Question

Harry runs a small movie theater, whose customers all have identical tastes. Each customer’s reservation price for the movie is $5, and each customer’s demand curve for popcorn at his concession stand is given by P_c=4-Q_c, where P_c is the price of popcorn in dollars and Q_c is the amount of popcorn in quarts. If the marginal cost of allowing another patron to watch the movie is zero, and the marginal cost of popcorn is $1, at what price should Harry sell tickets and popcorn if his goal is to maximize his profits? (Assume that Harry is able to costlessly advertise his price structure to potential patrons.)

Hint：please note that movie price alone is not the entry price (not the only fixed portion). You can actually offer a movie+popcorn bundle for a fixed fee, and charge lower unit prices for popcorn. So ideally, you should discuss three scenarios: (1) Single-price scheme in which movie and popcorn are priced separately based on their demand curves(consider the movie theater as a monopoly, not a price taker); (2) Two-part pricing scheme in which consumers pay a fixed fee (including the movie price) and pay a lower price of popcorn compared to (1); (3) Two-part pricing scheme in which consumers pay a fixed fee and do not pay a unit price. And then you find the relationship between (2) and (3), and then you will find the best plan. I highly recommend you guys do the actual calculations, rather than only argue theoretically and graphically.

Answer 1

This is a two part tariff problem.

He will be optimizing his profit if he charges the reservation price for the ticket and price for the popcorn where Marginal Revenue from the popcorn= Marginal Cost of the popcorn.

It is given that the demand function of the popcorns is given by P=4-Q. The marginal cost is given as 1.

Since we know that Revenue=Price*Q, it means revenue= 4Q-Q². Marginal revenue=4-2Q. Equating MR and MC, we get

4-2Q=1

Q=3/2=1.5. At this quantity,

P=4-1.5=2.5 per quart.

Since he will be selling 1.5 quarts per customer, the price of pop-corn per customer will be 2.5*1.5=3.75

Hence, he should charge a total charge of= Reservation price+price of pop-corn

=5+3.75=$8.75 for the ticket.