Question

Short Answer Section II. Do 6 of the following 7 questions. (36 marks) 1. Suppose the demand function facing a monopolist is given as p=50-q and the cost function is given as C=109. Determine the total revenue function for the monopolist, and graph the relationship between total revenue, total cost and output. In your answer determine the profit maximizing level of output that would be chosen by the monopolist, and discuss mathematically how you know it is a profit maximum.

Answer 1

Demand Function : P = 50- Q

Cost Function: TC = 10q

Total Revenue Function : TR = P*Q

Substituting P by the demand function

TR = (50 -Q)Q = 50Q - Q²

The following graph plot TR and TC against Q

700 600 500 TC 400 300 200 100 TR 0 0 10 20 30 40 50 60 Quantity

The profit maximizing level of monopolist is when MR =MC

MR =dTR/dQ

MR = 50 - 2Q

MC = dTC/dQ

MC = 10

Profit maximizing output

MR = MC

50-2Q = 10

Q = 20

P = 50 - 20 = 30

At this level the gap between TC and TR is maximum which is the profit

Profit = TR - TC

At Q = 20

TR = 600

TC = 200

Profit = 400

Profit Function = TR - TC

Here profit = 50Q - Q² - 10Q

Profit = 40Q - Q²

This function has a maxima at Q = 20

Hence proved q=20 is the profit maximizing quantity for this monopolist