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Chapter 12: Problem 11

Two firms compete by choosing price. Their demand functions are \\[ Q_{1}=20-P_{1}+P_{2} \\] and \\[ Q_{2}=20+P_{1}-P_{2} \\] where \(P_{1}\) and \(P_{2}\) are the prices charged by each firm, respectively, and \(Q_{1}\) and \(Q_{2}\) are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero. a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.) b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be? c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

Short Answer

Expert verified
The Nash equilibrium prices, quantities, and profits for each scenario are calculated by maximizing the profit functions using the demand functions provided. The preferred strategy depends on which scenario results in the highest profit for each firm.

Step by step solution

01

Calculating the Nash Equilibrium for Simultaneous Pricing

Firstly both firms maximize their profits, supposing the other firm's action is given. For zero marginal costs, the profit of a firm is just the firm’s price times the quantity it sells, namely \( \Pi_{i}=P_{i}Q_{i} \). Firm 1 maximizes: \( \Pi_{1}=P_{1}(20-P_{1}+P_{2}) \) and Firm 2 maximizes: \( \Pi_{2}=P_{2}(20+P_{1}-P_{2}) \). To find the best responses, take the first derivatives with respect to own price and setting equal to zero. This allows us to find the prices \( P_{1} \) and \( P_{2} \) that maximize the profit for each firm.
02

Substituting and Calculating the Equilibrium Prices and Profits

Solving the equations from step 1, allows us to find the Nash equilibrium which is the price combination ( \( P_{1} \), \( P_{2} \)) where neither firm would want to deviate, given the price of the other. The equilibrium quantities and profits can also be calculated using these equilibrium prices.
03

Calculating for Sequential Pricing

Here, we suppose Firm 1 sets its price first (it becomes the leader) and then Firm 2 sets its price. Find the best response of Firm 2 as in step 1 but by treating \( P_{1} \) as a given. Substitute this into the profit function of firm 1 and maximize it with respect to \( P_{1} \) to obtain \( P_{1} \). The resulting equilibrium prices, quantities and profits can also be calculated for this scenario.
04

Selecting Optimal Strategy Given three Options

Analyzing the results of the previous steps, decide which scenario yields the highest profit for a firm. Consider the three available options: (i) Both firms set price at the same time; (ii) Firm sets price first; or (iii) competitor sets price first.

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Most popular questions from this chapter

A monopolist can produce at a constant average (and marginal) cost of \(\mathrm{AC}=\mathrm{MC}=\$ 5 .\) It faces a market demand curve given by \(Q=53-P\) a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits. b. Suppose a second firm enters the market. Let \(Q_{1}\) be the output of the first firm and \(Q_{2}\) be the output of the second. Market demand is now given by \\[ Q_{1}+Q_{2}=53-P \\] Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of \(Q_{1}\) and \(Q_{2}\) c. Suppose (as in the Cournot model) that each firm chooses its profit- maximizing level of output on the assumption that its competitor's output is fixed. Find each firm's "reaction curve" (i.e., the rule that gives its desired output in terms of its competitor's output). d. Calculate the Cournot equilibrium (i.e., the values of \(Q_{1}\) and \(Q_{2}\) for which each firm is doing as well as it can given its competitor's output). What are the resulting market price and profits of each firm? *e. Suppose there are \(N\) firms in the industry, all with the same constant marginal cost, \(\mathrm{MC}=\$ 5 .\) Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much profit will each firm earn? Also, show that as N becomes large, the market price approaches the price that would prevail under perfect competition.

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