The Comparative Statics of Changes in Risk Revisited

doi:10.1006/jeth.1995.1052

Journal of Economic Theory

Volume 66, Issue 2, August 1995, Pages 522-535

https://doi.org/10.1006/jeth.1995.1052 Get rights and content

Abstract

In this paper, we consider the problem of determining the conditions under which a change in risk increases the optimal value of a decision variable for all risk-averse agents. For a large class of payoff functions, we obtain the least constraining (necessary and sufficient) condition on the change in risk for signing its effect without any additional restriction on the utility function than risk aversion. It entails all existing sufficient conditions as particular cases. Our results are applied to the linear model which describes the standard portfolio problem. The necessary and sufficient condition for unambiguous comparative statics in this class of problems is termed "greater central riskiness" (CR). It is shown that CR dominance is neither stronger nor weaker than second-degree stochastic dominance. Journal of Economic Literature Classification Number: D81.

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Cited by (82)

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It must be emphasized that, while CD implies a deterministic comparative static of a change in decision when the risk (distribution) changes, SD does not have a similar implication. It has also been shown that second-order SD is neither sufficient nor necessary for CD (Gollier, 1995). Despite the practical relevance of CD, testing CD has not been considered in the literature, to the best of our knowledge.
Central dominance (CD) introduced in Gollier (1995, Journal of Economic Theory) is a risk concept that differs from stochastic dominance (SD) in an important way. In particular, CD implies a deterministic comparative static of a change in decision when risk changes, but SD does not have such an implication. In this paper, we propose the first test of central dominance, which amounts to checking a functional inequality. We derive the asymptotic distribution of a lower bound of the proposed test and suggest a bootstrap procedure to compute the critical values. We also conduct simulations to evaluate the performance of this test. Our empirical study finds clear evidence of CD relations between the S&P 500 index return distributions during 2001–2013 and results in unambiguous implications for investment decisions.
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Under which condition does the set of desirable uncertain prospects expand when wealth increases? We show that the decreasing concavity (DC) of the utility function u is necessary and sufficient in the α-maxmin expected utility model. In the smooth ambiguity aversion model with the ambiguity valuation function ϕ, the DC of u and of $ϕ \circ u$ is necessary and sufficient. An alternative classical definition of decreasing aversion is based on the hypothesis that the investment in a risky asset is increasing in wealth. We show that this hypothesis does not hold in general under ambiguity aversion, and that one needs to constrain the structure of ambiguity to obtain unambiguous results.
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Gollier (2011) proves that an increase in ambiguity aversion does not necessarily imply a reduction in the demand for risky assets. Gollier (1995) characterized the necessary and sufficient condition on the change in the risky asset to guarantee that all risk averse expected utility agents will increase their demand of the risky asset. This condition is called Central Dominance (thereafter CD).
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The Comparative Statics of Changes in Risk Revisited