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

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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Pre 2010

Uploaded on 09/17/2009

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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, x2R,tH> tL: (x2x1)f(2, tH)
f(1, tH)(x2x1)f(2, tL)
f(1, tL)
f(x, t) is LSPM.
(by Condition 1 in Lemma 2)
1Lemma 2 Suppose that h:R2R+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, x0R, θH> θL: (xx0)[h(x, θH)/h(x0, θH)] (xx0)[h(x, θL)/h(x0, θL)].
2.[∂h(x, θ)/∂ x]/h(x, θ) is nondecreasing in θ.
1
pf2

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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, 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:

  1. ∀x, x 0 ∈ R, ∀θH > θL : (x − x 0 )[h(x, θH )/h(x 0 , θH )] ≥ (x − x 0 )[h(x, θL)/h(x 0 , θL)].
  2. [∂h(x, θ)/∂x]/h(x, θ) is nondecreasing in θ.

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.