Question

2. As the owner of a new expansion team in the fledgling Canadian Women’s Professional Hockey League (CWPHL), you have been granted exclusive territorial rights to operate in the city of Calgary. The team will play 35 home games in an arena that has a seating capacity of 10,000. Based on market research, you know that the demand curve for single game tickets in Calgary is given by the equation P = 75 – 0.004Q. You have had to incur a lot of fixed costs to get the operation going, including player salaries, facility rental, equipment and transportation costs, insurance and medical costs, totaling $10,000,000 for the season, but variable costs are virtually zero. a. Compute the profit-maximizing number of tickets to sell for each game, the average ticket price and the total profit for the season.

b. How many seats will be left unsold? Should the team lower the price to sell these seats? Explain.

c. Fixed costs have increased to $12,500,000 due to an increase in the rental fee for the facility. What is the new profit-maximizing quantity of tickets to sell and average ticket price? Show your work.

d. The first season of the CWPHL was a tremendous success and ticket demand for your team has increased to P = 100 – 0.004Q, with fixed costs left unchanged at $10,000,000 from part a. What is the new profit-maximizing quantity of tickets to sell? What will the average ticket price be? Show your work. Will you be happy in your arena? Why or why not? 3. As a professional hockey player, you have received a salary offer for

Answer 1

Part a) We have the following information

Demand equation: P = 75 – 0.004Q

P = Price

Q = Quantity

Total Revenue (TR) = P × Q

TR = (75 – 0.004Q)Q

TR = 75Q – 0.004Q²

Total Cost (TC) = $10,000,000

Profit = TR – TC

Profit = 75Q – 0.004Q² – 10,000,000

Taking the first derivative

ΔProfit/ΔQ = 75 – 0.008Q

Equating the first derivative to zero

75 – 0.008Q = 0

Q = 9375

Taking the second derivative

Δ²Profit/ΔQ² = – 0.008

Since the second derivative is less than zero so the profit is maximized at Q = 9375

Equilibrium number of tickets = 9375

Equilibrium price = 75 – 0.004Q

Equilibrium price = 75 – 37.5

Equilibrium price = $37.5

Total number of games in the season = 35

TR = 9375 × 37.5 × 35

TR = 12,304,687.5

Profit = TR – TC

Profit = 12,304,687.5 – 10,000,000

Profit = $2,304,687.5

Part b) Total number of seats sold during the season = 9375 × 35 = 328,125

Total number of seats available during the season = 10,000 × 35 = 350,000

Seats unsold = 350,000 – 328,125 = 21,875

Given the total cos the figure of 328,125 seats is the profit maximizing so there should not be any decrease in the price.

Part c) Now it is given that the fixed cost has increased to $12,500,000.

Profit = TR – TC

Profit = 75Q – 0.004Q² – 12,500,000

Taking the first derivative

ΔProfit/ΔQ = 75 – 0.008Q

Equating the first derivative to zero

75 – 0.008Q = 0

Q = 9375

Taking the second derivative

Δ²Profit/ΔQ² = – 0.008

Since the second derivative is less than zero so the profit is maximized at Q = 9375

Equilibrium number of tickets = 9375

Equilibrium price = 75 – 0.004Q

Equilibrium price = 75 – 37.5

Equilibrium price = $37.5

So, the profit maximizing price and quantity remains the same as the change has taken place in the fixed cost.

Part d) We have the following information

Demand equation: P = 100 – 0.004Q

P = Price

Q = Quantity

Total Revenue (TR) = P × Q

TR = (100 – 0.004Q)Q

TR = 100Q – 0.004Q²

Total Cost (TC) = $10,000,000

Profit = TR – TC

Profit = 100Q – 0.004Q² – 10,000,000

Taking the first derivative

ΔProfit/ΔQ = 100 – 0.008Q

Equating the first derivative to zero

100 – 0.008Q = 0

Q = 12,500

Taking the second derivative

Δ²Profit/ΔQ² = – 0.008

Since the second derivative is less than zero so the profit is maximized at Q = 12,500

Equilibrium number of tickets = 12,500

Equilibrium price = 100 – 0.004Q

Equilibrium price = 100 – 50

Equilibrium price = $50

However, given the fact that the seating capacity of the arena is 10,000 so the profit maximizing level of seats (12,500) cannot be sold. So, one cannot be happy with the arena.