Chapter 14, Problem 28RE

### 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 that the production function for a company is given by Q   =   80 K 1 / 4 L 3 / 4 where Q is the output (in hundreds of units), K is the capital expenditures (in thousands of dollars), and L is the work-hours. Find ∂ Q / ∂ K  and  ∂ Q / ∂ L when expenditures are $625,000 and total work-hours are 4096. Interpret the results. To determine To calculate: The values of QK and QL when expenditures are$625,000 and total work-hours are 4096 and interpret the results.

Explanation

Given Information:

The production function for a company is given as,

Q=80K1/4L3/4,

Where, Q is the output (in hundreds of dollars), and L is the work-hours and K is the capital expenditure.

Also, the expenditures are \$625,000 and total work-hours are 4096.

Formula used:

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

And, 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.

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Chain rule for function f(x)=u(v(x)) is fâ€²(x)=uâ€²(v(x))â‹…vâ€²(x).

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 production function for a company,

Q=80K1/4L3/4

Partial differentiate the above function with respect to K,

âˆ‚Qâˆ‚K=80L3/4(14Kâˆ’3/4)=20L3/4K3/4=20(LK)3/4

Thus, âˆ‚Qâˆ‚K=20(LK)3/4

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