Question

Nina has a convex utility of wealth function, u(x). She is contemplating two prospects, L and M, where L is a mean preserving spread of M. Which of the following statements is therefore true?

1. Nina will prefer M to L.

2. Nina will prefer L to M.

3. Nina’s ranking of L and M cannot be determined without more information.

4. Nina must be risk averse, but that is the most one can say.

Answer 1

Nina will prefer L to M .

This is because a mean preserving spread change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function or probability mass function while leaving the mean (the expected value) unchanged.

Ranking gambles by mean-preserving spreads is a special case of ranking gambles by second-order stochastic dominance – namely, the special case of equal means: If B is a mean-preserving spread of A, then A is second-order stochastically dominant over B; and the converse holds if A and B have equal means. Hence, in our case L is preferable to M.

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