Chapter 14.3, Problem 21E

### 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 the Cobb-Douglas production function for a company is given by z = 400 x 3 / 5 y 2 / 5 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 1024 work-hours, substitute y = 1024 in your answer to part (a) and graph the result.(c) Find the marginal productivity of y.(d) If the current capital investment is $59,049, substitute x = 59,049 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=400x3/5y2/5 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=400x3/5y2/5 where x is the companyâ€™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 âˆ‚zâˆ‚x and the marginal productivity of y is given by âˆ‚zâˆ‚y. 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 fâ€²(x)=nxnâˆ’1. Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0. The 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) (b) To determine To calculate: The marginal productivity of x if the current labor force is 1024 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=400x3/5y2/5 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$59,049 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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