A promoter decides to rent an arena for concert. Arena seats 20,000. Rental fee is 10,000. (This is a fixed cost.) The arena owner gets concessions and parking and pays all other expenses related to concert. The promoter has properly estimated the demand for concert seats to be Q = 40,000 - 2000P, where Q is the quantity of seats and P is the price per seat. What is the profit maximising ticket price?
As the promoter's marginal costs are zero, promoter maximises profits by charging a ticket price which will maximise revenue. Total revenue equals price, P, times quantity. Total revenue is represented as a function of quantity, so we need to work with the inverse demand curve:
P (Q) = 20 - Q / 2000
This gives total revenue as a function of quantity, TR (Q) = P (Q) x Q, or
TR (Q) = 20Q - Q^{2} / 2000
Total revenue reaches its maximum value when marginal revenue is zero. Marginalrevenue is first derivative of total revenue function: MR (Q) =TR'(Q). So
MR (Q) = 20 - Q / 1000
Setting MR (Q) = 0 we get
0 = 20 - Q / 1000
Q = 20,000
Recall that price is a function of quantity sold (inverse demand curve. So to sell this quantity, ticket price should be
P (20000) = 20 - 20,000 / 2,000 = 10
It may appear more natural to view the decision as price setting instead of quantity setting. Normally, this isn't a more natural mathematical formulation of profit maximisation since costs are generally a function of quantity (not of price). In this specific illustration, though, the promoter's marginal costs are zero. This means the promoter maximises profits simply by charging a ticket price that would maximise revenue. In this specific case, we characterise total revenue as a function of price:
TR2 (P) = (40,000 - 2000P)P = 40,000P - 2000 (P) 2
Total revenue reaches its maximum value when marginal revenue is zero. Marginal revenue is the first derivative of the total revenue function. So
MR2 (P) = 40,000 - 4000P
Setting MR2 = 0 we get,
0 = 40,000 - 4000P
P = 10
Profit = TR2 (P) -TC
Profit = [40,000P - 2000(P) 2] - 10,000
Profit = [40,000(10) - 2000(10)2] - 10,000
Profit = 400,000 - 200,000 - 10,000
Profit = 190,000
What, if the promoter had charged 12 per ticket?
Q = 40,000 - 2000P.
Q = 40,000 - 2000(12)
Q = 40,000 - 24,000 = 16,000 (tickets sold)
Profits at 12:
Q = 16,000(12) = 192,000 - 10,000 = 182,000