Uncertainty and Risk - DARP
Uncertainty and Risk - DARP
Uncertainty and Risk - DARP
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Microeconomics<br />
Exercise 8.8 An individual faces a prospect with a monetary payo¤ represented<br />
by a r<strong>and</strong>om variable x that is distributed over the bounded interval of the real<br />
line [a; a]. He has a utility function Eu(x) where<br />
u(x) = a 0 + a 1 x<br />
<strong>and</strong> a 0 ; a 1 ; a 2 are all positive numbers.<br />
1<br />
2 a 2x 2<br />
1. Show that the individual’s utility function can also be written as '(Ex; var(x)).<br />
Sketch the indi¤erence curves in a diagram with Ex <strong>and</strong> var(x) on the<br />
axes, <strong>and</strong> discuss the e¤ect on the indi¤erence map altering (i) the parameter<br />
a 1 , (ii) the parameter a 2 .<br />
2. For the model to make sense, what value must a have? [Hint: examine<br />
the …rst derivative of u.]<br />
3. Show that both absolute <strong>and</strong> relative risk aversion increase with x.<br />
Outline Answer:<br />
1. Clearly<br />
Eu(x) = a 0 + a 1 E(x) + a 2<br />
1<br />
2 [(E(x))2 var(x)]:<br />
Marginal utility is a 1 a 2 x:<br />
2. For this to be non-negative we must have E(x) a 1 =a 2 hence the indi¤erence<br />
curves are depicted with E(x) as good, var(x) as bad <strong>and</strong><br />
MRS = 2 [a 1 =a 2 E(x)] :<br />
3.<br />
u x (x) = a 1 + a 2 x<br />
u xx (x) = a 2<br />
(x) =<br />
(x) =<br />
%(x) =<br />
where <strong>and</strong> 0 x x max :=<br />
u xx (x)<br />
u x (x) = a 2<br />
a 1 + a 2 x<br />
1<br />
x max x<br />
1<br />
x max =x 1<br />
a1<br />
a 2<br />
:<br />
cFrank Cowell 2006 123