Uncertainty and Risk - DARP
Uncertainty and Risk - DARP
Uncertainty and Risk - DARP
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Microeconomics<br />
Exercise 8.3 Consider the following de…nition of risk aversion. Let P :=<br />
f(x ! ; ! ) : ! 2 g be a r<strong>and</strong>om prospect, where x ! is the payo¤ in state !<br />
<strong>and</strong> ! is the (subjective) probability of state ! , <strong>and</strong> let Ex := P !2 !x ! ,<br />
the mean of the prospect, <strong>and</strong> let P := f(x ! + [1 ]Ex; ! ) : ! 2 g be a<br />
“mixture” of the original prospect with the mean. De…ne an individual as risk<br />
averse if he always prefers P to P for 0 < < 1.<br />
1. Illustrate this concept in (x red ; x blue )-space <strong>and</strong> contrast it with the concept<br />
of risk aversion used in the text<br />
2. Show that this de…nition of risk aversion need not imply convex-to-theorigin<br />
indi¤erence curves.<br />
Outline Answer:<br />
1. See Figure 8.1.<br />
x BLUE<br />
P <br />
P λ<br />
P<br />
x RED<br />
Figure 8.1: Nonconvex indi¤erence curve<br />
2. Let P be the prospect <strong>and</strong> P its mean. P can be any point in the line<br />
joining them. The de…nition implies that moving along this line towards<br />
P puts the person on a successively higher indi¤erence curves. In Figure<br />
8.1 it is clear that this condition is consistent with there being indi¤erence<br />
curves that violate the convex-to-the-origin property locally.<br />
cFrank Cowell 2006 117
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