Math 135 Business Calculus Spring 2009

Project: Monopoly Pricing

Due Monday, April 20

IntroductionWhen 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 providedby 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. Aproduct 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 theconsumer if they were interested not in providing enough of their product for everyone but in providingjust enough to maximize their profits.

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

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

Fifty families live in houses and fifty families live in apartments. It has been estimated that peopleliving in houses are more desirous of having cable TV. If you charge a price of p per month for a cableTV hookup, the following piecewise-defined function gives the number of families living in houses, Nh,who will pay for a hookup.

Nh(p) =

50 if p ≤ 100100− 1

2p if 100 < p ≤ 2000 if p > 200

For instance, if p = 75, then Nh(75) = 50, meaning all 50 families living in houses will sign up. Ifp = 120, then Nh(120) = 100− 1

2 (120) = 40, meaning 40 families will sign up.The following piecewise-defined function for Na gives the number of families living in apartments

that will pay for a cable TV hookup if you charge a price p per month.

Na(p) =Ω

50− 13p if p ≤ 150

0 if p > 150

Economists usually call these demand curves (or demand functions). The town that is giving youthe franchise will only allow you to set one price for cable TV.

Your task is to determine what price you should charge for cable TV to maximize your profits bycarrying out the following steps.

• You need to first determine a demand function N(p) that tells you the total number of families thatwill buy a cable TV hookup if you charge a price p per month, using the two demand functionsNh(p) and Na(p). Plot the demand function N(p).

• Your revenue is the amount of money that you take in from cable TV hookups. It is equal to theprice you charge times the number of cable TV hookups you sell. Express the revenue from salesin terms of the price, p, you charge per month. Plot the revenue as a function of price.

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• You next must consider the costs incurred in providing cable service. The costs are $20 per hookupper month plus $2000 in maintenance costs. If you charge a price p for cable TV, what are thecosts to you in providing cable TV to every family that demands it? Plot total costs as a functionof the unit price charged.

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

• Since your profit function is defined in pieces, the derivative will also be a piecewise-defined function.Your profit is a continuous function of your price, so the Extreme Value Theorem guarantees thatthere is a price that will maximize your profits. Why is it continuous? What are the endpoints foryour maximization problem?

• Solve the maximization problem and determine the price you should charge to maximize yourprofits. Display the point of maximum profits on the graph of the profit function.

• At the price you are going to charge for cable TV, how many families in homes are paying forhookups? How many families in apartments are paying for cable TV?

Going into PoliticsSuppose instead of going into business, you are part of the government of the small town that has givensomeone the right to provide cable TV. The cable TV company has been in business several years andhas led to some complaints. The biggest complaint has come from people living in apartments. Theycomplain that the price of a cable TV hookup is so high that they can’t afford it, and they want youto do something that will allow them to enjoy cable TV as well. The monopolist claims that he mustcharge a high price to remain in business but suggests that you allow him to charge different prices toapartments and homes.• Suppose the monopolist can charge a price pa to families living in apartments and a price ph to

families living in houses. What pair of prices maximizes profit? How many houses and apartmentsare served at these prices?

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