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Chapter 14: Problem 2

A monopolist faces a market demand curve given by \\[Q=70-p.\\] a. If the monopolist can produce at constant average and marginal costs of \(A C=M C=6,\) what output level will the monopolist choose to maximize profits? What is the price at this output level? What are the monopolist's profits? b. Assume instead that the monopolist has a cost structure where total costs are described by \\[C(Q)=0.25 Q^{2}-5 Q+300.\\] With the monopolist facing the same market demand and marginal revenue, what price-quantity combination will be chosen now to maximize profits? What will profits be? c. Assume now that a third cost structure explains the monopolist's position, with total costs given by \\[C(Q)=0.0133 Q^{3}-5 Q+250.\\] Again, calculate the monopolist's price-quantity combination that maximizes profits. What will profit be? Hint: Set \(M C=\) \(M R\) as usual and use the quadratic formula to solve the second-order equation for \(Q\) d. Graph the market demand curve, the \(M R\) curve, and the three marginal cost curves from parts (a), (b), and (c). Notice that the monopolist's profit- making ability is constrained by (1) the market demand curve (along with its associated \(M R\) curve) and (2) the cost structure underlying production.

Short Answer

Expert verified
Answer: The main concept used to find the profit-maximizing output for a monopolist firm with different cost structures is the equalization of marginal cost (MC) and marginal revenue (MR).

Step by step solution

01

Calculate the marginal revenue (MR) from the demand curve

To do this, rewrite the demand curve as a function of price: \(p=70-Q\). Take the derivative of the total revenue with respect to quantity: \[MR = \frac{d(TR)}{dQ} = \frac{d(pQ)}{dQ} = \frac{d((70-Q)Q)}{dQ}\] Then, solve for MR.
02

Equalize MR and MC to find the profit-maximizing output

For this part, we know that MC=6. Set MR=MC and solve for Q.
03

Calculate the price using the demand curve

Plug the Q value obtained in step 2 into the demand curve to find the price.
04

Calculate the monopolist's profits

Profits = Total Revenue (TR) - Total Cost (TC). Calculate TR (from Step 3) and TC (using the given constant AC and MC) and then calculate the profit. #b. Monopolist with quadratic total cost function#
05

Calculate the marginal cost from the total cost function

Differentiate the total cost function with respect to quantity to obtain the MC: \[MC = \frac{d(TC)}{dQ} = \frac{d(0.25Q^2-5Q+300)}{dQ}\] Then, solve for MC.
06

Equalize MR and MC to find the profit-maximizing output

Set the MR calculated in part (a) equal to the MC calculated in this part and solve for Q.
07

Calculate the price using the demand curve

Plug the Q value obtained in step 2 into the demand curve to find the price.
08

Calculate the monopolist's profits

Calculate TR and TC, using the price and output values obtained in step 2, and then calculate the profit. #c. Monopolist with cubic total cost function#
09

Calculate the marginal cost from the total cost function

Differentiate the total cost function with respect to quantity to obtain the MC: \[MC = \frac{d(TC)}{dQ} = \frac{d(0.0133Q^3-5Q+250)}{dQ}\] Then, solve for MC.
10

Equalize MR and MC to find the profit-maximizing output

Set the MR calculated in part (a) equal to the MC calculated in this part and use the quadratic formula to solve the second-order equation for Q.
11

Calculate the price using the demand curve

Plug the Q value obtained in step 2 into the demand curve to find the price.
12

Calculate the monopolist's profits

Calculate TR and TC using the price and output values obtained in step 2, and then calculate the profit. #d. Graph the market demand curve, the MR curve, and the three marginal cost curves#
13

Plot the demand curve, MR curve, and the three MC curves

Use the equations found in parts (a), (b), and (c) to plot the demand curve, the MR curve, and the three different MC curves, respectively.
14

Observe the constraints for the monopolist's profit-making

Analyze the graph and identify how the market demand curve (along with its associated MR curve) and the cost structure underlying production constrain the monopolist's profit-making ability.

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Most popular questions from this chapter

A monopolist can produce at constant average and marginal costs of \(A C=M C=5 .\) The firm faces a market demand curve given by \(Q=53-P\). a. Calculate the profit-maximizing price-quantity combination for the monopolist. Also calculate the monopolist's profits. b. What output level would be produced by this industry under perfect competition (where price \(=\) marginal cost)? c. Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of the monopolist's profits and the consumer surplus received in case (a). What is the value of the "deadweight loss" from monopolization?

Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of \(\$ 10\) per unit. Monopolized marginal costs increase to \(\$ 12\) per unit because \(\$ 2\) per unit must be paid to lobbyists to retain the widget producers' favored position. Suppose the market demand for widgets is given by \\[Q_{D}=1,000-50 P.\\] a. Calculate the perfectly competitive and monopoly outputs and prices. b. Calculate the total loss of consumer surplus from monopolization of widget production. c. Graph your results and explain how they differ from the usual analysis.

Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price (P) but also on the amount of advertising the firm does ( \(A\), measured in dollars). The specific form of this function is \\[Q=(20-P)\left(1+0.1 A-0.01 A^{2}\right).\\] The monopolistic firm's cost function is given by \\[C=10 Q+15+A.\\] a. Suppose there is no advertising \((A=0) .\) What output will the profit- maximizing firm choose? What market price will this yield? What will be the monopoly's profits? b. Now let the firm also choose its optimal level of advertising expenditure. In this situation, what output level will be chosen? What price will this yield? What will the level of advertising be? What are the firm's profits in this case? Hint: This can be worked out most easily by assuming the monopoly chooses the profit-maximizing price rather than quantity.

Suppose a monopolist produces alkaline batteries that may have various useful lifetimes \((X) .\) Suppose also that consumers (inverse) demand depends on batteries' lifetimes and quantity (Q) purchased according to the function \\[P(Q, X)=g(X \cdot Q),\\] where \(g^{\prime} < 0 .\) That is, consumers care only about the product of quantity times lifetime: They are willing to pay equally for many short-lived batteries or few long-lived ones. Assume also that battery costs are given by \\[C(Q, X)=C(X) Q,\\] where \(C^{\prime}(X) > 0 .\) Show that, in this case, the monopoly will opt for the same level of \(X\) as does a competitive industry even though levels of output and prices may differ. Explain your result. Hint: Treat \(X Q\) as a composite commodity.

An alternative way to study the welfare properties of a monopolist's choices is to assume the existence of a utility function for the customers of the monopoly of the form utility \(=U(Q, X),\) where \(Q\) is quantity consumed and \(X\) is the quality associated with that quantity. A social planner's problem then would be to choose \(Q\) and \(X\) to maximize social welfare as represented by \(S W=U(Q, X)-C(Q, X)\). a. What are the first-order conditions for a welfare maximum? b. The monopolist's goal is to choose the \(Q\) and \(X\) that maximize \(\pi=P(Q, X) \cdot Q-C(Q, X) .\) What are the first-order conditions for this maximization? c. Use your results from parts (a) and (b) to show that, at the monopolist's preferred choices, \(\partial S W / \partial Q>0\). That is, as we have already shown, prove that social welfare would be improved if more were produced. Hint: Assume that \(\partial U / \partial Q=P\) d. Show that, at the monopolist's preferred choices, the sign of \(\partial S W / \partial X\) is ambiguous-that is, it cannot be determined (on the sole basis of the general theory of monopoly) whether the monopolist produces either too much or too little quality.
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