Question

N. If Parsifal does not have an accident, then he will earn $20,000, if he does, he will be reduced to earning only $4,000. The probability of having an accident is 0.02. His expected utility function is ur) = ve where I represents his earnings. Ch. 7 will cover the expected utility approach to choice under uncertainty, this is a first instance. 1. Give the cdf of Parsifal's random wealth. 2. Give the expected value and the variance of Parsifal's random wealth. 3. Give Parsifal's expected utility, that is, give Eu(X). 4. What income, W, if delivered to Parsifal for sure, would give him the same expected utility. 5. In expected terms, how much could an insurance company make off of Parsifal by offering him income replacement insurance?

Answer 1

1. Let X denote his wealth.

The cdf is given as follows:

P(X<4000) = 0, P(4000<X<20000) = 0.02, P(X>20000) =1

2. E(X) = 4000*0.02 + 20000*0.98 = 19680

V(X) =E($X^2$) -

EP(x

E($X^2$) = (20000)^2 *0.98 +(4000)^2 *0.02 = 10^4 * 39232

V(X) = 50176000

3. Eu(X) = 0.98 *
(20000)
+ 0.02 *
(4000)
= 98$\sqrt(2)$ + 0.4
(10)

4. W = Eu(X) ^2 = (98$\sqrt(2)$ + 0.4) ^ 2 =

(10)