CAS EC 501 Microeconomic Theory Final Exam 5/4/21 1

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CAS EC 501 Microeconomic Theory Final Exam 5/4/21 1. Consider an exchange economy where two agents i = A; B have the following preferences over two goods x and y : p uA (xA ; y A ) = xA y A p xB + y B : uB (xB ; y B ) = Let the initial endowment be e = (eA ; eB ) where eA = (2; 2) and eB = (2; 2): (a) Draw an Edgeworth box and clearly indicate the individuals’preferences and the initial allocation. (b) Normalize the prices so that px = p and py = 1: Find the equilibrium, and indicate the equilibrium allocation in the Edgeworth box. (c) Is the equilibrium Pareto e¢ cient? Is the initial allocation Pareto e¢ cient? Verify that both agents’utility levels are higher with the equilibrium allocation than with the initial endowment. (d) State Walras’law and verify that it holds in equilibrium. Does Walras’law hold only in equilibrium? 2. Let an investor have initial wealth w0 = 1 and the following von-Neumann-Morgenstern utility of wealth: u(w) = ln w: (a) Suppose that she can invest a fraction h of her wealth in a risky asset and the remainder 1 h in a risk-free asset. The only restriction on the weights h and 1 h is that they sum to one. The risky asset has a payo¤ of 3 in case of success and payo¤ 0 if it is a failure. The investor believes that success and failure are equally likely. The risk-free asset has a payo¤ of 1:2; regardless whether the risky investment is a success or a failure. Derive an expression of the expected wealth and expected utility if she invests h in the risky asset and 1 h in the risk-free asset. (b) Use the information above to derive her optimal investment portfolio. What is the expected wealth associated with this portfolio? (c) Assume now that everything is as above, but the risky asset is only paying o¤ 2 if it is a success (instead of 3): What is the optimal investment portfolio in this case? Explain brie‡y what this investment strategy entails. What is her expected wealth resulting from this portfolio? 1 3. There are two …rms i = 1; 2 in a market. They produce perfect substitutes using identical technologies: y2 c(y) = : 2 The inverse demand in this market is: p(y) = 1 y: (a) Compute the Cournot-Nash equilibrium quantities, pro…ts, and price. (b) Suppose now that …rm one only has the opportunity to sell output on another market as well. If it sells quantity x in this market its cost is: c(x; y1 ) = (x + y1 )2 ; 2 and inverse demand in the second market is: p (x) = x: Here, is a positive parameter. Consider the simultaneous move game in which …rm one chooses y1 and x and …rm two chooses y2 . What are the equilibrium output levels y1 and y2 ? (c) Show that a small increase in the parameter Provide an interpretation for this. evaluated at = 1 4 hurts …rm one. 4. Two identical …rms i = f1; 2g with zero cost produce a homogenous product. Inverse market demand in every period t is given by: p(y) = 12 y; where y is output and p is price. Both …rms seek to maximize the discounted sum of per-period pro…ts, and both discount future payo¤s at rate 2 (0; 1) : Assume that the stage game is repeated in…nitely many times (T = 1): (a) Suppose …rst that the …rms set prices in every period. Assuming they use a grimtrigger strategy (GTS), under what conditions can cooperation be sustained as a subgame perfect Nash equilibrium (SPNE)? (b) Suppose instead that the …rms set quantities in every period. Again assuming they employ a GTS, under what conditions can cooperation be sustained as a SPNE? (c) Is the SPNE in (b) di¤erent from that in (a)? Explain why/why not.