Write your answers to each exercise in a different page. Show all your work, and be as clear as
possible in your answer. You can work in groups, but each student must submit a copy of his/her
exam. The due date of this take-home exam is Tuesday, March 26th, in class. I strongly
recommend you to work a few exercises every day, rather than trying to solve all exercises in
one day. Late submission will be subject to significant grade reduction.
2
Exercise 1
Consider the entry-exit two-stage game represented in the figure below in which firm A is the incumbent
firm that faces potential entrant firm B. In stage I, firm B decides whether to enter into As market or
whether to stay out. The cost of entry is denoted by . In stage II, the established firm, firm A, decides
whether to stay in the market or exit.
The game tree reveals that firm A can recover some of its sunk entry cost by selling its capital for the
price , where 0 . Solve the two problems:
a) Compute the subgame-perfect equilibrium strategies of firms B and A assuming that < 60 . Prove your answer. Answer the above question assuming that 60 < <100 Exercise 2 Return to the game with two neighbors creating positive externalities on each other that you solve in Homework #4 (Exercise #3, see the exercise and its answer key on the course website). Continue to suppose that player is average benefit per hour of work on landscaping is 10 2 j i l l + Continue to suppose that player 2s opportunity cost of an hour of landscaping work is 4. Suppose that player ls opportunity cost is either 3 or 5 with equal probability (1/2), and that this cost is player 1s private information (player 2 does not observe this information). a) Find player 1s best response function when his opportunity cost is 3. Denote it as lL 1(l2). b) Find player 1s best response function when his opportunity cost is 5. Denote it as lH 1(l2). c) Find player 2s best response function. Denote it as lL 2(lL 1, lH 1). d) Solve for the Bayesian-Nash equilibrium. Exercise 3 Consider two consumers (1, 2), each with income M to allocate between two goods. Good 1 provides 1 unit of consumption to its purchaser and units of consumption to the other consumer, where 0 1. Each consumer i, i = 1,2, has the utility function 1 2 Ui = log(x j ) + xi where 1 1 1 xi = yi + y j is his consumption of good 1 (that takes into account the amount of good 1 purchased by individual i and j), and 2 i x is his consumption of good 2. 3 a) Provide a verbal interpretation of b) Suppose that good 2 is a private good. Find the Nash equilibrium levels of consumption when both goods have a price of 1. c) By maximizing the sum of utilities, show that the equilibrium is Pareto-efficient if = 0 but inefficient for all other values of . d) Now suppose that good 2 also provides 1 unit of consumption to its purchaser and units of consumption to the other consumer, where 0 1. For the same preferences, find the Nash equilibrium. e) Show that it is efficient for all values of . f) Explain the conclusion in part (d). Exercise 4 There is a given rent of R, for instance, the profits from a new contract with a government agency. Each of two players (e.g., two firms) spends resources competing for the rent (e.g., lobbying). If player 1 spends 1 x , the probability that he wins the rent is 1 1 1 2 p x x x = + when player 2 spends the amount 2 x , and where > 1 . [Note that parameter represents the higher productivity of every unit of effort of
player 1 relative to that of player 2, in terms of increasing the probability of him winning the rent.]
a) What is the optimal spending of player 1 in response to a given spending level of player 2 (best
response function of player 1)? What is the optimal response of player 2 to player 1 (best
response function of player 2)?
b) Draw the best-response function of player 1 and player 2 you found in part (a). Discuss the effect
of changing on the function.
c) How much will each player spend on lobbying in the equilibrium of this rent seeking game?
d) Which player is more likely to win the rent in equilibrium?
e) Compare the total equilibrium spending for > 1 and = 1 .
f) Should we expect more spending in rent-seeking activities when players are identical? Why and
why not?
Exercise 5
Answer the following exercises from the textbooks:
1. Harrington: exercise 1 from Chapter 14, and exercise 1 from chapter 15.
2. Watson: exercises 1 and 5 from Chapter 26.
Exercise 6 [See the last page of the exam].
EconS 424 Strategy and Game Theory
Exercise #6 Midterm Exam #2
6. Consider the following game with incomplete information. Two agents, A and B,
simultaneously decide whether to join or not join a common enterprise. The following
matrix represents the payos accruing to agents A and B under each strategy prole:
Agent B
Join Not Join
Agent A Join 1; 1