Explanation Given: Production function level is to be given as P ( x , y ) = 100 x 0.6 y 0.4 , where x represents the number of units of labor and y represents the number of units of capital. Cost of labor is $ 72 per unit and cost of capital is $ 80 per unit. Formula Used: Below are the equations to be solved by using Lagrange multipliers method for solving constrained optimization problems. f x ( x , y ) = λ g x ( x , y ) f y ( x , y ) = λ g y ( x , y ) g ( x , y ) = c Calculation: Use these facts to create the Constraint equation. P ( x , y ) = 100 x 0.25 y 0.75 = 50000 Cost of labor is $ 72 per unit and cost of capital is $ 80 per unit. Use these facts to create the objective function. C ( x , y ) = 72 x + 80 y Objective is to use the method of Lagrange multipliers for determining the minimum cost. Consider the objective function represented by graph z = f ( x , y ) and Constraint equation g ( x , y ) = c where c is constant. Differentiating the given objective function partially with respect to x , we get; g x ( x , y ) = ∂ ∂ x ( 72 x + 80 y ) = 72 …… (1) Differentiating the given objective function partially with respect to y , we get; g y ( x , y ) = ∂ ∂ y ( 72 x + 80 y ) = 80 …… (2) Differentiating the given Constraint function partially with respect to x , we get; P x ( x , y ) = ∂ ∂ x ( 100 x 0.6 y 0.4 ) = 100 ( 0.6 ) x 0.6 − 1 y 0.4 = 60 x − 0.4 y 0.4 = 60 ( y x ) 0.4 Differentiating the given Constraint function partially with respect to y , we get; P y ( x , y ) = ∂ ∂ x ( 100 x 0.6 y 0.4 ) = 100 ( 0.4 ) x 0

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 13.10, Problem 52E

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To calculate: The minimum cost of producing 50,000 units of a product, where cost of labor is $72 per unit and cost of capital is $80 per unit by using Lagrange multiplier.

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Chapter 13.10, Problem 51E

Chapter 13.10, Problem 53E

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Chapter 13 Solutions

Calculus: Early Transcendental Functions

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To calculate: The minimum cost of producing 50,000 units of a product, where cost of labor is $72 per unit and cost of capital is $80 per unit by using Lagrange multiplier.

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Chapter 13 Solutions