The Paper Dragon Limited Partnership makes two products, paper and cardboard. Th

Question

The Paper Dragon Limited Partnership makes two products, paper and cardboard. The relationship between , the firm’s annual profit (in thousands of dollars), and its output of each good is: = -50 + 40X + 30Y – 5X^2– 4Y^2– 3XY where X is the firm’s annual output of paper (in tons) and Y is the firm’s annual output of cardboard (in tons). a. find the output of each good that Paper Dragon LP should produce if it wants to maximize profit. b. if the community in which the firm is located imposed a flat tax of $5,000 per year on the firm, will this alter the answer to part a? If so, how will the answer change?

Explanation / Answer

dpi/dx = 40-10X-3Y = 0

Also dpi/dy = 30-8Y-3X = 0

Solve these two equations:

40 = 10X + 3Y => 120 = 30X + 9Y

30 = 3X + 8Y=> 300 = 30X + 80Y

=> - 180 = -71 Y => Y = 180/71 = 2.5352

And also 30 = 3X + 8(180/71) => X = 230/71 = 3.2394

d2pi/dxdy = -3,d2pi/dx2 = -10, d2pi/dy2 = -8

D = d2pi/dx2  d2pi/dy2 -(d2pi/dxdy)2 =(-10)(-8)-(-3) = 80+3 = 83>0

Also d2pi/dx2 < 0, then pi is maximum at (230/71, 180/71)

b) The answer will not be alterd as 5000 flat will be decreased from profit equation but then it is constant and so it will not affect the answer

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