SCHOOL OF ECONOMICS AND FINANCE
THE UNIVERSITY OF HONG KONG
ECON6021
MICROECONOMICS PROBLEM SET NO.4
OLIGOPOLY AND GAME THEORY
1) Consider the following strategic setting. There are three people: Amy, Bart, and Chris. Amy and Bart have hats. These three people are arranged in a room so that Bart can see everything that Amy does, Chris can see everything that Bart does, but the players can see nothing else. In particular, Chris cannot see what Amy does. First, Amy chooses either to put her hat on her head (abbreviated by H) or on the floor (F). After observing Amy's move, Bart chooses between putting his hat on his head or on the floor. If Bart puts his hat on his head, the game ends and everyone gets a payoff of 0. If Bart puts his hat on the floor, the Chris must guess whether Amy's hat is on her head by saying either "yes" or "no." This ends the game. If Chris guesses correctly, then he gets a payoff of 1 and Amy gets a payoff of -1. If he guesses incorrectly, then these payoffs are reversed. Bart's payoff is 0, regardless of what happens. Represent this game by the extensive form (draw the game tree).
2) Draw the normal form (matrix) of each of the following extensive form games.
3) Determine which strategies are dominated in the following normal form games (as usual, player 1 chooses row, player 2 chooses column.) Find the Nash equilibria for the games.
A / 3,3 / 2,0
B / 4,1 / 8,-1
L / C / R
U / 5,9 / 0,1 / 4,3
M / 3,2 / 0,9 / 1,1
D / 2,8 / 0,1 / 8,4
Game a
Game b
W / X / Y / ZU / 3,6 / 4,10 / 5,0 / 0,8
M / 2,6 / 3,3 / 4,10 / 1,1
D / 1,5 / 2,9 / 3,0 / 4,6
L / R
U / 1,1 / 0,0
D / 0,0 / 5,5
Game c Game d
4) There are two firms producing the same product. Each firm can produce 1, 2, or 3 units of output at a zero marginal cost. The (inverse) market demand function is given in the following table. How much should each firm produce to maximize their joint profits? How much will each firm produce based on individual’s incentive (i.e., what is the Nash equilibrium)?
Quantity demanded / Price2 / 16
3 / 10
4 / 6
5 / 3
6 / 1.5
5. (Partnership game) Suppose that two people decide to form a partnership firm. The gross profit of the firm depends on the amount of effort expended on the job by each person and is given by where is the effort level of person 1 and is the effort level of person 2. The numbers and are positive constants. The contract that was signed by the partners stipulates that person 1 receives a fraction (between 0 and 1) of the firm's profits and person 2 receives a fraction. That is, person 1 receives a profit of , and person 2 receives. Each person dislikes effort and this is measured by a personal cost of effort of for person 1 and for person 2. Person utility in this endeavor is the amount of the firm's gross profit that this person receives, minus the effort cost The effort levels (assumed nonnegative) are chosen by the people simultaneously and independently.
a. Define the normal form of this game (by describing the strategy spaces and payoff functions).
b. Using dominance, compute the strategies that the players rationally select (as a function of and ). Do this computation by solving the players’ maximization problems.
c. Suppose that you could set t before the players interact. How would you set t to maximize the profit of the firm? (Remember that t must be between 0 and 1. Your answer should depend on and .) What does this imply for optimal ownership of the firm?
6) Consider a market with 10 firms. Simultaneously and independently, the firms choose between locating downtown and locating in the suburbs. The profit of a firm that locates downtown is given by where denotes the number of firms that locate downtown. Similarly, the profit of a firm that locates in the suburbs is given by where denotes the number of firms that locates in the suburbs. Find the equilibrium distribution of firms between downtown and suburbs.
7) The game has two players: a criminal (C) and the government (G). The government selects a level of law enforcement, which is a number The criminal selects a level of crime, These choices are made simultaneously and independently. The government's payoff is given by
with the interpretation that is the negative effect of crime on society (moderated by law enforcement) and is the cost of law enforcement, per unit of enforcement. The number c is a positive constant. The criminal's payoff is given by
with the interpretation that is the value of criminal activity when the criminal is not caught, whereas is the probability that the criminal evades capture.
a. Find the best-response functions. Graph the best-response functions.
b. Compute the Nash equilibrium of this game.
c. Explain how the equilibrium levels of crime and enforcement change as c increases.
8) Compute the Nash equilibria and subgame perfect equilibria for the following games. Do so by writing the normal form matrices for each game and its subgames. Which Nash equilibria are not subgame perfect?
9) Consider the following market game. An incumbent firm, called firm 3, is already in an industry. Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 10. First, firm 1 decides whether to enter or not. Then, after observing firm l's choice, firm 2 decides whether to enter or not. Every firm, including firm 3, observes the choices of firms 1 and 2. After this, all of the firms in the industry (including firm 3) compete in a Cournot oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve, where Q is the total quantity produced in the industry. Assume that the firms produce at no cost in this Cournot game. Thus, if firm i is in the industry and produces then it earns a gross profit of in the Cournot phase. (Remember that firms 1 and 2 have to pay the fixed cost 10 to enter.)
a. Compute the subgame perfect equilibrium of this market game. Do so by first finding the equilibrium quantities and profits in the Cournot subgames. Show your answer by designating optimal actions on the tree and writing the complete subgame perfect equilibrium strategy profile. [Hint: In an n-firm Cournot oligopoly with demand and 0 costs, the Nash equilibrium entails each firm producing the quantity ]
b. In the subgame perfect equilibrium, which firms (if any) enter the industry?
10) [von Stackelberg duopoly model] Imagine a market in which two firms compete by selecting quantities and , respectively, with the market price given by Firm 1 (the incumbent) is already in the market. Firm 2 (the potential entrant) must decide whether or not to enter and, if she enters, how much to produce. First the incumbent commits to her production level, Then the potential entrant, having seen, makes her entry/output decision. Both firms have the cost function where is a constant fixed cost. If firm 2 decides not to enter, then it obtains a payoff of 0. Otherwise, it pays the cost of production, including the fixed cost. Note that firm i in the market earns a payoff of
a. What is firm 2's optimal quantity as a function of conditional on entry?
b. Suppose Compute the subgame perfect Nash equilibrium game. Report equilibrium strategies as well as the outputs, profits, and realized in equilibrium. This is called the Stackelberg or entry-accommodating outcome.
c. Now suppose Compute, as functions of F, the level of that would make entry unprofitable for firm 2. This is called the limit quantity.
d. Find the incumbent's optimal choice of output and the outcome of the in the following cases: (i) F = 18,723, (ii) F = 8,112, (iii) F = 1728, and (iv) F == 108. It will be easiest to use your answers from parts (b) and (c) here; in each case, compare firm l's profit from limiting entry to its profit accommodating entry.
11) A manufacturer of automobile tires produces at a cost of $10 per tire. It sells units to a retailer who in turn sells the tires to consumers. Imagine that the retailer faces the inverse demand curve That is, if the retailer brings tires to the market, then these tires will be sold at a price of The retailer has no cost of production, other than whatever it must pay to the manufacturer for the tires.
a. Suppose that the manufacturer and retailer interact as follows. First, the manufacturer sets a price x that the retailer must pay for each tire. Then, the retailer decides how many tires q to purchase from the manufacturer and sell to consumers. The manufacturer's payoff (profit) is whereas the retailer’s profit is Calculate the subgame perfect equilibrium of this game.
b. Next suppose that the manufacturer sells its tires directly to consumers, bypassing the retailer. Thus, the manufacturer can sell tires at price Calculate the manufacturer’s profit maximizing choice of in this case.
c. Compare the joint profit of the manufacturer and retailer in part (a) with the manufacturer’s profit in part (b). Explain why there is a difference. This is called the double marginalization problem.
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