[re] 대학 시험 예제? (경제)

저희학교
경제파트에 게임이론 기말고사 문제입니다.


1. ( 15 points ) Consider a Rubinstein-type bargaining game.
(a) Consider an alternate bargaining game with 2 periods, common discount factor   and payoffs ( , ) if the offer is rejected in the second period. Derive the unique SPNE. ( 5pts )
(b) Show that the infinitely repeated Rubinstein bargaining game has a unique SPNE. ( 10pts )

2. ( 15 points ) Consider an investment and trade game whereby player 1 (Su-mi) first must choose an investment level   at a cost of  . After Su-mi’s investment choice, which player 2 (Mi-young) observes, the two players negotiate over how to divide the surplus  . Negotiation is modeled by a demand game, in which the players simultaneously and independently make demands   and  . These numbers are required to be between zero and  . If   (compatible demands, given that the surplus to be divided equals  ), then Su-mi obtains the payoff   and Mi-young obtains  . On the other hand, if   (incompatible demands), then Su-mi gets   and Mi-young gets zero. Note that Su-mi must pay her investment cost even if the surplus is wasted owing to disagreement.
(a) Show that there is an equilibrium in which Su-mi chooses the efficient level of investment. Completely describe the equilibrium strategies. ( 7pts )
(b) Discuss the nature of the hold-up problem in this example. Offer an interpretation of the equilibrium of (a) in terms of the parties’ bargaining weights. ( 8pts )

3. ( 20 points) Consider the two player normal form game G shown below.

                Player 2
                C        D        E
Player1        C        3, 3        -3, 6        -2, -3
        D        6, -3        0, 0        -2, -3
        E        -3, -2        -3, -2        -5, -5


(a) Let   be the T-fold repetition of G, i.e. the game where player 1 and 2 play G in periods 1, 2, …, T observing the   actions before choosing their period t actions, and receiving as payoffs the simple undiscounted sum of their payoffs in each period, Find all of the SPNE of  . ( 3pts )
(b) Show that there is no NE of   where the players both play C in the first period, but that   does have a NE in which (C, C) is played in the first period. ( 3pts )
(c) Let   be the infinite repetition of G with discount factor  . In the limit as   what average per period payoffs can the players receive in a SPNE of  ? (Just giving a graph of the set of possible average payoffs is sufficient.) ( 7pts )
(d) Show that (C, C) for every period can be sustained as an SPNE of   for  . ( 7pts )

4. ( 25 points ) The following problem is common in Industrial Organization. We want to analyze under what circumstances firms can collude. Two firms produce a homogenous good with the same marginal cost c < 1, and face demand q = 1 - p. The firms engage in Bertrand competition in each period, e.g. the firm with the lower price gets the entire market (the market is equally divided in case of a tie). The Bertrand game is repeated T times where T can be finite or infinite. Each firm seeks to maximize the present discounted value of its profits; that is
                        
(a) Show that collusion is sustainable (i.e. an SPNE of the game) if T =   and  . In the optimal collusive regime both firms set prices equal to the monopoly price in each period and get per period profit  / 2 . ( 10pts )
(b) Assume that there are \'information lags\': each firm can detect defection only after two time periods. Show that optimal collusion is still achievable but is harder to sustain. Explain intuitively. ( 5pts )
(c) Assume that there are n   2 firms in the market. Show that collusion is only sustainable if  . What does this result tell us about the feasibility of collusion in the real world? ( 10pts )

5. ( 10 points ) Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak; she assigns probability   to person 2 being strong. Person 2 is fully informed. Each person can either fight or yield. Each person obtains a payoff of 0 if she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields. If both people fight, then their payoffs are (-1, 1) if person 2 is strong and (1, -1) if person 2 is weak. Formulate the situation as a Bayesian game and find its Bayesian equilibria when  < 1 / 2 and when   > 1 / 2

6. ( 20 points ) Whether Sungchan or Changmo is elected depends on the votes of two citizens, Miyoung and Jihoon. The economy may be in one of two states, A and B. The citizens agree that Sungchan is best if the state is A and Changmo is best if the state B. Each citizen’s preferences assigns a payoff of 1 if the best candidate for the state wins (obtains more votes than the other candidate), a payoff of 0 if the other candidate wins, and payoff of   if the candidates tie. Miyoung is informed of the state, whereas Jihoon believes it is A with probability 0.9 and B with probability 0.1. Each citizen may either vote for Sungchan, vote for Changmo, or not vote.
(a) Show that the game has exactly two pure Bayesian equilibria, in one of which Jihoon does not vote and in the second she votes for Sungchan. ( 10pts )
(b) Why doesn’t Jihoon vote in the first equilibrium? ( 10pts )

7. (15 points) Compute all the perfect Bayesian equilibria of the following game.

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이거도 역시 경제파트에 미시경제학 기말고사 문제^^


1. There are three consumers who have the same utility function  , where   is the consumption level of ice-cream, and   is the leisure time per day. The price of the ice-cream is fixed by the government at  , and the wage rate per hour is  . The consumer earns income by offering his/her labor to the firm that produces ice-cream. The production function is  , where   is the total working hours per day.

(1) Derive the market labor supply and the market demand for the ice-cream. [10]
(2) What\'s the equilibrium wage rate? [10]
(3) Assume that the number of firms is divisible, which means it can be any number in  , i.e., 0.3 firms, 1.7 firms, etc. To clear the ice-cream market, how many firms should the government allow to enter in the market? [10]

2. The following is the information about the demand of an individual consumer and the individual firm’s cost in a perfect competition market.

Individual consumer’s demand
Price        0        1        2        3        4        5        7        9        10        14        20
Consumption        20        19        18        17        15        13        11        10        9        6        0

Individual firm’s cost
Production        0        1        2        3        4        5        6        7        8
Cost        5        6        8        11        16        24        34        50        70

(1) There are 400 identical consumers, and 600 identical firms in the market. Find equilibrium price and quantity. [10]
(2) Is the market at the long-run equilibrium? If yes, explain why. If not, explain how the market will move toward a long-run equilibrium. [10]


3. A monopolist has the following weekly short-run total cost,  . In order to maximize profit he produces 40 per week, at which output his weekly profit is 1000.

(1) Calculate the point elasticity of the demand curve at the equilibrium output level. [10]
(2) Derive the equation of the demand curve on the assumption that it is linear. [10]
(3) A per-unit tax is now imposed on the good, as a consequence of which the monopolist’s profit-maximizing output falls to 39 units. How much is the tax per unit? [10]

4. Coca-Cola announced that it is developing a “smart” vending machine. Such machines are able to change prices according to the outside temperature.
Suppose for the purposes of this problem that the temperature can be either “High” or “Low”. On days of “High” temperature, demand is given by  , where  is number of cans of Coke sold during the day and  is the price per can measured in cents. On days of “Low” temperature, demand is only  . There is an equal number of days with “High” and “Low” temperature. The marginal cost of a can of Coke is 20 cents.

(1) Suppose that Coca-Cola indeed installs a “smart” vending machine, and thus is able to charge different prices of Coke on “Hot” and “Cold” days. What price should Coca-Cola charge on a “Hot” day? What price should Coca-Cola charge on a “Cold” day? [10]
(2) Alternatively, suppose that Coca-Cola continues to use its normal vending machines, which must be programmed with a fixed price, independent of the weather. Assuming that Coca-Cola is risk neutral, what is the optimal price for a can of Coke? [10]
(3) What are Coca-Cola’s profits under constant and weather-variable prices? How much would Coca-Cola be willing to pay to enable its vending machine to vary prices with the weather, that is, to have a “smart” vending machine?[10]



5. The market demand curve for mineral water is given by  . If there are two firms that produce mineral water, each with a constant marginal cost of 3 per unit. Find the equilibrium price, quantity, and profit for each firm when firms compete a la
(1) Cartel (Joint profit maximizing firms) [5]
(2) Cournot [5]
(3) Bertrand [5]
(4) Stackelberg, assuming that firm 1 is the leader [5]
(5) Rank the above four models (1)-(4) in terms of consumer surplus. [10]

6. Two used car dealerships compete side by side on a main road. The first, Harry’s Cars, always sells high-quality cars. On average, it costs Harry’s $8000 for each car that it sells. The second dealership, Lew’s Motors, always sells lower-quality cars. On average, it costs Lew’s only $5000 for each car that it sells. If consumers knew the quality of the used cars they were buying, they would pay $10000 on average for Harry’s cars and only $7000 on average for Lew’s cars. Suppose that consumers do not know the quality of each dealership’s cars. In this case, they would figure that they have a 50-50 chance of ending up with a high-quality. Assume that two dealerships can offer a bumper-to-bumper warranty for all cars that they sells. Harry’s Y year warranting cost is $500Y and Lew’s is $1000Y. Find all equilibria, separating and/or pooling. [20]

7. For those markets in which asymmetric information is prevalent, would you agree or disagree with each of the following? Explain briefly:

(1) The government should impose quality standards, for example, firms should not be allowed to sell low-quality items. [10]
(2) The producer of a high-quality good will probably want to offer an extensive warranty. [10]
(3) The government should require all firms to offer extensive warranties. [10]