Chapter 13, Problem 40RE

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) Consider the Cobb-Douglas production model given by the formula $P=L^{0.75}{K}^{0.25}.$ Sketch the level curves $P\left(L,K\right)=1.P(L,K)=2,\text{and }P\left(L,K\right)=3$ in an LK-coordinate system (L horizontal and K vertical). Your sketch need not be accurate numerically, but it should show the general shape of the curves and their relative positions.

(b) Use a graphing utility to make a more extensive contour plot of the model.