Risk, ambiguity, and state-preference theory
The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling "smooth" ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state dependence of utility and unobservable background risk. The decision maker's local revealed beliefs are encoded in her risk-neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk-neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow-Pratt measure for approximating local risk premia, and its structure reveals whether the decision maker's risk preferences are ambiguity averse as well as risk-averse. Two versions of the smooth ambiguity model are analyzed-the source-dependent risk aversion model and the second-order uncertainty (KMM) model-and it is shown that in both cases, the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium. © 2011 Springer-Verlag.
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