Question

(Production function) Condsider a representitive firm with a production function which is (i) twice continuously differentiable; (i) exhibits positive and diminishing marginal product and (iii) has constant return to scale: 1. Y = F(K, L) Given the capital rental price R and the wage w, and the good price P is normalized to 1, the firm can choose K and L to maximize its profit: max F(K, L) - RK -wI K,L (b) Do the following three production functions satisfy the three assumptions ((i),(ii),(iii) shown above? If not, please explain that which assumption(s) it violates (2) F(K, L)in2L) (3) F(KL)-K1/2 L1/3

Please explain and show all the steps instead of just giving an answer.

Answer 1

F(k, L) = alpha k + beta L, alpha beta > 0, Now partial differential F/partial differential k = alpha, partial differential^2 F/partial differential K^2 = 0, thus not twice differentiable MP_k = alpha, partial differential (MP_k)/partial differential k = 0, so MP is independent of change ink (No +ve or diminishing MP F(lambda k_1 lambda L) = alpha lambda k + beta lambda L = lambda (alpha k + beta L) = lambda F (k, L), 80 CRS F(k, L) = Min (k, 2L), not differentiable No +ve diminishing - MP F(lambda k_1 2 lambda L) = lambda min (k_1 L) hence CRS F(k_1 L) = k^1/2 L^1/3 MP_K partial differential F/partial differential k = (1/2) (k)^-1/2 partial differential MP_k/partial differential K = -1/4 (k)^1/2 L^1/3 = lambda^5/6 F(k_1, L) so doesn�t exhibit CRS Rather DRS F(k_1 L) = (k^sigma + L^sigma)^1/sigma partial differential F/partial differential k = MP_k = 1/sigma (k^sigma + L^sigma)^1-sigma/sigma sigma (k)^sigma - 1 = (k)^sigma - 1 (k^sigma + L^sigma)^1-sigma/sigma partial differential^2 F/partial differential k^2 = partial differential MP_k/partial differential K =