Textbook Problem

Least-Cost Rule The production function for a company is given by

$f\left(x,y\right)=100x^{0.7}{y}^{0.3}$ where x is the number of units of labor (at $50 per unit) and y is the number of units of capital (at $ 100 per unit). Management sets a production goal of 20,000 units.

(a) Find the numbers of units of labor and capital needed to meet the production goal while minimizing the cost.

(b) Show that the conditions of part (a) are met when

$\frac{\text{Marginal productivity of labor}}{\text{Marginal productivity of capital}}\text{=}\frac{\text{Unit price of labor}}{\text{Unit price of capital}}$

This proportion is called the Least-Cost Rule (or Equimarginal Rule).