Chapter 6 Problem Set Name: _______________________ 1. Suppose that a firm uses both labor (L) and capital (K)

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Chapter 6 Problem Set Name: _______________________ 1. Suppose that a firm uses both labor (L) and capital (K) as inputs and has the long-run production functionQ = F(L,K) = L*(L+K)1/2.a. If capital is fixed at K = 10 in the short run, what is its short-run production function? (2 points) b. Complete the table below (show your work for calculating total product). (6 points) L12345 K1010101010 TP MPL APL c. If K increases to K=20, what will happen to labor productivity? Be specific and discuss both average andmarginal product of labor. (4 points) d. Does the production function exhibit diminishing marginal productivity for labor in the short-run? (4pts) e. Does the long-run production function exhibit increasing, constant, or decreasing returns to scale? (4 points)

Answer 1

q = L(L+K)^1/2 APL =Q/L = (L+K)^1/2 when K= 10 Q = L(L+10)^1/2 APL =(L+10)^1/2

MPL = (L+10)^1/2 +(1/2)L/((L+10)^1/2) as L increases so so MPL so, No, MPL is not diminishing

Q = f(K,L) Now Increase Both K and L by some factor λ,, f(λK,λL) = (λL)(λK+λL)^1/2 = λL(λ^1/2)(K+L)^1/2

λ^3/2L(K+L)^1/2. So it is homogeneous of degree 3/2. Or the elasticity of output with respect to inputs is 3/2. Increasing returns to scale