# ECON 357 – Consider a Stackelberg game where the incumbent and entrant face

June 14, 2018

ECON 357 – Problem Set 3Due Thursday, June 30 at the beginning of class.1. (30 marks) Consider a Stackelberg game where the incumbent and entrant face ainverse demand function of p(Q) = 500 ? 4Q. The incumbent faces a cost functionof c(q1 ) = 10 + q1 and the entrant faces a cost function of c(q2 ) = A + q2 .a) Derive the best-response function of the entrant. (5 marks)b) Determine the incumbent’s choice of q1 . (10 marks)c) Determine the size of A necessary for the entrant to be deterred from enteringthe market under Stackelberg. (10 marks)d) Explain why the Stackelberg outcome requires knowing player 2’s best-responsefunction, but not player 1’s. (HINT: A complete answer considers the effect ofsequential movement and commitment.) (5 marks)2. (25 marks) Consider the following version of the battle of the sexes game:P layer1EFP layer2AB(5, 1)(0, 0)(0, 0)(2, 7)a) Find the Pure Strategy Nash Equilibrium to the game. (5 marks)b) Find the Mixed Strategy Nash Equilibrium to this game. (10 marks)c) Graph the best-response curves for each player. Show the pure and mixedstrategies found in a) and b) on your graph.(10 marks)3. (35 marks) Consider the penalty point game. Let player 1 be the “kicker” and player2 be the “goalie”. Both players can choose one of two actions: left or right. If bothplayers 1 and 2 go left, player 1 receives a payoff of 20 and player 2 receives a payoffof -20. If both players go right, player 1 receives a payoff of 30, and player 2 receives-30. If player 1 goes left and player 2 goes right, player 1 gets 90, player 2 gets -90.If player 1 goes right and player 2 goes left, player 1 gets 70, player 2 gets -70.a) Represent this as a normal-form game. (5 marks)b) Determine the Nash Equilibria to this game. (15 marks)c) Suppose player 2 can lower player 1’s payoff by 10 if opposite strategies areplayed (ie. for the states where player 1 plays left and 2 plays right, or player1 plays right and player 2 plays left). This also raises player 2’s payoff by 10if either state is realized. Suppose each player’s payoff is expressed in dollars.How much would player 2 be willing to pay to to cause this difference in payoffsif player 2 is risk-neutral? (15 marks)q4. (45 marks) A’s utility function is U (x1A , x2A ) = x1A +4 x2A while B’s utility function isqU (x1B , x2B ) = x1B + 16 x2B , where the superscript denotes the good and the subscriptdenotes the person. A’s initial endowment of goods 1 and 2, respectively, is given by1 = 12 and ? 2 = 20. B’s initial endowment is ? 1 = 18 and ? 2 = 4?AABBa) Derive an expression for each of the individual’s marginal rate of substitutiondx2(| dx1i |). (10 marks)ib) On the contract curve, A’s marginal rate of substitution equals B’s. Write anequation that states this condition. From this equation, find the value ofx2Ax2Batall points along the contract curve. (10 marks)2 + ? 2 Use thisc) Along the contract curve it is also the case that x2A + x2B = ?ABin conjunction with the value ofx2Ax2Bfrom part b) to show that x2A and x2B areconstant on the contract curve. (10 marks)d) Find the equilibrium prices and quantities. (10 marks)1e) Draw the Edgeworth box. Label the initial endowments with the letter W. Drawindifference curves for each person and show the contract curve for this exchangeeconomy. (5 marks)5. (35 marks) Robinson Crusoe spends 10 hours a day gathering food. He can eitherspend his time gathering coconuts or catching fish. He can catch 1 fish per hour andhe can gather 4 coconuts per hour.a) Show Robinson’s production possibility frontier between fish and coconuts perday with coconuts on the vertical axis. Write an equation for the line segmentthat is Robinsons production possibilities frontier. (5 marks)11b) Robinson’s utility function is U (f, c) = f 2 c 2 , where f is fish consumption and cis coconut consumption. How many fish will Robinson choose to catch per day?How many coconuts will he collect? (15 marks)c) Suppose that Robinson obtains a technology that allows him to catch more fish.Specifically, Robinson can now catch 3 fish per hour. Determine Robinson’s newproduction choice. (15 marks)2

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