Econ 440 Homework 4: Utility Functions, Risk, and Insurance - Prof. Peter Norman, Assignments of Economics

Download Econ 440 Homework 4: Utility Functions, Risk, and Insurance - Prof. Peter Norman and more Assignments Economics in PDF only on Docsity! Econ 440 November 25 2007 Peter Norman Homework 4 1. Consider a consumer with preferences over an uncertain income stream given by U (x1; x2) = u (x1) + (1 )u (x2) ; where  is the probability of state 1. For each of the following functional forms for u (x) ; sketch the indi¤erence curves and classify the preferences as risk averse, risk neutral or risk loving. (a) u (x) = p x (b) u (x) = x2 (you need only to consider (x1; x2)with x1  0 and x2  0 as negative consumptions don’t make much sense) (c) u (x) = ax+ b where a > 0: (d) u (x) = lnx 2. Lisa has decided to invest all her wealth in stocks. She is only considering 2 stocks suggested by her …nancial advisor: Mron and Nron. Stocks in both Mron and Nron cost 1 dollar each and Lisa has 40 dollars to invest. Each stock in Mron will be worth 10 cents with probability 12 and 10 dollars with probability 1 2 : Each stock in Nron will be worth 10 cents with probability 12 and 10 dollars with probability 1 2 : Moreover, the value of the Mron stock and the value of the Nron stock are independent outcomes, so to sum up we have that: probability dollar value of 1 Mron stock dollar value of 1 Nron stock 1 4 0.1 0.1 1 4 10 0.1 1 4 0.1 10 1 4 10 10 Label consumption when both stocks are worth 10 cents cLL: and consumption when both stocks are worth 10 dollars cHH : Next, let the consumption when Mron is worth 10 dollars and Nron is worth 10 cents be cHL, and the consumption when Mron is worth 10 cents and Nron is worth 10 dollars be cLH : Finally, assume that Lisas’expected utility function is U (cHH ; cHL; cLH ; cLL) = 1 4 p cHH + 1 4 p cHL + 1 4 p cLH + 1 4 p cLL a. Suppose Lisa puts all her wealth in Mron. Calculate cHH ; cHL; cLH ; cLL and Lisas expected utility. b. Suppose Lisa puts all her wealth in Nron. Calculate cHH ; cHL; cLH ; cLL and Lisas expected utility. c. Suppose Lisa puts 20 dollars each in Mron and Nron respectively. Calculate cHH ; cHL; cLH ; cLL and Lisas expected utility. Demonstrate that it cannot be optimal for Lisa to invest all her wealth in a single asset. This can be done by constructing an alternative portfolio that Lisa think is better. Explain as well as you can what is going on. 1 3. Gunnar is a farmer who owns 5 pigs. His neighbor Leif is willing to buy them all for 1000 Kronas each. If he doesn’t sell, the pigs will multiply and he will have 50 pigs to sell next year. However, pigs are costly to raise, so he’ll have to pay 800 Kronas per pig for food, sty-rental etc. Moreover, the pork price is uncertain, so with probability 12 he will get 800 Kronas per pig when selling in the next period, whereas with probability 12 he can sell them for 1000 Kronas each also in the next period. (a) Suppose that Gunnars’ex post utility function is u (c) = p c: Will Gunnar accept Leifs’o¤er? (b) Find some other function u (c) that gives a di¤erent answer in terms of the acceptance decision. 4. Consider the adverse selection in insurance problem discussed in class. Assume that there are two types, J = L;H; with preferences given by Ju (xB) + (1 J)u (xG) where L < H is interpreted as the probability of a loss, xB is the consumption in case of a loss, and xG is the consumption in case of no loss. The consumption in the absence of insurance is mB is the loss occurs and mG when there is no loss (for both type consumers). Assume that the consumer is risk averse unless anything else is stated. (a) In a carefully labeled graph, draw an indi¤erence curve for type L and an indi¤erence curve for type H that intersects some arbitrary point x = (xB ; xG) : (b) De…ne an “isopro…t”as a set of contracts where the expected pro…t is constant, that is (xB ; xG) such that JxB + (1 J)xG = k In one or two carefully labeled graphs, draw the isopro…t for types L and H and explain in words why they are di¤erent. (c) In a carefully labeled graph, depict the optimal contract that a monopolist would sell to type J = L;H if the monopolist would know that the consumer is of type J: Explain (using indi¤erence and isopro…t curves) why this contract is the best for the monopolist. (d) In a carefully labeled graph, depict the equilibrium contract for type J = L;H if there is a competitive insurance industry that can observe type J: What is the di¤erence with the monopoly solution? (e) Using yet another and even more carefully labeled graph, explain (with care) what would happen if the monopolist would try to sell the optimal contract for type L (call it xL) from above to type L and the optimal contract (xH) for type H to type H in a world where the monopolist cannot see who is of which type. (f) In a new and even prettier graph, show why it must be the case that type L will get full insurance in the optimal contract for the monopolist (still assuming that type is unobservable). (g) Depict the optimal contract for the monopolist (still assuming that type is unobservable). In partic- ular, it should be clear how the utility of each type compares to the utility of no insurance. Explain this property intuitively. 5. Consider a world with a single student/worker and two …rms that compete in Bertrand fashion for the worker. The student/worker can be of ability a 2 f1; 2g ; where “ability”is interpreted as the productivity the worker has when hired by one of the …rms. Let Pr [a = 2] =  > 0: Before going on the job market, 2