Proof: MLR Deterioration in Background Risk Increases Risk Aversion for DARA Utility - Pro, Assignments of Economics

Download Proof: MLR Deterioration in Background Risk Increases Risk Aversion for DARA Utility - Pro and more Assignments Economics in PDF only on Docsity! Answer to Gollier Problem 58 E. Kelly Chung (Athey 1999) Because the condition Ross-DARA that we obtained in the previous exercise is very restrictive, it may be interesting to relax it by constraining the set of FSD deterioration of the background risk. Let f(x, t) be the density function of background risk, x̃t. The marginal indirect utility function can therefore be written as v′t(z) =∫ u′(z+x)f(x, t)dx. Using the properties of log supermodular functions, prove that a MLR deterioration in the distribution of the background risk raises the degree of risk aversion of vt if u is DARA in the sense of Arrow-Pratt. Proof: Suppose u is DARA. Then, − u′′(z, x) u′(z, x) is nonincreasing in x ⇔ u′′(z, x) u′(z, x) is nondecreasing in x ⇔ u′(z, x) is LSPM. (by Condition 2 in Lemma 2)1 Next, we assume that x̃1 is dominated by x̃2 in the sense of the MLR order. Then, l(t) = f(1, t) f(2, t) is nonincreasing in t. ⇔ ∀x1, x2 ∈ R,∀tH > tL : (x2 − x1) f(2, tH) f(1, tH) ≥ (x2 − x1) f(2, tL) f(1, tL) ⇔ f(x, t) is LSPM. (by Condition 1 in Lemma 2) 1Lemma 2 Suppose that h : R2 → R+ is differentiable with respect to its first argument. Then h is LSPM if and only if one of the following two equivalent conditions holds: 1. ∀x, x0 ∈ R, ∀θH > θL : (x − x0)[h(x, θH)/h(x0, θH)] ≥ (x − x0)[h(x, θL)/h(x0, θL)]. 2. [∂h(x, θ)/∂x]/h(x, θ) is nondecreasing in θ. 1