth, but the reverse is not true. Our empirical results provide evidence in favor of the ‘finance causes growth’ view for the case of Korea while rejecting the ‘growth causes finance’ view. The policy implication is that Korea should give policy priority to financial reform rather than economic growth, because only a decisive and accelerated pace of financial restructuring can ensure a sustainable growth in the medium or long term.     

Towards a more precise decision framework

The terms negative utility of gambling and risk aversion conflate three things:

Perception of own death risk

Individuals’ perception of their own road-traffic and overall mortality risks are examined in this paper. Perceived risk is compared with the objective risk of the respondents’ peers, i.e. their own gender and age group, and the results suggest that individuals’ risk perception of their own risk is biased. For road-traffic risk we obtain similar results to what have been found previously in the literature, overassessment and underassessment among low- and high-risk groups, respectively. For overall risk we find that all risk groups underestimate their risk. The results also indicate that men's risk bias is larger than women’s.

Just Who Are You Calling Risk Averse?

This paper estimates individual risk preferences based upon data that are generated by the same individuals acting in different institutions. The results show that the (estimated) numerical values of individuals' implied risk parameters are not stable within individuals across institutions. Furthermore, the ranking across subjects of the numerical values of individuals' implied risk parameters is not preserved across institutions.     

A note on uncertainty and discounting in models of economic growth

The implications of uncertainty for appropriate discounting in models of economic growth have been studied at some length, notably, (Review of Economic Studies, 36:153–163; 1969) and (Journal of Public Economics, 85:149–166; 2002). A detailed account has now appeared in Journal of Risk and Uncertainty, 37:141–169; 2008, sections 4 and 5 (pp. 160–166). One interesting, if perhaps minor, aspect is that under certain circumstances, there appeared to be no solution or at least no satisfactory one. More importantly, the formulas are usually given for the log normal case and are somewhat complicated and hard to interpret intuitively. I show here that assuming a general distribution for returns to capital gives simpler and more understandable results.      

Horizon Length and Portfolio Risk

In this paper, we compare the attitude towards current risk of two expected-utility-maximizing investors who are identical except that the first investor will live longer than the second one. It is often suggested that the young investor should take more risks than the old investor. We consider as a benchmark the case of complete markets with a zero risk-free rate. We show that a necessary and sufficient condition to assure that younger is riskier is that the Arrow-Pratt index of absolute tolerance (T) be convex. If we allow for a positive risk-free rate, the necessary and sufficient condition is T convex, plus T(0) = 0. It extends the well-known result that rational investors can behave myopically if and only if the utility function exhibits constant relative risk aversion.     

The impact of risk sharing on efficient decision

A group of risk-averse members must choose among monetary risks and payoff-sharing rules. Departure from the status quo requires unanimous consent. Such groups drill for oil, bail out nations, and make hostile takeover bids. Assume agreement on probabilities. As is well known, if all members have identically shaped HARA utility functions, efficient group act-choices follow another such function independently of payoff sharing. We show that all other groups inevitably have complex efficient behavior, accepting gambles among individually unacceptable lotteries in almost every status quo position. We also develop proper risk aversion for groups, and treat disagreement on probabilities.Support of the Associates of the Harvard Business School (Pratt) and the Business and Government and Energy and Environmental Policy Centers (Zeckhauser) is gratefully acknowledged. We received helpful comments from Scott Johnson, Mark Machina, and a referee. Jay Patel provided assistance beyond the call of collegiality.     

Uncertainty aversion and aversion to increasing uncertainty

According to the original Ellsberg (1961) examples there is uncertainty version if the decision maker prefers to bet on an urn of known composition rather than on an urn of unknown composition. According to another definition (Schmeidler, 1989), there is uncertainty aversion if any convex combination of two acts is preferred to the least favorable of these acts. We show that these two definitions differ: while the first one truly refers to uncertainty aversion, the second one refers to aversion to increasing uncertainty. Besides, with reference to Choquet Expected Utility theory, uncertainty aversion means that there exists the core of a capacity, while aversion to increasing uncertainty means that the capacity is convex. Consequently, aversion to increasing uncertainty implies uncertainty aversion, but the opposite does not hold. We also show that a completely analogous situation holds for the case of risk and we define a set of risk and uncertainty premiums according to the previous analysis.     

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaari's (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaari's language, when subjective odds differ from the market odds).