Chapter 14.3, Problem 21E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

Suppose the Cobb-Douglas production function for a company is given by

*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 *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 *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 *x* is given by *y* is given by

For a function *f* with respect to *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant. The partial derivative of *f* with respect to *x* is denoted by

Power of

Constant function rule for a constant

The coefficient rule for a constant

**(b)**

To determine

**To calculate:** The marginal productivity of *x* if the current labor force is

**(c)**

To determine

**To calculate:** The marginal productivity of *y* if the Cobb-Douglas production function for a company is given by *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

**(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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