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 2

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?

Short Answer

Expert verified
Answer: The production function exhibits constant returns to scale.

Step by step solution

01

Total Productivity of Labor (TPL) with k=10

Insert k=10 into the production function: \(q = 10l - 0.8(10)^2 - 0.2l^2\) \(q = 10l - 80 - 0.2l^2\) TPL = \(q(l) = 10l - 0.2l^2 - 80\) Now, since we want to graph the TPL curve, rewrite the equation in terms of \(l\): \(l(q) = \frac{\sqrt{400 - 0.5(q + 80)}}{2}\)
02

Average Productivity of Labor (APL) with k=10

To find the average productivity of labor, divide the total productivity by the number of labor units: APL = \(\frac{q}{l} = \frac{10l - 0.2l^2 - 80}{l}\) To determine the maximum of APL, take the derivative of APL with respect to \(l\) and set it to 0: \(\frac{d(APL)}{dl} = \frac{10 - 0.4l}{l^2} = 0\) \(l = 25\) So, the maximum average productivity of labor occurs at 25 units of labor input. At this point, the number of widgets produced is: \(q(25) = 10(25) - 0.2(25)^2 - 80 = 245\)
03

Marginal Productivity of Labor (MPL) with k=10

To calculate the marginal productivity of labor, find the derivative of the total productivity with respect to \(l\): \(\frac{dq}{dl} = 10 - 0.4l\) To find the point where MPL = 0, \(10 - 0.4l = 0\) \(l = 25\)
04

Adjusting Analysis for k=20

Find the new production function with k=20: \(q = 20l - 0.8(20)^2 - 0.2l^2 = 20l - 0.2l^2 - 320\) Repeat Steps 1-3 as above for this new production function with k=20.
05

Analyze Returns to Scale

To determine whether there are constant, increasing, or decreasing returns to scale, consider the behavior of the production function when a constant (c) times the inputs are used: \(q(ck,cl) = c^2k^2 - 0.8(c^2k^2) - 0.2(c^2l^2).\) Now, divide both sides by c: \(\frac{q(ck,cl)}{c} = c(kl - 0.8k^2 - 0.2l^2)\) Compare this to the original production function: \(q(k,l) = kl - 0.8k^2 - 0.2l^2\) If \(\frac{q(ck,cl)}{c} > q(k,l)\), then there are increasing returns to scale. If \(\frac{q(ck,cl)}{c} = q(k,l)\), then there are constant returns to scale. If \(\frac{q(ck,cl)}{c} < q(k,l)\), then there are decreasing returns to scale. Since \(\frac{q(ck,cl)}{c} = q(k,l)\), the widget production function exhibits constant returns to scale.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Production Function
Understanding the production function is crucial for any economics or business student as it describes the relationship between inputs used in production and the output that is produced. To illustrate, the production function for widgets produced annually (\(q\)) is given by \[ q = k l - 0.8 k^{2} - 0.2 l^{2} \] Here, \(k\) represents the annual capital input and \(l\) the annual labor input. The function reveals that output depends positively on both labor and capital, but with diminishing returns as indicated by the negative squared terms.

When capital remains constant (\(k=10\)), the total productivity of labor shows how output varies with different levels of labor input. By graphing this relationship, one can visualize how output (\(q\)) changes as labor (\(l\)) increases or decreases. This provides valuable insights into the efficiency of resource allocation and helps businesses understand the impact of labor input on their production volume.
Average Productivity of Labor
The average productivity of labor (APL) is an essential measure for businesses as it indicates the output produced per labor unit. APL is calculated by dividing the total productivity by the number of labor units, represented mathematically as \[ APL = \frac{q}{l} \] When \(k=10\), the APL for widgets is described by \[ APL = \frac{10l - 0.2l^2 - 80}{l} \] Taking the derivative of APL with respect to labor and setting it to zero reveals the point at which APL is maximized. This occurs at 25 units of labor, yielding a production of 245 widgets, indicating the most efficient use of labor resources in terms of average output per unit of labor. Understanding where APL reaches its maximum can help businesses decide on the optimal level of labor employment to maximize efficiency and profitability.

Moreover, students should pay close attention to the behavior of APL as labor input changes. Observing the increase or decrease of APL as more labor units are added can indicate whether adding more labor is beneficial or if it leads to inefficiencies due to too many workers contributing to production.
Returns to Scale
Returns to scale is a concept in economics that describes how output responds when all input factors are scaled up or down. In general, there are three types of returns to scale:
  • Constant returns to scale: Proportional increases in input lead to proportional increases in output.
  • Increasing returns to scale: Proportional increases in input lead to more than proportional increases in output.
  • Decreasing returns to scale: Proportional increases in input lead to less than proportional increases in output.

For the widget production function given by \[ q(k,l) = kl - 0.8k^{2} - 0.2l^{2} \] when we scale both inputs by a constant (\(c\)), the output is \( c(kl - 0.8k^{2} - 0.2l^{2}) \), which is proportional to the scaling factor (\(c\)). This tells us the production function exhibits constant returns to scale.

Understanding whether a production function demonstrates constant, increasing, or decreasing returns to scale helps businesses and policymakers make informed decisions about resource allocation, investment strategies, and potential benefits or drawbacks of scaling production up or down. This knowledge aids in maximizing efficiency and cost-effectiveness within production processes.

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

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 \(\infty ?\)

Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we pointed out that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by \\[ \sigma=\frac{f_{k} f_{l}}{f \cdot f_{k l}} \\] Suppose now that we define the homothetic production function \(F\) as \\[ F(k, l)=[f(k, l)]^{\gamma} \\] where \(f(k, l)\) is a constant returns-to-scale production function and \(\gamma\) is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function \(f\) b. Show how this result can be applied to both the CobbDouglas and CES production functions.

As we have seen in many places, the general Cobb-Douglas production function for two inputs is given by \\[ q=f(k, l)=A k^{\alpha} l^{\beta} \\] where \(0 < \alpha < 1\) and \(0 < \beta < 1 .\) For this production function: a. Show that \(f_{k} > 0, f_{1} > 0, f_{k k} < 0, f_{i l} < 0,\) and \\[ f_{k l}=f_{l k} > 0 \\] b. Show that \(e_{q, k}=\alpha\) and \(e_{q, l}=\beta\) c. In footnote \(5,\) we defined the scale elasticity as \\[ e_{q, t}=\frac{\partial f(t k, t l)}{\partial t} \cdot \frac{t}{f(t k, t l)} \\] where the expression is to be evaluated at \(t=1 .\) Show that, for this Cobb- Douglas function, \(e_{q, t}=\alpha+\beta\) Hence in this case the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9 ). d. Show that this function is quasi-concave. e. Show that the function is concave for \(\alpha+\beta \leq 1\) but not concave for \(\alpha+\beta > 1\)

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 \(\ln w\)

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 \((l) ?\)
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