ps6_sol0-1 - Solutions to Practice Problem Set 6 ECON 100C...

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1 Solutions to Practice Problem Set 6 ECON 100C Perloff Chapter 13 13.14 What happens to the homogeneous-good Bertrand equilibrium price if the number of firms increases? Why? The equilibrium price in the homogeneous-good Bertrand equilibrium is equal to marginal cost, so the entry of additional firms will not affect the price (unless the new entrants have lower costs than the incumbents). 13.25 Suppose that identical duopoly firms have constant marginal costs of $10 per unit. Firm 1 faces a demand function of ݍ ൌͳͲͲെʹ݌ ൅݌ , where ݍ is Firm 1’s output, ݌ is Firm 1’s price, and ݌ is Firm 2’s price. Similarly the demand Firm 2 faces is ݍ ൌͳͲͲെʹ݌ ൅݌ . Solve for the Bertrand equilibrium. See solution in the back of the book. 13.26 Solve for the Betrand equilibrium for the firms described in Problem 25 if both firms have a marginal cost of $0 per unit. We can first find the profit-maximizing price for Firm 1 as a function of the price charged by Firm 2 (Firm 1’s best-response function): ߨ ൌ݌ ݍ ሺ݌ ǡ݌ ሻെ ܿ ሺݍ ൌ݌ ሺͳͲͲെʹ݌ ൅݌ ߲ߨ ߲݌ ൌͳͲͲെͶ݌ ൅݌ ؠͲ ՜݌ ൌʹͷ൅ ݌ Ͷ Note that this best-response function is upward sloping. If Firm 2 charges a higher price, Firm 1 should increase its price as well in order to maximize profits. By symmetry, Firm 2’s best-response function has the same functional form: ݌ ൌʹͷ൅ ݌ Ͷ Solving these two equations simultaneously to find the Nash equilibrium yields: ݌ ൌ݌ ͳͲͲ ͵ 13.27 Solve for the Betrand equilibrium for the firms described in Problem 25 if Firm 1’s marginal cost is $30 per unit and Firm 2’s marginal cost is $10 per unit. We can use the same approach—find both best-response functions (profit-maximizing price as a function of the other firm’s price), and then find the set of prices that satisfy
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2 both equations simultaneously: ߨ ൌ݌ ݍ ሺ݌ ǡ݌ ሻെ ܿ ሺݍ ൌ݌ ሺͳͲͲെʹ݌ ൅݌ ሻെ ͵ͲሺͳͲͲെʹ݌ ൅݌ ߲ߨ
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