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Chapter 12: Problem 14

A lemon-growing cartel consists of four orchards. Their total cost functions are $$\begin{array}{l} \mathrm{TC}_{1}=20+5 Q_{1}^{2} \\ \mathrm{TC}_{2}=25+3 Q_{2}^{2} \\ \mathrm{TC}_{3}=15+4 Q_{3}^{2} \\ \mathrm{TC}_{4}=20+6 Q_{4}^{2} \end{array}$$ TC is in hundreds of dollars, and \(Q\) is in cartons per month picked and shipped. a. Tabulate total, average, and marginal costs for each firm for output levels between 1 and 5 cartons per month (i.e., for \(1,2,3,4,\) and 5 cartons) b. If the cartel decided to ship 10 cartons per month and set a price of \(\$ 25\) per carton, how should output be allocated among the firms? c. At this shipping level, which firm has the most incentive to cheat? Does any firm not have an incentive to cheat?

Short Answer

Expert verified
a. The total, average and marginal costs can be calculated using the given total cost functions and arranged in a table format. b. Allocating 10 cartons should be done in accordance with the marginal costs of each firm until all cartons have been assigned to the firm with the lowest cost. c. The firm with the highest marginal cost will have the most incentive to cheat because it stands to gain the most profit from selling one extra unit at the market price.

Step by step solution

01

Calculate Total, Average, and Marginal Costs

The total cost for each firm is given. The average cost is obtained by dividing the total cost by the quantity, i.e. \(AC_{i} = \frac{TC_{i}}{Q_{i}}\). To find the marginal costs, differentiate the total cost function in respect to quantity, i.e. \(MC_{i} = \frac{d(TC_{i})}{d(Q_{i})}\) for each firm. Tabulate the values for quantities between 1 to 5.
02

Allocate Output Among The Firms

Next, it is required to determine how 10 cartons should be allocated among the firms. This implies, the allocation should be done by considering marginal cost of each firm. One carton will be assigned to the firm that has the lowest marginal cost, then the next carton is assigned to whichever firm has the lowest marginal cost after the first has been assigned, and so on until all 10 cartons have been assigned.
03

Determine The Firm With The Highest Incentive To Cheat

In a cartel, a firm may cheat the system by producing more output than allocated thus increasing its profits. The firm with the highest incentive to cheat will be the one with the highest marginal cost at this level of output because it would stand to gain the most from producing one extra unit as it could sell the extra unit at the market price and keep all of the profits.

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Most popular questions from this chapter

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.

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.

Suppose that two competing firms, \(A\) and \(B\), produce a homogeneous good. Both firms have a marginal cost of \(\mathrm{MC}=\$ 50 .\) Describe what would happen to output and price in each of the following situations if the firms are at (i) Cournot equilibrium, (ii) collusive equilibrium, and (iii) Bertrand equilibrium. a. Because Firm \(A\) must increase wages, its \(\mathrm{MC}\) increases to \(\$ 80\). b. The marginal cost of both firms increases. c. The demand curve shifts to the right.

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?

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.
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