Lecture 1 Expected utility and risk aversion.pdf

Lecture 1: Expected Utility and Risk Aversion 1 Expected utility Let X be the set of outcomes, where each outcome x 2 X ? R can be understood as an amount of money or of other consumption good. Assume X to be an interval on the real line. Let u : X ! R be a utility function on X, such that for each outcome x 2 X, u(x) represents the utility of consuming x. As a utility function, u(x) is expected to satisfy certain reasonable properties. For example, it should be strictly increasing in x, so that more money is always better, and it should be continuous, so that a small change in x only leads to small change in utility. A lottery is a probability distribution on the outcome space X. For simplicity, we consider only lotteries that assign probabilities on finitely many outcomes. A (finite) lottery, or a gamble, describes which outcomes (prizes) could occur, and each with what probability. Formally, a lottery is a vector of (nonnegative) probabilities L = (p1, p2, . . . , pN) such that p1+ p2+ · · ·+ pN = 1, and for all n, pn is the probability that some outcome xn 2 X occurs. A simple example of a lottery is the gamble (1 2 , 1 2 ) on outcomes 1 and ?1: if you face this lottery, then you receive 1 and ?1 each with half probability. People have preferences over lotteries, just as they have preferences over out- comes. The expected utility theory o§ers a simple treatment of preferences over lotteries. Definition 1.1. Given her utility function u(x) on outcomes, an agent’s (von Neumann- Morgenstein, vNM) expected utility at a lottery L = (p1, . . . , pN) is defined as U(L) = NP n=1 pnu(xn) That is, for any lotteries L and L0, an agent weakly prefers L to L0 i§ U(L) ≥ U(L0). There are intuitive axioms (independence and continuity) that ensure the existence u(x) and such a representation of preferences on lotteries; we don’t go into that. In probability theory, the monetary return x = x (!) is a random variable which is defined from the state space ? to X. Put di§erently, th