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Chapter 14: Problem 9

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.

Short Answer

Expert verified
Based on the analysis and solution above, it can be concluded that the monopolist and the competitive industry will choose the same battery lifetime, \(X\). This is because the consumers care only about the composite commodity \((XQ)\), allowing both the monopolist and competitive firms to opt for the same battery lifetime. Although the monopolist may restrict output and charge a higher price compared to the competitive market, the optimal battery lifetime will remain the same in both cases.

Step by step solution

01

Demand function

Given the inverse demand function, \(P(Q, X) = g(XQ)\). Since \(g'(XQ) < 0\), it means that the demand function is inversely related to the composite commodity \((XQ)\).
02

Cost function

The cost function is given by \(C(Q, X) = C(X)Q\). Since \(C'(X)>0\), the cost of production increases with the battery lifetime.
03

Monopolist's profit function

The monopolist's profit function can be written as: \\[\pi(Q, X) = P(Q, X) Q - C(Q, X),\\] Substitute the given demand and cost functions to obtain the profit function: \\[\pi(Q, X) = g(XQ) Q - C(X) Q.\\]
04

Optimal values of \(Q\) and \(X\) for the monopolist

To find the optimal values of \(Q\) and \(X\), we will take the partial derivatives of the profit function with respect to \(Q\) and \(X\), and set them equal to zero: \\[\frac{\partial \pi}{\partial Q} = g'(XQ) X Q + g(XQ) = 0,\\] \\[\frac{\partial \pi}{\partial X} = g'(XQ) Q^2 - C'(X) Q = 0.\\] From the first equation, we can find the optimal value of \(Q\) in terms of \(X\) as: \\[Q^*(X) = -\frac{g(XQ)}{g'(XQ)X},\\] Substitute the optimal value of \(Q\) in the second equation to solve for the optimal value of \(X\): \\[g'(XQ)Q^{*2} - C'(X)Q^* = g'(XQ)\left(-\frac{g(XQ)}{g'(XQ)X}\right)^2 - C'(X)\left(-\frac{g(XQ)}{g'(XQ)X}\right) =0.\\] We can simplify this equation to get the optimal value of \(X\): \\[\frac{g(XQ)}{(g'(XQ)X)^2} = \frac{C'(X)}{g'(XQ)X}.\\] Since both sides are equal, the value of \(X\) that maximizes the monopolist's profit will be the same as that chosen by a competitive industry.
05

Comparison with competitive market output

In a competitive market, the price is equal to the marginal cost. So, the competitive firms will select the value of \(X\) that equates the marginal benefit \((g'(XQ)XQ)\) with the marginal cost \((C'(X))\). As shown in step 4, the monopolist also ends up choosing the same value of \(X\). So, even though the monopolist may charge a higher price and produce a different quantity, the battery lifetime chosen by the monopolist and the competitive industry will be the same. This result can be explained by the fact that consumers care only about the composite commodity \((XQ)\), i.e., they treat the product of battery lifetime and quantity purchased as a single commodity, and this allows both monopolist and competitive firms to choose the same optimal lifetime for the batteries. However, the monopolist may still restrict the output and charge a higher price compared to the competitive market.

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Most popular questions from this chapter

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.

The taxation of monopoly can sometimes produce results different from those that arise in the competitive case. This problem looks at some of those cases. Most of these can be analyzed by using the inverse elasticity rule (Equation 14.1 ). a. Consider first an ad valorem tax on the price of a monopoly's good. This tax reduces the net price received by the monopoly from \(P\) to \(P(1-t)-\) where \(t\) is the proportional tax rate. Show that, with a linear demand curve and constant marginal cost, the imposition of such a tax causes price to increase by less than the full extent of the tax. b. Suppose that the demand curve in part (a) were a constant elasticity curve. Show that the price would now increase by precisely the full extent of the tax. Explain the difference between these two cases. c. Describe a case where the imposition of an ad valorem tax on a monopoly would cause the price to increase by more than the tax. d. A specific tax is a fixed amount per unit of output. If the tax rate is \(\tau\) per unit, total tax collections are \(\tau Q .\) Show that the imposition of a specific tax on a monopoly will reduce output more (and increase price more) than will the imposition of an ad valorem tax that collects the same tax revenue.

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.

Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of a subsidy. a. Why would a lump-sum subsidy not achieve the government's goal? b. Use a graphical proof to show how a per-unit-of-output subsidy might achieve the government's goal. c. Suppose the government wants its subsidy to maximize the difference between the total value of the good to consumers and the good's total cost. Show that, to achieve this goal, the government should set \\[\frac{t}{P}=-\frac{1}{e_{Q, P}},\\] where \(t\) is the per-unit subsidy and \(P\) is the competitive price. Explain your result intuitively.

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.
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