Introduction

In Chapter 9 we defined an agent as risk averse if he or she prefers the expectation of a consumption plan to the consumption plan itself. The consumption plan is obviously riskier than its expectation, and a risk-averse agent prefers the latter.

A natural extension of this discussion is to consider a risk-averse agent who compares two consumption plans, neither of which is deterministic. In general, without more information about an agent's preferences, two risky consumption plans cannot be ranked: some risk-averse agents prefer one and some the other. However, in the spirit of the discussion of Chapter 9, it is appropriate to ask whether there is some condition on the distribution of two consumption plans such that if the two consumption plans have the same expectation, then all risk-averse agents do prefer one to the other. In Section 10.2 an ordering on consumption plans is defined which, as will be seen in Section 10.5, has the desired property.

In this chapter, we assume that date-0 consumption does not enter the utility functions.

Greater Risk

Let y and z be two (date-1) consumption plans. As in Chapter 9, these consumption plans can be viewed narrowly as random variables on the set of states S with given probabilities or broadly as arbitrary random variables (with finite expectations).