# Problem Set 1 1. Find the set of subgame perfect equilibria in pur...

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Problem Set 1 1. Find the set of subgame perfect equilibria in pure strategies in the following game. (3, 2, 0) II L2 R2 I L1 R1 II l2 r2 (5, 2, 6) • • • • • • III III (4, 7, 2) (6, 1, 5) (0, 3, 4) (7, 4, 1) L3 R3 l3 r3 A1 (0, 5, 2) (4, 0, 3) U3 D3 III 1 2. (a) Find the complete set of pure strategy Nash equilibria (Hint: write the game in strategic form.) (b) Find the complete set of pure strategy subgame perfect equilibria (4, 1, 3) II L2 R2 I L1 R1 II l2 r2 (0, 0, 0) • • • • III • (2, 2, 4) (4, 2, 0) (1, 1, 1) (3, 4, 5) L3 R3 L3 R3 3. Suppose that a parent and a child have fixed incomes of Ip and Ic, respectively, Ic < Ip < 2Ic. The child first decides how much of income Ic to save (S) for the future, consuming the rest (Ic − S) now. The parent observes the child’s choice of S and chooses a bequest B. The child’s payoff is the sum of current and future utilities U c = ln(Ic − S) + 2 ln(S + B). The parent cares about both his wealth and his child’s happiness: U p = 2 ln(Ip − B) + [ln(Ic − S) + 2 ln(S + B)]. Show that the child saves too little as to induce the parent to leave a larger bequest in the subgame-perfect equilibrium. (Hint: compare equilibrium payoffs in simultaneous and sequential games). 2 Problem Set 2 1.Solve for the Nash equilibrium in the Bertrand model with 2 firms, linear demand and different marginal costs. q1= a − bp1+ ep2 q2= a − bp2+ ep1 b > e > 0 C1(q1) = c1q1, C2(q2) = c2q2, c1 6= c2 2. Suppose two firms compete in quantity in a market with demand P = a − bQ, a, b > 0, where P is the market price and Q is the total output. Both firms have the same constant marginal cost of production ci = c, i = 1, 2. Each firm i is run by a manager who chooses quantity qito maximize his objective function Oi= αiπi+ (1 − αi)Si , where πi is firm i’s profit, Si is firm i’s sales pqi , and αi is a constant chosen by firm i’s owner prior to the quantity setting stage. (a) Suppose managers simultaneously set quantities to maximize Oi , i = 1, 2 for a given pair of contants (α1, α2). Find manager i’s best response function Ri(qj ). Find the Nash equilibrium in the simultaneous move quantity setting game. Find the equilibrium payoffs Oi , i = 1, 2 and profits πi , i = 1, 2 (if you can’t find the exact values, you can write them as functions of the parameters, a, b, c, α1, α2). (b) Suppose that prior to the managers’ simultaneous choice of quantities the owners of the two firms simultaneously and non-cooperatively choose α1 and α2 to maximize their respective firms’ profits, π1 and π2. Find a subgame perfect equilibrium pair of parameters (α1, α2). What are the profits to the owners in this equilibrium? 1 3. Two firms compete in quantity in a market of homogeneous good with demand P = a−bQ, a, b > 0, where P is the market price and Q is the total output. The duopolists play the following game: • At stage one, the duopolists simultaneously choose the amount of research and development to undertake, x1 and x2 for firm 1 and firm 2, respectively. R&D is measured in quality normalized units and the cost of undertaking an amount xi is k 2 x 2 i , i = 1, 2, where we assume k > 8 9b . • At stage two, the firms simultaneously choose quantity given the R&D (x1, x2). If firm i has an R&D level of xi , its total cost at the quantity setting stage is ci(qi , xi) = ( (c − xi)qi if xi ≤ c; 0 if xi > c, where c > 0 is a base per-unit cost that is available to a firm engaging in no research. Hence, for a given profile of choices (x1, q1, x2, q2) of the two firms, the profit of firm i in the overall two-stage game is πi = [a − b(q1 + q2)]qi − (c − xi)qi − kx2 i 2 . (a) For given choices x1 and x2 such that xi≤ c, i = 1, 2, solve for the Nash equilibrium of the second stage quantity setting game. Find the profits of the two firms in this equilibrium in terms of a, b, c, x1, and x2. (b) Find firm 1’s R&D best response function x1= R1(x2) in the first stage of the game. (Remember the assumption that 8 < k so that the second order condition for maximization holds) Is this best response function upward or downward sloping? 9b (c) Find the subgame perfect equilibrium of the two-stage game by finding the first-stage equilibrium levels of research x1 and x2. What are the resulting quantities in the market (in terms of a, b, c, and k) and profits of the firms? (d) Suppose that the two firms could reach an agreement in stage one to slightly reduce their R&D levels below the subgame perfect equilibrium levels, while still maintaining x1 = x2. Is there a joint reduction below the SPNE levels that would benefit both firms? Demonstrate why. 2 Problem Set 3 1. Four firms trying to decide whether to produce in one or more of two independent markets, A and B. Each firm has identical discount factor δ, 0 < δ < 1 and stationary cost function ci A(q) = cq, i = 1, 2, 3, 4. Demand in market A is stationary: the time t demand function (t = 0, 1, 2, . . .) in market A is q t A = D(p t A) where q t A is period t quantity demanded in market A and p t A is period t price in market A. Demand in market B grows over time: the time t demand function is q t B = D(p t B)θ t where θ is a demand growth factor with θ > 1 and δθ < 1. Assume the base demand function D(p) is the same in both markets and there exits a price p¯ such that D(p¯) = 0 for all p ≥ p¯. (a) Suppose firms 1 and 3 enter market A, but not in market B, and firms 2 and 4 do not enter market A. Firms 1 and 3 simultaneously set price in each of an infinite number of periods, t = 0, 1, 2, . . . For what values of δ can the monopoly price be sustained as a subgame perfect equilibrium outcome of the infinitely repeated game? (Assume that when firms tie in price they share the demand.) (b) Suppose that firms 2 and 4 enter market B, but not in market A and firms 1 and 3 do not enter market B. firms 2 and 4 are simultaneous price setters. For what values of δ and θ can the monopoly price be sustained as a subgame perfect equilibrium? (c) Suppose firms 1 and 2 enter both markets and firms 3 and 4 do not enter either market. Firms 1 and 2 simultaneously set price in both markets in each period. For what values of δ and θ can the monopoly price be sustained in both markets as a subgame perfect equilibrium outcome of the infinitely repeated game? 2. Consider a Cournot duopoly operating in a market with inverse demand P(Q) = a − Q, where Q = q1 + q2 is the aggregate quantity in the market. Each firm has total costs Ci(qi) = cqi . Demand is uncertain: it is high (a = aH) with probability θ and low (a = aL) with probability 1 − θ, aH > aL > c. There is information asymmetry: firm 1 knows whether demand is high or low, but firm 2 does not. The two firms simultaneously choose quantities. What is the Bayesian Nash equilibrium to this game? What restrictions concerning aH, aL, θ, and c to ensure positive equilibrium quantities? 1 3. Solve for the perfect Bayesian Nash equilibrium in the following game. D U L t2 Sender Sender Nature 0.5 D 0.5 U L t1 Receiver R D U R Receiver D U • • • • • • • • • • • • • • • 1 − p p 1 − q q 3, 1 0, 0 2, 0 1, 2 2, 2 1, 0 3, 0 0, 1 2

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