Question

Suppose that Total revenue of the firm is
TR = 5Q + 3Q2 and TC is TC = Q + 4Q2.
Obtain the quantity that maximizes profit and hence maximum profit.

Answer 1

Given these equations the profit maximizing quantity is determined through the following steps

i) Determine marginal revenue by taking the derivative of total revenue with respect to quantity

MR= $\delta$TR/$\delta$Q=5Q+3Q²

ii) Determine the marginal cost by taking the derivative of total cost with respect to quantity

MC = $\delta$TC/$\delta$Q=Q+4Q²

iii) Set marginal revenue equal to marginal cost solve for Q

MR=MC

5Q+3Q²=Q+4Q²

5Q-Q=4Q²-3Q²

4Q=Q²

look for a square root for both 4Q and Q² which is equal to

2Q=Q

$\because$ Q = 2

iv)Substitute 2 for Q in the equation enables you to determine the price thus the quantity that nmaximizes profit is 2

TR = 5Q+3Q² = 5(2) + 3(2)²

= 22

TC = Q+4Q² = 2+4(2)²

=18

Thus the quantity that maximizes profit is 2 thus the maximum profit is

TR-TC

=22-18

$4 per unit.