Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 9: Problem 4

Suppose that the production of crayons ( \(q\) ) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by \(q_{1}=10 l_{1}^{0.5}\) and in location 2 by \(q_{2}=50 l_{2}^{0.5}\) a. If a single firm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations to do so? Explain precisely the relationship between \(l_{1}\) and \(l_{2}\) b. Assuming that the firm operates in the efficient manner described in part (a), how does total output ( \(q\) ) depend on the total amount of labor hired \((l) ?\)

Short Answer

Expert verified
The relationship between labor input in location 1 (l1) and location 2 (l2) required to maximize output is given by: \(l_{1}=\frac{1}{25}l_{2}\). b. Write down the total production function of crayons depending on the total amount of labor hired while assuming the firm operates efficiently. The total output (q) depends on the total labor input (l) through the following total production function: \(q=\frac{1}{2\sqrt{26}}l^{0.5}+\frac{25\sqrt{10}}{2\sqrt{26}}l^{0.5}\).

Step by step solution

01

Analyze the production functions

We are given the production functions \(q_{1}=10 l_{1}^{0.5}\) and \(q_{2}=50 l_{2}^{0.5}\) for location 1 and location 2, respectively. Our first task is to find the relationship between \(l_{1}\) and \(l_{2}\). To maximize its output, a firm should allocate labor where marginal productivities are equal, which means the marginal product of labor in both locations should be the same.
02

Equate the marginal products of labor in both locations

To equate marginal products, we need to find the marginal product of labor (MPL) for both locations. The MPL is the first derivative of the production function with respect to labor input. For location 1, calculate the MPL: \(\frac{dq_{1}}{dl_{1}} = 5l_{1}^{-0.5}\) For location 2, calculate the MPL: \(\frac{dq_{2}}{dl_{2}} = 25l_{2}^{-0.5}\) Now equate both MPL to find the relationship between \(l_{1}\) and \(l_{2}\): \(5l_{1}^{-0.5} = 25l_{2}^{-0.5}\)
03

Solve for \(l_{1}\) in terms of \(l_{2}\)

From the equation above, we can solve for \(l_{1}\) in terms of \(l_{2}\): \(l_{1}^{-0.5} = 5l_{2}^{-0.5}\) \(l_{1}^{0.5} = \frac{1}{5}l_{2}^{0.5}\) \(l_{1} = \left(\frac{1}{5}l_{2}^{0.5}\right)^{2}\) \(l_{1} = \frac{1}{25}l_{2}\) Hence, the relationship between the labor input in location 1 (\(l_{1}\)) and the labor input in location 2 (\(l_{2}\)) is \(l_{1}=\frac{1}{25}l_{2}\).
04

Calculate the total production function

Now let's calculate the total production function in terms of the total labor input (l): The total labor input is \(l = l_{1} + l_{2}\) and according to our relationship from part (a) we have \(l_{1} =\frac{1}{25}l_{2}\). We can rewrite this as \(l = \frac{1}{25}l_{2} + l_{2}\) and thus calculate \(l_{2}=\frac{25}{26}l\) Now substitute \(l_{1}\) and \(l_{2}\) in terms of the total labor input (l) back into the production functions for location 1 and location 2: \(q_{1}=10 \left(\frac{1}{25}\cdot\frac{25}{26}l\right)^{0.5}\) \(q_{1}= \frac{1}{2\sqrt{26}}l^{0.5}\) \(q_{2}=50 \left(\frac{25}{26}l\right)^{0.5}\) \(q_{2}=\frac{25\sqrt{10}}{2\sqrt{26}}l^{0.5}\) Finally, we can write down the total output (q) as: \(q=q_{1}+q_{2}=\frac{1}{2\sqrt{26}}l^{0.5}+\frac{25\sqrt{10}}{2\sqrt{26}}l^{0.5}\)
05

Write down final results

We found the following results: a. The relationship between labor input in location 1 (\(l_{1}\)) and location 2 (\(l_{2}\)) required to maximize output is given by: \(l_{1}=\frac{1}{25}l_{2}\) b. The total output (q) depends on the total labor input (l) through the following total production function: \(q=\frac{1}{2\sqrt{26}}l^{0.5}+\frac{25\sqrt{10}}{2\sqrt{26}}l^{0.5}\)

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

Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by \\[ q=0.1 k^{0.2} l^{0.8} \\] where \(q\) is the number of bar stools produced during the renovation week, \(k\) represents the number of hours of bar stool lathes used during the week, and \(l\) represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of \(\$ 10,000\) for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount (\$50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, how much of each input will he hire and how much will the renovation project cost? b. Norm (who knows something about bar stools) argues that once again Sam has forgotten his microeconomics. He asserts that Sam should choose quantities of inputs so that their marginal (not average) productivities are equal. If Sam opts for this plan instead, how much of each input will he hire and how much will the renovation project cost? c. On hearing that Norm's plan will save money, Cliff argues that Sam should put the savings into more bar stools to provide seating for more of his USPS colleagues. How many more bar stools can Sam get for his budget if he follows Cliff's plan? d. Carla worries that Cliff's suggestion will just mean more work for her in delivering food to bar patrons. How might she convince Sam to stick to his original 10 -bar stool plan?

Suppose the production function for widgets is given by \\[ q=k l-0.8 k^{2}-0.2 l^{2} \\] where \(q\) represents the annual quantity of widgets produced, \(k\) represents annual capital input, and \(l\) represents annual labor input. a. Suppose \(k=10 ;\) graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(k=10,\) graph the \(M P_{1}\) curve. At what level of labor input does \(M P_{l}=0 ?\) c. Suppose capital inputs were increased to \(k=20 .\) How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

A local measure of the returns to scale incorporated in a production function is given by the scale elasticity \\[ e_{q, t}=\partial f(t k, t l) / \partial t \cdot t / q \text { evaluated at } t=1 \\] a. Show that if the production function exhibits constant returns to scale, then \(e_{q, t}=1\) b. We can define the output elasticities of the inputs \(k\) and \(l\) as \\[ \begin{aligned} e_{q, k} &=\frac{\partial f(k, l)}{\partial k} \cdot \frac{k}{q} \\ e_{q, l} &=\frac{\partial f(k, l)}{\partial l} \cdot \frac{l}{q} \end{aligned} \\] Show that \(e_{q, t}=e_{q, k}+e_{q, 1}\) c. A function that exhibits variable scale elasticity is \\[ q=\left(1+k^{-1} l^{-1}\right)^{-1} \\] Show that, for this function, \(e_{q, t} > 1\) for \(q < 0.5\) and that \\[ e_{q, t} < 1 \text { for } q > 0.5 \\] d. Explain your results from part (c) intuitively. Hint: Does \(q\) have an upper bound for this production function?

Suppose we are given the constant returns-to-scale CES production function \\[ q=\left(k^{\rho}+l^{\rho}\right)^{1 / \rho} \\] a. Show that \(M P_{k}=(q / k)^{1-\rho}\) and \(M P_{l}=(q / l)^{1-\rho}\) b. Show that \(R T S=(k / l)^{1-\rho} ;\) use this to show that \\[ \sigma=1 /(1-\rho) \\] c. Determine the output elasticities for \(k\) and \(l ;\) and show that their sum equals 1 d. Prove that \\[ \frac{q}{l}=\left(\frac{\partial q}{\partial l}\right)^{\sigma} \\] and hence that \\[ \ln \left(\frac{q}{l}\right)=\sigma \ln \left(\frac{\partial q}{\partial l}\right) \\] Note: The latter equality is useful in empirical work because we may approximate \(\partial q / \partial l\) by the competitively determined wage rate. Hence \(\sigma\) can be estimated from a regression of \(\ln (q / I)\) on \(\ln w\)

Consider a generalization of the production function in Example 9.3: \\[ q=\beta_{0}+\beta_{1} \sqrt{k l}+\beta_{2} k+\beta_{3} l \\] where \\[ 0 \leq \beta_{i} \leq 1, \quad i=0, \ldots, 3 \\] a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters \(\beta_{0}, \ldots, \beta_{3} ?\) b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0 c. Calculate \(\sigma\) in this case. Although \(\sigma\) is not in general constant, for what values of the \(\beta\) 's does \(\sigma=0,1\), or \(\infty ?\)
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