Consider a homogeneous-good duopoly with linear demand P(q) = 1-q, where q is the total industry...

 

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Consider a homogeneous-good duopoly with linear demand P(q) = 1-q, where q is the total industry output. Suppose that firms are quantity-setters and firms incur constant marginal costs of production ci. 1. (8 points) Consider the duopoly with c₁ = 0 and c2 = c = [0, 1]. Determine the equilibrium (price, quantities, profit). Hint: The active firms are different for different levels of c: there are two different cases. 2. (5 points) Consider now an extended model in which only firm 1 is necessarily present. At stage 1, firm 1 can make an investment I after which firm 2's marginal cost is c2 = 1/2 instead of c₂ = 0. Afterwards, firm 2 observes the investment decision of firm 1 and, at stage 2, decides whether to enter at a negligible entry cost e > 0. At stage 3, active firms set quantities simultaneously. Determine the subgame-perfect equilibrium of this game. 3. (6 points) Consider now a different entry model. Both firms have zero marginal costs of production but consumers have become accustomed to product 1 (even if they did not consume it themselves). Therefore, consumers are willing to pay 1/2 money units less for product 2 than for product 1. The inverse demand function P(q) = 1-q gives demand for product 1. At stage 1, the potential entrant, firm 2, considers entering the market at an entry cost e > 0. (There is no investment stage in this game.) At stage 2, firms set quantities simultaneously. Report the profit function for each firm. Determine the equilibrium in case firm 2 has entered (report prices, quantities, profit, welfare). Determine the subgame-perfect equilibrium and comment on your result. You may want to reuse some of the results derived above. Consider a homogeneous-good duopoly with linear demand P(q) = 1-q, where q is the total industry output. Suppose that firms are quantity-setters and firms incur constant marginal costs of production ci. 1. (8 points) Consider the duopoly with c₁ = 0 and c2 = c = [0, 1]. Determine the equilibrium (price, quantities, profit). Hint: The active firms are different for different levels of c: there are two different cases. 2. (5 points) Consider now an extended model in which only firm 1 is necessarily present. At stage 1, firm 1 can make an investment I after which firm 2's marginal cost is c2 = 1/2 instead of c₂ = 0. Afterwards, firm 2 observes the investment decision of firm 1 and, at stage 2, decides whether to enter at a negligible entry cost e > 0. At stage 3, active firms set quantities simultaneously. Determine the subgame-perfect equilibrium of this game. 3. (6 points) Consider now a different entry model. Both firms have zero marginal costs of production but consumers have become accustomed to product 1 (even if they did not consume it themselves). Therefore, consumers are willing to pay 1/2 money units less for product 2 than for product 1. The inverse demand function P(q) = 1-q gives demand for product 1. At stage 1, the potential entrant, firm 2, considers entering the market at an entry cost e > 0. (There is no investment stage in this game.) At stage 2, firms set quantities simultaneously. Report the profit function for each firm. Determine the equilibrium in case firm 2 has entered (report prices, quantities, profit, welfare). Determine the subgame-perfect equilibrium and comment on your result. You may want to reuse some of the results derived above.

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Solution Answer 1 In a homogeneousgood duopoly with linear demand Pq 1q and constant marginal costs of production c 0 the equilibrium is determined by the following conditions 1 The market clearing co... View the full answer