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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

Production The production function for a manufacturer is given by

f ( x , y ) = 100 x 0.8 y 0.2 where x is the number of units of labor (at $40 per unit) and y is the number of units of capital (at $35 per unit). The total cost for labor and capital cannot exceed $110,000.

(a) Use Lagrange multipliers to 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 $120,000 is available for labor and capital.

(a)

To determine

To calculate: The maximum production level for the manufacturer by the application of Lagrange multipliers whose production function is given as f(x,y)=100x0.8y0.2 where x is the number of units of labor at $40 per unit and y is the number of units of capital at $35 per unit and the total cost for labor and capital cannot exceed $110,000.

Explanation

Given Information:

The production function for a manufacturer is given by f(x,y)=100x0.8y0.2 where x is the number of units of labor at $40 per unit and y is the number of units of capital at $35 per unit and the total cost for labor and capital cannot exceed $110,000.

Formula used:

To find the maximum or minimum values of function f follow the steps as,

Step 1: Solve the provided system of equations as,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Step 2: Next calculate the function f at each of the solution point obtained from first step where the greatest value obtained gives the maximum of function f subject to constraint g(x,y)=0 and the least value gives the minimum of function f subject to constraint g(x,y)=0.

Calculation:

The provided function is f(x,y)=100x0.8y0.2 and the subject to constraint is 40x+35y=110,000

Now let f(x,y)=100x0.8y0.2 and g(x,y,z)=40x+35y110,000.

Then define a new function F as,

F(x,y,λ)=f(x,y)λg(x,y)=100x0.8y0.2λ(40x+35y110,000)

Now solve the equation by finding the partial derivatives of F with respect to x, y and λ as,

Fx(x,y,λ)=80x0.2y0.240λFy(x,y,λ)=20x0.8y0.835λFλ(x,y,λ)=(40x+35y110,000)

Then set each of the obtained partial derivatives equal to 0 as,

80x0.2y0.240λ=020x0.8y0.835λ=0(40x+35y110,000)=0

Now from the first equation,

80x0.2y0.240λ=080x0.2y0.2=40λλ=80x0.2y0.240λ=2x0

(b)

To determine

To calculate: The marginal productivity of money when production function of manufacturer is given as f(x,y)=100x0.8y0.2 where x is the number of units of labor at $40 per unit and y is the number of units of capital at $35 per unit and the total cost for labor and capital cannot exceed $110,000.

(c)

To determine

To calculate: The maximum number of units that can be produced by using marginal productivity of money when $120,000 is available for labor and capital and production function of manufacturer is given as f(x,y)=100x0.8y0.2 where x is the number of units of labor at $40 per unit and y is the number of units of capital at $35 per unit and the total cost for labor and capital cannot exceed $110,000.

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