Given below are some questions on microeconomics, do try and provide solutions for these

(30 points) Consider an expected profit maximizing monopolist who faces an uncertain demand. He supplies q units of goods at zero cost and sells it at price e - q, where e is unknown. [The price and the supply level can be negative.] (a) Assuming that # ~ N (y. o'), compute the monopolist's optimal supply q and his expected profit under the optimal supply. (b) Suppose that, through market research, the monopolist can learn about 0. In particular, by investing c', he can learn the value of a random variable Y before choosing his supply q, such that 0 = X + Y, X ~ N (0, 1 - c) and Y ~ N (0, c). How much should the monopolist invest? [Note that the utility function of the monopolist is (0 - q) q - c2.] 3. (40 Minutes - 20 Points) Harvard and MIT are both considering whether to admit a particular student to their economics Ph.D. programs. Assume that MIT has read the student's application carefully and knows the quality q of the student. Assume that Harvard faculty members are too busy to read applications carefully. Instead they must base their decisions on their prior about the student's ability. Harvard's prior is that q may be 1, 2 or 3 and that each of these values is equally likely. Assume that each school must make one of two decisions on the student: admit with financial aid or reject (the student has no source of support and could not attend graduate school without financial aid). The schools make these decisions simultaneously. Assume that each school's payoff in the game is 0 if they do not offer the student admission, -1 if the student is offered admission and turns them down (this is costly both because the school loses prestige and because the slot could have been given to another student), and q - 1.5 if the student is offered admission and decides to come. Assume that if the student is admitted to both schools she chooses to come to MIT with probability 0.65 and to go to Harvard with probability 0.35. In the following questions treat this as a two player game between Harvard (player 1) and MIT (player 2). (a) What type spaces Oj and Oz would you use to represent this situation as a static game of incomplete information? How many elements are in each set? Write down the val- ues of the utility functions w, (a1, 02; 61, 02) for a couple values of i, a1, a2, 01, 02 to illustrate how to compute them. How many pure Bayesian strategies does each player have? (b) What actions are strictly (conditionally) dominated for each possible type of each player? (c) Find the Bayesian Nash equilibrium of this game. (d) Would Harvard be any better off if it could observe MIT's admission decision before making its decision?Alice has A dollars and has a constant absolute risk aversion a (i.e. u (r) = =e "?) for some a > 0. With some probability # 6 (0, 1) she may get sick, in which case she would need to spend L dollars on her health. There is a health-insurance policy that fully covers her health care expenses in case of sickness and costs P to her. (If she buys the policy, she needs to pay P regardless of whether she gets sick.) (a) Find the set of prices P that she is willing to pay for the policy. How does the maximum price P she is willing to pay varies with the parameters M. L. o, and a? (b) Suppose now that there is a test te {-1, + 1} that she can take before she makes her decision on buying the insurance policy. If she takes the test and the test t is positive, her posterior probability of getting sick jumps to * * > * and if the test is negative, then her posterior probability of getting sick becomes 0. What is the maximum price c she is willing to pay in order to take the test? Take P " (E) >0 > > > u (D). Find the condition under which she takes the test. (b) (4 minutes) Calculate consumer and producer surplus under trade. (5) (15 minutes) The US government is unhappy with steel imports and decides to impose a 200 percent tariff on imported steel so that the price of imported steel is now 3 when importing from abroad. (Continue to assume that the US domestic steel market operates in perfect competition with production function S(L) = ;1) (a) (2 minutes) What is the price of domestic steel? Will car manufacturers choose to use domestic or foreign steel? (b) (5 minutes) Calculate the new equilibrium in the US market for cars, continuing to assume that cars are traded freely at a world price of 100. Does the US still export cars