Question

Consider the following data: Price per Unit $ 45 $ 42 $ 40 $ 38 Quantity Demand (Qd) 0 6 10 14 In a monopoly market. With function AC=Q? -89 +57 + On Determine: a. AR and TR functions! b. Find the maximum total profit?

Answer 1

(a) AR and TR are determined as follows

we have

P = 45 - 0.5Q

Thus, TR = PQ

= (45 - 0.5Q)*Q

= 45Q - 0.5Q²

Average revenue = TR/Q = (45Q - 0.5Q²)/Q = 45 - 0.5Q

(b) Profit is maximized when MR = MC

MR = dTR/dQ = 45 - Q

MC = dTC/dQ

TC = Q*AC = Q³ - 8Q² + 57Q + 2

=> MC = 3Q² - 16Q+ 57

Thus, in equilibrium we have,

45 - Q = 3Q² - 16Q+ 57

=> 3Q² - 15Q + 12 = 0

=> Q² - 5Q + 4 = 0

=> Q²-Q-4Q + 4 =0

=> Q(Q-1) -4(Q-1)

=> Q = 1 Or Q = 4

Profits with Q =1: (45 - 0.5) - (1-8+57+2) = -7.5

Profits with Q - 4: (45*4 - 0.5*4*4) - (4*4*4 -8*4*4+57*4+2) = 6

Thus, the maximum profit is 6