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Exercise 9.3 The aim of this exercise is to show that the independence assumption of the functions in (9.7) cannot be omitted in Proposition 9.3. 9.6 Exercises 139 (a) Show first that the time-homogeneous version of Proposition 5.2 can be derived as a corollary of Proposition 9.3 for m = 1. (b) Consider the two-factor state process dZ, = Zadi + min[1. ZiIdw*. dZ2 = min[1. Z1] di with state space Z = R. You can assume, without proving it, that there exists an Ra-valued solution Z = 2: with Z(0) = z, for every z e R.. Show that this state process is not affine but is nevertheless consistent with the ATS (x, () = Exercise 9.4 Extend Proposition 9.3 to the case of a time-inhomogeneous ATS: $(x. 2) = 80(x. 20) + 81 (x, 20)21 + + + 8m (1, 20)2m. where the state vector : = (zo. 21. .... [m) E R. x Z is extended accordingly by calendar time zo. Exercise 9.5 Find the general form of a one-factor (m = 1) QTS model f(, 7) = 501 -781\\ -1)2()+2(T -1) 2() for some real-valued (Z = R) diffusion state process Z. Show there are specifications which yield an ATS. Exercise 9.6 Consider the Nelson Siegel family ous(x. 2) = 2 + (22 + zur)e -Lax (a) Check whether the linear independence assumption of Proposition 9.2 is satis- fied. (b) Give a full proof of Proposition 9.4. Exercise 9.7 Consider the Hull-White extended Vasicek short-rate model dr(1) = (zizs + zie"til + ze list - zar(1) )di + vzzgetst aw*(). which is consistent with the Svensson family given in Proposition 9.5. Show that the zero-coupon bond price equals P(. T)=e-Au.)-80.)rif) where r() = 21 + Zz(t) with Z2(1) =e (22 + zyr + #4 (1 -e-zor) ) + /zazge"=> W"() and A(, 7) = =(e-23(7-0) - 1+ 25(7 - 1))+ 23e aT (ets(7-0) -1-25(7 -1)) 2(c s(7-0) - 1 -225(7 -1)). Bu. 7) =-(1-e-3(-m.5. Assume the inverse demand function for a Bertrand oligopoly is P = 300 -Q/4, and that the firm's total cost function is TC = 200 + 100Q. What is the firm's profit maximizing output? MC = 250 a. 400 b. 800 c. 1,200 d. 1,600 6. In a finite repeating game with a definite end, collusion is possible if a. the interest rate is very high b. laws prohibit firms from communicating with each other c. there is a high probability of the game continuing. d. This question makes no sense since collusion is not possible in a finite repeating game with a definite end. 7. Firms A and B compete in an industry. Suddenly, Firm A lowers its price so that is lower than the price charged by Firm B. Firm B files a complaint with the government that Firm A is engaging in predatory pricing. What additional information would the government need to know to determine if Firm A is guilty of predatory pricing? a. whether or not Firm A's profits increased b. whether or not Firm B's profits decreased c. how the selling price of Firm A compares to its average total cost at its chosen output d. how the selling price of Firm A compares to the marginal cost of Firm B c. how the selling price of Firm A compares to its marginal cost at its chosen output 8. Suppose two types of consumers buy suits. Consumers of type A are willing to pay $120 for a coat and $90 for pants. Consumers of type B are willing to pay $80 for a coat and $100 for pants. The firm selling suits faces no competition and has a marginal cost of zero. What is the optimal commodity bundling strategy? a. charge $165 for a suit # 120 90 120 b. charge $170 for a suit - c. charge $180 for a suit YO 80x2 = Ka d. charge $210 for a suit 9: In a finite repeating game with an indefinite end, collusion is most likely if a. there is a low probability of the game continuing there is a high probability of the game continuing This question makes no sense since collusion is not possible in a finite repeating game with an indefinite end. 10, Penetration pricing is a strategy used by new firms in an industry that charge a low inition price to gain a critical mass of consumers, . True b. False11. A mobile phone provider offers two contracts. Contract I offers 100 voice minutes for $25 per month, plus 10 cents per minute for each additional minute beyond that. Contract II offers unlimited talking minutes for a flat fee of $50 per month (i.e., a unit price of zero). Assuming that the firm has designed these contracts optimally to price discriminate between high-value and low- value customers, which of the following statements is true? (a) High-value customers prefer the bundle with the lower monthly fee. (b) The firm's marginal cost of providing an additional voice minute is zero. (c) The firm's marginal cost of providing an additional voice minute is 10 cents. (d) The quantity purchased by low value customers is efficient. 12. A firm attempting to price discriminate through self selection faces two types of constraints: incentive compatibility and self-selection. Which of the following statements best describes the economic content of these constraints? (a) High value customers must prefer the expensive bundle to the cheap bun- dle, while low value customers must prefer purchasing the cheap bundle to not purchasing. (b) High value customers are indifferent between purchasing and not purchas- ing, while low value customers prefer not to purchase. (c) Low value customers are indifferent between the cheap bundle and the expensive bundle, while high type consumers strictly prefer the expensive bundle to the cheap bundle.15. In Industry A (Airlines) firms must decide on capacity well in advance of set- ting prices, while in Industry B (Banking) firms can quickly adjust production to meet any quantity demanded. Suppose that the number of firms in each industry increases from a monopoly to a duopoly. In which industry would you expect this change to produce a larger price change, and why? (a) There would be a larger price change in Industry A, since Industry A is Bertrand while Industry B is Cournot. (b) There would be a larger price change in Industry B, since Industry A is Bertrand while Industry B is Cournot. (c) There would be a larger price change in Industry A, since Industry A is Cournot while Industry B is Bertrand. (d) There would be a larger price change in Industry B, since Industry A is Cournot while Industry B is Bertrand. 16. Which of these statements best explains the game-theoretic relationship be- tween repeated interaction and collusion, as developed in lecture? (a) Repeated interaction can be used to sustain collusion, unless defection is a dominant strategy for some player in the one-shot (static) game. (b) With repeated interaction, the threat of future competition can be used to sustain collusion, as long as both players assign sufficient value to gains from future cooperation. (c) With repeated interaction, each player may eventually learn to anticipate their rival's moves. This can be used for either competition or cooperation. (d) With repeated interaction, one player may cooperate to lull their rival into a false sense of security. This accomplished, they can then strike.13. Which of the following is NOT a feature of a Nash equilibrium? (a) No player can gain by unilaterally changing their strategy. (b) The actions chosen maximize the joint payoffs for all players. (c) Best responses for all players are compatible (i.e., intersect). (d) Each player maximizes their own payoff, taking rival strategies as given. 14. What is the Nash equilibrium of the following simultaneous-move game? P1 \\ P2 | L R (6, 6) (2, 8) R (8, 2) (1, 1) (a) (L, L) only. (b) (R, R) only. (c) Both (L, L) and (R, R). (d) Both (L, R) and (R, L)
