On Exact Index Numbers

W E shall motivate our investigation of exact index numbers by presenting an example of application of index number theory. We consider a profit-maximizing, price-taking firm endowed with a continuously differentiable non-decreasing and concave production function F(q). Suppose we have observations on the normalized prices p (nominal prices divided by the price of output) and the quantities of the inputs q at two distinct points in time, but not the outputs. An index number formula for the unobserved output is a function of the normalized prices and the quantities of the inputs at the two points, say I(P1,P2,q1,q2), which gives a measure of the ratio of the outputs at the two points, that is,