PB = $140 – 4AB where PB is the ticket price paid by businesses, measured in dol

Question

PB = $140 – 4AB
where PB is the ticket price paid by businesses, measured in dollars, and AB is
their attendance measured in thousands of fans.
a. Draw this attendance demand function.
b. Using this demand function, find the total revenue function. What
is the shape of the total revenue function? What is the highest
possible total revenue that the team can hope to collect? At what
level of attendance? At what price?
c. Using this demand function and your answer to part (b), what is
the elasticity of demand at the revenue maximizing attendance
level?
d. Using your answers to a-c, if capacity at the team's stadium is
25,000 seats, should the team owner fill the stands with business
buyers? Why or why not?

Explanation / Answer

a)
Let us write PB = P and AB = A for ease.A is in thosands.
So, P = 140 - 4A ($)
A = (140 - P)/4 (demand function) (ANSWER)

b)
R = P*A
= 140A - 4A^2 (revenue function) (ANSWER)
This function is parabolic.
R max occurs at dR/dA = 0
=> 140 -8A = 0
=> A = 17.5
-> so, attendance = 17500 (ANSWER)
and P = 140 - 4*17.5 = 140 -70 = 70 ($) ( ANSWER)
and R = 70*17.5 = 1225 thousand $

c)
Elasticity of demand = (P/A)(dA/dP)= (70/17.5)(-1/4) = -1 (ANSWER)

d)
If A = 25 (i.e. 25000)
P = 140 -4*25 = 40 $
R = P*A = 40*25 = 1000 thousand $
As R is decreased, hence not advisable.

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