Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 10

A chair manufacturer hires its assembly-line labor for \(\$ 30\) an hour and calculates that the rental cost of its machinery is \(\$ 15\) per hour. Suppose that a chair can be produced using 4 hours of labor or machinery in any combination. If the firm is currently using 3 hours of labor for each hour of machine time, is it minimizing its costs of production? If so, why? If not, how can it improve the situation? Graphically illustrate the isoquant and the two isocost lines for the current combination of labor and capital and for the optimal combination of labor and capital.

Short Answer

Expert verified
The chair firm is not minimizing its production costs. For optimal production, it needs to use twice as much machinery per hour as labor. This change would reduce costs per chair from \$105 to \$90. A graphical illustration of the isoquant and the two isocost lines would depict this optimization.

Step by step solution

01

Define the costs

The company is currently spending \(3 \times \$30 + 1 \times \$15 = \$105)\ per chair, where \$30 is the cost per hour of labor and \$15 is the cost per hour of machinery.
02

Establish optimal production levels

To optimally produce a chair with 4 hours labor or machinery, the firm should equate the marginal rate of technical substitution (MRTS) to the ratio of the input costs. The MRTS is the amount of one input that can be replaced with another input, while holding output constant. In this case, since it takes the same amount of time to produce a chair whether using labor or machinery, the MRTS equals 1.
03

Balance Cost and Production

Given the MRTS, we find the cost ratio of inputs to be \(\$30/\$15 = 2\). Therefore, for every hour of labor used, the company should use two hours of machinery to equate the marginal productivities per dollar of all inputs.
04

Graphical Representation

The isoquant is a curve showing all possible combinations of inputs that result in the production of a given level of output. Given the information, a 45-degree line is obtained from the origin: all points along this line indicate combinations where the total time is always 4 hours. For the isocost lines, plot one line depicting the current scenario at \(\$105) and another indicating the optimal scenario where for every hour of labor, two hours of machinery are used leading to a total cost of \(\$90). The point where the isocost line is tangential to the isoquant line gives the optimal combination of labor and machinery.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Joe quits his computer programming job, where he was earning a salary of \(\$ 50,000\) per year, to start his own computer software business in a building that he owns and was previously renting out for \(\$ 24,000\) per year. In his first year of business he has the following expenses: salary paid to himself, \(\$ 40,000 ;\) rent, \(\$ 0 ;\) other expenses, \(\$ 25,000 .\) Find the accounting cost and the economic cost associated with Joe's computer software business.

Suppose the long-run total cost function for an industry is given by the cubic equation \(\mathrm{TC}=\mathrm{a}+\mathrm{bq}+\mathrm{cq}^{2}+\mathrm{d} q^{3}\) Show (using calculus) that this total cost function is consistent with a U-shaped average cost curve for at least some values of \(a, b, c,\) and \(d\).

A computer company's cost function, which relates its average cost of production AC to its cumulative output in thousands of computers \(Q\) and its plant size in terms of thousands of computers produced per year \(q\) (within the production range of 10,000 to 50,000 computers \(),\) is given by \\[ \mathrm{AC}=10-0.1 \mathrm{Q}+0.3 \mathrm{q} \\] a. Is there a learning-curve effect? b. Are there economies or diseconomies of scale? c. During its existence, the firm has produced a total of 40,000 computers and is producing 10,000 computers this year. Next year it plans to increase production to 12,000 computers. Will its average cost of production increase or decrease? Explain.

The short-run cost function of a company is given by the equation \(\mathrm{TC}=200+55 q\), where \(\mathrm{TC}\) is the total cost and \(q\) is the total quantity of output, both measured in thousands. a. What is the company's fixed cost? b. If the company produced 100,000 units of goods, what would be its average variable cost? c. What would be its marginal cost of production? d. What would be its average fixed cost? e. Suppose the company borrows money and expands its factory. Its fixed cost rises by \(\$ 50,000,\) but its variable cost falls to \(\$ 45,000\) per 1000 units. The cost of interest ( \(i\) ) also enters into the equation. Each 1-point increase in the interest rate raises costs by \(\$ 3000 .\) Write the new cost equation.

The cost of flying a passenger plane from point \(A\) to point \(B\) is \(\$ 50,000\). The airline flies this route four times per day at \(7 \mathrm{AM}, 10 \mathrm{AM}, 1 \mathrm{PM}\), and \(4 \mathrm{PM}\). The first and last flights are filled to capacity with 240 people. The second and third flights are only half full. Find the average cost per passenger for each flight. Suppose the airline hires you as a marketing consultant and wants to know which type of customer it should try to attract-the off-peak customer (the middle two flights) or the rush-hour customer (the first and last flights). What advice would you offer?
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free