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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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Least-Cost Rule Repeat Exercise 37 for the production function given by

f ( x ,   y ) =   100 0.4 y 0. 6 .

Least-Cost Rule The production function for a company is given by

f ( x ,   y ) =   100 x 0.7 y 0.3 where x is the number of units of labor (at $50 per unit) and y is the number of units of capital (at $ 100 per unit). Management sets a production goal of 20,000 units.

(a) Find the numbers of units of labor and capital needed to meet the production goal while minimizing the cost.

(b) Show that the conditions of part (a) are met when

Marginal productivity of labor  Marginal productivity of capital = Unit price of labor Unit price of capital

This proportion is called the Least-Cost Rule (or Equimarginal Rule).

(a)

To determine

To calculate: The number of units of labor and capital needed to meet a production goal while minimize the cost from the function f(x,y)=100x0.4y0.6.

Explanation

Given Information:

The production function for a manufacturer is given by f(x,y)=100x0.4y0.6x 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

2. Determine the value of the function f at each solution obtained from the step 1.

The largest value gives the maximum value of function f subject to the constraint g(x,y)=0 and the lowest value gives the minimum value of function f subject to the constraint g(x,y)=0.

Calculation:

Consider the function,

f(x,y)=100x0.4y0.6

Minimize f(x,y)=50x+100y subject to constraint 100x0.4y0.620,000=0. now solve the constraint.

Fx(x,y,λ)=50x+100yλ(100x0.4y0.620,000)

Thus,

Fx=5040λx0.6y0.6Fy=10060λx0.4y0.4Fλ=(100x0.4y0.620,000)

Now, set all derivatives equal to 0:

5040λx0

(b)

To determine

To prove: marginal productivity of labourmarginal productivity of capital=unit price of labourunit price of capital by using the function f(x,y)=100x0.4y0.6.

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