Question

5.  A monopoly company faces the demand curve given by the equation P = 300 - 2Q. Its production process is characterized by the total cost function TC=4Q2. The company charges a single price in the market.
a. What is the marginal cost (as a function of output Q)?

b. What is the total revenue (as a function of output Q)?

c. What is the marginal revenue (as a function of output Q)?

d. What levels of output and price would maximize total revenue of the firm?

e. Now calculate profit-maximizing output, price, and the maximum profit of the company. Illustrate your solution graphically.

f. If a $500 lump-sum tax is levied on this firm, what will be the output and price of its product, and the firm’s profit?

g. If a $60 excise (per unit) tax is levied on the product of the company, how much will the monopolist produce, what price will be charged, and what profit will be made?

Answer 1

a) Total Cost = 4Q²

Marginal cost is calculated by differentiating the total cost function with respect to Q. Marginal cost is the net addition to the total cost when an additional unit of quantity is produced.

So, differentiating the total cost function with respect to Q,

marginal cost = 8Q is the answer.

b) Total Revenue = price x quantity

= (300 - 2Q) x Q

= 300Q - 2Q² is the answer.

c) Marginal revenue is calculated by differentiating the total revenue function with respect to Q. It is the addition to the total revenue when an additional unit of output is sold.

So, differentiating the total revenue function with respect to Q,

Marginal Revenue = 300 - 4Q is the answer.

d) The quantity and price which maximizes the total revenue can be found out by equating the first derivative of the total revenue function that marginal revenue with 0,

So, 300 - 4Q = 0

300 = 4Q

Q = 300 / 4 = 75 is the answer.

Price = 300 - 2Q

= 300 - 2(75)

= 300 - 150 = 150 is the answer.