Uncertainty and Risk - DARP

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