Chapter 13, Problem 42RE

In economics, a production model is a mathematical relationship between the output of a company or a country and the labor and capital equipment required to produce that output. Much of the pioneering work in the field of production models occurred in the 19205 when Paul Douglas of the University of Chicago and his collaborator Charles Cobb proposed that the output P can be expressed in terms of the labor L and the capital equipment K by an equation of the form

$P=c{L}^{\alpha }{K}^{\beta }$

where c is a constant of proportionality and $\alpha \text{and}\beta$ are constants such that $0<\alpha <1\text{and}0<\beta <1.$ This is called the Cobb-Douglas production model. Typically, $P,L,andK$ are all expressed in terms of their equivalent monetary values. These exercises explore properties of this model.

Consider the Cobb-Douglas production model

$P=1000L^{0.6}{K}^{0.4}$

(a) Find the maximum output value of P if labor costs $\$50.00$ per unit, capital costs $\$100.00$ per unit. and the total cost of labor and capital is set at $\$200,000.$

(b) How should the $\$200,000$ be allocated between labor and capital to achieve the maximum?