Question

to the company oficer of an 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 r dollars in case of an accident, he must pay rr 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. In particular, she sets a price r and the consumer pays re dollars for x dollars coverage, as before. What price would the monopolist charge this consumer? (c) Now suppose the monopolist offers an insurance policy (P, 2) to the consumer with r dollars of coverage for a total price of P. What insurance policy will the monopolist choose to maximize expected profits?

Answer 1

Answer :

The following information is given

vN - M utility function : U(x) = log (x)

W = $500,000

Probability of loss = 0.2

Loss (d) = $200,000

Price of insurance = $r

(a). Suppose the price of  insurance, r = 0.25. The probability of 0.8 that no loss occurs and probability 0.2 that loss of $200,000 occur and consumer left with $ 300,000 in case of loss

The consumer's expected utility is,

E (U) = 0.8U ($500,000) + 0.2U ($300,000)

= 0.8 log ($500,000) + 0.21 log($300,000)

  = 0.8(13.1224) + 0.2(12.61154)

= 10.4979 + 2.5223

= 13.0202

Thus, the expected utility of consumer is 13.0202.

Now, suppose that y be the insurance consumer willing to buy. Compute the amount paid by consumer,

E(U) = U(500,000 - y)

= log(500,000 - y)

13.0202 = log(500,000 - y)

500,000 - y = antilog (13.0202)

y = 500,000 - 451,441.0136

= $48,558.9864

Hence, the consumer is willing to buy $48,558.9864 of insurance

(b). Suppose the insurance company acts like a monopolist, that is aims at maximizing profits. In such case, monopolist charge rx from consumer. When consumer's loss is $200,000, the amount charge by monopolist is ,

Amount charge by monopolist = rx

= 0.25 * 200,000

= $50,000

Hence, the amount charged by monopolist is $50,000