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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

Referring to Exercise 27, we now suppose that the production is fixed at bLα K1 - α = Q, where Q is a constant. What values of L and K minimize the cost function C(L, K) = mL + nK?

To determine

To find: The values of L and K which minimize the cost function C(L,K)=mL+nK .

Explanation

Given:

The cost function is C(L,K)=mL+nK and the production function is Q(L,K)=bLαK1α where b, α are positive constants, L is labor, K is capital investment, Q is constant.

Definition used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx fy=λgy fz=λgz and g(x,y,z)=k ”.

Calculation:

The given function is C(L,K)=mL+nK and Q(L,K)=bLαK1α .

The Lagrange multipliers C(L,K)=λQ(L,K) is,

C(L,K)=λQ(L,K)CL,CK=λQL,QKCL(mL+nK),CK(mL+nK)=λQL(bLαK1α),QK(bLαK1α)m,n=λbαK1αLα1,b(1α)LαKα

Thus, the value of C(L,K)=λQ(L,K) is m,n=λbαK1αLα1,b(1α)LαKα .

By the definition, m,n=λbαK1αLα1,b(1α)LαKα can be express as,

m=λbαK1αLα1 (1)

n=λb(1α)LαKα (2)

From the equation (1),

m=λbαK1αLα1λbαK1αL(1α)=mλ(KL)1α=mbαλ=mbα(LK)1α

From the equation (2),

λb(1α)(LK)α=nλ(LK)α=nb(1α)λ=nb(1α)(KL)α

Equating the λ values,

mbα(LK)1α=nb(1α)(KL)αmα(LK)1α=n(1α)(KL)αm(1α)nα=(KL)α(KL)1α

Simplify further as follows

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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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