443- Economics of Law and Regulation

443: Economics of Law and RegulationHomework 2 Lecture 1-4Chapters 1 to 4 This is Homework #2. You need to explain your answers. Each problem has 20 points. Theanswers are due Wednesday February 8 in class. If you want to submit them electronically(via email or Canvas), you should do so before the class.Problem 1: R&D Rivalry. Consider the R&D model discussed in class. A firm needsto choose how quickly to finish the development project of its next product. The presentvalue of the cost to the firm of finishing the project in T years is 1/T million dollars. Thebenefit depends on the number of rivals as follows. At year T , the firm is going to enjoy abenefit of 1 million dollars. Every year after that, the firm shares the benefit of 1 milliondollars equally with all its rivals (so if there is one rival, the firm gets 500 thousand dollars.)111+ (1+r)Assume that the discount rate is r = 1. Recall the formula 1+r2 + . . . = r.(a) Assume that the firm has one rival. What is the present value of the benefit that thefirm gets by choosing T = 1? How about T = 2?(b) Assume that the firm has three rivals. What is the present value of the benefit thatthe firm gets by choosing T = 1? How about T = 2?(c) On a picture draw the benefit curve of the firm when there is 1 rival. Draw anothercurve when there is three rivals. Draw the firm’s cost curve. On the picture specifythe optimal T when there is one rival and when there is three rivals (you do not needto calculate optimal T exactly).Reminder. If the benefit of an investment is 1 a year from now, and is 4 two years from now,1414+ (1+r)then with discount rate r = 1, the present value of the benefit is 1+r2 = 2 + 4 = 1.5.Problem 2: Game Theory. Consider a game with the following payoff matrix.UpDown Left Right1,10,00,02,2 (a) How many players does this game have? What are the strategies of each player?1 (b) What are the Nash equilibria of this game?(c) What are the Pareto Efficient outcomes?(d) Design the payoff matrix of a game with no Nash Equilibria. The game should have 2players, 2 strategies for each player, and the payoffs for each player should be either 0or 1.Disclaimer. If you know game theory, you may know that every game has a mixed (randomized) Nash equilibrium. Part (d) of problem 2 wants you to design a game with no pure(deterministic) Nash equilibrium. If you do not know game theory, ignore this disclaimer.Problem 3: Monopoly vs. Oligopoly. The marginal cost of a product is fixed atM C = 20. The demand for the product is Q = 100 ? 2P .(a) What is the profit maximizing quantity for a monopolist? What is the price? What isthe total surplus?(b) Now consider a Cournot model with two firms that are choosing quantities simultaneously. What is the best reply (best response) function for each firm? What is theNash equilibrium? What is the total surplus?(c) What do you expect the total surplus would be with three firms? Why? (You do notneed to calculate an exact value. You can say ”total surplus is at least 100”, or ”totalsurplus is at most 80”).Problem 4: Stackelberg Bertrand Game. Two firms are producing identical products,and the marginal cost is fixed at M C = 20. The firms choose prices sequentially. Firm 1, the”leader”, moves first and chooses price p1 . Firm 2, the ”follower”, observes p1 and choosesprice p2 . There are 100 consumers. All of them will buy from the firm with lower price. Ifthe prices are equal, 50 consumers buy from firm 1, and 50 consumers from firm 2. Assumethat the prices are integers.(a) If p1 = 50, what is follower’s best reply?(b) Specify follower’s best reply for any value of p1 .(c) Given the follower’s best reply, what price should the leader set to maximize its payoff? 2 Problem 5: Repeated Cournot and Collusion. Consider the repeated Cournot gamewe discussed in class. The marginal cost is fixed at M C = 40. Demand is P = 100?Q. Firm1 chooses a quantity q1t to produce at each year t = 1, 2, . . . (similarly firm 2 chooses q2t ).Quantities are either 20 or 15. Consider the following strategies: in year 1, firm 1 produces20. In all following years, firm 1 produces 20 units, unless firm 2 produced 15 units in anyyear before, in which case firm 1 produces 15 units. Firm 2’s strategy is identical to firm 1’s.(a) Write down the formula for firm 1’s strategy.(b) What is the present value of the payoff of each firm, given the above strategies? Assumethe discount rate is r = 1.(c) Is the above list of strategies a Nash equilibrium? Why? 3