Consider a Cobb-Douglas production function with three inputs: K is capital (e.g

Question

Consider a Cobb-Douglas production function with three inputs: K is capital (e.g., number of machines), L is labor (e.g., number of workers), and H is "human capital" (e.g., number of college degrees among the workers). Normalizing the scale factor A to one gives us: Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor? Derive an expression for the marginal product of human capital. How does an increase in the amount of human capital affect the marginal product of human capital? What is the income share paid to labor? What is the income share paid to human capital? In the National Income Accounts of this economy, what share of total income do you think workers would appear to receive? (Hint Consider where the return to human capital shows up

Explanation / Answer

Cobb-Douglas Production Function If you have not already done so, look at how the parameters of a Cobb-Douglas production function can be estimated: Estimating a Cobb-Douglas production function. The three factor Cobb-Douglas production function is: q = A * (L^alpha) * (K^beta) * (M^gamma) = f(L,K,M). where L = labour, K = capital, M = materials and supplies, and q = product. The symbol "^" means "raise to the power," i.e. L^alpha means "raise the value of L to the power of the value of alpha." Production functions need to have certain properties, to ensure that we can solve the least-cost problem: Check any of the many textbooks. If for given values of L,K, and M, the Hessian of the production function f is negative definite, then its isoquants at that point are concave to the origin. I. Decreasing returns to scale: alpha + beta + gamma < 1 With decreasing returns to scale, a proportional increase in all inputs will increase output by less than the proportional constant. When we estimated the Cobb-Douglas production function, we found that: A = 1.01278, alpha = .317, beta = .417, and gamma = .186. alpha + beta + gamma = .317 + .417 + .186 = .92 < 1 Then, q = A * (L^alpha) * (K^beta) * (M^gamma) --> q = 1.01278 * (L^.317) * (K^.417) * (M^.186) Suppose the firm can buy its factors at the prices: wL = 7, wK = 13, wM = 6. Its costs will be: c(q) = wL * L + wK * K + wM * M = 7 * L + 13 * K + 6 * L Then to produce 35 units of product at minimum cost, it should use: L = 59.36, K = 42.05, and M = 40.64 units of inputs. Notes: 1. 35 = 1.01278 * (59.36^.317) * (42.05^.417) * (40.64^.186) 2. c(q) = 7 * L + 13 * K + 6 * M --> 1205 = 7 * 59.36 + 13 * 42.05 + 6 * 40.64 3. Average cost = c(q)/q --> 1205.95 / 35 = 34.46 Other combinations of factor inputs will also produce 35 units of product, like L = 74.01, K = 37.44, and M = 36.19. But, these combinations will be more costly at the given factor prices. With these inefficient input combinations: 1. 35 = 1.01278 * (74.01^.317) * (37.44^.417) * (36.19^.186) 2. 1221.9 = 7 * 74.01 + 13 * 37.44 + 6 * 36.19 3. Average cost = 1221.9 / 35 = 34.91 The inputs L = 59.36, K = 42.05, and M = 40.64 are the least-cost combination of inputs that will produce q = 35 units of product at the input prices wL = 7, wK = 13, and wM = 6. If for each feasible amount of product, we compute the cost of producing the product using the cost minimizing combination of inputs, we obtain the cost function, from which the average cost and marginal cost functions can be obtained.

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