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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 $$Q=(20-\mathrm{P})\left(1+0.1 \mathrm{A}-0.01 \mathrm{A}^{2}\right)$$ The monopolistic firm's cost function is given by $$T C=10 g+15+A$$ a. Suppose there is no advertising \((\mathrm{A}=0) .\) 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

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#tag_title#Short Answer:#tag_content# When there is no advertising (A=0), the profit-maximizing output is 15, the market price is 5, and the monopoly's profits are -90. To determine the optimal level of advertising, output, and price, it is necessary to use numerical optimization techniques, solving the system of equations derived from finding the profit-maximizing levels of Q and A. After finding these optimal values, they can be substituted into the price expression and profit function to find the firm's profits.

Step by step solution

01

Find the demand, cost, revenue, and profit functions

We are given the demand function and cost function as: $$Q=(20-P)(1+0.1 A-0.01 A^2)$$ $$TC=10Q+15+A$$ Since A=0 in this case: $$Q=20-P$$ Revenue is found by multiplying the price P and quantity Q: $$R = P * Q$$ Substitute the demand function into the revenue function: $$R = P*(20-P)$$ $$R=20P - P^2$$ Profit is revenue minus total costs: $$\pi = R-TC$$ Substituting the revenue and cost functions, and the fact that A=0, we have: $$\pi = (20P - P^2) - (10Q+15)$$ Now we can replace Q with its expression in terms of P: $$\pi = (20P - P^2) - (10(20-P)+15)$$
02

Determine the profit-maximizing output and price

To find the profit-maximizing output and price, we can take the first derivative of the profit function with respect to P, set it equal to 0, and solve for P. $$\frac{d\pi}{dP} = 20-2P -10 = 0$$ Solving for P, $$P^* = 5$$ Now that we have the profit-maximizing price, we can find the corresponding output. Substitute P into the demand function: $$Q^* = 20 - P^* = 20 - 5 = 15$$
03

Calculate the monopoly's profits

Now that we have the profit-maximizing price and output, we can calculate the monopoly's profits by plugging these values into the profit function: $$\pi = (20P - P^2) - (10Q+15)$$ Substitute the profit-maximizing values of P and Q: $$\pi = (20*5 - 5^2) - (10*15+15) = 100-25-165$$ $$\pi = -90$$ Thus, the profit-maximizing output is 15, the market price is 5, and the monopoly's profits are -90. b. Optimal level of advertising, output, price, and profits
04

Determine the profit-maximizing price with advertising

As the hint suggests, we will assume that the monopolist chooses profit-maximizing price. So, we will rewrite the demand function in terms of P given the advertising level A: $$Q=(20-P)(1+0.1 A-0.01 A^{2})$$ Solve for P: $$P = 20 - \frac{Q}{(1+0.1 A-0.01 A^{2})}$$
05

Determine the revenue function with advertising

Now, using this price expression and the demand function, we will find the revenue function: $$R = PQ=Q(20 - \frac{Q}{(1+0.1A-0.01 A^2)})$$
06

Determine the profit function with advertising

To find the profit function, subtract the total costs: $$\pi = R - TC = Q(20 - \frac{Q}{(1+0.1A-0.01 A^2)}) - (10Q+15+A)$$
07

Find the profit-maximizing levels of advertising, output, and price

To maximize profits, we should take the partial derivatives of the profit function with respect to Q and A and set them equal to 0. Then, solve the system of equations for Q and A. Afterward, apply the values to the price expression, P. Note: The solution to this step might be too complex to be solved analytically. It is suggested to use numerical optimization techniques like the Newton-Raphson method to find the optimal values. Once we find the optimal levels of advertising, output, and price, we can substitute them into the profit function to find the firm's profits.

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Most popular questions from this chapter

Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of \(\$ 5\) per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the first market is given by $$a,=55-p_{n}$$ and the demand curve in the second market is given by $$\mathrm{Q}_{2}=70-2 \mathrm{P}_{2}$$ a If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total profits in this situation? b. How would your answer change if it only cost demanders \(\$ 5\) to transport goods between the two markets? What would be the monopolist's new profit level in this situation? c. How would your answer change if transportation costs were zero and the firm was forced to follow a single-price policy? d. Suppose the firm could adopt a linear two-part tariff under which marginal prices must be equal in the two markets but lump-sum entry fees might vary. What pricing policy should the firm follow?

Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of S10 per unit. Monopolized marginal costs rise 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 $$d_{D}-1,000-50 P$$ 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.

A monopolist faces a market demand curve given by $$d=70-P$$ a. If the monopolist can produce at constant average and marginal costs of \(\mathrm{AC}=M C=6\) what output level will the monopolist choose in order 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 $$T C=.25\left(?^{2}-5 g+300\right.$$ With the monopolist facing the same market demand and marginal revenue, what pricequantity 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 $$T C=.0133 Q^{3}-5 Q+250$$ Again, calculate the monopolist's price-quantity combination that maximizes profits. What will profit be? (Hint: Set \(M C=\)MRas 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 Mi? curve) and (2) the cost structure underlying production.

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?

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 cal culate the monopolist's profits. b. What output level would be produced by this industry under perfect competition (where price \(=\text { marginal cost }) ?\) 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?
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