Chapter 7.4, Problem 25E

(a)

To determine

The amount of labor and capital that should be utilized in order to minimize the cost, that is the value of $x,y$ that minimizes $96x+162y$, subject to the constraint where the production function for a firm is $f\left(x,y\right)=64x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{y}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

(b)

To determine

The value of the $\lambda$ at the optimal level of production, where the production function for a firm is $f\left(x,y\right)=64x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{y}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$, $x$ and $y$ are the number of units of labor and capital utilized

(c)

To determine

To prove: That at the optimal level of production

$\frac{\left[\text{marginal}\text{productivity}\text{of}\text{labor}\right]}{\left[\text{marginal}\text{productivity}\text{of}\text{capital}\right]}\text{=}\frac{\left[\text{unit}\text{price}\text{of}\text{labor}\right]}{\left[\text{unit}\text{price}\text{of}\text{capital}\right]}$, where the production function for a firm is $f\left(x,y\right)=64x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{y}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$, $x$ and $y$ are the number of units of labor and capital utilized.