Uncertainty and Risk - DARP

Microeconomics CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.12 A risk-averse person has wealth y 0 and faces a risk of loss
L < y 0 with probability . An insurance company o¤ers cover of the loss at
a premium > L. It is possible to take out partial cover on a pro-rata basis,
so that an amount tL of the loss can be covered at cost t where 0 < t < 1.
1. Explain why the person will not choose full insurance
2. Find the conditions that will determine t , the optimal value of t.
3. Show how t will change as y 0 increases if all other parameters remain
unchanged.
Outline Answer:
1. Consider the person’s wealth after taking out (partial) insurance cover
using the two-state model (no loss;loss). If the person remained uninsured
it would be (y 0 ; y 0 L); if he insures fully it is (y 0 ; y 0 ). So
if he insures a proportion t for the pro-rata premium wealth in the two
states will be
which becomes
([1 t] y 0 + t [y 0 ] ; [1 t] [y 0 L] + t [y 0 ])
So expected utility is given by
Therefore
@Eu
@t
(y 0 t; y 0 t [1 t] L)
Eu = [1 ] u (y 0 t) + u (y 0 t [1 t] L)
= [1 ] u y (y 0 t) + [L ] u y (y 0 t [1 t] L)
Consider what happens in the neighbourhood of t = 1 (full insurance).
We get
@Eu
@t = [1 ] u y (y 0 ) + [L ] u y (y 0 )
t=1
= [L ] u y (y 0 )
We know that u y (y 0 ) > 0 (positive marginal utility of wealth) and, by
assumption, L < . Therefore this expression is strictly negative which
means that in the neighbourhood of full insurance (t = 1) the individual
could increase expected utility by cutting down on the insurance cover.
2. For an interior maximum we have
@Eu
@t
which means that the optimal t is given as the solution to the equation
= 0
[1 ] u y (y 0 t ) + [L ] u y (y 0 t [1 t ] L) = 0
cFrank Cowell 2006 130