Problem 1. The Stock Corporation makes two products, paper and cardboard. The relationship between Π the firm's annual profit (in thousands of dollars), and its output of each good is:
Π = -50 + 40 Q1 + 30 Q2 - 5Q12 - 4Q22 - 3Q1 Q2,
where Q1 is the firm's annual output of paper (in tons) and Q2 is the firm's annual output of cardboard (in tons). Find the output of each good that the Stock Corporation should produce if it wants to maximize profit.
Problem 2. The Miller Company uses skilled and unskilled labor to do a particular construction project. The cost of doing the project depends on the number of hours of skilled labor and the number of hours of unskilled labor used, the relationship between hours of labor used and the cost is estimated as:
C = 4 - 3 X1 - 4 X2 + 2X12 + 3X22 + X1 X2,
where C is cost (in thousands of dollars), X1 is the number of hours (in thousands) of skilled labor, and X2 is the number of hours (in thousands) of unskilled labor. Find the number of hours of skilled labor and the number of hours of unskilled labor that minimizes the cost of doing the project.
Source: Bruce Allen, K. Weigelt , N. Doherty, and E. Mansfield (2005). Managerial Economics: Theory, Application, and Cases. Sixth Edition. W. W. Norton.