Risk averters that love risk? Marginal risk aversion in comparison to a reference gamble.

Applied economists have developed a vibrant literature that seeks to quantify the impacts of risk on production decisions. This technical literature posits an expected utility objective function and relies on advanced econometric techniques to back risk preferences out of observed input decisions. The accuracy of the assumption that individuals maximize expected utility is often threatened by pervasive and arbitrary decision heuristics, which could therefore cause severe bias when estimating risk preferences. Consider, for example, an investor who receives and evaluates new information, then makes slight adjustments in her investment positions accordingly. Such an incremental approach with marginal adjustments could imply different behavior than if she reevaluated her entire portfolio from scratch in response to new information or new investment options. Although applied economists generally assume that individuals globally reevaluate their alternatives when estimating risk preferences, individuals often seem to favor incremental adjustments based on comparisons between options. In such contexts methods that explicitly assess changes in risk preferences on the margin may yield more robust estimates of behavioral parameters.

In this article, we distinguish between standard risk aversion and marginal risk aversion. The conceptual distinction between these measures of risk aversion is simple. Standard measures of risk aversion such as Arrow-Pratt coefficients are generally based on isolated, stand-alone gambles and indicate risk aversion whenever an individual's certainty equivalent for a gamble is below its expected value. Marginal risk aversion, on the other hand, involves a comparison between gambles and is displayed when the individual responds conservatively to changes in the gamble. Thus, an individual is risk averse on the margin if her valuation increases (decreases) when the gamble changes to become less (more) risky.

In an expected utility maximization world the distinction between standard and marginal measures of risk aversion may be trivial, but these differences quickly become important when behavior deviates from expected utility. To illustrate, consider one of the best-known expected utility deviations: the tendency to overweight small probabilities and underweight large probabilities, which is responsible in part for the certainty effect and other cognitive biases. Such probability weighting drives a wedge between standard and marginal risk aversion. To see how, suppose an individual has a simple linear utility function, which implies risk neutrality in the expected utility framework but systematically underweights large probabilities. This individual will value a gamble with a large probability of gain below its expected value, suggesting risk aversion. The same individual, however, may be indifferent between this gamble and another one with the same expected value and similar probabilities but greater variability in potential outcomes, suggesting risk neutrality. Such a disparity between behavior taken on average and behavior on the margin could help to explain why empirical estimates of risk aversion are widely variant and inconsistent (see Just and Peterson 2003). In particular, slightly different approaches to the same problem may produce very different risk estimates if one approach considers behavioral responses in aggregate, and the other implies marginal behavioral adjustments.

We derive simple measures of standard and marginal risk aversion and apply these measures to data collected from an economic experiment conducted among Indian farmers in the state of Tamil Nadu. In this experiment farmers stated their willingness-to-pay (WTP) for several different payoff distributions. Our derived measures show a substantial discrepancy between standard and marginal risk attitudes, with many farmers simultaneously appearing to be risk averse on average and risk loving on the margin. The evidence points to an anchoring-and-adjustment process in which individuals anchor on their willingness to pay for a specific gamble, then make adjustments to this value when similar gambles are presented. Rather than using a reference point (e.g., Kahneman and Tversky 1979), (1) individuals appear to use a reference gamble. Importantly, it is impossible to reconcile this behavior with a probability weighted model, since the estimates of a value function would have to account for both local convexity and global concavity.

Literature Review

To the best of our knowledge, no one has yet distinguished between standard and marginal risk aversion or suggested that both of the aversions may be the result of separate cognitive processes. There are, however, a couple of good reasons in the existing risk literature for why this distinction may matter to applied economists. First, expected utility theory attributes risk aversion entirely to the curvature of the utility function. Empirical applications in agriculture have generally sought to determine the level of curvature that best describes the input decisions of farmers (see Just and Just forthcoming). The outcomes of such estimation rely heavily on the assumption of expected utility maximization. Anomalies such as the certainty effect--the discounting of uncertain outcomes more than the probability would imply--could therefore cause severe bias in risk aversion estimates, because the psychological effect can be interpreted as severe risk aversion. Alternatively, the certainty effect should be absent when comparing two nontrivial gambles, potentially leading to very different estimates of risk aversion. This is fundamentally a failure of the independence axiom, which supposes that preferences should be independent of the addition of identical lottery components.

A second reason why the standard versus marginal risk aversion distinction may matter is highlighted by Rabin (2000) and Just and Peterson (2003), who show that an absurd degree of utility function curvature is required in an expected utility framework to rationalize many individuals' responses to relatively small gambles. Responses to small changes in gambles may not elicit the same radical responses. Thus, distinguishing between risk aversion and marginal risk aversion may provide insight into this apparent anomaly.

In the balance of this review section, we highlight in greater detail four strands in the risk literature that are relevant to this distinction. The first relates to Arrow's (1971) hypothesis that absolute risk aversion decreases as wealth increases--identical to shifting the mean of a distribution holding all else constant. Second, we discuss the certainty effect that implies a distinction between standard and marginal risk aversion with the point of departure being certainty. Third, the probability distortion literature highlights a potential practical benefit of considering risk aversion on the margin. Since the certainty effect can be understood as a result of a systematic misperception of probabilities, these second and third strands are closely related. Finally, we discuss a pair of papers that hypothesize a delineation between diminishing marginal utility of wealth and risk preferences.

Friedman and Savage's (1948) explanation of risk taking behavior supposed that the utility of wealth function changes its degree of concavity (or convexity) dramatically as one moves from a situation with a low level of wealth to a situation in which she/he has large amounts of wealth. In particular, they suppose that utility of wealth functions must display risk-loving behavior (convexity) for intermediate amounts of wealth and risk-averse behavior (concavity) for large or small amounts of wealth. A utility function of this shape would explain why poor individuals may buy insurance and lottery tickets simultaneously, while wealthier individuals would reject lottery tickets. Later, Arrow (1971) addressed the relationship between wealth and risk behavior, taking the mathematical theory to its limits. Arrow hypothesized that utility of wealth functions would demonstrate decreasing absolute risk aversion (DARA) and increasing relative risk aversion as wealth increased. The intuition behind these relationships is simple. As their wealth increases, individuals should be more willing to take any particular risk and less willing to risk any percentage of their wealth.

These two hypotheses, while simple, have fueled many applied studies with mixed results, although they generally support DARA. These studies employ econometric estimation to examine both production decisions facing profit risk (e.g., Bar-Shira, Just, and Zilberman 1997; Chavas and Holt 1990, 1996; Lins, Gabriel, and Sonka 1981; Pope and Just 1991; Saha, Shumway, and Talpaz 1994) and more conventional individual decisions (e.g., Bellante and Saba 1986; Cohn et al. 1975; Landskroner 1977; Morin and Fernandez-Suarez 1983; Siegel and Hoban 1982). Thus, according to the extant theoretical and empirical literature, one would expect individuals to be risk loving on the margin as the expected value of a gamble increases. DARA relates to the third derivative of utility of wealth function and occurs if and only if - u"'(w)/u"(w) > - u"(w)/u'(w). The right-hand side of this inequality represents absolute risk aversion--a measure of one's willingness to take on risk, while the left-hand side represents absolute prudence (Kimball 1990)--a measure of how sensitive choice is to risk. (2) Thus, absolute prudence measures how absolute risk aversion changes as wealth level changes and may rightly be thought of as a marginal risk-aversion concept.

Next, the certainty effect may shed light on how risk aversion changes as variance of a prospect changes. The certainty effect commonly occurs when individuals must choose between some certain outcome and at least one risky choice. Individuals behave as if the probability assigned to the sample space for the risky choice sums to less than one. (3) In this case, the individual penalizes all risky choices in a way that is inconsistent with the independence axiom. In particular, the certainty effect can be found when a choice between lotteries, one being a certain outcome and the other a lottery with greater expected value, is compared to the same lotteries compounded with another lottery that receives a majority of the probability. When choosing between the original lotteries, a majority of individuals will choose the certain outcome. When presented with the compound lottery, the differences in probabilities for the best outcome become disproportionately small in the minds of individuals choosing between the lotteries, leading them to choose the riskier outcome--a violation of the independence axiom.

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