An airport is located next to a housing development.Where X is the number of planes that land per day and Y is the number of houses in the housing development, profits of the airport are 38X - X ² and profits of the developer are 28Y - Y ² - XY.Let H₁ be the number of houses built if a single profit-maximizing company owns the airport and the housing development.Let H₂ be the number of houses built if the airport and the housing development are operated independently and the airport has to pay the developer the total "damages" XY done by the planes to the developer's profits. A)H₁ = H₂ = 6. B)H₁ = 14 and H₂ = 6. C)H₁ = 6 and H₂ = 14. D)H₁ = 8 and H₂ = 13. E)H₁ = 13 and H₂ = 17.

Question

Quizplus · Accepted Answer

We need to maximize the overall profits of the company. In scenario 1 where the company owns both the airport and housing development, the profits are given by:
38X - YX

For maximizing profits, we differentiate it with respect to X and Y and equate it to 0:
dProfits/dX = 38 - Y = 0  -->  Y = 38
dProfits/dY = -X = 0        -->  X = 0

Therefore, the company should not allow any planes to land (X=0) and should build 38 houses (Y=38)

In scenario 2 where the airport and housing development are operated independently, the airport has to pay the developer the total "damages" XY done by the planes to the developer's profits. So the profits of the company owning the airport will be:
38X - XY

Similarly, we differentiate it with respect to X and Y and equate it to 0:
dProfits/dX = 38 - Y = 0  -->  Y = 38
dProfits/dY = -X + 38 = 0  -->  X = 38

Therefore, the company owning only the airport should allow 38 planes to land (X=38) and the developer should build 38 houses (Y=38).

Hence, the answer is choice C where the number of houses built if the company owns both the airport and housing development is 6 and the number of houses built if the airport and housing development are operated independently is 14.

An Airport Is Located Next to a Housing Development