Chapter 14.3, Problem 16E

### 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

# Suppose that a company’s production function for a certain product is z = ( x + 1 ) 1 / 2 ln ( y 2 +   1 ) where x is the number of work-hours of unskilled labor and y is the number of work-hours of skilled labor.Find the marginal productivity of (a) x. (b) y.

(a)

To determine

To calculate: The marginal productivity of x if the production function for a certain product is z=(x+1)1/2ln(y2+1), where x is the number of work-hours of untrained labor and y is the number of work-hours of skilled labor.

Explanation

Given Information:

A company’s production function for a certain product is z=(x+1)1/2ln(y2+1), where x is the number of work-hours of untrained labor and y is the number of work-hours of skilled labor.

Formula used:

For a production function of the form z=f(x,y), the marginal productivity of x is given by zx and the marginal productivity of y is given by zy.

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. The partial derivative of f with respect to x is denoted by fx.

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

Chain rule for the function f(x)=u(v(x)) is f(x)=u(v(x))v(x).

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

The 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:

A company’s production function for a certain product is z=(x+1)1/2ln(y2+1), where x is the number of work-hours of untrained labor and y is the number of work-hours of skilled labor

(b)

To determine

To calculate: The marginal productivity of y if the production function for a certain product is z=(x+1)1/2ln(y2+1), where x is the number of work-hours of untrained labor and y is the number of work-hours of skilled labor.

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