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 = 220−40PS, where QS is the number of movie tickets demanded by students at price PS. The nightly demand for nonstudent moviegoers is QN = 140−20PN.
(a) If the Grand Theater charges a single price, PT , to everybody, then at prices between 0 and $5.50, the aggregate demand function for movie tickets is ________. Over this range of prices, the inverse demand function is then ________.
(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? $3. 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? ________.
(d) If you know calculus, see if you can do this part. Suppose that the Grand Theater can hold only 150 people and that the manager wants to maximize profits by charging separate prices to students and to nonstudents. If the capacity of the theater is 150 seats and QS tickets are sold to students, what is the maximum number of tickets that can be sold to nonstudents? ________. Write an expression for the price of nonstudent tickets as a function of the number of student tickets sold. ________. Write an expression for Grand Theater profits as a function of the number QS only. __________________. How many student tickets should the Grand sell to maximize profits? ________. What price is charged to students? ________. How many nonstudent tickets are sold? ________. What price is charged to nonstudents? ________. How much profit does the Grand make under this arrangement? ________.