The Grand Theater is a movie house in a medium-sized college town. This theater

Question

The Grand Theater is a movie house in a medium-sized college town. This theater shows unusual

films and treats early-arriving movie goers to live organ music and Bugs Bunny cartoons. If the

theater is open, the owners have to pay a fixed nightly amount of $500 for films, ushers, and

so on, regardless of how many people come to the movie. For simplicity, assume that if the

theater is closed, its costs are zero. The nightly demand for Grand Theater movies by students

is QS = 220%u221240PS, where QS is the number of movie tickets demanded by students at price PS.

The nightly demand for nonstudent moviegoers is QN = 140%u221220PN .

(a) If the Grand Theater charges a single price, P, to everybody, then at prices between 0 and

3

$5.50, the aggregate demand function for movie tickets is Q(P) = . Over this range of prices,

the inverse demand function is then P(Q) = .

(b) What is the profit-maximizing number of tickets for the Grand Theater to sell if it charges

one price to everybody? . At what price would this number of tickets be sold? . How much

profits would the Grand make? . How many tickets would be sold to students? . To

nonstudents? .

(c) Suppose that the cashier can accurately separate the students from the nonstudents at the

door by making students show their school ID cards. Students cannot resell their tickets and

nonstudents do not have access to student ID cards. Then the Grand can increase its profits by

charging students and nonstudents different prices. What price will be charged to students? .

How many student tickets will be sold? . What price will be charged to nonstudents? . How

many nonstudent tickets will be sold? . How much profit will the Grand Theater make?

Explanation / Answer

a. In this case, the total demand will be Q = 360 60p, marginal revenue will be 6-(1/30)Q, and so prot-maximizing quantity will be 180 patrons, at a price of $3. Prots are $40 ($3*180-$500).

b. Treating students and non-students as two separate groups, the theater will charge students $2.75 and

sell them 110 tickets, while charging non-students $3.50 and selling them 70 tickets. Total prots are $47.50.

c. We know that the number of students admitted plus the number of non-students admitted must equal

150. We also know that to be maximizing prots, it must be that the marginal revenue from the last student

admitted must equal that from the last non-student admitted. Were this not so, the theater could admit one

less from the low marginal revenue group and one more from the high marginal revenue group, and increase

prots. Thus, the following two equations dene the solution:

qs + qn= 150

5.5-(1/20)qs= 7-(1/10)qn

which has solution qn= 60, qs= 90, meaning ps= $3:25 and pn= $4

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