Airline pricing is a good example of price discrimination. Airlines set different prices for first-class and excursion. Suppose the economics division of a major airline company estimates the demand and marginal revenue functions for first-class and excursion fares from Los Angeles to Beijing as:
First Class
Qa = 2100 - 0.5 Pa
MRa = 4200 - 4 Qa
Excursion
Qb = 8800 - 4 Pb
MRb =2200 - 0.5 Qb
where Q = number of passengers and P = ticket price
a. If the marginal cost of production is $200 per passenger, what fare and what number of passengers will maximize profit?
b. Would the airline make more profit by charging a single price? (If a single price is to be set, the demand equations from each market segment have to be combined)
To find the fare and number of passengers that will maximize profit, we need to determine the optimal price and quantity in each market segment separately and then calculate the overall profit for each combination.
a. To find the optimal fare and number of passengers for each market segment, we need to set the marginal revenue (MR) equal to the marginal cost (MC) and solve for the corresponding price and quantity. Let's start with the First Class segment:
MRa = MC
4200 - 4 Qa = 200
Solving for Qa:
4 Qa = 4200 - 200
4 Qa = 4000
Qa = 1000
Substituting the value of Qa back into the demand equation:
Qa = 2100 - 0.5 Pa
1000 = 2100 - 0.5 Pa
0.5 Pa = 2100 - 1000
0.5 Pa = 1100
Pa = 1100 / 0.5
Pa = 2200
So, in the First Class segment, the fare that maximizes profit is $2200 and the number of passengers is 1000.
Now let's move on to the Excursion segment:
MRb = MC
2200 - 0.5 Qb = 200
Solving for Qb:
0.5 Qb = 2200 - 200
0.5 Qb = 2000
Qb = 4000
Substituting the value of Qb back into the demand equation:
Qb = 8800 - 4 Pb
4000 = 8800 - 4 Pb
4 Pb = 8800 - 4000
4 Pb = 4800
Pb = 4800 / 4
Pb = 1200
So, in the Excursion segment, the fare that maximizes profit is $1200 and the number of passengers is 4000.
b. To calculate the overall profit, we need to consider both market segments. If a single price is to be set, the demand equations from each segment need to be combined. The total demand equation would be:
Q = Qa + Qb
Q = (2100 - 0.5 Pa) + (8800 - 4 Pb)
Now we need to find the fare and number of passengers for the combined demand equation that maximizes profit. We can do this by comparing different combinations of fares and passengers and calculating the profit for each combination.
However, without specific profit and cost functions, it is not possible to determine exactly which combination will maximize profit without further information.
In summary, to find the fare and number of passengers that will maximize profit, we set the marginal revenue equal to the marginal cost for each market segment. By solving these equations, we found the optimal fare and number of passengers for the First Class segment and the Excursion segment separately. To determine whether a single price would lead to higher profit, we would need more information about the total cost function and additional analysis of the overall profit given different combinations of fares and passengers.