Question

The Miller Corporation 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 π = -80 + 50 Q1 + 40 Q2 -8 Q12 – 6 Q22 -2 Q1Q2, where Q1 is the firm’s annual output of paper (in tons), and Q2 is the firm’s annual output of cardboard (in tons). How much paper should the Miller Corporation produce in order to maximize profit?

Answer 1

π = -80 + 50 Q1 + 40 Q2 -8 Q12 – 6 Q22 -2 Q1Q2

We find the first order conditions with respect to Q1 and Q2 to maximize the profit.

So, 2Q2 = 50 - 16Q1
So, Q2 = (50/2) - (16/2)Q1
So, Q2 = 25 - 8Q1

So, 40 - 12Q2 - 2Q1 = 0
So, 40 - 12(25-8Q1) - 2Q1 = 0
So, 40 - 300 + 96Q1 - 2Q1 = 0
So, 94Q1 = 260
So, Q1 = 260/94

So, Q1 = 2.766 tons

The Miller Corporation should produce 2.766 tons paper in order to maximize profit.