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Homework answers / question archive / You must allocate the 70,000 seats in Reliant Stadium (in Houston) among Texan (Houston) and Cowboy (Dallas) fans for an upcoming game between the two footfall teams
You must allocate the 70,000 seats in Reliant Stadium (in Houston) among Texan (Houston) and Cowboy (Dallas) fans for an upcoming game between the two footfall teams. You can set different prices for seats in the Dallas and Houston sections of the stadium. Suppose you can obtain $40/ticket from Houston fans irrespective of the number of seats you allocate to Houston fans. You must drop price in order to sell more tickets to Dallas fans, however. Let Q be the number of tickets you allocate to Dallas fans. Assume that the maximum price you can charge for these tickets is given by the following inverse demand function Q P=80- 500 (a) Express the total revenue (on all 70,000 seats) from ticket sales as a function of Q; (b) Derive the first-order condition of the revenue-maximizing problem (it's a function about Q); (c) What is the optimal number of seats allocated to Dallas fans?
a.
Given demand function of Dallas fans
P = 80 - Q/500
Revenue from Dallas fans = P*Q = 80Q - Q??????2???/500
After selling Q tickets to Dallas fans, tickets left to sell to Houston fans = 70000-Q.
Price charged to Houston fans is at $40.
So, revenue from Houston fans = 40(70000-Q) = 2,800,000 - 40Q
Total revenue = sum of revenue from Dallas fans and Houston fans.
Total revenue (TR) = 80Q - Q??????2 /500 + 2,800,000 - 40Q
--> TR = 2,800,000 + 40Q - Q??????2 /500.
So, Total revenue = 2,800,000 + 40Q - (Q??????2???/500).
b.
Revenue is maximized when dTR/dQ = 0
dTR/dQ = 40 - Q/250
dTR/dQ = 0 ----> 40 - (Q/250) = 0
So, the first order condition :
40-(Q/250) = 0
??????c. Optimal number of seats is determined when first order condition is met.
---> 40-(Q/250) = 0
---> Q = 10,000
So, optimal amount of Q = 10,000.
Therefore, optimal number of seats allocated to Dallas fans is 10,000.
