   Chapter 14.8, Problem 28E

Chapter
Section
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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