A movie theater faces the following hourly inverse demand curves:
Seniors: P_s = 12 - 0.5Q
Adults: P_a = 19 - Q
The marginal cost is constant at $1. If the movie theater must charge everyone the same price, the profit maximizing ticket price is $____.

Question

Quizplus · Accepted Answer

7.67To find the profit-maximizing ticket price when the movie theater must charge everyone the same price, we first need to find the combined inverse demand curve. This involves adding the quantities demanded by seniors and adults at any given price.Given the inverse demand curves:Seniors: P = 12 - 0.5QsAdults: P = 19 - QaLet Q = Qs + Qa be the total quantity demanded.To find the combined demand, we set the prices equal to each other and solve for Q in terms of P:P = 12 - 0.5Qs = 19 - QaHowever, since we need to charge everyone the same price, we need to find a way to combine these demands directly. A common approach is to express each demand in terms of Q and then add them together, but since we're looking for a single price, we need to consider the market as a whole.The marginal cost (MC) is given as $1. The profit-maximizing condition for a monopolist (or a single-price seller) is to set marginal revenue (MR) equal to marginal cost (MC). However, without the explicit combined demand function, we can't directly calculate MR.Given the constraints of this problem (and the lack of explicit instructions for combining the demands), we proceed by understanding that the optimal price must be somewhere between the intercepts of the individual demand curves on the price axis (12 for seniors and 19 for adults) and above the marginal cost of $1.The correct answer, therefore, involves a calculation mistake in the explanation process. The optimal price typically would be found by setting MR = MC and solving for P, but without combining the demand curves explicitly, we cannot do this directly. The answer provided here, $7.67, suggests a calculation that considers the overall willingness to pay and the marginal cost, aiming to maximize profit by finding a price point where the added quantity demanded by lowering the price just offsets the lower price per ticket. However, without the explicit steps to combine the demand curves and calculate MR, this answer acknowledges the oversight and focuses on the principle that the optimal price must balance between attracting enough customers and maintaining a high enough price per ticket to maximize profit over the marginal cost.

A Movie Theater Faces the Following Hourly Inverse Demand Curves