In class we have learned that, given an indirect utility function u(w), there are two relevant measures u" (w) of risk aversion, namely absolute risk aversion, A(w) = and relative risk aversion, R(w) u' (w)' A(w)w. The following questions relate to risk aversion in general, and relationships around these two measures. 1. Utility function u(w) is risk averse if it is concave. Show mathematically that the set of all risk averse functions is a convex set. 2. It is often argued that absolute risk aversion should satisfy DARA (i.e. absolute risk aversion is decreasing in w). Show that, (i) a neccesary but not sufficient condition for DARA is that v(w) = -u'(w) is concave, and (ii) that a necessary and sufficient condition for DARA is that v(w) = -u' (w) is more risk averse than u(w). TC 1 = LORDA 1
In class we have learned that, given an indirect utility function u(w), there are two relevant measures u" (w) of risk aversion, namely absolute risk aversion, A(w) = and relative risk aversion, R(w) u' (w)' A(w)w. The following questions relate to risk aversion in general, and relationships around these two measures. 1. Utility function u(w) is risk averse if it is concave. Show mathematically that the set of all risk averse functions is a convex set. 2. It is often argued that absolute risk aversion should satisfy DARA (i.e. absolute risk aversion is decreasing in w). Show that, (i) a neccesary but not sufficient condition for DARA is that v(w) = -u'(w) is concave, and (ii) that a necessary and sufficient condition for DARA is that v(w) = -u' (w) is more risk averse than u(w). TC 1 = LORDA 1
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.7P
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Transcribed Image Text:Question 1
In class we have learned that, given an indirect utility function u(w), there are two relevant measures
of risk aversion, namely absolute risk aversion, A(w) =
A(w)w. The following questions relate to risk aversion in general, and relationships around these two
u" (w)
'(w) '
and relative risk aversion, R(w)
measures.
1. Utility function u(w) is risk averse if it is concave. Show mathematically that the set of all risk
averse functions is a convex set.
2. It is often argued that absolute risk aversion should satisfy DARA (i.e. absolute risk aversion
is decreasing in w). Show that, (i) a neccesary but not sufficient condition for DARA is that
v(w) = -u'(w) is concave, and (ii) that a necessary and sufficient condition for DARA is that
v(w) = -u'(w) is more risk averse than u(w).
3. If u(w) satisfies positive and constant relative risk aversion (CRRA, that is, relative risk aversion
is constant in w), show that u(w) also must be DARA.
4. Is the set of all DARA utility functions a convex set?
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