(5) Find each concessions reaction function (i.e. the optional choice of own price as a function of rival price.) (6) Solve for the equilibrium prices. (7) Solve for the equilibrium values for q1 and q2. Solve for equilibrium profits for the two concessions. (8) Given your answers to (1) and (6), what values will the two firms choose for V1 and V2? Explain.
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Consider a duopoly game in which two firms simultaneously and independently select prices. Assume that prices cannot be negative. Let p1 denote the price set by firm 1 and p2 denote the price set by firm 2. Unlike Bertrand competition (see Chapter 10), we assume that products are differentiated. To be precise, once prices are set by both firms, consumers demand 10 - p1 + p2 units from the good that firm 1 produces, and they demand 10 - p2 + p1 units from the good that firm 2 produces. Assume that each firm must supply the number of units demanded. Also assume that the cost of producing qi units is equal to q for firm i=1,2. a) Write the payoff functions for both players (as functions of their strategies p1 and p2). b) Characterize each player's best response function (as a function of the price set by the other player). That is, characterize BR1(p) and BR2(p). Are prices strategic substitutes or complements in this game? c) Can you determine the set of rationalizable strategies in this game by inspection of players' best-response functions? What is the set of rationalizable strategies?
Amman Z.
Consider a duopoly game in which two firms simultaneously and independently select prices. Assume that prices cannot be negative. Let p1 denote the price set by firm 1 and p2 denote the price set by firm 2. Unlike Bertrand competition (see Chapter 10), we assume that products are differentiated. To be precise, once prices are set by both firms, consumers demand 10 - p1 + p2 units from the good that firm 1 produces, and they demand 10 - p2 + p1 units from the good that firm 2 produces. Assume that each firm must supply the number of units demanded. Also assume that the cost of producing qi units is equal to 1/2 * qi for firm i = 1, 2. (a) Write the payoff functions for both players (as functions of their strategies p1 and p2). (b) Characterize each player's best response function (as a function of the price set by the other player). That is, characterize BR1(p2) and BR2(p1). Are prices strategic substitutes or complements in this game? (c) Can you determine the set of rationalizable strategies in this game by inspection of players' best-response functions? What is the set of rationalizable strategies?
Akash M.
Two firms (A and B) play a quantity competition game (i.e. Cournot) in which they can choose any Qi from 0 to ∞. The firms have the same cost functions C(Qi) = 10Qi + 0.5Qi^2, and thus MCi = 10 + Qi. They face a market demand curve of P = 220 – (QA + QB). Assume the firms choose quantity simultaneously. 1.1. What is firm A’s profit as a function of QA and QB? (Hint: substitute the inverse demand function P into the typical profit formula to get a function with both QA and QB.) Since costs are symmetric, firm B’s profit function is identical to firm A’s profit function with QA and QB flipped. 1.2. The marginal revenues are given by MRA = 220 – 2QA – QB and MRB = 220 – 2QB – QA. What is firm A’s best response to an arbitrary QB selected by firm B? (Firm B’s best response function is identical to firm A’s best response function with QA and QB flipped.) 1.3. What are the equilibrium QA and QB selected in this game? 1.4. What is the equilibrium price, and how much profit does each firm collect? Now assume firm A chooses quantity first. Firm B observes this choice and then chooses its own quantity. (Otherwise everything is the same as in Q1). 1.5. What is firm B’s profit as a function of QA and QB? 1.6. Firm B has MRB = 220 – 2QB – QA. What is firm B’s best response to an arbitrary QA selected by firm A? 1.7. Given that firm A expects firm B’s best response, what is firm A’s profit as a function of QA? (Hint: the only unknown variable in the profit function should be QA. This profit function will be different from Q1.1 and Q1.5). 1.8. Firm A has MRA = 150 – 4QA/3. What are the equilibrium QA and QB selected in this game? 1.9. What is the equilibrium price, and how much profit does each firm collect?
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