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

PB = 540 - 0.5QB and PE = 380 - QE.

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.

1. What is the difference between constrained optimization and unconstrained optimization.

2. Write down the optimization problem this airline is faced with. Set it up as an unconstrained optimization problem with two variables.

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

4. Solve the first order condition(s) to find out the optimal number of flights.

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

Answer 1

a) Constrained Optimization is the process of optimiziming any given function subject to some variable constraints. The function can be of utility, profit, revenue etc having different constraints to them. On the other hand unconstrained optimization is the process of optimizing any function in absence of any variable constraints.

b) Unconstrained optimization problem is -

Maximize Profit = -0.5QB^2 - QE^2 + 540QB + 380QE

In the photo attached below all the calculations have been done for the optimal number of flights.