Ennio Stacchetti
December 2, 2022
Microeconomic Analysis
Homework #9
Due Date: Thursday, December 8
(1) A consumer with vN-M utility function U (x) = log(x) and initial wealth W = $500, 000
faces a probability p = 0.2 of incurring a monetary loss of d = $200, 000 in an accident. An
insurance company offers him insurance at a price r for each dollar of coverage. That is, if
he wants to get back x dollars in case of an accident, he must pay rx dollars for insurance
to the company up front.
(a) Assume r = 0.25. How much insurance does he buy?
(b) Assume now that the insurance company is a monopolist that wants to maximize
expected profits. What price would the monopolist charge this consumer?
(2) An investor with a strictly concave and increasing utility function u(w) for final wealth w
wants to invest in a stock. The current price of the stock is p = 1 and the price tomorrow
will be ph > 1 with probability 12 or p` < 1 with probability 21 . His initial wealth is W .
He must decide how many shares s to buy. Assume that the expected price of the stock
tomorrow is
1
1
ph + p` > 1,
2
2
so an investment in the stock is a better than fair bet.
(a) Write the optimization problem of the investor and the corresponding first-order conditions for optimality.
(b) Assume u(w) = log(w), ph = 2 and p` = 1/2. How many shares should the investor
buy?
1