   Chapter 14.8, Problem 27E

Chapter
Section
Textbook Problem

The total production P of a certain product depends on the amount L of labor used and the amount K of capital investment, in Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model P = bLαK1-α follows from certain economic assumptions, where b and α are positive constants and α < 1. If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint mL + nK = p. Show that the maximum production occurs when L   =   α P m   and   K   =   ( 1   −   α ) p n

To determine

To show: The maximum production of the function P(L,K)=bLαK1α subject to the constraint mL+nK=p .occurs when L=αpm and K=(1α)pn .

Explanation

Solution:

Given:

The Cobb-Douglas model is P(L,K)=bLαK1α where b, α are positive constants, L is labor, K is capital investment, P is total production and the function is subject to the constraint p=mL+nK where m is cost of the unit labor, n is cost of the unit capital, p is total budget.

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 function is P(L,K)=bLαK1α and p(L,K)=mL+nK .

The Lagrange multipliers P(L,K)=λp(L,K) is computed as follows.

P(L,K)=λp(L,K)PL,PK=λpL,pKPL(bLαK1α),PK(bLαK1α)=λpL(mL+nK),pK(mL+nK)bαK1αLα1,b(1α)LαKα=λm,n

Thus, the value of P(L,K)=λp(L,K) is bαK1αLα1,b(1α)LαKα=λm,n .

By the definition, bαK1αLα1,b(1α)LαKα=λm,n can be express as,

bαK1αLα1=λm (1)

b(1α)LαKα=λn (2)

From the equation (1),

bαK1αL(1α)=λmbα(KL)1α=λmλ=bαm

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