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A proof that a markovian lognormal (mlr) deterioration in the distribution of background risk raises the degree of risk aversion of the marginal indirect utility function vt, under the assumption that the utility function u is dara in the sense of arrow-pratt. The proof is based on lemma 2 and the properties of log supermodular functions.
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(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, x˜t. 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 ˜x 1 is dominated by ˜x 2 in the sense of the MLR order. Then,
l(t) =
f (1, t) f (2, t)
is nonincreasing in t.
⇔ ∀x 1 , x 2 ∈ R, ∀tH > tL : (x 2 − x 1 )
f (2, tH ) f (1, tH )
≥ (x 2 − x 1 )
f (2, tL) f (1, tL) ⇔ f (x, t) is LSPM.
(by Condition 1 in Lemma 2) (^1) Lemma 2 Suppose that h : R (^2) → 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:
Since the product of two LSPM functions is LSPM, u′(z+x)f (x, t), the integrand of v t′(z), is also LSPM. By Proposition 20^2 , v′ t(z) =
u′(z + x)f (x, t)dx =
Eu′(z + x) is LSPM, i.e., v
′′(z,t) v′(z,t) is nondecreasing in^ t^. So, we now know that when l(t) is nonincreasing in t, v
′′ v′^ is nondecreasing in^ t.
Therefore, a MLR deterioration in the distribution of the background risk raises the degree of risk aversion of vt. ¥
(^2) Proposition 20 H(z) = Eh(z, ˜θ) is LSPM if h is LSPM.