Consider weighted utility. Let have zero mean and unit variance. For a constant , denote the certainty equivalent of w + by w

Consider weighted utility. Let ε˜ have zero mean and unit variance. For a constant σ, denote the certainty equivalent of w + σε˜ by w − π(σ ). Assume

π(·) is twice continuously differentiable. By differentiating v(w −π(σ ))E[λ(w+ σε)˜ ] = E[λ(w+σε)˜ v(w+ σε)˜ ], assuming differentiation and expectation can be interchanged, show successively that π

(0) = 0 and

π(0) = −v(w)

v

(w) − 2λ

(w)

λ(w) .

Note: This implies that for CRRA weighted utility and small σ, π(σ )/w ≈ (ρ −

2γ )var(σε/˜ w)/2.