advanced microeconomic theory

ECO533 Advanced Mircroeconomic Theory
11 Questions1. QUESTION I: Can you write down a CES production function? Which parameter determines the (constant) elasticity of substitution? And the labor income share? Which assumptions would you need to make in the evolution of wages and rental rates in order to replicate the declining US labor income share? 2. QUESTION II: Suppose that we have increasing returns to scale. Would the cost function be concave or convex in output? Would the marginal cost be increasing or decreasing? Would the optimization problem be well defined? 3. QUESTION III: In some textbooks we are told that @y @w = @L @p . This looks counterintuitive. How would you explain this? 4. QUESTION IV: What does it mean third-degree price discrimination? Would you provide some examples? 5. QUESTION V: Please, discuss. Does it make sense for Zara to have these short selling cycles? How Zara grew into the World’s Largest Retailer. ”When you went to Gucci or Chanel in October, you knew the chances were good that clothes would still be there in February,”BUT ”With Zara, you know that if you don’t buy it, right then and there, within 11 days the entire stock will change. You buy it now or never. And because the prices are so low, you buy it now.” 6. QUESTION VI: Could you write down the first-order condition of the prototypical moral hazard model (hidden e↵ort) as a function of the likelihood ratio? How would you interpret this condition to provide optimal incentives for rewarding e↵ort? 7. QUESTION VII In the adverse selection model discussed in class, we encountered the following condition: ✓L C(0, ✓L) ✓H C(t,ˆ ✓L). Please, explain this condition. Moreover, how does this condition get translated in an insurance market with two types of consumers (the bad type and the good type). Hint: Mas-Colell, 13.D.2. 1 8. QUESTION VIII In our corruption paper, is Equation (5) an IR constraint or an IC constraint? Explain the meaning of this constraint (is it a moral hazard model?). Would you suggest any changes to this constraint? 9. PROBLEM 1: A monopolist (AT&T) is facing the following demand schedule P = 243Q. That is, Q = 0 implies P = 24, then Q = 1 implies P = 21, and Q = 2 implies P = 18, and so one. Fixed costs will be neglected in this analysis. The marginal cost is constant and equal to 6 for every unit produced. Determine: (i) The quantity produced corresponding to the amount of maximum profit. (ii) Equilibrium price if a new competitor, Vodafone, enters the market with a MC =6 under both Cournot and Bertrand Competition. (iii) Equilibrium price if a new competitor, Vodafone, enters the market with a MC=7 under Bertrand Competition. (iv) Equilibrium price if a new competitor, Vodafone, enters the market with a MC=5 under Bertrand Competition. 10. PROBLEM 2: Spotify o↵ers a hassle-free (no-commercials) program. Here, at UM, a professor would be willing to subscribe for a monthly free of $30, and a student would be willing to subscribe for $10. There is a second program in which commercials take ten percent of the time. The company will get one-dollar revenue for running these commercials in all cases. However, the disutility of the commercials to each student would be $2, and to a professor would be $8.00. Assume that half of the population are professors and half are students, and Spotify is committed to serve both markets. A. (Market segmentation) Compute the optimal price(s) for each type in the case that Spotify has access to the UM records and is able to tell who is a professor and a student. B. (Hidden Types). Compute optimal price(s) in the case that Spotify has no access to UM records, and cannot identify types. C. (Hidden Types). Assume linear preferences and linear revenues. An undergrad student from ECO 533 has suggested to Spotify to change the second program so that commercials should take 20 percent of the time. Again redo part B, assuming that in the second program commercials take 20 percent of the time. D. (Hidden Types) Actually, our student goes further and claims to have computed the optimal percentage amount of advertising for the second program under the linearity assumption on preferences and commercial revenues. Following part B, are you able to compute the optimal amount of advertising for Spotify serving UM? 11. PROBLEM 3: This is Mas-Collel, 13.C.6. Let us assume the following condition: pG⇧ (1 + r) > pB⇧ (1 + r) > 0. Now, following this problem in the book: (i) Find the level of Ri (i = 1, 2) in a separating equilibrium. (ii) Find the level of R in a pooling equilibrium. 2 (iii) How a separating equilibrium will look like in case that the bank can also o↵er a new contract in which the entrepreneur is required to contribute a fraction x of the one dollar initial outlay. Assume that the entrepreneur can get this money at an interest rate ⇢ slightly higher than r. (Else, assume that the entrepreneur has the cash.)

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