Chapter 14.5, Problem 20E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

** Production** Suppose a company has the Cobb-Douglas production function

(a) Find the number of units of labor and the number of units of capital that maximize production.

(b) Find the marginal productivity of money and interpret it.

(c) Graph the constraint with the optimal value for production and with two other *z*-values (one smaller than the optimal value and one larger).

**(a)**

To determine

**To calculate:** The number of units of labor and the number of units of capital that maximize production. If a company has the Cobb-Douglas production function

Explanation

**Given Information:**

A company has the Cobb-Douglas production function *x* is the number of units of labor, *y* is the number of units of capital, and *z* is the units of production. Also, labor costs

**Formula used:**

According to the Lagrange multipliers method to obtain maxima or minima for a function

Step 1: Find the critical values of

Step 2: The critical points of

Step 3: The critical points of

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

Power of

Constant function rule for a constant

Coefficient rule for a constant

**Calculation:**

Consider the problem, A company has the Cobb-Douglas production function *x* is the number of units of labor, *y* is the number of units of capital, and *z* is the units of production.

Consider the function,

Also, if number of units of labor is *x*, which costs *y*, which costs

Thus, the constraint is

According to the Lagrange multipliers method,

The objective function is

Thus,

Substitute

Since, the critical points of

Recall that, for a function *f* with respect to *y* is calculated by taking the derivative of *y* and keeping the other variable *x* constant

**(b)**

To determine

**To calculate:** The marginal productivity of money If a company has the Cobb-Douglas production function

**(c)**

To determine

**To graph:** The constraint with the optimal value for production and with two other *z*-values (one smaller than the optimal value and one larger.) If a company has the Cobb-Douglas production function

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