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# chapter3 - Chapter 3 Numerical Problems 1 a. Y=AK.3N.7 In...

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Chapter 3 Numerical Problems 1 a. Y=AK .3 N .7 In order to find the growth of total factor productivity, we start by calculating the value of A in the production function. A = Y / K .3 N .7 . We then calculate the growth rate of A as: Growth rate = (A t+1 - A t )/ A t x 100 %. We get the following values: A % increase in A 1960 11.706 - 1970 13.800 17.9% 1980 14.315 3.7% 1990 15.441 7.9% b. We calculate the marginal product of labor (MPN) by seeing what are the changes in the level of output (Y) when you add one unit of labor (increase N by 1): Y 1 = AK .3 N .7 ex: Y 1 = A(1879) .3 65.8 .7 = 2262 Y 2 = AK . 3(N+1) . 7 ex: Y 2 = A(1879) .3 66.8 .7 = 2286 MPN=( Y 2 - Y 1 )/[ (N+1)-N] =( Y 2 - Y 1 ) ex: MPN = 2286- 2262=24 The exact method to compute the MPN is to take the derivative of output with respect to N; dY / dN = 0 .7A(K/ N) .3 . Y 1 Y 2 MPN 1960 2263 2287 24 1970 3398 3428 30 1980 4615 4648 33 1990 6136 6172 36 2. Y=.2(K+N .5 ) a. N=100; Y=.2K+2 Y K Slope = .2 2 22 100

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For each additional unit of capital, output increases by 0.2 units. The MPK is 0. 2 The slope of the production function line is 0.2. The MPK is constant regardless of the level of K. So there is no diminishing marginal productivity of capital. The production function is a straight line, that is it has a constant slope. b. K=100; Y=.2 N .5 +20 When N is 100, output is Y - 0.2(100 + 100 .5 )= 22. When N is 110, Y is 22.0976. So the MPN for raising N from 100 to 110 is (22.0976 - 22)/ 10 = 0.00976. When N is 120, Y is 22.1909. So the MPN for raising N from 110 to 120 is (22.1909 -22.0976) / 10 = 0.00933. There is diminishing marginal productivity of labor because the MPN decreases as N increases. In the graph this is shown as a decline in the slope of the production function as N increases. 3. a. MPN = ( Y 2 - Y 1 )/[ (N+1)-N] = ( Y 2 - Y 1 ) MRPN = ( PY 2 -P Y 1 )/[ (N+1)-N] = ( PY 2 - PY 1 )= P( Y 2 - Y 1 ) = P MPN N Y MPN MRPN MRPN (P=5) (P= 10) 1 8 8 40 80 2 15 7 35 70 3 21 6 30 60 4 26 5 25 50 5 30 4 20 40 6 33 3 15 30 b. P =\$5. Y K 2 22 10
W=\$38. The firm hires one worker, since MRPN (\$40) is greater than W (\$38) at N = 1. It does not hire the second worker, since MRPN (\$35) is less than W (\$38) at N = 2. Or in real terms: w=W/P=\$38/\$5=7.6. The firm hires one worker, since MPN (8) is greater than W (7.6) at N = 1. It does not hire the second worker, since MPN (15) is less than W (7.6) at N = 2. W = \$27. The firm hires the second worker, since MRPN (\$35) is greater than W (\$27) at N = 2. It hires the third worker, since MRPN (\$30) is higher than W (\$27) at N = 3. It does not hire the fourth worker, since MRPN (\$25) is less than W (\$27) at N = 4. (3) W = \$22. The firm hires four workers, since MRPN (\$25) is greater than W (\$22) at N = 4. It does not hire five workers, since MRPN (\$20) is less than W (\$22) at N = 5 . c.

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## This note was uploaded on 07/24/2008 for the course ECON 360 taught by Professor Wang during the Fall '06 term at SUNY Buffalo.

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chapter3 - Chapter 3 Numerical Problems 1 a. Y=AK.3N.7 In...

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