What is the profit maximising ticket price, Managerial Economics

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 - Q2 / 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

Posted Date: 8/12/2013 2:43:57 AM | Location : United States







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