Example

A perfect competitive firm faces P=4 and TC = Q3 –

7Q2 + 12Q+5Determine by using calculus the best level of out put of the firm by the marginal approach.

TR = PQ = 4QMR = d(TR) = 4 = P

dqMC = d(TC) = 3Q2 – 14Q +12

dq

4 = 3Q2 – 14Q +12

0 = 3Q2 – 14Q +8

0 = (3Q – 2)(Q-4)

So that , Q = 2/3 and Q = 4

Thus, MR =MC at Q =2/3 and Q=4

In order for profit maximized rather than minimized, the MCcurve should be rising(slope must be positive) at the point whereMR =MC.

Slope of the MC curve = d(MC) = 6Q -14

dq

At Q=2/3, the slope is -10(the firm minimizes total profit)

At Q=4, the slope is 10 so the firm maximizes its total profits.

• Find the total profit of the firm at this level of out put

Profit = TR –TC

= 4Q - 3Q2 – 14Q +12

= - Q3 – 7Q2 +8Q-5

= -64+112-32-5

= 11