Question

The total production of a product depends on the amount of labour, L, and the amount of capital, K, according to the Cobb-Douglas model. If the cost of a unit of labour is m and the cost of a unit of capital is n, and the total budget for the manufacturer is p($), determine the amount of labour and capital that maximise production, in terms of m, n and the two constants in the model.

Answer 1

amount of labour = L Capital Investemnet = K Total production, P = bL^alphaK^1 - alpha = P[L, K] where 0 < alpha < 1, b > 0 cost of the unit labour = m Cost of the unit Capital = n Total budget = B = mL + nK = In order to maximize P subject to constraint mL + nK = B G(L, K) = mL + nK - P = 0. nabla P = (alpha b L^alpha - 1 K^1 - alpha, b(1 - alpha) K^alphaL^alpha) = (P_L, P_K) nabla = (partial differential G/partial differential L, partial differential G/partial differential K) = (m, n) method nablaP = lambda nabla b alpha (L/K) ^-1 + alpha = lambda m b(1 - alpha) (L/K) ^alpha = lambda n lambda = b/m alpha (L/K) -1 + alpha = b/n(1 - alpha) (L/K) ^alpha labour, L = alpha B/m, Capital Investement K = B/n(1 - alpha) maximum production P(L, K) = b(alpha B/m) ^alpha [B/n(1 - alpha)]^1 - alpha