Q5505PGiven the firms demand function Q5505P where P price

Q=55-0.5PGiven the firm’s demand function Q=55-0.5P (where P = price and Q = rate of output), and the total cost function TC=20+Q+0.2Q^2 where TC = total cost, determine a. The total revenue function for the firm. (Hint: To find the total revenue function, solve the demand function for P and then multiply both sides of the equation by Q). b. The marginal revenue and marginal cost functions and find the rate of output for which marginal revenue equals marginal cost. c. An equation for profit by subtracting the total cost function from the total revenue function. Find the level of output that maximizes total profit. Compare your answer to that obtained in part (b). Is there any correspondence between these answers?

Solution

a.

Q = 55 – 0.5P

P = 110 – 2Q

PQ = Q(110 – 2Q)

Total revenue, PQ = 110Q – 2Q^2

b.

Marginal revenue (MR) is the derivative of total revenue

MR = 110 – 4Q

Marginal cost is the derivative of total cost, TC.

MC = 1 + 0.4Q

MR = MC

110 – 4Q = 1 + 0.4Q

4.4Q = 109

Q = 109/4.4

c.

Profit = TR – TC = (110Q – 2Q^2) – (20 + Q + 0.2Q^2) = 109Q – 2.2Q^2 – 20

The profit would maximum if “Q” comes from MR = MC. Therefore, the required quantity is 109/4.4.