Question

An airport is located next to a large tract of land owned by a housing developer. The
developer would like to build houses on this land, but noise from the airport reduces the
value of the land. The more planes that fly, the lower is the amount of profits that the
developer makes. Let X be the number of planes that fly per day and let Y be the number of
houses that the developer builds. The airport’s total profits are 48X−X2

, and the developer’s

total profits are 60Y −Y2−XY.
A. Suppose the airport and developer act independently of each other. How many
planes will fly every day and how many houses will be built? Find the profits of the
airport and developer.
B. Suppose that a single firm bought the developer’s land and the airport. What would
be the optimal numbers of planes and houses? What is the total profit?
C. Which of the two outcomes (parts A and B) is socially optimal? Why? Explain.

Answer 1

A. If the airport and the developer acts independently, then they will individually choose the optimal quantity so that they can maximize their profits. Then for the airport, the profit function will be maximized at the q where the first order differential with respect to quantity is equal to 0. So we have,
48X -X^2 differtiated wrt to X,
48-2X = 0,
or, X = 24.
So the airport will fly 24 planes.
Profit = 48.24 - 24.24 = 576
putting the value of X in the developer's profit function, we have, 60Y - Y^2 - 24Y. The function will be maximized when the first order differential with respect to Y  will be equated to 0.
36 - 2Y = 0
or, Y = 18.
So the developer will build 18 houses.
Profit of the developer = 60.18-18.18-24.18 = 324

B. Total profit function of the single firm when airport and land is combined = 60Y -Y^2 -XY + 48X - X^2
Differentiating with respect to X and Y, we get two equations which must be equated to 0 as we are trying to maximize the function.
so we have,
60- 2Y -X = 0 ---- (1)
48- 2X - Y = 0 --- (2)
equation 1 and 2 form simultaneous systems and solving simultaneously ( multiplying eqn 1 by 2 and subtracting it from eqn 2 to get value of Y and then substituting it to get value of X )
We get the value of Y = 32 ad X = -4
However, X cannot be negative and at the most it will be 0.
Thus it is clear that the new firm will shut down the airport and will only be using the land to build houses as it generates more profit.
So, effectively, the new firm will have the profit function : 60Y -Y^2
By differentiating with respect to Y and equating to 0, we get, 60-2Y = 0 or, Y = 30
Profit of the new firm = 30.60-30.30 = 900.

C. Both the outcomes generate the same level of profit, which is 900. So one can be indifferent between both the scenarios as both can be counted as socially optimal, if we assume there is no greater social preference for constructing houses instead of having airport service.