When there is only one firm that produces a certain product, that one firm is called a monopoly. Monopolies are familiar to most consumers. Local phone service and electric power are often provided by monopolies. Monopolists have the advantage over firms that must compete since, without regulation, they might have the ability to control the price of their service by controlling the quantity produced. A product that is in short supply will fetch a high price if people are demanding that product. Conversely, if the product is easy to come by, the price of it will be low. Monopolies could be detrimental to the consumer if they were interested not in providing enough of their product for everyone but in providing just enough to maximize their profits.

Some local monopolies that have received a lot of attention lately are cable TV franchises. People have been dissatisfied with having to pay extra fees for special cable channels. The question of regulating the cable companies has been a matter of some concern. In this project, you will be able to examine how monopolies set prices and answer for yourself whether or not the cable companies should be regulated by the government.

28.1 Going into Business

Suppose a small town has offered to give you the rights to provide cable TV service to families in the town. As a merchandiser you are interested in maximizing your profits. You are told that there are 100 families in the town who do not have cable TV. The cost to you of providing cable TV is $20 per month per family as well as $2000 in monthly overhead that is related to maintenance of your equipment and does not depend on how many families you service.

Fifty families live in houses and fifty families live in apartments. It has been estimated that people living in houses are more desirous of having cable TV. If you charge a price of p for a cable TV hookup, the following expression gives the number of families living in houses, q_h, who will pay for a hookup.

The following expression for q_a gives the number of families living in apartments that will pay for a cable TV hookup if you charge a price p.

Economists usually call these demand curves (or demand functions). The town that is giving you the franchise will only allow you to set one price for cable TV. These "piecewise defined" functions may be defined on the computer, and there is help in the program MonopolyHelp on our website.

% Enter the computer commands in MonopolyHelp to make a plot of the demands by households and apartments as in Figure 28.2.

Figure 28.1: Demands for Cable TV

Your first task as monopolist is to determine what price you would charge for cable TV. We will determine this by solving a maximization problem in the exercises below. If you decide to charge a price p for cable TV, how many families in houses will buy a cable TV hookup? How many families living in apartments will buy a hookup? Determine expressions in terms of the price, p, of cable TV. These expressions will depend on what interval the price p lies in.

Your revenue is the amount of money that you take in from cable TV hookups. It is equal to the price you charge times the number of cable TV hookups you sell. Express the revenue from sales to houses in terms of the price, p, you charge for hookups. What is the revenue from sales to apartments? Plot revenue as a function of price as in Figure 28.4.

Figure 28.2: Cable Revenue

You next must consider the costs incurred in providing cable service. The costs are $20 per hookup per month plus $2000 in maintenance costs. If you charge a price, p, for cable TV, what are the costs to you in providing cable TV to every family that demands it? Express your costs in terms of the price of cable TV, p. Plot total costs as a function of the unit price charged (Figure 28.6).

Figure 28.3: Cost for Providing Cable TV

You are now ready to go into business (or are you?). The profit from running your cable company is your total revenue from sales minus the cost to you of providing the service. Express you profit as a function of the price, p, you charge for a cable TV hookup. Plot this. You wish to find the price to charge that will maximize your profit.

Since your profit function is defined in pieces, you must solve more than one maximization problem. Your profit is a continuous function of your price, so the Extreme Value Theorem guarantees that there is a price that will maximize your profits. Why is it continuous? What are the endpoints for your maximization problems?

Solve each maximization problem and determine the price you would charge to maximize your profits. How does the Extreme Value Theorem guarantee that you have found the point of maximum profits? You may wish to include a graph of your profit function to display the point of maximum profits (Figure 28.9).

Figure 28.4: Profit for Providing Cable TV at a Single Price