Chapter 14, Problem 34RE

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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

*y* is the number of units of capital, and *z* is the units of production. Suppose labor costs are $50 per unit, capital costs are $50 per unit, and total costs are limited to $75,000.

(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 your result.

(c) Graph the constraint with the production function when *z* = 180,000, z = 300,000, and when the z-value is optimal.

**(a)**

To determine

**To calculate:** The number of units of labor and the number of units of capital that maximize production if the production function is

Explanation

**Given Information:**

A company has the production function,

Where, *x* is the number of units of labor, *y* is the number of units of capital, and *z* is the units of production.

The 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 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

**(b)**

To determine

**To calculate:** The marginal productivity of money and interpret the answer if the production function is

**(c)**

To determine

**To graph:** The constraint with the production function when

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