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Chapter 5: Problem 78

Do Exercise 77 when the equation relating output to resources is \(Q=L^{1 / 4} C^{1 / 2}\)

Short Answer

Expert verified
Question: Find the marginal product of labor and the marginal product of capital for the given Cobb-Douglas production function: \(Q = L^{\frac{1}{4}} C^{\frac{1}{2}}\). Answer: The marginal product of labor is \(\frac{\partial Q}{\partial L} = \frac{1}{4} L^{-\frac{3}{4}}C^{\frac{1}{2}}\), and the marginal product of capital is \(\frac{\partial Q}{\partial C} = \frac{1}{2} L^{\frac{1}{4}}C^{-\frac{1}{2}}\).

Step by step solution

01

Identify the production function

The given production function is \(Q = L^{\frac{1}{4}} C^{\frac{1}{2}}\). This function is used to describe the relationship between the output (Q) given the inputs Labor (L) and Capital (C).
02

Calculate the partial derivative of the production function with respect to Labor (L)

To find the marginal product of labor, we will calculate the partial derivative of the production function Q with respect to L: \(\frac{\partial Q}{\partial L} = \frac{\partial}{\partial L} (L^{\frac{1}{4}}C^{\frac{1}{2}})\). Applying the power rule for taking the derivative, we get: \(\frac{\partial Q}{\partial L} = \frac{1}{4} L^{-\frac{3}{4}}C^{\frac{1}{2}}\).
03

Calculate the partial derivative of the production function with respect to Capital (C)

To find the marginal product of capital, we will calculate the partial derivative of the production function Q with respect to C: \(\frac{\partial Q}{\partial C} = \frac{\partial}{\partial C} (L^{\frac{1}{4}}C^{\frac{1}{2}})\). Applying the power rule for taking the derivative, we get: \(\frac{\partial Q}{\partial C} = \frac{1}{2} L^{\frac{1}{4}}C^{-\frac{1}{2}}\).
04

Interpret the results

We have found two partial derivatives: a) \(\frac{\partial Q}{\partial L} = \frac{1}{4} L^{-\frac{3}{4}}C^{\frac{1}{2}}\) : This is the marginal product of labor. It shows how much the output (Q) would change with a small change in labor (L) keeping capital (C) constant. b) \(\frac{\partial Q}{\partial C} = \frac{1}{2} L^{\frac{1}{4}}C^{-\frac{1}{2}}\) : This is the marginal product of capital. It shows how much the output (Q) would change with a small change in capital (C) keeping labor (L) constant.

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Key Concepts

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

Partial Derivative
In economics, a partial derivative is used to understand how a particular variable changes in relation to another while keeping all other influencing factors constant. This is crucial because real-world production often depends on several variables. For a production function like the one given in the exercise, which is \( Q = L^{\frac{1}{4}} C^{\frac{1}{2}} \), calculating partial derivatives helps isolate the effect of each input on the output.
  • To find the partial derivative with respect to labor \( L \), we differentiate the production function \( Q \) treating \( C \) as constant. This provides insights into how changes in labor affect production.
  • Similarly, the partial derivative with respect to capital \( C \) reveals how changes in capital, while keeping labor constant, influence the output.
Understanding partial derivatives is a fundamental skill for analyzing and optimizing production processes.
Marginal Product
The concept of Marginal Product refers to the additional output generated by adding one more unit of a particular resource while keeping other resources constant. In our context, the marginal product is important for guiding decisions on resource allocation. In the solved exercise:
  • The marginal product of labor is expressed as: \( \frac{\partial Q}{\partial L} = \frac{1}{4} L^{-\frac{3}{4}} C^{\frac{1}{2}} \). This indicates how the output changes with a slight increase in labor, given a fixed amount of capital.
  • Similarly, the marginal product of capital is \( \frac{1}{2} L^{\frac{1}{4}} C^{-\frac{1}{2}} \). This reflects the change in output with a small increase in capital, with labor held constant.
Analyzing these marginal products helps businesses determine the most efficient way to allocate their resources.
Labor as a Variable
Labor, denoted as \( L \), serves as a crucial variable in production functions. Its variability allows businesses to adjust the number of workers or the amount of time worked to meet production needs. In the production function given:
  • The exponent \( \frac{1}{4} \) represents the share of labor in production, implying decreasing returns to scale where each additional unit of labor adds less output than the previous one.
  • Understanding labor's role and its effects on production can lead to optimal staffing and efficient labor use, especially when labor costs form a significant part of overall expenses.
Flexibility in labor allocation can significantly impact production levels and cost efficiency.
Capital as a Variable
Capital, represented by \( C \), is another essential variable in production functions. It implicates the machinery, buildings, and technology used in production processes. For the given production function:
  • The exponent \( \frac{1}{2} \) suggests a more substantial impact on the output compared to labor, indicating higher returns on capital investments.
  • The concept of capital as a variable involves adapting capital investments to complement labor and maximize output. Careful capital allocation can ensure that resources are not wasted and that production processes are efficient.
By examining capital's role, businesses can plan better for long-term asset investments, which are often crucial for productivity and growth.

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Most popular questions from this chapter

The number of children who were home schooled in the United States in selected years is shown in the table. (a) Sketch a scatter plot of the data, with \(x=0\) corresponding to 1980 (b) Find a quadratic model for the data. (c) Find a logistic model for the data. (d) What is the number of home-schooled children predicted by each model for the year \(2015 ?\) (e) What are the limitations of each model? $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Fall of } \\ \text { School Year } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Children (in thousands) } \end{array} \\ \hline 1985 & 183 \\ \hline 1988 & 225 \\ \hline 1990 & 301 \\ \hline 1992 & 470 \\ \hline 1993 & 588 \\ \hline 1994 & 735 \\ \hline 1995 & 800 \\ \hline 1996 & 920 \\ \hline 1997 & 1100 \\ \hline 1999 & 1400 \\ \hline 2000 & 1700 \\ \hline 2005 & 1900 \\ \hline \end{array}$$

In the year 2009 , Olivia's bank balance is 1000 dollars. In the year 2010 , her balance is 1100 dollars. (a) If her balance is growing exponentially, in what year will it reach 2500 dollars? (b) If her balance is instead growing linearly, in what year will it reach 2500 dollars ?

(a) Graph \(f(x)=x^{5}\) and explain why this function has an inverse function. (b) Show algebraically that the inverse function is \(g(x)=x^{1 / 5}\) (c) Does \(f(x)=x^{6}\) have an inverse function? Why or why not?

The population of St. Petersburg, Florida (in thousands) can be approximated by the function $$ g(x)=-127.9+81.91 \ln x \quad(x \geq 70) $$ where \(x=70\) corresponds to 1970 (a) Estimate the population in 1995 and 2003 . (b) If this model remains accurate, when will the population be \(260,000 ?\)

Rationalize the denominator and simplify your answer. $$\frac{1+\sqrt{3}}{5+\sqrt{10}}$$
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