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# answerstohomework3summer2006 - Economics 302 4 Week Summer...

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Economics 302 4 Week Summer Session 2006 Answers to Homework #3 Due Monday, June 12, 2006 5/30/06 Homework will be graded for content as well as neatness. Sloppy or illegible work will not receive full credit. This homework requires the use of Excel which is a spreadsheet program. For this homework please round to two places past the decimal. 1. (5 points) Suppose you are given the production function Y = F(K, L) = AK 1/4 L 3/4 Where Y is real GDP, K is capital, L is labor, and A is a parameter expressing the level of technology available in this economy. a. Suppose K and L are both tripled in this economy. Does this production function exhibit constant returns to scale? Provide a proof of your answer. If this production function exhibits constant returns to scale then F(zK, zL) = zF(K, L) = zY. So, we look at the question, what happens to Y if K and L are increased by the same proportion? Let’s define a new level of K, K’ = 3K, and a new level of L, L’ = 3L and consider what happens to our level of output, Y’. Y’ = AK’ 1/4 L’ 3/4 = A(3K) 1/4 (3L) 3/4 Y’ = A K 1/4 (3) 1/4 L 3/4 (3) 3/4 Y’ = A(3) K 1/4 L 3/4 Y’ = 3Y, since Y = AK 1/4 L 3/4 b. Rewrite the above production function in terms of output per worker, y. That is, normalize the function so that it can be written as y = f(k) where k is capital per worker. Provide a verbal interpretation of your mathematical result. Y = F(K, L) = AK 1/4 L 3/4 Y/L = y y = (1/L)( AK 1/4 L 3/4 ) = AK 1/4 L -1/4 = A(K/L) 1/4 = Ak 1/4 = f(k) Output per worker, y, depends upon the amount of capital per worker (k) and the level of technology (A). c. Using the expression you found in part (b) and assuming labor is equal to 16 units, fill in the following table. Recall that k, the capital to labor ratio (K/L), measures capital intensity: that is, it provides a measure of how much capital each unit of labor has to work with in the economy. Assume 1

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that A equals 1 in this problem. Hint: this might be a good time to pull out your Excel spreadsheet program. L K k y Y MPk 16.00 0.00 0.00 0.00 0.00 16.00 20.00 1.25 1.06 16.92 0.85 16.00 40.00 2.50 1.26 20.12 0.16 16.00 60.00 3.75 1.39 22.27 0.11 16.00 80.00 5.00 1.50 23.93 0.08 d. Use your Excel spreadsheet program to graph k and y, placing k on the horizontal axis and y on the vertical axis. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.00 2.00 4.00 6.00 k = K/ L y e. Let’s define s as the saving rate per worker, i as the amount of investment per worker, and c as consumption per worker. Suppose that you are told that the saving rate in this economy is constant and equal to .2. Use this
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