You are organizing a rock concert to benefit Amnesty International. The concert

Question

You are organizing a rock concert to benefit Amnesty International. The concert is to be held in a football stadium that has a seating capacity of 56,000. There are two types of seating: reserved and general. Demand for these seats are estimated as:

PR = 100-.00125 QR and PG = 60-.00125 QG  

where PR and PG denote ticket prices for reserved and general seating and QR and QG are the number of those seats

All artists and support staff are donating their time, but there is a fixed cost of $100,000.

(a) Using the method of the Lagrange multiplier calculate the prices that you will charge to so that the profit is maximized and at the same time the stadium will be filled to capacity.

Calculate the number of each type of tickets sold, as well as the consequent profit level.

(b) Calculate the value of the Lagrange multiplier. Explain its economic significance in this case.

Explanation / Answer

(iii) (a) TC=100,000

TR=QR(100-.00125QR)+QG(60-.00125QG) =

100QR-.00125QR^2+60QG-.00125QG^2

Profit=(100QR-.00125QR2+60QG-.00125QG2)-100,000

Constraint: QR+QG=56,000 = 56,000-QR-QG=0

L(QR,QG, ?)=(100QR-.00125QR2+60QG-.00125QG2)-100,000+ ?[56,000-QR-QG]

LQR= - ?-.0025QR+100 then ?=100-.0025QR

LQG= - ?-.0025QG+60 then ?=60-.0025QG

L ?= 56,000-QR-QG then 56,000=QR+QG

LQR=LQG then 1=(100-.0025QR)/(60-.0025QG) then 100-.0025QR=60-.0025QG then

40-.0025QR=-.0025QG then 40=.0025QR-.0025QG and 16,000=QR-QG

56,000=QR+QG - 16,000=QR-QG then 2QG=40,000 then QG=20,000

56,000=QR+20,000 so QR=36,000

The Demand functions are: PR=100-.00125QR PG=60-.00125QG

PR=100-.00125(36,000) then 100-45 and 55

PG=60-.00125(20,000) then 60-25 and 35

TC=100,000

TR=(36000*55)+(20,000*35) then 1,980,000+700,000 and 2,680,000
Profit= 2,680,000-100,000 which is $2,580,000

Therefore, the profit that will be charged so the prices are maximized will be $55 for Reserved seating and $35 for General seating. There will be 36,000 Reserved seats sold and 20,000 General seats sold. The profit will be $2,580,000.

(b) ?=100-.0025QR then100-.0025(36,000) then 100-90 which is 10

The Lagrange multiplier is 10. This means that if the total number of reserved and general seating that is sold could be increased then the total profit would increase by 10.

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