1.     Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 – 8Q + 36  + 3/Q, calculate the level of output Q which  a)  maximizes total revenue       b) maximizes profits

30 Points
13 years ago

Dear Sanjay,

Solution:- (a) Total Revenue (TR) = P*Q = (24 - 0.5Q) * Q

To find the max value of Total Revenue, differentiate it and equate it to zero.

d(TR)/dQ = 0  or  d(24Q - 0.5Q2)/dQ = 0

24 - Q = 0 or Q = 24 (ANS)

(b) Total cost (TC) = (AC) * Q = (Q2 - 8Q +36 +3/Q) * Q = (Q3 - 8Q2 +36Q +3)

Profit (P) =  [TR - TC] = [(24Q - 0.5Q2) - (Q3 - 8Q2 +36Q +3)] = (-12Q +7.5Q2 -Q3 -3)

To find the max value of Profit, differentiate it and equate it to zero.

d(P)/dQ = 0  or  d(-12Q +7.5Q2 -Q3 -3)/dQ = 0

-12 +15Q -3Q2 = 0 or Q2- 5Q+4 =0

Q =1 or 4

At Q =1, P = - 8.5 and at Q = 4, P = 5

So, 'P' is maximum at Q =4 (ANS)

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All the best Sanjay!!!

Regards,