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Chapter 12: Problem 3

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.

Short Answer

Expert verified
a) For a monopolist, the profit-maximising quantity and price can be found by equating MR and MC, resulting in a profit assurance that depends on the chosen quantities. \n b) In a duopoly, each firm's profits depend on its own production level and its competitor's, resulting in reaction curves specific to each firm. \n c) The reaction curve of each firm is derived from maximising its profits on the assumption of its competitors' output being fixed. \n d) The Cournot equilibrium refers to the point where neither firm can better its profit given its competitor's output. \n e) For N firms, the solution is analogous to the case with two firms. As N grows large, firms act similar to perfect competition where price equals marginal cost.

Step by step solution

01

Calculate the Profit-Maximizing Price and Quantity

To find the profit-maximizing price and quantity, we need to set Marginal Revenue (MR) equal to Marginal Cost (MC). First, the total revenue (TR) function is determined by multiplying price (P) by the quantity (Q), which is calculated using the inverse demand curve. The MR is the derivative of the TR function. Since MC is a constant 5, we equate this to the MR and solve for the quantity. This can then be substituted back into either the demand function or price function to find the corresponding price.
02

Profits of Each Firm as Functions of \(Q_1\) and \(Q_2\)

In this step, we determine the profit functions of both firms. Both firms have the same cost structures and thus their profit functions will be similar. The profit function is typically given by \(\pi = TR - TC\), where \(\pi\) represents profit, TR is total revenue, and TC is total cost. Total cost is computed by multiplying the quantity for each firm ( \(Q_1\) or \(Q_2\) ) by the cost per unit. Total revenue for each firm can be computed by multiplying the quantity for each firm ( \(Q_1\) or \(Q_2\) ) by the price, which is determined by the market demand function shared by both firms.
03

Find the Reaction Curve with Cournot Model

The Cournot Model suggests that each firm chooses its output levels based on its competitor's output. To find the reaction functions, differentiate the profit function of each firm with respect to its own output and equate that to zero.
04

Calculate the Cournot Equilibrium

The Cournot equilibrium can be found by simultaneously solving the two reaction function equations obtained in Step 3. This gives the optimal output for each firm. The resulting market price can be computed by substituting either \(Q_1\) or \(Q_2\) into the inverse demand function.
05

Extend the Cournot Model to N firms

The Cournot equilibrium needs to be found in case of N firms. Similar logic as the duopoly case needs to be applied. Another aspect to be shown is that as N becomes large, the market price approaches the price under perfect competition, which is equal to the marginal costs in this case.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Profit-Maximizing Price
When a firm aims to maximize its profits, it sets a specific price that aligns with this goal. The profit-maximizing price is determined where the difference between the firm's total revenue (TR) and total costs (TC) is at its greatest. A fundamental principle in economics is setting the Marginal Revenue (MR) equal to the Marginal Cost (MC).

Considering a monopolistic firm's scenario with a constant marginal cost of \(5, finding the profit-maximizing price involves calculus. One must first establish the TR by multiplying the price, which is derived from the inverse demand function, by the produced quantity, Q. Taking the derivative of the TR function gives us MR. Equating the MR to the MC of \)5, and solving for Q, we find the profit-maximizing quantity. This quantity is essential to pinpoint the profit-maximizing price, after which the firm calculates its profits by subtracting the total cost from the total revenue at this price point.
Reaction Curve
The reaction curve in the context of oligopoly markets, such as the Cournot duopoly, represents a firm's optimal response to the quantity produced by its rival.

In the Cournot model, each firm assumes that its competitor's output is fixed and decides its quantity to maximize profits based on this assumption. The reaction function is mathematically derived by differentiating the profit function with respect to the firm's output and setting the derivative equal to zero. This process reveals how much one firm will produce in response to various quantities produced by the other firm. When dealing with multiple firms, as in the extended Cournot model, the reaction curve for each firm will be influenced by the collective output of all its competitors.
Marginal Cost
Marginal Cost (MC) is a critical concept in economics and refers to the additional cost incurred for producing one more unit of a good. In our textbook example, the constant MC is denoted by $5. This implies that no matter the level of output, each additional unit's cost remains the same.

In a profit-maximization context, the MC plays a vital role as firms equate this to their marginal revenue to decide on their production level. If MC were to rise with increased output, it would complicate the profit-maximization calculus, potentially requiring more intricate strategies for setting prices and output.
Perfect Competition
Under perfect competition, numerous small firms compete against each other, none of them having any significant market power to set prices. In such markets, the price of goods tends toward the marginal cost of production. This is a consequence of the fact that if firms set prices above the marginal cost, they would not be able to successfully sell their products as consumers would switch to a different seller offering a lower price.

In the textbook exercise, it's shown that as the number of firms increases in the Cournot model, reaching a large number N, the resulting equilibrium price approaches the price that would exist under perfect competition conditions. This convergence illustrates the relationship between oligopoly dynamics and perfect competition, highlighting that the more competitors there are in a market, the more the outcome resembles that of a perfectly competitive market.

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Most popular questions from this chapter

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?

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.

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.

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.

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?
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