Chapter 6.5, Problem 43E

Economics: the Law of Equimarginal Productivity. Suppose $p\left(x,y\right)$ represents the production of a two-product firm. The company produces x units of the first product at a cost of $c_1$ each and y unit of the second product at a cost of $c_2$ each. The budget constraint. B, is

$B=c_{1}x+c_{2}y.$

Use the method of Lagrange multipliers to find the value of λ in terms of $p_x,p_y,c_1,\text{ and }{c}_2$. The resulting equation holds for any production function p and is called the Law of Equimarginal Productivity.

$\lambda =\frac{p_x}{c_1}=\frac{p_y}{c_2}$