Question

Sal’s satellite company broadcasts TV to subscribers in Los Angeles and New York. The demand functions for each of these two groups are

Q_NY = 60 - 0.25P_NY

Q_LA = 100 - 0.50P_LA

where Q is in thousands of subscriptions per year and P is the subscription price per year. The cost of providing Q units of service is given by

C = 1000 + 40Q

where Q = QNY + QLA.

1. What are the profit-maximizing prices and quantities for the New York and Los Angeles markets?
2. As a consequence of a new satellite that the Pentagon recently deployed, people in Los Angeles receive Sal’s New York broadcasts and people in New York receive Sal’s Los Angeles broadcasts. As a result, anyone in New York or Los Angeles can receive Sal’s broadcasts by subscribing in either city. Thus Sal can charge only a single price. What price should he charge, and what quantities will he sell in New York and Los Angeles?
3. In which of the above situations, (a) or (b), is Sal better off? In terms of consumer surplus, which situation do people in New York prefer and which do people in Los Angeles prefer? Why?

Short answer

1. The profit-maximizing quantity will be 25 units at $140 for New York and 40 units at $120 for Los Angeles.
2. The quantity will be 28.3 units for New York and 36.7 units for Los Angeles at $126.68.
3. Sal is better off in situation (a). The consumer in New York will prefer situation (b) as the equilibrium price is lower than situation (a); the consumer in Los Angeles will prefer situation (a) as the equilibrium price is lower than situation (b).

Step-by-step solution

Explanation for part (a)

The profit-maximizing output level will be attained when the marginal revenue is equal to the marginal cost in the market.

The profit-maximizing price and quantity for New York are calculated below:

$$\begin{gathered} {\text{P}}_{\text{NY}}{\text{= 240 - 4Q}}_{\text{NY}} \\ {\text{MR}}_{\text{NY}}{\text{= 240 - 8Q}}_{\text{NY}} \\ \text{MC = 40} \\ {\text{MR}}_{\text{NY}}\text{= MC} \\ {\text{240 - 8Q}}_{\text{NY}}\text{= 40} \\ {\text{8Q}}_{\text{NY}}\text{= 200} \\ {\text{Q}}_{\text{NY}}\text{= 25} \\ {\text{P}}_{\text{NY}}\text{= 240 - 4}\left(\text{25}\right) \\ \text{= 240 - 100} \\ \text{= \$ 140} \end{gathered}$$

The output level will be 25 units at $140.

The profit-maximizing price and quantity for Los Angeles are calculated below:

$$\begin{gathered} {\text{P}}_{\text{LA}}{\text{= 200 - 2Q}}_{\text{LA}} \\ {\text{MR}}_{\text{LA}}{\text{= 200 - 4Q}}_{\text{LA}} \\ \text{MC = 40} \\ {\text{MR}}_{\text{LA}}\text{= MC} \\ {\text{200 - 4Q}}_{\text{LA}}\text{= 40} \\ {\text{4Q}}_{\text{LA}}\text{= 160} \\ {\text{Q}}_{\text{LA}}\text{= 40} \\ {\text{P}}_{\text{LA}}\text{= 200 - 2}\left(\text{40}\right) \\ \text{= 200 - 80} \\ \text{= \$ 120} \end{gathered}$$

The output level will be 40 units at $120.

Explanation for part (b)

The total demand function is calculated below:

$$\begin{gathered} {\text{Q}}_{\text{NY}}{\text{= 60 - 0.25P}}_{\text{NY}} \\ {\text{Q}}_{\text{LA}}{\text{= 100 - 0.5P}}_{\text{LA}} \\ {\text{Q =Q}}_{\text{NY}}{\text{+Q}}_{\text{LA}} \\ {\text{Q = 60 - 0.25P}}_{\text{NY}}{\text{+ 100 - 0.5P}}_{\text{LA}} \\ \text{Q = 160 - 0.75P} \\ \text{P = 213.33 - 1.33Q} \end{gathered}$$

The price and quantity are calculated below:

$$\begin{gathered} \text{MR = 213.33 - 2.667} \\ \text{MC = 40} \\ \text{MR = MC} \\ \text{213.33 - 2.667Q = 40} \\ \text{2.667Q = 173.33} \\ \text{Q = 65} \\ \text{P = 213.33 - 1.333}\left(\text{65}\right) \\ \text{= 213.33 - 86.645} \\ \text{= \$ 126.68} \end{gathered}$$

At $126.68, the quantity in each market is calculated below:

$$\begin{gathered} {\text{Q}}_{\text{NY}}\text{= 60 - 0.25}\left(\text{126.68}\right) \\ \text{= 60 - 31.67} \\ \text{= 28.3} \\ {\text{Q}}_{\text{LA}}\text{= 100 - 0.5}\left(\text{126.68}\right) \\ \text{= 100 - 63.34} \\ \text{= 36.7} \end{gathered}$$

The quantity will be 28.3 units for New York, and the quantity will be 36.7 units for Los Angeles at $126.68.

Explanation for part (c)

To check whether Sal is better off in situation (a) or (b), the profit in both situations needs to be checked. Thus, the profit in both the situation is calculated below:

$$\begin{gathered} \mathrm{\pi }=\left(140\times 25\right)+\left(120\times 40\right)-1000-40\left(25+40\right) \\ =3500+4800-1000-3600 \\ =\$3700 \\ \mathrm{\pi }=1\left(26.67\times 65\right)-1000-40\left(65\right) \\ =8233.55-1000-2600 \\ =\$463.55 \end{gathered}$$

Sal is better off in situation (b) as profit is greater than situation (a).

The consumer surplus in the situation (a) for both the market is calculated below:

$$\begin{gathered} {\text{CS}}_{\text{NY}}\text{= 0.5}\left(\text{25}\right)\left(\text{240 - 140}\right) \\ \text{= 0.5}\left(\text{25}\right)\left(\text{100}\right) \\ \text{= \$ 1250} \\ {\text{CS}}_{\text{LA}}\text{= 0.5}\left(\text{40}\right)\left(\text{200 - 120}\right) \\ \text{= 0.5}\left(\text{40}\right)\left(\text{80}\right) \\ \text{= \$ 1600} \end{gathered}$$

The consumer surplus in situation (b) for both the market is calculated below:

$$\begin{gathered} {\text{CS}}_{\text{NY}}\text{= 0.5}\left(\text{28.3}\right)\left(\text{240 - 126.67}\right) \\ \text{= 0.5}\left(\text{28.3}\right)\left(\text{113.33}\right) \\ \text{= \$ 1603.62} \\ {\text{CS}}_{\text{LA}}\text{= 0.5}\left(\text{36.7}\right)\left(\text{200 - 126.67}\right) \\ \text{= 0.5}\left(\text{36.7}\right)\left(\text{73.33}\right) \\ \text{= \$ 1345.61} \end{gathered}$$

The consumer in New York will prefer situation (b) as the equilibrium price is lower than situation (a); the consumer in Los Angeles will prefer situation (a) as the equilibrium price is lower than situation (b).