microecon,,

Question 1 Let an individual's utility function be given as u(x1, x2) = 2 x1 x2 . a) Compute the Marginal Rate of Substitution. b) Initially, the individual consumes bundle (x1 = 100, x2 = 12.5). Then, the individual's consumption of the first good is cut to x 0 1 = 50. What is the new level of consumption of good 2, x 0 2 , that the individual needs to consume in order to reach the same utility level as before? c) Given the prices p1 = 1 and p2 = 2 for the first and the second good, respectively, and a budget of m = 100, what is the best consumer choice? d) Find the individual's general demand function for good 2. e) If the price for the first good rises to p 0 1 = 50, how much less of good 2 will the individual conusme? f) Assuming the demand function for good 1 is x1(p1) = 1 2 m p1 , what is the inverse demand funtion, and what is the own-price elasticity of demand for good 1! g) Assuming the demand function for good 1 is x1(p1) = 1 2 m p1 , show mathematically that the good is not inferior.

An individual's preferences over consumption bundles A, B, and C are given as A B B C A C Are these preferences transitive? Explain why or why not!

The demand function is given by x = A p with x giving the demand, p the price and a and as positive parameters. a) Derive the price elasticity of demand, . What is the economic meaning of the price elasticity of demand? What is elastic, what is inelastic demand? b) Denote revenues as a function of demand x and price p. How do revenues change as a reaction to an increase of the price, if demand is inelastic? c) Is the good in focus a Giffen good? Explain your answer both verbally and analytically.

Consider a firm that produces output q 0 at a cost of c(q), where c 0 (q) > 0 and c 00(q) > 0. Also assume that there is a probability 1 > > 0 that the firm experiences an equipment failure and incurs additional repair costs equal to cR > 0 per unit of output or cRq in total. The competitive price of output is p > 0. (a) Derive the firm's first-order condition for an interior solution assuming its objective is to maximize expected profit. What is the economic intuition of this condition? (b) Derive the firm's first-order condition for an interior solution assuming its objective is to maximize its expected utility of profit, where the strictly increasing and strictly concave function u() characterizes its risk preferences. (c) Will the firm produce more if its objective is to maximize expected profit or if its objective is to maximize the expected utility of profit? Justify your answer and provide the economic intuition for your result.

There are N > 2 profit-maximizing firms that compete as Cournot oligopolists in a market with inverse demand given by P(Q) = a Q, where P is price, Q = PN i=1 qi and qi is the output of firm i for i = 1, . . . N. Assume that all firms have the same cost function: C(qi) = cqi . (a) Solve for Cournot equilibrium. Show that the profit for each firm in Cournot equilibrium, C i , is equal to the square of output: C i = q 2 i . (b) Suppose that prior to choosing quantities, two firms have the option to merge into a single firm. Compare the profit of the single merged firm in this new Cournot equilibrium, with a total of N 1 firms, to the sum of the two firms' individual profits in the original Cournot equilibrium, where there were N firms. Show whether the firms should merge or not. (c) Now suppose N firms compete in an infinitely repeated game in which in each period they simultaneously choose qi . Let be the discount factor between periods (0 < < 1). Let t = 0, 1, 2, . . . represent the time period and let QM denote the monopoly level of output. Suppose that each firm plays the following simple trigger strategy: Set qi = QM/N in period t = 0 and in all t > 0 for which aggregate output was QM in all preceding periods, and Set qi equal to the Cournot equilibrium output for all t > 0 for which aggregate output was not QM in all preceding periods. For what range of the discount factor can the trigger strategy support collusion at the monopoly output level as a subgame-perfect equilibrium outcome?