Multivariate risk aversion with applications

Abstract

Risk aversion has played a fundamental part in applied decision models under uncertainty. The measure which has found wide applications is the Arrow-Pratt (r _A) measure of absolute risk aversion [1,10]:

$$ \mathop r\nolimits_A = - \mathop \partial \nolimits^2 u(z)/\partial u(z)$$

((1))

defined on the space of real-valued utility functionsu(z). Herez may be a scalar i.e. wealth or income, or it may be a vector of goods over which the scalar utility function is defined. If$z = z(\tilde c,x)$ represents the consequences of a lottery generated by the random variables${\tilde c}$ with a probability distribution$\begin{gathered}F(\tilde c\left| {\theta )} \right. \hfill \\{\tilde z} \hfill \\\end{gathered}$ indexed by its parametersθ (e.g. mean, variance), then the decision-maker (DM) has a problem of optimal decision-making under the uncertain environment. Two types of uses are then usually made. One is the notion of certainty equivalence of a lottery which has random outcomes denoted by${\tilde z}$. For all monotonic utility functions$u(\tilde z)$ defined on the space of${\tilde z}$, a DM is said to be risk averse, if he prefers$u(E(\tilde z))$ over$E(u(\tilde z))$ whereE is the expectation operator over the nondegenerate distribution of the random variable${\tilde z}$ or, of${\tilde c}$ givenθ. If these expectations are finite, then the certainty equivalent (CE) of the lottery is defined by an amount${\hat z}$ such that

$$u(\hat z) = E[u(\tilde z)]$$

((2))

i.e. the DM is in different between thelottery and the amount${\hat z}$ for certain.