The Cobb-Douglas production function for a company building widgets is given by Y = AL K*- where Y is the total number of widgets produced, L is the number of units of labor (in hours, for instance), and K is the amount of capital invested in equipment and so on (in dollars, for instance). Both A and ß are constants (real numbers), which depend on what company is being modeled. The constant 3 is between 0 and 1. The constants A and 8 are typically found by fitting the equation to past data. Below, you will study an imaginary company where B is the following. The blank spot in the table below is the last digit of your student ID. If your discussion section is with use B = 0.4 +0.0_ use 8= 0.5 + 0.0_ use B = 0.6 + 0.0 use 8=0.7+ 0.0_ Aaron Victorin-Vangerud use 3 = 0.8+ 0.0_. Cruz Godar Elisha Hulbert Jacob Lebovic Bo Phillips For example, if my ID were 951234567 and my discussion section were with Jacob Lebovic then I would use B = 0.67. We will consider a company with the following numbers for 2019: Annual labor of 20 million hours (which corresponds to about 10,000 full-time work- ers), • Capital investment of 100 million dollars, and Annual output of 1 million widgets. (a) What is your value of 8? (b) Use the data aboe and the Cobb-Douglas production function to compute A to five decimal places. (c) Assume that the total output of the company is fixed at 1 million widgets (i.e., Y = 1000000). Use the value of A you found in the previous part to write the Cobb-Douglas production equation only in terms of L and K. (d) Solve the equation you found above for K as a function of L, ie., as K = f(L). (Hint: you will need to use rules of exponents. Your answer should be of the form F(L) = cLª for some constants e and d.)
The Cobb-Douglas production function for a company building widgets is given by Y = AL K*- where Y is the total number of widgets produced, L is the number of units of labor (in hours, for instance), and K is the amount of capital invested in equipment and so on (in dollars, for instance). Both A and ß are constants (real numbers), which depend on what company is being modeled. The constant 3 is between 0 and 1. The constants A and 8 are typically found by fitting the equation to past data. Below, you will study an imaginary company where B is the following. The blank spot in the table below is the last digit of your student ID. If your discussion section is with use B = 0.4 +0.0_ use 8= 0.5 + 0.0_ use B = 0.6 + 0.0 use 8=0.7+ 0.0_ Aaron Victorin-Vangerud use 3 = 0.8+ 0.0_. Cruz Godar Elisha Hulbert Jacob Lebovic Bo Phillips For example, if my ID were 951234567 and my discussion section were with Jacob Lebovic then I would use B = 0.67. We will consider a company with the following numbers for 2019: Annual labor of 20 million hours (which corresponds to about 10,000 full-time work- ers), • Capital investment of 100 million dollars, and Annual output of 1 million widgets. (a) What is your value of 8? (b) Use the data aboe and the Cobb-Douglas production function to compute A to five decimal places. (c) Assume that the total output of the company is fixed at 1 million widgets (i.e., Y = 1000000). Use the value of A you found in the previous part to write the Cobb-Douglas production equation only in terms of L and K. (d) Solve the equation you found above for K as a function of L, ie., as K = f(L). (Hint: you will need to use rules of exponents. Your answer should be of the form F(L) = cLª for some constants e and d.)
Principles of Economics (MindTap Course List)
8th Edition
ISBN:9781305585126
Author:N. Gregory Mankiw
Publisher:N. Gregory Mankiw
Chapter18: The Markets For The Factor Of Production
Section: Chapter Questions
Problem 9PA
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Im working on a study guide for our midterm and I am stuck on the last question. My value of B is .81 and I got 3.69946 for question b. I don't know where to go from there and I'm confused on how to use the value of 3.69946 to write a new equation only in terms of L and K. I'm also confused on how I would approach part d. when it asks to solve for K as a function of L.
Transcribed Image Text:INSTRUCTOR: ROBERT LIPSHITZ
The Cobb-Douglas production function for a company building widgets is given by
Y
AL K'-8
where Y is the total number of widgets produced, L is the number of units of labor (in hours,
for instance), and K is the amount of capital invested in equipment and so on (in dollars,
for instance). Both A and B are constants (real numbers), which depend on what company
is being modeled. The constant B is between 0 and 1. The constants A and B are typically
found by fitting the equation to past data.
Below, you will study an imaginary company where B is the following. The blank spot
in the table below is the last digit of your student ID. If your discussion section is with
use B = 0.4 + 0.0
use B = 0.5 +0.0
use B = 0.6 + 0.0
use B = 0.7 +0.0,
Aaron Victorin-Vangerud use ß = 0.8 + 0.0_.
Cruz Godar
Elisha Hulbert
Jacob Lebovic
Bo Phillips
For example, if my ID were 951234567 and my discussion section were with Jacob Lebovic
then I would use 3 = 0.67.
We will consider a company with the following numbers for 2019:
• Annual labor of 20 million hours (which corresponds to about 10,000 full-time work-
ers),
• Capital investment of 100 million dollars, and
• Annual output of 1 million widgets.
(a) What is your value of 3?
(b) Use the data above and the Cobb-Douglas production function to compute A to five
decimal places.
(c) Assume that the total output of the company is fixed at 1 million widgets (i.e.,
1000000). Use the value of A you found in the previous part to write the
Cobb-Douglas production equation only in terms of L and K.
(d) Solve the equation you found above for K as a function of L, i.e., as K
(Hint: you will need to use rules of exponents. Your answer should be of the form
f(L) = cLd for some constants c and d.)
(e) Using the function f from the previous part, compute f'(L). (This is called the
marginal rate of technical substitution.)
(f) What does the fact that f'(L) is always negative imply about labor and capital? (Re-
member that we are assuming the total output is constant.) (One or two sentences.)
(g) Compute and write a sentence or two interpreting the valu
106). Include units in your computation and sentence.
Email address: lipshitz@uoregon.edu
Y
f(L).
f'(18000000) = f'(18×
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