Question

Professor Smith and Professor Jones are going to produce a new introductory textbook. As true scientists, they have laid out the production function for the book as \\[ q=S^{1 / 2} J^{1 / 2} \\] where \(q=\) the number of pages in the finished book, \(S=\) the number of working hours spent by Smith, and \(J=\) the number of hours spent working by Jones. Smith values his labor as \(\$ 3\) per working hour. He has spent 900 hours preparing the first draft. Jones, whose labor is valued at \(\$ 12\) per working hour, will revise Smith's draft to complete the book. a. How many hours will Jones have to spend to produce a finished book of 150 pages? Of 300 pages? Of 450 pages? b. What is the marginal cost of the 150 th page of the finished book? Of the 300 th page? Of the 450 th page?

Short answer

Jones needs 25, 100, and 225 hours for 150, 300, and 450 pages respectively. Marginal costs for 150th, 300th, and 450th pages are $300, $900, and $1500.

Step-by-step solution

Understanding the Production Function

The production function is given by \( q = S^{1/2} J^{1/2} \). Here, \( q \) is the number of pages, \( S \) is the hours worked by Smith (900 hours already used), and \( J \) is the hours worked by Jones. Our task is to find \( J \) for given \( q \) values (150, 300, and 450 pages).

Solving for Jones's Hours (\(J\))

To express \( J \) in terms of \( q \), use the formula \( q = (S^{1/2} J^{1/2}) \). Rearrange it to find \( J \). First, square both sides: \( q^2 = S J \). Then, solve for \( J \): \( J = \frac{q^2}{S} \).

Calculate \( J \) for Different Values of \( q \)

Using \( S = 900 \), calculate \( J \) for each \( q \):- For \( q = 150 \): \( J = \frac{150^2}{900} = 25 \) hours.- For \( q = 300 \): \( J = \frac{300^2}{900} = 100 \) hours.- For \( q = 450 \): \( J = \frac{450^2}{900} = 225 \) hours.

Understand Marginal Cost Calculation

The marginal cost of the \( q^{th} \) page is the additional cost of producing one more page at that level of production. Since Jones's time is the variable cost that changes, we need to find the cost associated with the additional hours for producing additional pages.

Calculate Marginal Cost for the Given Pages

Given Jones's labor cost of \\(12 per hour, calculate the marginal cost:- For the 150th page: The cost of 25 hours (since \( J = 25 \) at 150 pages) is \( 25 \times 12 = \\)300 \).- For the 300th page: Incremental hours from 25 to 100 (75 hours) at \\(12 per hour gives an additional cost of \( 75 \times 12 = \\)900 \).- For the 450th page: The further increment of 125 hours (from 100 to 225) will cost \( 125 \times 12 = \$1500 \).

Key concepts

Marginal Cost

Understanding marginal cost is crucial in production and economics. It refers to the cost of producing one additional unit of output. In this exercise, the output is the number of pages in the book. The cost comes from Jones's labor hours, which vary with each additional page produced.

When Jones worked to increase the number of pages from 150 to 300, the hours worked jumped from 25 to 100, showing a significant increase. Similarly, going from producing 300 to 450 pages requires increased hours on Jones's part, reflecting higher marginal costs. As you can see, with each increase in production level, the marginal cost reflects the added cost of labor Jones needed to invest to achieve it. His labor cost is critical because it's variable and influences the total production cost. By calculating the marginal cost at each production level, students can understand how changes in production volume affect cost.

Labor Economics

Labor economics deals with the study of labor forces as an element in the process of production. This exercise showcases a simple example of labor economics by analyzing how the use of two labor inputs—Smith and Jones's hours—can affect book production.

Smith has already invested 900 hours, representing a sunk cost in this scenario. These hours are set and influence the production function. Jones, however, represents the adjustable labor input. By altering his hours, you can observe shifts in production outputs, i.e., the number of book pages.

This situation is typical in labor economics studies, where aspects such as labor costs, hours worked, and productivity are analyzed to optimize production. Understanding labor economics is essential for making decisions that enhance productivity while managing costs.

Cost Analysis

Cost analysis is a systematic approach to estimating the strengths and weaknesses of various options, typically of business transactions or projects. Here, the focus is on Jones's contribution to the book project and how it affects overall costs.

Cost analysis begins with understanding fixed and variable costs. Smith's time is a sunk fixed cost. Jones's time is variable, meaning it's flexible and can change. His labor cost influences the book's overall production cost.

In this exercise, the total costs are closely tied to Jones's hours worked. By calculating these hours and multiplying by Jones's hourly rate, we can determine the total labor cost for different production levels: 150, 300, and 450 pages. You can spot trends such as increasing marginal cost as more pages are produced. Cost analysis like this aids in making informed decisions about resource allocations.