   Chapter 14.5, Problem 42E

Chapter
Section
Textbook Problem

A manufacturer has modeled its yearly production function P (the value of its entire production, in millions of dollars) as a Cobb-Douglas functionP(L, K) = 1.47L0.65K0.35where L is the number of labor hours (in thousands) and K is the invested capital (in millions of dollars). Suppose that when L = 30 and K = 8, the labor force is decreasing at a rate of 2000 labor hours per year and capital is increasing at a rate of \$500,000 per year. Find the rate of change of production.

To determine

To find: The rate of change of production.

Explanation

Given:

The Cobb-Douglas function is P(L,K)=1.47L0.65K0.35 where L is the number of labor hours (in thousands) and K is the invested capital (in millions of dollars).

The values of L=30,K=8,dLdt=2 and dKdt=0.5 .

Chain Rule:

“Suppose that z=f(x,y) is a differentiable function of x and y , where x=g(t)andy=h(t) are both differentiable functions of t then, z is differentiable function of t and dzdt=zxdxdt+zydydt ”.

Calculation:

The Cobb-Douglas function is P(L,K)=1.47L0.65K0.35 .

By using chain rule, obtain the value of dPdt .

dPdt=L(1.47L0.65K0.35)dLdt+K(1.47L0.65K0.35)dKdt=1.47K0.35L(L0.65)dLdt+1.47L0.65K(K0.35)dKdt=1

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