Submission to the Journal of Answers
Benjamin J. Keys
Submitted January 27, 2005
Revision submitted February 9, 2005
Exercise 46
Use Proposition 16 to prove that an increase in wealth increases the demand for the risky
asset if and only if the investor is DARA.
Solution 46
We want to determine the conditions under which the following holds:
∀˜x, E ˜xu0(wL+ ˜x) = 0 ⇒ ∀wH> wL:E˜xu0(wH+ ˜x)≥0
This is a straightforward restatement of Gollier equation (7.1),
∀˜x∈[a, b], Eg (˜x)h(˜x, wL) = 0 ⇒ ∀wH> wL:Eg(˜x)h(˜x, wH)≥0
replacing g(˜x) = ˜xand h(˜x, w ) = u0(w+ ˜x).
Proposition 16 states that for any real valued function gthat satisfies the single-crossing
condition: ∃x0:∀x: (x−x0)g(x)≥0, the above equation holds if and only if his log-
supermodular (LSPM).
By Proposition 16, the above equation holds if and only if h(x, w) = u0(w+x) is LSPM.
Furthermore, from Lemma 2, h(x, w) is LSPM if and only if [∂h(x, w)/∂x]/h(x, w) is non-
decreasing in w. That is,
∂
∂w (∂h(x, w)/∂x
h(x, w))≥0
Replacing h(x, w) with u0(w+x) yields
∂
∂w (u00 (w+x)
u0(w+x))≥0
∂
∂w (−u00 (w+x)
u0(w+x))≤0
∂
∂w A(w)≤0
Which is the definition of Decreasing Absolute Risk Aversion (DARA).
We can see that the reverse is true as well because all of our results are ”if and only if”
results. u0(w+x) exhibits DARA if and only if u0(w+x) is LSPM. u0(w+x) is LSPM if
and only if the above statement of Gollier equation (7.1) is true, which is the condition we
intended to satisfy. This completes both directions of the proof.
