Question

Microeconomics:

SG Express has a monopoly on shipping delivery in a local community. The demand for SG Express is Q = 600 – 5P.

SG Express' cost of providing shipping delivery is C = 200 + Q²

a. What are the profit-maximizing price and output level? What is the profit? Show your workings.

b. Find the Lerner Index of SG Express. Show your workings.

c. Use the result to (b), estimate the price elasticity of demand. Show your workings.

Answer 1

a) Q = 600 - 5P

5P = 600 - Q

P = 120 - 0.2Q (this is inverse demand function)

TR = P * Q = 120Q - 0.2Q^​​​2

MR = 120 - 0.4Q

TC = 200 + Q^{​​​​​​2}

MC = 2Q

The profit maximization condition is

MR = MC

120 - 0.4Q = 2Q

2.4Q = 120

Q = 120 / 2.4 = 50

P = 120 - 0.2(50) = $110

TR = P * Q = $110 * 50 = $5,500

TC = 200 + (50)² = $2,700

Profit = TR - TC = $5,500 - $2,700 = $2,800

Thus, the profit maximizing price is $110, output is 50 and profit is $2,800.

b) P = $110

MC = 2Q = 2 * 50 = $100

Lerner Index = (P - MC) / P

= (110 - 100) / 110

= 0.091

c) we know, the Lerner index is equal to the inverse of the elasticity in its absolute value

Learner Index = 1 / |E|  

0.091 = 1 / |E|

|E| = 1 / 0.091 = 10.99

Thus, the elasticity is 10.99