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# ps2 - Economics 202 Principles of Macroeconomics Name...

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Economics 202 Name: _______________ Principles of Macroeconomics Professor Melick Problem Set #2 Due Wednesday January 30, 2008 This problem set is meant to provide you with a solid understanding of production functions by using as an example the Cobb-Douglas production function - the most popular functional form in economics and the cornerstone of most models of economic growth (a topic we will cover in several weeks.) The function is named after the two economists who popularized it in the 1920s. 1 I. Theory Abel and Bernanke write a general production function as ( ) N K F A Y , = . The Cobb- Douglas functional form is given by Y A K N = α β where output (GDP) total factor productivity capital stock labor force parameters, elasticity of output with respect to capital and labor, respectively. Y A K N = = = = = α β , Most commonly, it is assumed that α β + = 1 and hence Y A K N = - α α 1 We now want to convince ourselves of several properties of the Cobb-Douglas function. We will use mathematical derivations as well as a particular form of the Cobb-Douglas function where we set α = = 1 3 10 and A giving Y K N = 10 1 3 2 3 A. Constant Returns to Scale 1. In the space below, use the properties of exponents ( ) ( ) x y x y and x x x z z z a b a b = = + , to demonstrate that the following is true ( ) ( ) Y A K N Y ' = = - λ λ λ α α 1 1 Charles W. Cobb; Paul H. Douglas, “A Theory of Production”, The American Economic Review , Vol. 18, No. 1, Supplement, Papers and Proceedings of the Fortieth Annual Meeting of the American Economic Association. (Mar., 1928), pp. 139-165.

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Viola!! Constant returns to scale. You have demonstrated that increasing both K and N by some factor λ , increases output by the same factor. With Excel, we can confirm this for our particular sample version of Y K N = 10 1 3 2 3 . We are going to set-up a spreadsheet that solves for Y, given K and N . and the first several rows of the end result should look like this 0.333333 0.666667 10 K N Y 0 0 0 1 1 10 Put the parameters 1/3 , 2/3, and 10 in the top row, and put values for K in column A and the values for N in column B starting in the second row. The production function formula Y K N = 10 1 3 2 3 can be translated into an Excel formula as =\$C\$1*(A3)^(\$A\$1)*(B3)^(\$B\$1) in order to solve for Y in column C. Fill in the rest of the rows by having K and N increase by units of 1 until they reach 16 and answer the following questions 2. As K and N double from 1 to 2, output _____________ from ______ to __________. 3. As K and N double from 2 to 4, output _____________ from ______ to __________. 4. As K and N double from 4 to 8, output _____________ from ______ to __________. Voila! Constant returns to scale. Now try changing the parameters so that both exponents α β and are equal to 2/3. We find that if the coefficients sum to more than one, we get increasing returns to scale. Set them both equal to 1/3 to find a production function with decreasing returns to scale.
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ps2 - Economics 202 Principles of Macroeconomics Name...

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