Derivatives In this section we solve problems on partial derivatives which have direct impact on Industrial areas. . . Production function and marginal productivities of two variables (i) Production function: Production P of a firm depends upon several economic factors like capital ( K ), labour ( L ), raw materials ( R ), machinery ( M ) etc… Thus P = f ( K,L,R,M, … ) is known as production function.
If P depends only on labour ( L ) and capital ( K ), then we write P = f ( L,K ). (ii) Marginal productivities: Let P = f ( L,K ) be a production function. Then L P is called the Marginal productivity of labour and K P is called the Marginal productivity of capital . Euler’s theorem for homo- geneous production function P ( L,K ) of degree states that L L P K K P = P .
. Partial elasticity of demand Let q = f p p h be the demand for commodity A , which depends upon the prices. p and p of commodities A and B respectively. - - Applications of Differentiation The partial elasticity of demand q with respect to p is defined to be η qp Eq Ep q q = − ∂ ∂ The partial elasticity of demand q with respect to p is defined to be η qp Eq Ep q q = − ∂ ∂ Example .
Find the marginal productivities of capital ( K ) and labour ( L ) if P = L L K . K KL when K = L = . We have P = L L K K KL ∂ ∂ P L L K ∂ ∂ P K K L Marginal productivity of labour: ∂ ∂ P L = + + = Marginal productivity of capital: ∂ ∂ P K = − +