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Chapter 12: Problem 10

Two firms produce luxury sheepskin auto seat covers: Western Where (WW) and B.B.B. Sheep (BBBS). Each firm has a cost function given by \\[ C(q)=30 q+1.5 q^{2} \\] The market demand for these seat covers is represented by the inverse demand equation \\[ P=300-3 Q \\] where \(Q=q_{1}+q_{2},\) total output. a. If each firm acts to maximize its profits, taking its rival's output as given (i.e., the firms behave as Cournot oligopolists), what will be the equilibrium quantities selected by each firm? What is total output, and what is the market price? What are the profits for each firm? b. It occurs to the managers of WW and BBBS that they could do a lot better by colluding. If the two firms collude, what will be the profit-maximizing choice of output? The industry price? The output and the profit for each firm in this case? c. The managers of these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce the Cournot quantity or the cartel quantity. To aid in making the decision, the manager of \(\mathrm{WW}\) constructs a payoff matrix like the one below. Fill in each box with the profit of \(\mathrm{WW}\) and the profit of BBBS. Given this payoff matrix, what output strategy is each firm likely to pursue? d. Suppose WW can set its output level before BBBS does. How much will WW choose to produce in this case? How much will BBBS produce? What is the market price, and what is the profit for each firm? Is WW better off by choosing its output first? Explain why or why not.

Short Answer

Expert verified
The calculated values for equilibrium quantities, total output, market price and profits under Cournot, collusion and Stackelberg competition conditions would be the final answer. Since the specific constants are not given in the exercise, the solution would be represented as formulas and depend on the given functions earlier.

Step by step solution

01

Calculate Cournot Equilibrium output

Start with the profit maximization equation for Cournot competitors and set them for each firm. The profit function for each firm i (i can be 1 or 2) is given by \[ \Pi_i = P \cdot q_i - C(q_i) \] where \( P = 300-3Q = 300 - 3(q_1 + q_2) \), and the cost function \( C(q) = 30q + 1.5q^{2} \). Equate the first-order derivative of the profit function with respect to quantity to zero for each firm, and solve the resulting equations simultaneously for \( q_1 \) and \( q_2 \).
02

Find total output and market price

Find the total output \( Q \) by adding \( q_1 \) and \( q_2 \) obtained in Step 1. Then calculate the market price \( P \) using the inverse demand equation given by \( P = 300 - 3Q \).
03

Determine the firm's profits

Substitute the obtained values of total output \( Q \) and quantities \( q_1 \) and \( q_2 \) into the profit function \( \Pi = P \cdot q_i - C(q_i) \) to calculate the profits of each firm.
04

Find Collusion Output

Calculate the collusion output by treating both firms as a single monopolist and maximizing the joint profits. This means that the total cost function is now double the given cost function, and the first order condition derived from joint profit function (sum of the individual profit functions) is set to zero. Solve for \( Q \) to get the collusion output.
05

Determine output under Stackelberg competition

This involves considering a sequential game where one firm (WW) moves first. WW's profit-maximizing output is found by considering the reaction function of BBBS, where the reaction function is the quantity BBBS would optimally produce for each possible quantity WW might produce. WW takes this into account and then chooses its own output to maximize profit. Find this output of WW, and then substitute this value into BBBS's reaction function to find BBBS's output.
06

Calculate Stackelberg market price and profits

Substitute the obtained values of WW and BBBS's output into the inverse demand function to get the market price. Then substitute these values along with market price into each firm's profit function to get their respective profits.

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Most popular questions from this chapter

Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by \(C_{1}=60 Q_{1}\) and \(C_{2}=60 Q_{2},\) where \(Q_{1}\) is the output of Firm 1 and \(Q_{2}\) the output of Firm 2. Price is determined by the following demand curve: \\[ \begin{aligned} P &=300-Q \\ \text { where } Q=Q_{1}+Q_{2} \end{aligned} \\] a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium. b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm's profit. c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1's profit differ from that found in part (b) above? d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm's profits?

Suppose all firms in a monopolistically competitive industry were merged into one large firm. Would that new firm produce as many different brands? Would it produce only a single brand? Explain.

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.

Two firms compete in selling identical widgets. They choose their output levels \(Q_{1}\) and \(Q_{2}\) simultaneously and face the demand curve \\[ P=30-Q \\] where \(Q=Q_{1}+Q_{2}\). Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm 2 's marginal cost to \(\$ 15 .\) Firm 1 's marginal cost remains constant at zero. True or false: As a result, the market price will rise to the monopoly level.

Consider two firms facing the demand curve \(P=50-5 Q\), where \(Q=Q_{1}+Q_{2}\). The firms' cost functions are \(C_{1}\left(Q_{1}\right)=20+10 Q_{1}\) and \(C_{2}\left(Q_{2}\right)=10+12 Q_{2}\) a. Suppose both firms have entered the industry. What is the joint profit- maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? b. What is each firm's equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms' reaction curves and show the equilibrium. c. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but a takeover is not?
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