= Example . The production function for a commodity is P = L L K K KL where L is labour and K is Capital. (i) Calculate the marginal products of two inputs when units of each of labour and Capital are used (ii) If units of capital are used, what is the upper limit for use of labour which a rational producer will never exceed? (i) Given the production is P = L L K K KL ∂ ∂ P L L K ∂ ∂ P K K L When L = K = units, Marginal productivity of labour: ∂ ∂ P L = − + = Marginal productivity of capital : ∂ ∂ P K = − + = (ii) Upper limit for use of labour when K = is given by L P k ≥ − .
L + ≥ ≥ . L i.e., L ≤ Hence the upper limit for the use of labour will be units. Example . For the production function, P L K = verify Euler’s theorem.
P = L K is a homogeneous function of degree . Marginal productivity of labour is ¶ ¶ P L = ´ L K − = K L Marginal productivity of capital is ¶ ¶ P K = L K × L K - - L P L K P K ∂ ∂ ∂ ∂ = 3L K L +K L K = L K + L K = L K = P Hence Euler’s theorem is verified. Example . The demand for a commodity x is q = p p