   Chapter 14.3, Problem 22E 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

Suppose the Cobb-Douglas production function for a company is given by z = 300 x 2 / 3 y 1 / 3 where x is the company’s capital investment and y is the size of the labor force (in work-hours).(a) Find the marginal productivity of x.(b) If the current labor force is 729 work-hours, substitute y = 729 in your answer to part (a) and graph the result.(c) Find the marginal productivity of y.(d) If the current capital investment is $27,000, substitute x = 27,000 in your answer to part (c) and graph the result.(e) Interpret the graphs in parts (b) and (d) with regard to what they say about the effects on productivity of an increased capital investment [part (b)] and of an increased labor force (part (d)]. (a) To determine To calculate: The marginal productivity of x if the Cobb-Douglas production function for a company is given by z=300x2/3y1/3 where x is the company’s capital investment and y is the size of the labor force (in work-hours). Explanation Given Information: The Cobb-Douglas production function for a company is given by z=300x2/3y1/3 where x is the company’s capital investment and y is the size of the labor force (in work-hours). 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. 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) (b) To determine To calculate: The marginal productivity of x if the current labor force is 729 work-hours and also plot the result. (c) To determine To calculate: The marginal productivity of y if the Cobb-Douglas production function for a company is given by z=300x2/3y1/3 where x is the company’s capital investment and y is the size of the labor force (in work-hours). (d) To determine To calculate: The marginal productivity of y if the current capital investment is$27,000 and also plot the result.

(e)

To determine

The interpretation of graphs in parts (b) and (d) with regard to what they say about the effects on productivity of increased capital investment and of an increased labor force.

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