Question

A football club is the only one in its region and is therefore able to behave like a monopolist. It sells tickets to Adults (a) and Juniors (j), whose demand curves are given by:

Pa = 200 − Qa
Pj = 160 − Qj

Additionally, the club’s total costs are given by = 20Qi

The club hires an economist to consider its pricing strategy and receives the advice that it should charge different prices to each type of supporter. Find the prices it sets in both markets, the sales (output) in each and its overall level of profit and illustrate profit maximising outputs and prices of each consumer type on a graph (one graph for each type of consumer). Calculate consumer surplus in this case.

Answer 1

Adult market

• price: 110
• sales: 9900
• profit: 8100
• consumer surplus: 9000

Junior market

• price: 90
• sales: 6300
• profit: 4900
• consumer surplus: 5600

Total profit: 13000

Total consumer surplus: 14600

Explanation:

Given Adult (a) and Junior (j) demand equations Pa = 200 -Qa and Pj = 160 -Qj, and cost equation C = 20Q, you want to find the price in each market that maximizes profit, the sales in each market, and the consumer surplus, and a graph of profit-maximizing sales and prices.

Revenue

Each demand equation is of the form P = Pmax -Q, where P is the price that will result in sales of Q tickets. The revenue (R) in each case is the product of numbers of tickets sold (Q) and the price at which they are sold (P).

R = QP = Q(Pmax -Q)

Profit

The profit is the difference between revenue and cost.

Profit = R - C = Q(Pmax -Q) -20Q

Profit = Q(Pmax -20 -Q)

Writing the demand equation in terms of P, we find ...

Q = Pmax -P

Substituting this into the Profit equation gives ...

Profit = (Pmax -P)(Pmax -20 -(Pmax -P))

Profit = (Pmax -P)(P -20)

The profit function describes a downward-opening parabolic curve with zeros at P=Pmax and P=20. The maximum profit is on the line of symmetry of this curve, halfway between these values of P:

Price for maximum profit = (Pmax +20)/2 = Pmax/2 +10

Prices

In the adult market, Pmax = 200, so the profit-maximizing ticket price is ...

Pa = 200/2 +10 = 110 . . . . price for maximum profit in Adult market

In the Junior market, Pmax = 160, so the profit-maximizing ticket price is ...

Pj = 160/2 +10 = 90 . . . . price for maximum profit in Junior market

Sales

Using the revenue equation, we find the sales in each market to be ...

Qa = 200 -Pa = 200 -110 = 90

Ra = Qa·Pa = 90(110) = 9900 . . . . sales in Adult market

Qj = 160 -Pj = 160 -90 = 70

Rj = Qj·Pj = 70(90) = 6300 . . . . sales in Junior market

Overall Profit

The profit in each market is ...

Adult market profit = 90(110 -20) = 8100

Junior market profit = 70(90 -20) = 4900

The overall profit will be the sum of the profits in each market:

Overall profit = 8100 +4900 = 13000

Consumer surplus

The consumer surplus in each market is the area below the demand curve and above the price point. It is half the product of the maximum price and the quantity actually sold.

CSa = (1/2)(200)Qa = 100(90) = 9000

CSj = (1/2)(160)(Qj) = 80(70) = 5600

The total consumer surplus is ...

CS = CSa +CSj = 9000 +5600 = 14,600 . . . . total consumer surplus

Graph

The first attachment shows the sales (output) in each market (red=Adult, purple=Junior) as a function of ticket price. It also shows the corresponding profit (orange=Adult, blue=Junior). The profit-maximizing price point is marked on each curve. You will note that it is different from the output-maximizing price point.

The second attachment illustrates the consumer surplus in each market. That graph has price on the vertical axis, and quantity on the horizontal axis. The colors correspond to the colors on the graph in the first attachment.