Question

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

#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

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 \\]

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) \\]

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).

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.

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).