Question

A standard tool of economic analysis is the Cobb-Douglas production function. This function shows how much output (Q) is produced with a given amount of labor (L) and capital (K) as follows: Q = AK^alpha L^(1-alpha)....The parameter A represents the efficiency of the economy, so A increases with technological change. Suppose that, over time, efficiency, capital and labor each grow at the rates, A(t) = Aoe^(ct), K(t) = Koe^(ft), and L(t) = Loe^(gt), where Ao, Ko, and Lo are initial values for technology, capital, and labor, respectively, and c, f, and g are their respective rates of growth. What is the percentage growth rate of output in terms of the production parameters and the growth parameters c, f, and g?The derivative of a natural logarithm is close to a percentage change. 

          A standard tool of economic analysis is the Cobb-Douglas production function. This function shows how much output (Q) is produced with a given amount of labor (L) and capital (K) as follows: Q = AK^alpha L^(1-alpha)....The parameter A represents the efficiency of the economy, so A increases with technological change. Suppose that, over time, efficiency, capital and labor each grow at the rates, A(t) = Aoe^(ct), K(t) = Koe^(ft), and L(t) = Loe^(gt), where Ao, Ko, and Lo are initial values for technology, capital, and labor, respectively, and c, f, and g are their respective rates of growth. What is the percentage growth rate of output in terms of the production parameters and the growth parameters c, f, and g?The derivative of a natural logarithm is close to a percentage change.
Show more…

Added by Monique R.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
AceChat toggle button
Close icon
Ace pointing down

Please give Ace some feedback

Your feedback will help us improve your experience

Thumb up icon Thumb down icon
Thanks for your feedback!
Profile picture
A standard tool of economic analysis is the Cobb-Douglas production function. This function shows how much output (Q) is produced with a given amount of labor (L) and capital (K) as follows: Q = AK^alpha L^(1-alpha)....The parameter A represents the efficiency of the economy, so A increases with technological change. Suppose that, over time, efficiency, capital and labor each grow at the rates, A(t) = Aoe^(ct), K(t) = Koe^(ft), and L(t) = Loe^(gt), where Ao, Ko, and Lo are initial values for technology, capital, and labor, respectively, and c, f, and g are their respective rates of growth. What is the percentage growth rate of output in terms of the production parameters and the growth parameters c, f, and g?The derivative of a natural logarithm is close to a percentage change.
Close icon
Play audio
Feedback
Powered by NumerAI
Danielle Fairburn Jennifer Stoner
Kathleen Carty verified

Sri K and 53 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question
*

Labs

-

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs
*

Key Concepts

-
Key Concept
Premium Feature
Explore the core concept behind this problem.
Play button
Key Concept
Premium Feature
Explore the core concept behind this problem.
Your browser does not support the video tag.
*

Recommended Videos

-
2-suppose-that-you-have-a-standard-solow-model-with-a-cobb-douglas-produc-tion-function-and-both-labor-augmenting-productivity-growth-and-population-growth-the-central-equation-of-the-model-93433

Suppose that you have a standard Solow model with a Cobb-Douglas production function and both labor-augmenting productivity growth and population growth. The central equation of the model is: a. Derive an expression for the steady-state capital stock per efficiency unit of labor. b. Use your answer from the previous part to derive an expression for the steady-state value of consumption per effective worker. c. Use calculus to derive an expression for the value of s which maximizes steady-state consumption per worker. Does the expression for this s depend at all on the values of z or n?

Akash M.
cobb-douglas-production-function-the-output-q-of-an-economic-system-subject-to-two-inputs-such-as--2

Cobb-Douglas production function The output $Q$ of an economic system subject to two inputs, such as labor $L$ and capital $K,$ is often modeled by the Cobb-Douglas production function $Q(L, K)=c L^{a} K^{b},$ where $a, b,$ and $c$ are positive real numbers. When $a+b=1,$ the case is called constant returns to scale. Suppose $a=\frac{1}{3}, b=\frac{2}{3},$ and $c=40.$ a. Graph the output function using the window $[0,20] \times[0,20] \times[0,500].$ b. If $L$ is held constant at $L=10,$ write the function that gives the dependence of $Q$ on $K.$ c. If $K$ is held constant at $K=15,$ write the function that gives the dependence of $Q$ on $L.$

Calculus for Scientists and Engineers: Early Transcendental

Functions of Several Variables

Graphs and Level Curves

cobb-douglas-production-function-quad-during-the-1920-s-charles-cobb-and-paul-douglas-modeled-total-

Cobb-Douglas production function $\quad$ During the 1920 s, Charles Cobb and Paul Douglas modeled total production output $P$ (of a firm, industry, or entire economy) as a function of labor hours involved $x$ and capital invested $y$ (which includes the monetary worth of all buildings and equipment). The Cobb-Douglas production function is given by $$P(x, y)=k x^{\alpha} y^{1-\alpha}$$ where $k$ and $\alpha$ are constants representative of a particular firm or economy. a. Show that a doubling of both labor and capital results in a doubling of production $P$ b. Suppose a particular firm has the production function for $k=120$ and $\alpha=3 / 4 .$ Assume that each unit of labor costs $\$ 250$ and each unit of capital costs $\$ 400,$ and that the total expenses for all costs cannot exceed $\$ 100,000 .$ Find the maximum production level for the firm.

University Calculus: Early Transcendentals

Partial Derivatives

Lagrange Multipliers
*

Recommended Textbooks

-
Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart 8th Edition
achievement 1,929 solutions
Calculus: Early Transcendentals

Calculus: Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet 3rd Edition
achievement 1,235 solutions
Thomas Calculus

Thomas Calculus

George B. Thomas Jr. 14th Edition
achievement 1,313 solutions
*

Transcript

-
00:01 Hi in the given problem we are required to find the the maximize function for x1 x2 x3 and x4 so here we will be finding x1 so x1 is the number of logs shipped from forest 1 to mill a so this is from forest 1 to mill a so that many number of logs x2 is the number of logs shipped from forest to forest to mill a similarly let's take x3 as there are number of logs shipped from forest one to mill b and x4 finally is the number of logs shifted from forest 1 forest 2 to mill b so here we have a constraint that the constraints are that x1 plus x2 is less equal to 240 because the capacity constraint of mill a because of mill a this constraint is obtained from will a and the next constraint is that x3 plus x4 is less equal to 300 and that is from the capacity constraint from will be and then the constraint that is we having x1 plus x3 is less equal to 200 is the yield constraint of forest one this is from the yield constraint of forest 1 similarly we have a constraint x2 plus x4 is less equal to 200 that is again the yield constraint from forest 2 so this is because of the yield constraint from forest 2 next we have x1 plus x2 plus x3 plus x4 should be greater equal to 300 that is the constraint on the number of logs needed for a doubt and x1 is greater equal to zero x i is greater equal to zero because i all the the values are greater equal to zero where i is equal to one two three or four so we have to minimize the objective function would be objective function is given as to minimize function f x1 x2 x3 comma x4 which is equal to the cost function the cost that is equal to 24 times 0 .51 x1 plus 20 .5 times 0 .15 x2 plus 17 .2 times 0 .15 x2 plus 17 .2 times 0 .15 x3 plus 18 times 0 .15 x4.
04:08 So if we simplify this further, this is equal to 3 .6 x1 plus 3 .075 x2 plus 2 .56 x3 plus 2 .7x4.
04:26 So that is the objective function which is the cost function...
Need help? Use Ace
Ace is your personal tutor. It breaks down any question with clear steps so you can learn.
Start Using Ace
Ace is your personal tutor for learning
Step-by-step explanations
Instant summaries
Summarize YouTube videos
Understand textbook images or PDFs
Study tools like quizzes and flashcards
Listen to your notes as a podcast
Continue solving this problem
Create a free account to:
  • View full step-by-step solution
  • Ask follow-up questions with Ace AI
  • Save progress and study later
Continue Free
Numerade

Get step-by-step video solution
from top educators

Continue with Clever
or



By creating an account, you agree to the Terms of Service and Privacy Policy
Already have an account? Log In

A free answer
just for you

Watch the video solution with this free unlock.

Numerade

Log in to watch this video
...and 100,000,000 more!


EMAIL

PASSWORD

OR
Continue with Clever