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Chapter 18: Problem 5

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 \\[ T 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: Part (b) can be worked out most easily by assuming the monopoly chooses the profit-maximizing price rather than quantity.

Short Answer

Expert verified
#Answer# Part (a) - No Advertising: By solving the equations, we find that the profit-maximizing output is: \\[ Q = 5 \\] And the market price is: \\[ P = 15 \\] The firm's profits are: \\[ \pi = 50 \\] Part (b) - With Advertising: By solving the system of equations with respect to P and A, we find that the optimal advertising level is approximately: \\[ A \approx 1.96 \\] The profit-maximizing output and market price are: \\[ Q \approx 6.13 \\] \\[ P \approx 13.87 \\] The firm's profits with advertising are: \\[ \pi \approx 59.42 \\] By choosing the optimal level of advertising, the monopolistic firm can increase its output, decrease the market price, and ultimately increase its profits.

Step by step solution

01

Setup the demand and cost functions

The demand function is given by: \\[ Q=(20-P)(1+0.1A-0.01A^2) \\] And the total cost function is given by: \\[ TC=10Q+15+A \\]
02

Find the revenue function

The revenue function is given by price multiplied by quantity, so we have: \\[ R = P \cdot Q \\] We plug in the demand function and obtain: \\[ R = P(20-P)(1+0.1A-0.01A^2) \\]
03

Calculate the profit function

The profit function is given by revenue minus total cost: \\[ \pi = R - TC \\] Substitute the revenue function and total cost function: \\[ \pi = P(20-P)(1+0.1A-0.01A^2) - (10Q+15+A) \\] We can also plug in the demand function for Q to have a profit function in terms of (P, A).
04

Solve Part (a) (No Advertising)

Since A=0: \\[ Q = (20-P) \\] The total cost function becomes: \\[ TC = 10 Q + 15 \\] The profit function becomes: \\[ \pi = P(20-P) - (10Q+15) \\] To find the profit-maximizing output, we need to take the first derivative of the profit function with respect to P, set it equal to 0, and solve for P: \\[ \frac{d\pi}{dP} = 20 - 2P - 10 \frac{dQ}{dP} \\] Now, set the derivative equal to 0 and solve for P: \\[ 20 - 2P -10\frac{dQ}{dP} = 0 \\] We can plug in the value of Q and determine the profit-maximizing output and market price.
05

Solve Part (b) (With Advertising)

To find the optimal level of advertising, we need to find the first derivative of the profit function with respect to A, and set it equal to 0 while also taking the first derivative with respect to P: \\[ \frac{d\pi}{dA} = 0 \\] \\[ \frac{d\pi}{dP} = 0 \\] By solving this system of equations, we can obtain the optimal levels of output (Q), market price (P), advertising level (A), and the firm's profits. Ultimately, solving these equations will provide the answers to the variables of interest: profit-maximizing output, market price, advertising level, and monopoly profits for both situations (no advertising and with advertising).

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Most popular questions from this chapter

A single firm monopolizes the entire market for widgets and can produce at constant average and marginal costs of \\[ A C=M C=10 \\] Originally, the firm faces a market demand curve given by \\[ Q=60-P \\] a. Calculate the profit-maximizing price-quantity combination for the firm. What are the firm's profits? b. Now assume that the market demand curve shifts outward (becoming stccper) and is given by \\[ Q=45-.5 P \\] What is the firm's profit-maximizing price-quantity combination now? What are the firm's profits? c. Instead of the assumptions of part (b), assume that the market demand curve shifts outward (becoming flatter) and is given by \\[ Q=100-2 P \\] What is the firm's profit-maximizing price-quantity combination now? What are the firm's profits? d. Graph the three different situations of parts (a), (b), and (c). Using your results, explain why there is no real supply curve for a monopoly.

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.)

Suppose the market for Hula Hoops is monopolized by a single firm. a. Draw the initial cquilibrium for such a market. b. Now suppose the demand for Hula Hoops shifts outward slightly. Show that, in general (contrary to the competitive case), it will not be possible to predict the effect of this shift in demand on the market price of Hula Hoops. c. Consider three possible ways in which the price elasticity of demand might change as the demand curve shifts- -it might increase, it might decrease, or it might stay the same. Consider also that marginal costs for the monopolist might be rising, falling, or constant in the range where \(M R=M C\). Consequently, there are nine different combinations of types of demand shifts and marginal cost slope configurations. Analyze each of these to determine for which it is possible to make a definite prediction about the cffcct of the shift in demand on the price of Hula Hoops.

Suppose a monopoly produces its output in several different plants and that these plants have differing cost structures. How should the firm decide how much total output to produce? How should it distribute this output among its plants to maximize profits?

Suppose the government wished to combat the undesirable allocation 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 wishes 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 it 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.
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