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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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 function

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). 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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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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