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.)
Chapter11: Profit Maximization
Section: Chapter Questions
Problem 11.9P
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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.
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