Question

A marketing analyst has estimated a firm’s demand function to be ?? = 40 ? 2?? + 20??, where X is an indicator for whether the economy is in a boom (X = 1) or recession (X = 0). Marginal cost of producing the good is 10. i) Write down the inverse demand, i.e. P as a function of Q and X. ii) Write down the marginal revenue function during a boom as a function of Q. iii) What is the firm’s optimal price during a boom? iv) Write down the marginal revenue function during a recession as a function of Q. v) What is the firm’s optimal price during a recession?

Answer 1

A marketing analyst has estimated a firm’s demand function to be ?? = 40 ? 2?? + 20??, where X is an indicator for whether the economy is in a boom (X = 1) or recession (X = 0).

Marginal cost of producing the good is 10.

i) Write down the inverse demand, i.e. P as a function of Q and X.

2P = 40 - Q + 20X

P = 20 - 0.5Q + 10X

ii) Write down the marginal revenue function during a boom as a function of Q.

First find TR (X = 1) = PQ = (20 - 0.5Q + 10)Q

= 30Q - 0.5Q^2

MR = dTR/dQ = 30 - Q.

iii) What is the firm’s optimal price during a boom?

MR = MC

30 - Q = 10

Q = 20

P = 30 - 0.5*20 = $20

iv) Write down the marginal revenue function during a recession as a function of Q.

First find TR (X = 0) = PQ = (20 - 0.5Q)Q

= 20Q - 0.5Q^2

MR = dTR/dQ = 20 - Q.

v) What is the firm’s optimal price during a recession?

MR = MC

20 - Q = 10

Q = 10 and so P = 20 - 0.5*10 = $15