Exercise 12

The Cobb-Douglas production function is used in economics to model production levels based on labor and equipment. Suppose we have a specific Cobb-Douglas function of the form $$\begin{gathered} f(x, y) = 50 x^{0.4}y^{0.6}, \end{gathered}$$ where $x$ is the dollar amount spent on labor and $y$ the dollar amount spent on equipment. Use the method of Lagrange multipliers to determine how much should be spent on labor and how much on equipment to maximize productivity if we have a total of 1.5 million dollars to invest in labor and equipment.