Eco 908

14.04 - Problem Set 1 Due Sept 22nd in recitation 1) Start with an arbitrary utility function u(x]..*2) that is differentiable. Let v(#) be a monotonic transformation of u. a) Solve: max w (x1,-*2) ST : PIXI + PzX2 = m b) Solve: max v( u (x1,x2)) ST : PIXI + PzX2 = m c) Discuss the relationship between these problems. What characteristics of the utility function is generating this result? 2) Consider the following problem: maxx"y Subject to: x + py = 10,x 2 0,y 20 a) Show formally that the utility function a"y is at least weakly monotonic and strongly convex for a > 0. You may use ideas from problem 1 to simplify the problem. b) Find V(a.p),x(a.p). y(a.p) 3) Solve the following: max Inx + y ST : 2x+ y = 10,x 2 0,y 20 4) One way to rule out the potential that the non negativity constraints aren't binding is to look at the marginal rate of substitution (MRS) when one of the factors gets arbitrarily close to zero. Suppose that we have a function f(x1,12). The MRS12(x1,x2) is the amount of x1 required to keep the function f the same when x, changes by a small amount. MRS12(x1.X2) is read "the marginal rate of substitution of good 1 for good 2 at (ri,x2)" Formally: MRS 12 (X1.X2) = dx1 a) Consider the function f(x, y) = xy. Starting from a point where x,y > 0. what happens to the MRS. as y grows smaller and approaches zero (ie lim, .MRS,,(x.y)) ? What happens to lim-MRS.? b) Consider the function x + y. What is lim, MRS.,? What is lim,-MRSy.? c) Consider Inx + y. What is limy DMRS.,? What is lime-oMRS,x?i. Suppose the hospitals objective function is to maximize the util- ity of its micro fibralgia patients. Argue that the only possible optimal wait time they could choose to impose is w=0,4. (5 points) Hi. Calculate the Expected Utility of the micro fibralgia patients and drug addicts at each of the wait times as a function of N and K. (5 points) Hi. Suppose that K = 10, N=60 - should the hospital screen its patients? Will there be a shortage of medication? (5 points) 4. "Consider a world in which state contingent contracts can be written. There are two people (Chris and Tatiana) and two goods - umbrellas and swim suits. When it is raining, both people like having as many um- brellas as possible. If it is sunny, both people like having as many swim suits as possible. Suppose each person has an identical utility function U(u, spraining) = u, U(u, ssunny) = s where u and s are the number of umbrellas and swimming suits the agent possesses in the given state (they get more utility from having more umbrellas). Assume that at 1=0 each agent is endowed with 1 umbrella and 1 swim- suit for the next period. The two agents disagree on the probability of whether it will rain in the future. Tatiana is an optimist and believes it is going to be sunny with probability .75 and rainy with probability .25. Chris is a pessamist and believes it is going to be sunny with probability 5 and rain with probability .5. Both people are vNM utility maximizers, thus: UTatiana (14, 8) = .750(u, s[sunny) + .250 (u, straining), Tatiana'sendowment : u = 1,s = 1 Uchris (1, s) = .5U(u, s(sunny) + .50(u, straining), Chris'sendowment : u = 1.s = 1 (a) Draw the edgeworth box for this problem. What do indifference curves for the two individuals look like? (b) Before doing any math - why is it that we expect our final allocation to be on the boundary? (c) Why in this problem do we not have a unique price on which all markets clear? (d) Suppose that I am a social planner trying to efficiently allocate um- brellas and bathing suits to my housemates. I solve: max AUTatiana (Ut: ;)) + (1- A)(UChris(Uc, Se)) at le + 1 = 2, se + 1 = 2 Show that all the pareto optimal outcomes are on the boundary of the edgeworth box. 5. Suppose that we are in an economy with two individuals i = 1,2 each with identical utility function u'(11, 12) = (2172) . (a) Suppose that agent 1 is endowed with If = 1, 17 = 0 and agent 2 is endowed with If = 0, ry = 2. At what price ratio P will prices clear in this market? Determine the allocation in this situation. (b) Prove that in this economy, the price found in part (a) will clear the market regardless of the initial allocation as long as the total endowment stays the same. What will the contract curve look like? (c) Show that the contract curve and the price function are orthoginal in this problem.2. "(From McAfee 1996) In May 1990, IBM announced the introduction of the LaserPrinter E, a lower cost alternative to its popular LaserPrinter. The LaserPrinter E was identical to the original Laser Printer except that it printed at 5 ppm instead of 10 ppm. According to Jones(1990), the engine and parts of the printer were virtually identical to the faster printer except that the controller for the slower printer had firmwear that inserted wait states to slow the print speed of the printer. This problem is designed to see how such damaging of a product may make everyone better off. (a) Assume first that you have a single high quality product that you are selling. The product has constant marginal cost to produce of $1. There are two types of consumers who are willing to buy a single unit of your good. 25% of the population is type 1 and are willing to buy your product if the cost of producing it is less than or equal to $11. 75% of the population is type 2 and are willing to buy your product 2 if it is priced at $3.00. Suppose that you must sell your product at a constant price (you can not screen your consumers) - show that you will only sell to the high types. (b) Now assume that you can produce a broken version of your product. Let s be a measure of how broken your product is. The high types get utility Up(x,,s) for consuming one unit of type s good. Thus Un(1, 0) is the value to the high type of buying one unit of the original quality good and Up (1, s) would be the utility of the high type for buying a good with quality s. Similarly, the low type get Uz(1,0) for the original good and Up (1, s) for consuming a low good of value s. Assume that a 0 (it is costly to produce an inferior good). i. Again assume that there is no screening other than offering a low quality good. Set up the monopolists problem. ii. Suppose that Un (1, a) = 11 -26, UL(1, a) =3-.250, and c(s) = 1. Find the optimal quality of the lower good and the amount charged for the two goods. ili. Suppose by law the monopolist can only reduce quality down to s = 3. Show that in this case, everyone is at least as well off due to the creation of the damaged good. 3. Suppose that a hospital carries K doses of an opium based medicine used to help patients suffering from micro fibralgia. N 2 K patients come seeking the drug at the same time. 6 of these actually suffer from the disease while N-6 of them are drug addicts looking for a fix. (a) Suppose that the hospital does not care about the utility of the drug addicts. The utility of a micro fiberalgia agent receiving the drug is 10. The utility of a micro fibralgia agent who does not receive the drug is zero. Calculate the total expected utility the hospital can provide to its micro fibralgia patients if it has no way to screen between drug addicts and true patients. (b) Suppose that the Hospital can force agents to waste time before being treated. The utility of a micro fibralgia user who must wait and receives the medication with probability p is: UMicro(w, p) = 10p - 2w Drug addicts have a utility function of UDA(w,P) =4-w The addicts are not rational - as long as they stay in the hospital they believe that the probability of receiving treatment is 1. Both agents outside option is zero, that is UMiers(0,0) = UpA (0,0) =0