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)
First, lets re-write the demand equations, to be P=f(Q).
First class:
Pa = 4200 - 2Qa
MRa = 4200 - 4Qa
Excursion:
Pb = 2200 - .25*Qb
MRb = 2200 - .5*Qb
a) set MC = MR in each equation, then solve for Qa and Qb. I get Qa=1000, Qb=4000. Ergo, Pa=2200, Pb=1200.
b) Undoubtedly, the airline will make more profit by charging two prices instead of one. The two prices I calculated are the profit-maximizing prices in each market; so the firm cant do any better than that.
(Do you need to calculate the profit-maximizing price if the firm was forced to charge a single price??. It's a little tricky operation, but very do-able.)
To determine the fare and number of passengers that will maximize profit, we need to find the point where the marginal revenue equals the marginal cost.
a. Let's start with the first-class fares. The marginal cost of production is $200 per passenger, so we need to find the point at which the marginal revenue (MRa) equals $200.
MRa = 4200 - 4Qa
Set MRa = 200:
4200 - 4Qa = 200
-4Qa = 200 - 4200
-4Qa = -4000
Qa = -4000 / -4
Qa = 1000
Now that we have the number of passengers for first-class (Qa = 1000), we can substitute it back into the demand equation for first-class to find the fare (Pa) that maximizes profit.
Qa = 2100 - 0.5Pa
1000 = 2100 - 0.5Pa
0.5Pa = 2100 - 1000
0.5Pa = 1100
Pa = 1100 / 0.5
Pa = 2200
So, to maximize profit in the first-class segment, the fare should be $2,200 and there should be 1,000 passengers.
Now, let's move on to the excursion fares. Following the same process, we need to find the fare (Pb) and number of passengers (Qb) that maximize profit.
MRb = 2200 - 0.5Qb
Set MRb = 200:
2200 - 0.5Qb = 200
-0.5Qb = 200 - 2200
-0.5Qb = -2000
Qb = -2000 / -0.5
Qb = 4000
Now that we have the number of passengers for the excursion segment (Qb = 4000), we can substitute it back into the demand equation for the excursion fares to find the fare (Pb) that maximizes profit.
Qb = 8800 - 4Pb
4000 = 8800 - 4Pb
4Pb = 8800 - 4000
4Pb = 4800
Pb = 4800 / 4
Pb = 1200
So, to maximize profit in the excursion segment, the fare should be $1,200 and there should be 4,000 passengers.
b. To determine if the airline would make more profit by charging a single price, we need to combine the demand equations from each market segment.
Total demand equation:
Q = Qa + Qb
Q = 1000 + 4000
Q = 5000
To determine the fare that maximizes profit with a single price, we need to find the fare (P) that maximizes profit.
Now that we have the total number of passengers (Q = 5000), we can substitute it back into the total demand equation to find the fare (P) that maximizes profit.
Q = 2100 - 0.5P + 8800 - 4P
5000 = 10900 - 4.5P
4.5P = 10900 - 5000
4.5P = 5900
P = 5900 / 4.5
P ≈ 1311.11
So, to maximize profit with a single price, the fare should be approximately $1,311.11 and there should be 5,000 passengers.
To determine if the airline would make more profit by charging a single price, compare the total profit with a single price to the combined profit from separate prices for each segment. Whichever is higher would result in more profit for the airline.