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, ˜xt. The
marginal indirect utility function can therefore be written as v0
t(z) =
Ru0(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 vtif uis DARA in the sense
of Arrow-Pratt.
Proof: Suppose uis DARA. Then,
−u00(z, x)
u0(z, x)is nonincreasing in x
⇔u00(z, x)
u0(z, x)is nondecreasing in x
⇔u0(z, x) is LSPM.
(by Condition 2 in Lemma 2)1
Next, we assume that ˜x1is dominated by ˜x2in 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 his 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.
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