Question

On a popular air route, an airline offers two classes of service: business class (B) and economy class (E). The respective inverse demands are given by

P_B = 540 - 0.5Q_B      and     P_E = 380 - Q_E.

Because of ticketing regulations, business passengers cannot take advantage of economy’s low fares.

Each flight has a capacity of 200 passengers, The cost per full flight is $20,000, so you may assume the marginal cost per passenger is $100.

Consider the scenario of the previous problem. Because of regulations, the airline has to operate two flights daily.

1. Write down the optimization problem faced by this firm as a constrained optimization problem with two variables.

2. Write down the first order condition(s).

3. What fares should the airline charge, and how many passengers will buy tickets of each type?

4. Write down the second order condition(s) and check whether it/they is/are satisfied

Answer 1

(1)

Profit (Z) = Total revenue from Class B + Total revenue from Class E - Total cost = TRB + TRE - TC

Z = PB x QB + PE x QE - MC x (QB + QE)

Z = 540QB - 0.5QB2 + 380QE - QE2 - 100 x (QB +QE)

Z = 540QB - 0.5QB2 + 380QE - QE2 - 100QB - 100QE

Z = 440QB - 0.5QB2 + 280QE - QE2

Optimization problem:

Maximize Profit (Z) = 440QB - 0.5QB2 + 280QE - QE2

Subject to the non-negativity constraints: QB >=0, QE >= 0

(2)

First order conditions are:

Z/QB = 0 (I)

Z/QE = 0 (II)

(3)

When Z/QB = 0,

440 - QB = 0

QB = 440

PB = 540 - 0.5 x 440 = 540 - 220 = 320

When Z/QE = 0,

Z/QE = 280 - 2QE = 0

2QE = 280

QE = 140

PE = 380 - 140 = 240

(4)

Second order conditions are:

2Z/QB2 < 0 (I)

2Z/QE2 < 0 (II)

2Z/QB2 = Z/QB(Z/QB) = - 1 < 0

2Z/QE2 = Z/QE(Z/QE) = - 2 < 0

Hence, Second order conditions are satisfied.