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 = 60 - 0.4P:

a.   What is the total revenue function for this firm in terms of Q?

b.   What is the average revenue function for this firm in terms of Q?

c.   What is the MR function for this firm in terms of Q?

d.   Show that MR will be less than AR for any positive level of Q. Why is that?

e.   What is the quantity level that maximizes total revenue? What is the price?

Answer 1

The revenue function for a company can be defined as; TR = Price (P) x Quantity Demanded (Q). We are given an ordinary demand function Q = 60 - 0.4P

a) What is the total revenue function for this firm in terms of Q?

Total revenue = P x Q

Find the inverse demand function

Q = 60 – 0.4P

0.4P = 60 – Q

P = 60/0.4 – Q/0.4

P = 150 – 2.5Q

Now find the TR = (150 – 2.5Q)Q

= 150Q – 2.5Q^2

b) What is the average revenue function for this firm in terms of Q?

Average revenue = Total revenue / Quantity

= (150Q – 2.5Q^2)/Q

= 150 – 2.5Q

c) What is the MR function for this firm in terms of Q?

MR = dTR/dQ

= d(150Q – 2.5Q^2)/Q

= 150 – 5Q

d) Show that MR will be less than AR for any positive level of Q. Why is that?

Let Q = 10. MR = 150 – 5*10 = 100 and AR = 150 – 2.5*10 = 125. Hence MR is less than AR. The reason includes price effect and output effect. To sell more, output is produced in a higher quantity and to attract consumers to buy it, the seller requires to offer a lower price. While AR is the price function, MR lies below it because to sell more of the additional product, price has to be reduced.

e) What is the quantity level that maximizes total revenue? What is the price?

It is the quantity at which MR = 0. This implies

150 – 5Q = 0 or Q = 30. At this quantity, price is P = 150 – 2.5*30 = $75.