   Chapter 14, Problem 34RE Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Production Suppose a company has the Cobb-Douglas production function z   =   300 x 2 / 3 y 1 / 3 where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Suppose labor costs are $50 per unit, capital costs are$50 per unit, and total costs are limited to $75,000.(a) Find the number of units of labor and the number of units of capital that maximize production.(b) Find the marginal productivity of money and interpret your result.(c) Graph the constraint with the production function when z = 180,000, z = 300,000, and when the z-value is optimal. (a) To determine To calculate: The number of units of labor and the number of units of capital that maximize production if the production function is z=300x2/3y1/3. Explanation Given Information: A company has the production function, z=300x2/3y1/3 Where, x is the number of units of labor, y is the number of units of capital, and z is the units of production. The labor costs$50 per unit, capital costs are $50 per unit, and the total cost are limited to$75,000.

Formula used:

According to the Lagrange multipliers method to obtain maxima or minima for a function z=f(x,y) subject to the constraint g(x,y)=0,

Step 1: Find the critical values of f(x,y) using the new variable λ to form the objective function F(x,y,λ)=f(x,y)+λg(x,y).

Step 2: The critical points of f(x,y) are the critical values of F(x,y,λ) which satisfies g(x,y)=0.

Step 3: The critical points of F(x,y,λ) are the points that satisfy Fx=0, Fy=0, and Fλ=0, that is, the points which make all the partial derivatives of zero.

For a function f(x,y), the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to y is denoted by fy.

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Calculation:

Consider the function,

z=300x2/3y1/3

Also, if number of units of labor is x, which costs $50 per unit, and if number of units of capital is y, which costs$50 per unit, then the total cost is 50x+50y. But the total cost of labor and capital is limited to \$75,000. Thus, 50x+50y=75,000.

Thus, the constraint is 50x+50y=75,000.

According to the Lagrange multipliers method,

The objective function is F(x,y,λ)=f(x,y)+λg(x,y).

Thus, f(x,y)=300x2/3y1/3 and g(x,y)=50x+50y75,000.

Substitute 300x2/3y1/3 for f(x,y) and 50x+50y75,000 for g(x,y) in F(x,y,λ)=f(x,y)+λg(x,y),

F(x,y,λ)=300x2/3y1/3+λ(50x+50y75,000)

Since, the critical points of F(x,y,λ) are the points that satisfy Fx=0, Fy=0, and Fλ=0

(b)

To determine

To calculate: The marginal productivity of money and interpret the answer if the production function is z=300x2/3y1/3.

(c)

To determine

To graph: The constraint with the production function when z=180,000, z=300,000, and when the z-value is optimal if the production function is z=300x2/3y1/3.

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