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 28X  X² and profits of the developer are 20Y  Y²  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 profits of the developer.

A) H₁  H₂  4.
B) H₁  4 and H₂  10.
C) H₁  10 and H₂  4.
D) H₁  6 and H₂  9.
E) H₁  9 and H₂  13.

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Jefferson Gilles · Accepted Answer

Final Answer : B, Explanation :  To maximize profits, the single profit-maximizing company owning both the airport and the housing development will choose the level of X and Y that maximizes the total profit function, which is:28X - X(Y/10) + 20YThis function can be represented graphically as a downward sloping curve. To find the optimal values of X and Y, we need to find the point where the slope of the curve is zero. This happens where the partial derivatives of the function with respect to X and Y are equal to zero. Solving these two equations simultaneously for X and Y, we get:X = 100Y = 400Therefore, H 1  = 400.Now, if the airport and the housing development are operated independently, the airport has to pay the developer the total damages XY done by the planes to the profits of the developer. This means that the profit function for the airport is:28X - XYTo find the optimal value of X, we need to take the partial derivative of this function with respect to X and set it equal to zero:28 - Y = 0Therefore, the optimal value of X is:X = Y/28Substituting this value of X in the profit function for the developer, we get:20Y - XY/28To find the optimal value of Y, we need to take the partial derivative of this function with respect to Y and set it equal to zero:20 - X/28 = 0Substituting the value of X in this equation, we get:20 - Y/(28*28) = 0Therefore, the optimal value of Y is:Y = 560However, this is not the number of houses built, as the airport has to pay the developer for damages. Therefore, the actual number of houses built is:H 2  = Y - X*Y/10Substituting the values of X and Y, we get:H 2  = 400Therefore, the best choice is B, where H 1     4 and H 2     10.

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