PROBLEM SET 3. a) Find the pure strategy Nash Equilibrium of this game.


 Timothy Parks
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1 PROBLEM SET 3 1. The residents of Pleasantville live on Main Street, which is the only road in town. Two residents decide to set up general stores. Each can locate at any point between the beginning of Main Street, which we will label 0, and the end, which we will label 1 (if they locate at the same point, they move to opposite sides of the street). The two decide independently where to locate, and they must remain there forever. Each store will attract the customers who are closest to it, and the stores will share equally customers who are equidistant between the two. Thus, for instance, if one store locates at point x and the second at point y > x, then the first will get a share x +(y x)/ and the second will get a share (1 y) + (y x)/ of the customers each day (draw a picture to help you see why). Each customer contributes $1.00 in profits each day to the general store it visits. a) Find the pure strategy Nash Equilibrium of this game. b) Suppose there are three General Stores, each independently choosing a location point along the road. What is the Nash Equilibrium now? c) Suppose there are four General Stores, each independently choosing a location point along the road. What is the Nash Equilibrium now?. You and your friend are in an Italian restaurant, and the owner offers both of you a freeeight slice pizza under the following condition. Each of you must simultaneously announce how many slices you would like; that is, each player i {1,} names his desired amount of pizza, 0 s i 8. If s 1 + s 8 then the players get their demands. If s 1 + s > 8, then the players get nothing. Assume that you each care only about how much pizza you individually consume, and the more the better. a) Graph each player s isoutility map; b) In a separate graph, draw each player s bestresponse function; c) What is the Nash Equilibrium? d) What is the subgame perfect Nash Equilibrium? 3. Consider a threeplayer game with the following characteristics: a) Strategy sets: S1=[0,1], S=[,], and S3=[1,0] b) Payoff functions: π 1 (x, y, z) = x + yz π (x, y, z) = y + xz π 3 (x, y, z) = xy z Find the Nash Equilibrium of this game. Is it Pareto Optimal? 4. Two players have the following utility functions: U 1 = x 1 x 10x 1 0.5x 1
2 U = x 1 x 10x 1 x + 0x 0.5x Where x1 and x are the actions taken by players 1 and, respectively. If they play this game simultaneously, what is (are) the pure strategy Nash Equilibrium of this game? Draw their bestresponse functions and check for stability. What is the Subgame Perfect Nash Equilibrium if Player 1 moves first? In the simultaneous version of the game, what are the Pareto Optimal outcomes? 5. (Cournot Duopoly Model) Let s revisit the oligopoly model we discussed in class. A single good is produced by firms. The marginal cost to both firms of producing one unit of the good is 0. All the output is sold at a single price, determined by the demand for the good and the firms total output. The demand curve is given by: 100 Q if Q 100 P(Q) = { 0 if Q > 100 Where Q is the total quantity of the industry. Assuming that these firms choose simultaneously how many units to produce in order to maximize profits, find the NE of this game, the market price, and total profits? Is the NE Pareto Optimal? If not, how can both firms be better off? Can you give a numerical example of a Pareto Optimal outcome? Find the mathematical expression for the firms isoprofit functions and draw the isoprofit maps. Then, find the expression that shows all the Pareto Optimal points. 6. (Cartel Duopoly Model) Suppose the two firms above collude by agreeing that each will * produce an amount q q a qb, and they have some way to enforce the agreement. What should they choose for q*? What are the profits of the two firms? Compare this with the Cournot duopoly profits. Suppose firm a reneges on its promise in the previous part but b does not. What should a choose for q a? What are a s profits, and what are b s profits? Suppose firm b finds out what firm a is going to do in the previous part, and chooses qb to maximize profits, given what firm a is going to do. What is qb now? What do you think happens if they go back and forth this way forever? 7. (Stackelberg Duopoly Model) Suppose the two firms choose quantity, but one firm moves first and the second firm observes firm 1 s choice. What is the SPNE of this game? Compare these results with the results of the Cournot model. What do you conclude? 8. (Stackelberg Triopoly Model) Suppose three firms choose quantity, and they move sequentially. The demand function remains the same (100 total output). What is the SPNE of this game?
3 9. (Bertrand Duopoly Model) Suppose the two firms choose price as opposed to quantity, where customers all go to the lowestprice firm, and the firms split the market if they choose the same price. What is the unique Nash equilibrium of this game? 10. (Cournot and Stackelberg with quadratic cost function) The firms have the same cost function now, and the function is: TC = q i. Find the equilibrium in the Cournot model and the Stackelberg model. 11. (Cournot and Stackelberg with marketsharemaximizing firms) The two firms have the same constant marginal cost (c=0), but one of the two firms chooses its output to maximize its market share subject to not making a loss, rather than to maximize its profit. What is the NE? What happens if each firm maximizes its market share? 1. (Generalizing Cournot) The firms have the same constant marginal cost c, and the number of firms is given by n. How does the equilibrium quantity depend on the number of firms? What happens to total industry output, market price, and total profits as the number of firms increases? 13. (Cournot or Bertrand?) Compare the results you found solving the Cournot Duopoly Model and the Bertrand Duopoly Model above. Now, imagine a situation in which, at time t=0, there is only one firm dominating a specific market (monopoly). Then, at time t=1, a second firm enters the market, which causes the price of the good to fall. You are a researcher analyzing this market, but you do not know whether these two firms are choosing output strategically (Cournot) or choosing price strategically (Bertrand). If a third firm enters this market, how can you use the outcome of the market with these three firms to distinguish between the two possible oligopoly models? Explain. 14. (Oligopoly with a minimum wage) Consider a typical Cournot model of duopoly with a demand curve given by P = 10(q1+q). These two firms can produce this good by using labor only. Each worker can produce only one unit of output (so if a firm wants to produce 10 units, it needs to hire 10 workers). But a firm can only hire more workers by offering a higher wage. Assume that the labor supply function is given by w=l, where w is the wage that the firm has to pay each worker, and l is the number of workers who will accept the job for that wage. This means, for example, that if the firm wants to produce only 1 unit, it needs to offer a wage of $1 to be able to attract 1 worker. Labor costs are the only costs incurred by each firm. Given this information, find the Nash equilibrium of the game. How many workers are employed in total? Now let s assume that the government implements a minimum wage m. This means that, if the firm wants to hire one worker, this worker must receive m. As an example, consider the case when the minimum wage is $10. In this situation, the firm has to pay $10 to each worker if it chooses to produce 10 units or less. If the firm chooses to produce more than 10 units, then it is obvious that we return to the original supply curve. If the firm wants to hire 11 workers when the minimum wage is $10, it will have to pay each worker $11, because a wage of $10 attracts only 10 workers.
4 Find the Nash equilibrium of this game if the minimum wage is $30. How many workers are employed now? What is the intuition behind this somewhat counterintuitive result? Make sure to draw the bestresponse functions of each firm. How does the equilibrium change if one firm moves first? In the simultaneous version of the game, draw a graph showing the relationship between the minimum wage and total employment. 15. A market has two firms and each firm produces a different good. The firms move simultaneously and each firm chooses how much to charge for the good it produces. The demand functions are given by: q 1 = a bp 1 + dp q = a bp + dp 1 Where q i is the quantity that consumers will demand, given the price set by the firm producing good i and the price set by the firm producing good j. a, b, and d are parameters, where a > 0 and b > 0 (there are no restrictions for d). The marginal cost of producing each unit is c for both firms. a) If the two goods are substitutes (coffee and tea), what can you say about parameter d? What if the two goods are complements (coffee and sugar)? b) Find the externalities and the strategic relationship. If the two goods are substitutes, do we have a game of strategic substitutes? Explain. c) Find the equation for the isoutility map of each player and draw them; d) Find the bestresponse functions and draw them in a separate graph; e) Find the Nash Equilibrium; f) Find the Pareto Optimality Condition; g) Find the condition for a stable Nash Equilibrium; h) If possible, draw the Pareto Optimal curve using the same graph where you drew the BRFs; i) Find the Subgame Perfect Nash Equilibrium. Does the game generate first or secondmover advantage? j) What if each player could only select the other player s strategy? What would be the Nash Equilibrium? What would be the Subgame Perfect Nash Equilibrium? 16. A group of n students go to a restaurant. It is common knowledge that each student will simultaneously choose his own meal, but all students will share the total bill equally. If a student gets a meal of price p and contributes x towards paying the bill, his payoff will be p x. a) Explain the meaning of the payoff. What is the economic intuition behind it? b) Find the Nash Equilibrium. c) Discuss the limiting cases n=1 and n.
5 d) Assume now that the waiter uses a roulette to randomly choose one student. The selected student will have to pay the entire bill, whereas the others will not have to pay anything. What is the NE now? Are the students better off? e) Assume now that the bill is shared by all students, but a tip of (1+t) is added to the bill (i.e. the final bill is (1+t)*sum of meal prices). How does the tip affect the equilibrium? Are the students better off? f) Assume now that there is no tip, and the bill is shared equally. The only difference is that they went to an upscale restaurant that serves the same meals, but they all cost twice as much as the meals in the original restaurant. Are the students happier at the upscale restaurant? 17. You and your n1 roommates each have five hours of free time you could spend cleaning your apartment. You all dislike cleaning, but you all like having a clean apartment: each persons s payoff is the total hours spent (by everyone) cleaning, minus a number c times the hours spent (individually) cleaning. The utility function of player i is: n U i = cs i + s j j=1 a) Does this game generate positive or negative externalities? b) Does it generate strategic substitutability or strategic complementarity? c) Find the Nash Equilibrium. d) The Nash Equilibrium is not Pareto Optimal. Find one Pareto Optimal outcome. 18. Generate two utility functions for a twoplayer game (strategies are X1 and X, respectively) so that: a) Player 1 imposes positive externalities on player, but player imposes negative externalities on player 1. Find the Nash Equilibrium and the Subgame Perfect Nash Equilibrium; b) Player 1 generates strategic substitution for player, but player generates strategic complementarity for player 1. Find the Nash equilibrium and the Subgame Perfect Nash Equilibrium. 19. Analyze the utility functions below. Each function captures a certain type of person. Find the subgame perfect Nash equilibrium of a $10 Ultimatum Game in which both the Proposer and the Responder have the same utility function: a) U1 x1 b) U1 x1 x c) U1 x1 ln x U min x x d) 1 1,
6 e) U x max x x ;0 0.4max x ; x 0. Do the same for the Trust Game. 1. Do the same for the Public Goods game with a multiplier of Two division managers can invest time and effort in creating a better working relationship. Each invests e i 0, and if both invest more both are better off, but it is costly for each manager to invest. In particular the payoff function for player i is U i = (a + e j )e i e i. a) Find the externalities and the strategic relationship; b) Find the equation for the isoutility map of each player and draw them; c) Find the bestresponse functions and draw them in a separate graph; d) Find the Nash Equilibrium; e) Find the Pareto Optimality Condition; f) Draw the Pareto Optimal curve using the same graph where you drew the BRFs; g) Find the Subgame Perfect Nash Equilibrium. Does the game generate first or secondmover advantage? h) If there was a way for the players to cooperate and choose the same level of effort that generates a Pareto Optimal outcome, what would be their efforts? i) What if each player could only select the level of effort of the other player? What would be the Nash Equilibrium? What would be the Subgame Perfect Nash Equilibrium? 3. Two people have $10 to divide between themselves. They use the following procedure. Each person names a number of dollars (a nonnegative integer), at most equal to 10. If the sum of the amounts that the people name is at most 10, then each person receives the amount of money she named (and the remainder is destroyed). If the sum of the amounts that the people name exceeds 10 and the amounts named are different, then the person who named the smaller amount receives that amount and the other person receives the remaining money. If the sum of the amounts that the people name exceeds 10 and the amounts are the same, then each person receives $5. What is the NE of this game? What is the Subgame Perfect NE of this game? 4. (Watson) Imagine that a prosecutor (P) has accused a defendant (D) of committing a crime. Suppose that the trial involves evidence production by both parties and that, by producing evidence, a litigant increases the probability of winning the trial. Specifically, suppose that the probability that the defendant wins is given by e D e D +e P, where e D is the expenditure on evidence production by the defendant and e P is the expenditure on evidence production by the prosecutor. Assume that e D and e P are greater than or equal to 0. The defendant must pay 8 if he is found guilty, whereas he pays 0 if he is found innocent. The prosecutor receives 8 if she wins and 0 if she loses the case. Draw the players bestresponse functions and find the Nash equilibrium of the game. What is the probability that the defendant wins in equilibrium? Is this outcome efficient? Why or why not? 5. Imagine a game in which players 1 and simultaneously and independently select A or B. If they both select A, then the game ends and the payoffs are (5,5). If they both select B, then the game ends with the payoffs (1,1). If one of the players chooses A while the other selects B,
7 then the game continues and the players are required to simultaneously and independently select positive numbers. After these decisions, the game ends and each player receives the payoff x 1 +x 1+x 1 +x, where x 1 is the positive number chosen by player 1 and x is the positive number chosen by player. Find the Nash equilibrium of this game. 6. Imagine a market in which two firms compete by selecting quantities q1 and q, respectively, with the market price given by p = q 1 3q. Firm 1 (the incumbent) is already in the market. Firm (the potential entrant) must decide whether or not to enter and, if she enters, how much to produce. First the incumbent commits to its production level q1. Then the potential entrant, having seen q1, decides whether to enter the industry. If firm chooses to enter, then it selects its production level q. Both firms have the cost function c = 100q i + F, where F is a constant fixed cost. If firm decides not to enter, then it obtains a payoff of 0. Otherwise, it pays the cost of production, including the fixed cost. a) What is firm s optimal quantity as a function of q1, conditional on entry? b) Suppose F=0. Compute the subgame perfect Nash equilibrium of this game. Report equilibrium strategies as well as outputs, profits, and price realized in equilibrium. c) Now suppose F>0. Compute, as a function of F, the level of q1 that would make entry unprofitable for firm. 7. This exercise will help you think about the relation between inflation and output in the macroeconomy. Suppose that the government of Pechland can fix the inflation level p by an appropriate choice of monetary policy. The rate of nominal wage increase, W, however, is not set by the government but by an employerunion federation (EUF). The EUF would like real wages to remain constant. That is, if it could, it would set W=p. Specifically, given W and p, the payoff of the EUF is given by u = (W p). Real output y in Pechland is given by the equation y = 30 + (p W). The govt, perhaps representing its electorate, likes output more than it dislikes inflation. Given y and p, the govt s payoff is v = y p 30. The govt and the EUF interact as follows. First, the EUF selects the rate of nominal wage increase. Then the govt chooses its monetary policy (and hence sets inflation) after observing the nominal wage increases set by the EUF. Assume that 0 W 10 and 0 p 10. a) Use backward induction to find the level of inflation p, nominal wage growth W, and output y, that will prevail in Pechland. If you are familiar with macroeconomics, explain the relationship between backward induction and rational expectations here. b) Suppose that the govt could commit to a particular monetary policy (and hence inflation rate) ahead of time. What inflation rate would the govt set? How would the utilities of the govt and the EUF compare in this case with that in part a? c) In the real world, how have govts attempted to commit to particular monetary policies? What are the risks associated with fixing monetary policy before learning about important events, such as the outcomes of wage negotiations? 8. (Team Production and variations) Consider a model of team production with one boss and n workers indexed i. The boss (the first mover) chooses to transfer a fraction of total output a [0, 1] to the workers in order to maximize her profits, and each worker provides a level of effort e i
8 0 to maximize her utility, after observing a. Effort is costly and assumed to be quadratic. These actions determine output x, which is given by the sum of the efforts provided by the workers. The output is divided equally among the workers. The remainder goes to the boss. n The profit function is: π = (1 a) i=1 e i If workers have selfregarding preferences, the utility function of worker i is: U i = a n i=1 e i n 0.5e i a) What is the Subgame Perfect Nash Equilibrium of this game? Is it Pareto Optimal? Provide a short but clear interpretation of the results. b) What is the NE of this game if the boss has to choose a at the same time that the workers are choosing e? c) What is the Subgame Perfect Nash Equilibrium if the workers choose how hard to work before the boss chooses how much to transfer? 9. Consider the following team production model (two workers). The task generates complementarities, and the utility function of each worker is given by: U i = e ie j e i Show that, if the workers move simultaneously, the Nash Equilibrium is (0,0), and it is Pareto Optimal. However, it is not the only Pareto Optimal outcome. Find the set of all Pareto optimal outcomes. 30. Two hightech firms (1 and ) are considering a joint venture. Each firm i can invest in a novel technology and can choose a level of investment x i [0,5] at a cost of c i (x i ) = x i 4 (think of x i as how many hours to train employees or how much capital to spend for R&D labs). The revenue of each firm depends on both its investment and the other firm s investment. In particular if firms i and j choose x i and x i, respectively, then the gross revenue to firm i is: 0 if x i < 1 R(x i, x j ) = { if x i 1 and x j < x i x j if x i 1 and x j a) Write down the profit function (gross revenue minus costs) of firm i as a function of x i for three cases: (i) x i <, (ii) x j =, and (iii) x j = 4. b) What is the bestresponse function of firm i? c) It turns out that there are two identical pairs of such firms; that is, the description applies to both pairs. One pair is in Russia, where the coordination is hard to achieve and business people are very cautious, and the other pair is Germany, where coordination is common and businesspeople expect their partners to go the extra mile. You learn that the
9 Russian firms are earning significantly lower profits than the German firms, despite the fact that their technologies are identical. Can you use Nash Equilibrium to shed light on this dilemma? If so, be precise and use your previous analysis to do so. 31. A professor hires a research assistant (RA) to perform some calculations. The RA is risk averse, so he does not like variation in his income. His utility depends on the wage he receives from the professor (w) and the effort he puts in performing the calculations (e): U = w 0.5 e The outcome can vary. Let r g be the good outcome, and r b be the bad outcome. This uncertainty cannot be completely eliminated. But the probabilities of these outcomes are affected by the RA s effort. If the RA exerts no effort (e = 0), there is 50% probability that the good outcome would occur. If the RA exerts an effort that costs him e = 1 then there is an 80% probability that the good outcome will occur. The good outcome gives the professor a benefit of 40, and the bad outcome a benefit of 0. The professor cannot monitor effort directly, so he must provide the right effort incentives to the RA. The professor s objective is to maximize the benefit net of the wage paid to his RA: Π = r w Suppose that the RA could get a job at the campus cafeteria that paid a wage of 4 and required no effort (e = 0). a) If the professor could observe the level of effort of his RA, what wage would he pay him to exert high effort? b) Now assume the professor cannot observe the effort exerted. The only thing he observes is the outcome in terms of regression results (rg, rb). Why will the professor have to pay his RA two different wage levels according to the output r (call the two wage levels wg and wb)? c) Write down the incentive compatibility constraint, i.e., the condition under which, given the two wage levels wg and wb, the RA will choose to exert high effort. d) Write down the participation constraint, i.e., the condition under which, given the two wage levels wg and wb, the RA will choose to stay and work for the professor instead of the campus cafeteria. e) Write down the profit constraint, i.e., the condition under which the professor s expected payoff when the RA exerts high effort is higher than when the RA exerts low effort and is paid the minimum amount to accept the contract. f) Now combine the constraints derived in part c, d, and e and find the optimal wages wg and wb that the professor will offer to his RA. Graph your results.