Consider CRRA weighted utility.

(a) Show thatg in (25.13) is strictly monotone in y > 0—so the preferences are monotone with regard to stochastic dominance—if and only if

γ ≤ 0 and ρ ≤ γ + 1 with at least one of these being a strict inequality.

(b) Show thatg in (25.13) is strictly monotone and concave if and only if

γ ≤ 0 and γ ≤ ρ ≤ γ + 1 with either γ < 0 or ρ<γ +1.

(c) Consider a lognormal gamble: w˜ = w(1+ ˜ε) where log(1+ ˜ε) is normally distributed with variance σ2 and mean −σ2/2 (implying E[˜ε] = 0). Show that the certainty equivalent is w(1− π ), where

π = 1− e−(ρ−2γ )σ 2/2 .

Note: This implies that π ≈ (ρ − 2γ )σ2/2 for small σ. Compare Exercise 1.5.