Question

Given Q = 300 – 5P and TC = 100 + 10Q for an oligopolistic firm, determine mathematically the price and output at which the firm maximizes its:

A. Total profits and calculate those profits

B. Total revenues and calculate the profits are that price and quantity

C. Total revenue in the presence of a $2980 profit constraint

TO HELP SOLVE:

Part (a) is the standard MR = MC procedure.

For part (b) you are looking for the turning point of the TR function. Note that the derivative of a function measures its rate of change and when a that derivative equals 0 it is at its turning point (in this case, maximum)

The profit constraint for the sales maximizer means that he must earn at least $2980 and cannot maximize sales (he will come as close as he can given the constraint). To do this write a total profit equation (TR minus TC) and set it equal to $2980.

Solve the equation by turning it into a quadratic equation and use the quadratic formula to find the quantities that satisfy the equation. Since the firm is a sales maximizer, the larger of the two will be the one chosen.

Answer #1

Q = 300 – 5P and TC = 100 + 10Q

a)

Writing inverse demand function we get,

P = 60 - 0.2Q

Total Revenue = P*Q = (60 - 0.2Q)*Q

Marginal Revenue = dTR/dQ = 60 - 0.4Q

MC = dTC/dQ = 10

Set MR = MC

60 - 0.4Q = 10

0.4Q = 50

Q = 125

P = 60 - 0.2Q = 35

Profit = TR - TC = 125*35 - [100 + 10*125] = 4375 - 1350 = 3025

b)

Total Revenue TR = P*Q = (60 - 0.2Q)*Q

Now the firm wants to maximise total revenue,

dTR/dQ = 60 - 0.4Q = 0

60 = 0.4Q

Q = 150

P = 60 - 0.2Q = 30

Total Revenue = P*Q = 150*30 = 4500

Profit = TR - TC = 150*30 - [100 + 10*150] = 4500 - 1600 = 2900

c)

Now the firm wants to earn a minimum profit of 2980

Profit = TR - TC

P = (60 - 0.2Q)*Q - 100 - 10Q = 2980

60Q - 0.2Q^{2} - 100 - 10Q = 2980

0.2Q^{2} - 50Q + 3080 = 0

Solving we get, Q = 140 or 110

Hence Q = 140

P = 60 - 0.2Q = 32

Total Revenue = P*Q = 140*32 = 4480

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