91Ó°ÊÓ

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.

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.

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:

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

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).