Eco 601E: Advanced Microeconomics II (Spring, 2nd, 2012)
Final Exam
1. True or False (10 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) The first-degree price discrimination is also called the “perfect price discrimi- nation” because the price charged by the monopolist is the same as the price determined by the perfectly competitive market.
(b) If a good is non-rival and excludable, then it is called (pure) public good. (c) A Lindahl equilibrium ALWAYS satisfies the Samuelson condition.
(d) Zermelo’s theorem assures that the first mover has a winning strategy in ANY perfect information game with strictly opposite interests.
(e) The weak perfect Bayesian equilibrium puts NO restriction on beliefs at the information sets that are not reached in equilibrium.
2. Duopoly (15 points)
Consider a duopoly game in which two firms, denoted by Firm 1 and Firm 2, simultaneously and independently select their own prices, p1 and p2, respectively. The firms’ products are differentiated. After the prices are set, consumers demand A− pi + pj
2 units (i 6= j, i = 1, 2) of the good that firm i produces. Assume that firm’s marginal costs ci (< A), i = 1, 2 are constant, and the payoff for each firm is equal to the firm’s profit, denoted by πi, i= 1, 2.
(a) Write the payoff functions π1 and π2 (as a function of p1 and p2).
(b) Derive the best response functions and solve the pure-strategy Nash equilib- rium of this game.
(c) Derive the prices (p1, p2) that maximize joint-profit, i.e., π1+ π2.
(d) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.
(e) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c1 = c2 = 8. Then, derive the condition under which the trigger strategy sustains the joint-profit maximizing prices you derived in (c) (as a subgame perfect Nash equilibrium).
1
3. Auction (9 points)
Consider a “common-value auction” with two players, where the value of the object being auctioned is identical for both players. Call this value V and suppose that V = v1+ v2, where vi is independently and uniformly distributed between 0 and 1, and player i can observe only vi. The players simultaneously submit bids, b1 and b2. If player i bids higher than does player j, then player i wins the auction and gets the payoff V − bi whereas player j gets 0. What is a symmetric Bayesian Nash equilibrium in this game?
Hint: You can assume the equilibrium strategy is linear and symmetric, i.e., bi = α+ βvi for i = 1, 2.
4. Screening (16 points)
A monopolist can produce a good in different qualities. The cost of producing a unit of quality s is s2. Consumers buy at most one unit and have utility function u(s|θ) = θs if they consume one unit of quality s and 0 if they do not consume. The monopolist decides on the quality and price that it is going to produce. Con- sumers observe qualities and prices and decide which quality to buy if at all.
(a) Characterize the first-best solution.
(b) Suppose that the seller cannot observe θ: θ ∈ {θL, θH} and Pr[θ = θL] = β with 0 < θL< θH. Set up the seller’s optimization problem under this asymmetric information structure.
Hint: By the taxation principle, you can focus on the pair of contracts: one is for the low type and the other is for the high type.
(c) Show that the low type consumers (θL) receive no rents under the second-best solution.
(d) Characterize the second-best solution. You can ignore possibility of a corner solution such that the low type consumers are excluded.
2