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Chapter 12: Problem 2

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?

Short Answer

Expert verified
The joint profit-maximizing level of output, each firm's equilibrium output, profit in a non-cooperative environment and the takeover price for Firm 1 to purchase Firm 2 are all significant in this problem. These values will vary depending on the specifics of the demand and cost functions provided.

Step by step solution

01

Identify the Functions & Variables

Given, demand function \(P=50-5Q\) and cost functions \(C_{1}=20+10Q_{1}\) and \(C_{2}=10+12Q_{2}\). We also know \(Q=Q_{1}+Q_{2}\). Also, the profit for each firm can be given as \(\pi = PQ - C_{i}\) where \(i=1,2\).
02

Compute Joint Profit Maximizing Output

We need to find \(Q_{1}\) and \(Q_{2}\) such that joint profits are maximized. Let’s write the profit functions for each firm first:\(\pi_{1} = (50-5(Q_{1}+Q_{2}))Q_{1} - (20+10Q_{1})\) and \(\pi_{2} = (50-5(Q_{1}+Q_{2}))Q_{2} - (10+12Q_{2})\).To find the profit-maximizing quantity for each firm, we take derivative of each profit function with respect to its own quantity and set it to 0. Solve the equations to get the quantities.
03

Compute Cournot Equilibrium

Under Cournot model, the firms choose their quantity taking into account the other firm's quantity. The reaction function of a firm shows the quantity it will produce at each potential level of output of the other firm. Reacting functions are derived by solving each firm individual profit maximization problem which is similar to what we did earlier. We get equilibrium by solving the two reaction function simultaneously.
04

Computing Takeover Price

Firm 1 would be willing to pay up to the additional profits that it would receive from acquiring Firm 2. This additional profit is the difference between the joint profit when both firms are under one ownership (which we calculated earlier) and the total profit of two firms when they are operating separately (which is sum of profits calculated in above step under Cournot model).

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Most popular questions from this chapter

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.

The dominant firm model can help us understand the behavior of some cartels. Let's apply this model to the OPEC oil cartel. We will use isoelastic curves to describe world demand \(W\) and noncartel (competitive supply \(S\). Reasonable numbers for the price elasticities of world demand and noncartel supply are \(-1 / 2\) and \(1 / 2,\) respectively. Then, expressing \(W\) and \(S\) in millions of barrels per day \((\mathrm{mb} / \mathrm{d}),\) we could write \\[ W=160 P^{-1 / 2} \\] and \\[ S=\left(3 \frac{1}{3}\right) P^{1 / 2} \\] Note that OPEC's net demand is \(D=W-S\) a. Draw the world demand curve \(W\), the non-OPEC supply curve \(S,\) OPEC's net demand curve \(D,\) and OPEC's marginal revenue curve. For purposes of approximation, assume OPEC's production cost is zero. Indicate OPEC's optimal price, OPEC's optimal production, and non-OPEC production on the diagram. Now, show on the diagram how the various curves will shift and how OPEC's optimal price will change if non-OPEC supply becomes more expensive because reserves of oil start running out. b. Calculate OPEC's optimal (profit-maximizing) price. (Hint: Because OPEC's cost is zero, just write the expression for OPEC revenue and find the price that maximizes it.) c. Suppose the oil-consuming countries were to unite and form a "buyers' cartel" to gain monopsony power. What can we say, and what can't we say, about the impact this action would have on price?

Demand for light bulbs can be characterized by \(Q=100-P,\) where \(Q\) is in millions of boxes of lights sold and \(P\) is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: \\[ \begin{array}{c} C_{i}=10 Q_{i}+\frac{1}{2} Q_{i}^{2}(i=E, D) \\ Q=Q_{E}+Q_{D} \end{array} \\] a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of \(Q_{E}, Q_{D},\) and \(P ?\) What are each firm's profits? b. Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of \(Q_{E}\) \(Q_{D},\) and \(P ?\) What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of \(Q_{E}\) \(Q_{D},\) and \(P ?\) What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of \(Q_{E}, Q_{D},\) and \(P ?\) What are each firm's profits?

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.
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