Question

Douglas production function F(x,, x)- xg, where X1, xl are Consider the Cobb- values of generic inputs, while α marginal product of input i? For any i, for what parameter values is there diminishing marginal product of inpu increasing, constant, and decreasing returns to scale? While a general answer is preferable, you can answer these questions for 1-3. 2. a, are constant nt parameters. Forthe , t i? Under what parameter values does the production fu

Answer 1

Cobb Angular production function: F (x_1, x_2, . . . x_I) = X_1 alpha_1 x_2 alpha_2 . . . x_2 alpha_I Marginal product of input I = partial differential F (X_1 X_2. . . x_I)/partial differential X_I = alpha_i x_1^alpha 1 x_2^alpha_2 - X_I - 1^alpha I -1 x_1^alpha I - 1 . . x_I^alpha I = (alpha_I x_I^alpha I -1) I/pi_j = 1 j notequalto I x_j alpha_j For I = 2, this becomes MP_I = (alpha_i X_i^alpha_I - 1) 3/pi_j = 1 j notequalto I X_j alpha_j For Marginal product of input I, d (MP_i)/d X_I < 0 alpha_i (alpha_i - 1) X_j alpha^I -2 I/pi_j = 1