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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Production The production function for a manufacturer is given by

f ( x ,   y ) =   100 x 0.25 y 0.75 where x is the number of units of labor (at $48 per unit) and y is the number of units of capital (at $36 per unit). The total cost for labor and capital cannot exceed $100,000.

(a) Find the maximum production level for this manufacturer.

(b) Find the marginal productivity of money.

(c) Use the marginal productivity of money to find the maximum number of units that can be produced when $125,000 is available for labor and capital.

(d) Use the marginal productivity of money to find the maximum number of units that can be produced when $350,000 is available for labor and capital.

(a)

To determine

To calculate: The maximum production level of the manufacturer from the function

f(x,y)=100x0.25y0.75.

Explanation

Given Information:

The production function for a manufacturer is given by f(x,y)=100x0.25y0.75x is the no of unit of labor and y is the no of unit of capital

Formula used:

Method of Lagrange multipliers,

If the function f(x,y) contains a maximum or minimum subject to the constraint g(x,y)=0 then the maximum or minimum can occur at one of the critical numbers of the function F is,

F(x,y,λ)=f(x,y)λg(x,y) where, λ is a Lagrange multiplier.

Steps to determine the minimum or maximum of the function f.

1. Solve the system of equations,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Calculation:

Consider the function,

f(x,y)=100x0.25y0.75

Maximize f(x,y)=100x0.25y0.75 subject to constraint 48x+36y=100000. now solve the

constraint.

Fx(x,y,λ)=100x0.25y0.75λ(48x+36y100000)

Since

Fx=25x0.75y0.7548λ=0

And

Fy=75x0.25y0.2536λ=0

And

Fλ=(48x+36y100000)=0

Using F=xFy

F,xλ=25x

(b)

To determine

To calculate: The marginal productivity of money from the function f(x,y)=100x0.25y0.75.

(c)

To determine

To calculate: The maximum number of units that can be produce when $125000 is available for labor and capital r from the function f(x,y)=100x0.25y0.75.

(d)

To determine

To calculate: The maximum number of units that can be produce when $350,000 is available for labor and capital from the function f(x,y)=100x0.25y0.75.

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