443- Economics of Law and Regulation
Subject: Economics    / General Economics
Question
443: Economics of Law and Regulation
Homework 2 Lecture 1-4
Chapters 1 to 4 This is Homework #2. You need to explain your answers. Each problem has 20 points. The
answers 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&amp;D Rivalry. Consider the R&amp;D model discussed in class. A firm needs
to choose how quickly to finish the development project of its next product. The present
value of the cost to the firm of finishing the project in T years is 1/T million dollars. The
benefit depends on the number of rivals as follows. At year T , the firm is going to enjoy a
benefit of 1 million dollars. Every year after that, the firm shares the benefit of 1 million
dollars equally with all its rivals (so if there is one rival, the firm gets 500 thousand dollars.)
1
1
1
+ (1+r)
Assume that the discount rate is r = 1. Recall the formula 1+r
2 + . . . = r.
(a) Assume that the firm has one rival. What is the present value of the benefit that the
firm 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 that
the 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 another
curve when there is three rivals. Draw the firm’s cost curve. On the picture specify
the optimal T when there is one rival and when there is three rivals (you do not need
to calculate optimal T exactly).
Reminder. If the benefit of an investment is 1 a year from now, and is 4 two years from now,
1
4
1
4
+ (1+r)
then with discount rate r = 1, the present value of the benefit is 1+r
2 = 2 + 4 = 1.5.
Problem 2: Game Theory. Consider a game with the following payoff matrix.
Up
Down Left Right
1,1
0,0
0,0
2,2 (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 2
players, 2 strategies for each player, and the payoffs for each player should be either 0
or 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 at
M 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 is
the 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 the
Nash equilibrium? What is the total surplus?
(c) What do you expect the total surplus would be with three firms? Why? (You do not
need to calculate an exact value. You can say ”total surplus is at least 100”, or ”total
surplus 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 chooses
price p2 . There are 100 consumers. All of them will buy from the firm with lower price. If
the prices are equal, 50 consumers buy from firm 1, and 50 consumers from firm 2. Assume
that 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 game
we discussed in class. The marginal cost is fixed at M C = 40. Demand is P = 100?Q. Firm
1 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 produces
20. In all following years, firm 1 produces 20 units, unless firm 2 produced 15 units in any
year 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? Assume
the discount rate is r = 1.
(c) Is the above list of strategies a Nash equilibrium? Why? 3