Question

(7 pts) A railroad which runs between two cities offers two products: passenger and freight service. The marginal cost of carrying an extra ton of freight is $0, and the marginal cost of carrying an extra passenger is $1. There are joint, fixed costs of $19,000 per day. There is no other competitor in this market. The daily demand for passenger service is with Qp the number of passengers and P the price of a two-way ticket. The daily demand for freight service is P10 0.0010 with Qf in tons and P the price per ton. a. (2 pts) Currently, Pp $5 per passenger and P 8 per metric ton. Please calculate the railroad's revenues from passenger service and freight service, respectively. In addition, calculate the railroad's overall profit. (3 pts) A manager at the railroad argues that prices should be raised on both products to increase profit. Do you agree with the argument? What pricing would you recommend for each product? (2 pts) Another manager argues that the firm can use price differentiation to improve profit. Please recommend a specific price differentiation scheme for the railroad's passenger and/or freight service, and explain how it will work. b. c.

Answer 1

(a)

(i) In passenger market,

Pp = 8 - 0.005Qp

0.005Qp = 8 - Pp

Qp = 1600 - 200Pp

when Pp = $5, Qp = 1600 - (200 x 5) = 1600 - 1000 = 600

Revenue (TRp) ($) = Pp x Qp = 5 x 600 = 3000

(ii) In freight market,

Pf = 10 - 0.001Qf

0.001Qf = 10 - Pf

Qf = 10000 - 1000Pf

When Pf = 8, Qf = 10000 - (1000 x 8) = 10000 - 8000 = 2000

Revenue (TRf) ($) = Pf x Qf = 8 x 2000 = 16000

(iii) Total profit (Z) = Total revenue (TR) - Total cost (TC)

TR ($) = TRp + TRf = 3000 + 16000 = 19000

TC ($) = Fixed cost + MC x (Qp + Qf) = 19000 + 1 x (600 + 2000) = 19000 + 2600 = 21600

Profit ($) = 19000 - 21600 = -2600 (Loss)

(b)

In passenger market, Elasticity = (dQp/dPp) x (Pp/Qp) = -200 x (5/600) = -1.67

In freight market, Elasticity = (dQf/dPf) x (Pf/Qf) = -1000 x (8/2000) = -4

Since absolute value of elasticity is higher than 1 in both markets, demand is elastic. With elastic demand, an increase in price will decrease total revenue, thus lowering profit. To find optimal prices, we use Lerner Index (LI) where

LI = -1 / Elasticity = (P - MC)/P

(i) In passenger market,

-1 / -1.67 = (P - 1)/P

1 / 1.67 = (P - 1)/P

P = 1.67P - 1.67

0.67P = 1.67

P = $2.5

(ii) In freight market,

-1 / -4 = (P - 1)/P

1 / 4 = (P - 1)/P

P = 4P - 4

3P = 4

P = $1.33

(c) With price discrimination, higher price should be charged in less elastic market and lower price should be charged in more elastic market. In each market, Marginal revenue should equal MC.

(i) In passenger market,

TRp = Pp x Qp = 8Qp - 0.005Qp²

MRp = dTRp / dQp = 8 - 0.01Qp

8 - 0.01Qp = 1

0.01Qp = 7

Qp = 700

Pp = 8 - (0.005 x 700) = 8 - 3.5 = $4.5

(ii) In freight market,

TRf = Pf x Qf = 10Qf - 0.001Qf²

MRf = dTRf/dQf = 10 - 0.002Qf

10 - 0.002Qf = 1

0.002Qf = 9

Qf = 4,500

Pf = 10 - (0.001 x 4,500) = 10 - 4.5 = $5.5

(iii) Profit = TRp + TRf - TC

TRp + TRf ($) = (4.5 x 700) + (5.5 x 4,500) = 3,150 + 24,750 = 27,900

TC ($) = 19,000 + 1 x (700 + 4,500) = 19,000 + 5,200 = 24,200

Profit ($) = 27,900 - 24,200 = 3,700

Therefore, compared to case (a), profit will increase by $(3,700 + 2,600) = $6,300.