Chapter 14.5, Problem 20E

### Mathematical Applications for the ...

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

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 =   100 0.75 y 0.25 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 further that labor costs $90 per unit, capital costs$150 per unit, and the total costs of labor and capital are limited to $90,000.(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 it.(c) Graph the constraint with the optimal value for production and with two other z-values (one smaller than the optimal value and one larger). (a) To determine To calculate: The number of units of labor and the number of units of capital that maximize production. If a company has the Cobb-Douglas production function z=100x0.75y0.25. Explanation Given Information: A company has the Cobb-Douglas production function z=100x0.75y0.25 where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Also, labor costs$90 per unit, capital costs $150 per unit, and the total cost of labor and capital is limited to$90,000.

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 Î» to form the objective function F(x,y,Î»)=f(x,y)+Î»g(x,y).

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

Step 3: The critical points of F(x,y,Î») are the points that satisfy âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=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 fâ€²(x)=nxnâˆ’1.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

Consider the problem, A company has the Cobb-Douglas production function z=100x0.75y0.25 where x is the number of units of labor, y is the number of units of capital, and z is the units of production.

Consider the function, z=100x0.75y0.25.

Also, if number of units of labor is x, which costs $90 per unit, and if number of units of capital is y, which costs$150 per unit, then the total cost is 90x+150y. But the total cost of labor and capital is limited to \$90,000. Thus, 90x+150y=90,000.

Thus, the constraint is 90x+150y=90,000.

According to the Lagrange multipliers method,

The objective function is F(x,y,Î»)=f(x,y)+Î»g(x,y).

Thus, f(x,y)=100x0.75y0.25 and g(x,y)=90x+150yâˆ’90,000.

Substitute 100x0.75y0.25 for f(x,y) and 90x+150yâˆ’90,000 for g(x,y) in F(x,y,Î»)=f(x,y)+Î»g(x,y).

F(x,y,Î»)=100x0.75y0.25+Î»(90x+150yâˆ’90,000)

Since, the critical points of F(x,y,Î») are the points that satisfy âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=0.

Recall that, 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

(b)

To determine

To calculate: The marginal productivity of money If a company has the Cobb-Douglas production function z=100x0.75y0.25.

(c)

To determine

To graph: The constraint with the optimal value for production and with two other z-values (one smaller than the optimal value and one larger.) If a company has the Cobb-Douglas production function z=100x0.75y0.25.

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