Chapter 13, Problem 41RE

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.

(a) Find $\frac{\partial P}{\partial L}\text{and}\frac{\partial P}{\partial K}$ for the Cobb-Douglas production model $P=c{L}^{\alpha }{K}^{\beta }.$

(b) The derivative $\frac{\partial P}{\partial L}$ is called the marginal productivity of labor, and and the derivative of $\frac{\partial P}{\partial K}$ is called the marginal productivity of capital. Explain what these quantities mean in practical terms.

(c) Show that if $\beta =1-\alpha ,$ then P satisfies the partial differential equation

$K\frac{\partial P}{\partial K}+L\frac{\partial P}{\partial L}=P$