Question

The revenue function for a company can be defined as; TR = Price (P) ´ Quantity Demanded (Q). If the ordinary demand function for your firm is Q = 75 - 0.4P:

1.    What is the total revenue function for this firm in terms of Q?
2.    What is the average revenue function for this firm in terms of Q?
3.    What is the MR function for this firm in terms of Q?
4.    Show that MR will be less than AR for any positive level of Q. Why is that?
5.    What is the quantity level that maximizes total revenue? What is the price?

Answer 1

Q = 75 - 0.4 P
=> P = 187.5 - 2.5 Q

a) TR = P x Q = (187.5 - 2.5 Q) x Q

TR = 187.5 Q - 2.5 Q²

b) AR = TR/Q = (187.5 Q - 2.5 Q²) / Q

AR = 187.5 - 2.5 Q

(Notice that AR = P)

c) MR = dTR/dQ = d (187.5 Q - 2.5 Q²) / dQ

MR = 187.5 - 5 Q

d) MR = 187.5 - 5 Q

AR = 187.5 - 2.5 Q

187.5 - 2.5 Q > 187.5 - 5 Q for all Q > 0.

That is, AR > MR for all Q > 0

This happens because, to increase the sale of an additional good, the company will have to reduce its price (which is equal to AR). thus, the additional revenue generated (MR) is less than the price charged because to increase sales, the prices of all units will have to be reduced which could earlier be sold at a higher price.

e) Maximising total revenue implies dTR/dQ = 0 (which is the first order maximisation condition).

Notice that, dTR/dQ = MR. Thus, TR is maximised where MR = 0.

=> MR = 187.5 - 5 Q = 0

=> 5Q = 187.5

=> Q = 37.5

Thus, total revenue is maximised when quantity sold is 37 units approximately.

At Q = 37.5,

P = 187.5 - 2.5 Q = 187.5 - 2.5 (37.5)

=> P = 93.75