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Midterm 2 Answer Key April 8, 2010 1) Graphically, &gure 6.2 in the text is an example of a production func- tion with constant returns to scale, and diminishing returns to both labor and capital. Using calculus it is apparent that any Cobb-Douglass production function in which the exponents add up to one has these properties. For example, a production function q=5L : 3 K : 7 demonstrates diminishing returns to labor and diminishing returns to capital. Taking second derivatives, @MP L @L = @ 2 q @L 2 = 1 : 05 L 1 : 7 K : 7 @MP K @K = @ 2 q @K 2 = 1 : 05 L : 3 K 1 : 3 (1) Since both second derivatives are less than zero, this production function demonstrates diminishing marginal returns for each factor. The function also demonstrates constant returns to scale. 2 q = 10 L : 3 K : 7 q (2 L; 2 K ) = 5(2 L ) : 3 (2 K ) : 7 = 5 ± 2 : 3+ : 7 L : 3 K : 7 = 10 L : 3 K : 7 (2) 2 q = q (2 L; 2 K ) (3) 2) The wage/rent ratio is 300/1000. The marginal rate of technical sub- stitution = the marginal product of labor (50)/the marginal product of capital (200). The wage/rent ratio of 6/20 is greater than the MRTS of 5/20. Because If marginal

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