Question

A freight firm holds a monopoly over rail freight transport in Australia. The freight firm has two types of customers:

• Coal mining companies’ demand for freight services is Q_c = 38 − P_c .
• Grain farmers’ demand for freight services is Q_g = 56 − 4P_g .

All quantities are measured in tonnes. The firm faces a constant marginal cost of $10 per tonne, regardless of which commodity is being transported. The firm has no fixed costs.

a) Express the demand for each group of customers as an inverse demand function. Sketch the inverse demand functions of the two groups in a single diagram (with P on the vertical axis and Q on the horizontal axis). Which is the larger market segment? What is the “choke price”, i.e. the price above which there is no demand, for each group of customers?

b) Write a total cost function for the firm as a function of quantity of goods transported.

c) If the firm can discriminate between coal miners and grain farmers, what price will it charge to each group of customers?

d) What is the firm’s profit under price discrimination?

e) Now suppose that the government regulates, requiring the monopoly to charge a uniform price. Find the optimal uniform price and the corresponding firm profit. [Hints: First derive the combined demand of the two market segments, and then express it as an inverse demand function. Note that this combined inverse demand function is NOT a continuous straight line. It may be instructive to graph the combined inverse demand function before deriving the freight firm’s optimal uniform price.]

f) Which, if any, group of customers benefits from price discrimination in this instance? What does it tell you about price discrimination in general?

Answer 1

Demand functions as given-

Coal Miners: Q_c = 38 − P_c ...... (i)

Grain farmers: Q_g = 56 − 4P_g ..... (ii)

a. Inverse demand functions are-

Coal Miners: P_c = 38 - Q_c ..... (iii)

Grain farmers: P_g= (56 − Q_g ) / 4 ..... (iv)

Larger market segment is that for the coal miners of demand equal to 14 tonnes.

Choke Price, the price above which the demand is zero-

P_g = 14

P_c = 38

lower the choke price, more elastic the demand curve. Hence the demand curve of Grain farmers is more elastic as compared to coal miners.

Total demand (horizontal summing of demand by coal miners and grain farmers) = 94 - 5P

P = 18.8 when Q = 0

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b. Total Cost function-

TC = 10 (Q_c + Q_g)

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c. Coal Miners

TR_c = (38 - Q_c) * Q_c

TR_c = 38 Q_c - Q_c²

therefore, MR_c = 38 - 2Q_c .... (v)

MC = MR_c

10 = 38 - 2Q_c

Q_c = 14

P_c = 38 - 14 = 24

Grain Framers

TR_g = (14 - 1/4*Q_g) * Qg

TR_g = 14 Qg - 1/4*Qg ²

therefore, MR_g = 14 - 1/2*Qg .... (vi)

MC = MR_g

10 = 14 - 1/2*Qg

Q_g = 8

P_g = (56 − Q_g ) / 4

P_g = 12

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d. Profit

Profit = TR_c + TR_g - TC

Profit = 38 Q_c - Q_c² + 14 Qg - 1/4*Qg ² - (10 (Q_c + Q_g))

substituting values

Profit = 38 * 14 - 196 + 14(8) - 16 - 220

= 532-196 + 112 -16 -220

= 644 - 432

Profit = 212

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