10 Partial differentiation Derive the first-order partial derivatives of multi-variable functions. Apply the concept of partial differentiation to production functions, utility functions and the Keynesian macroeconomic model. Derive second-order partial derivatives and interpret their meaning. | 10 Partial differentiation Learning objectives After completing this chapter students should be able to Derive the first-order partial derivatives of multi-variable functions. Apply the concept of partial differentiation to production functions utility functions and the Keynesian macroeconomic model. Derive second-order partial derivatives and interpret their meaning. Check the second-order conditions for maximization and minimization of a function with two independent variables using second-order partial derivatives. Derive the total differential and total derivative of a multi-variable function. Use Euler s theorem to check if the total product is exhausted for a Cobb-Douglas production function. Partial differentiation and the marginal product For the production function Q f K L with the two independent variables L and K the value of the function will change if one independent variable is increased whilst the other is held constant. If K is held constant and L is increased then we will trace out the total product of labour TPL schedule TPL is the same thing as output Q . This will typically take a shape similar to that shown in Figure . In your introductory microeconomics course the marginal product of L MPl was probably defined as the increase in TPL caused by a one-unit increment in L assuming K to be fixed at some given level. A more precise definition however is that MPl is the rate of change of TPl with respect to L. For any given value of L this is the slope of the TPL function. Refer back to Section if you do not understand why. Thus the MPl schedule in Figure is at its maximum when the TPL schedule is at its steepest at M and is zero when TPL is at its maximum at N. Partial differentiation is a technique for deriving the rate of change of a function with respect to increases in one independent variable when all other independent variables in the function areheld constant. Therefore if the production function Q f K L is differentiated .
