Notes on some inequalities in economics

Inequalities created by relating gauge (distance) functions and their dual support functions have been used in economics and operations research/management science to measure efficiency and productivity. The most familiar example is the Farrell (1957) measure of cost efficiency and its decomposition into technical and allocative components, which can be shown to be an application of the Mahler (1939) inequalities, although Farrell was probably unaware of the connection. In this short paper we add to this literature by providing explicit relationships between distance functions and support functions to form five different inequalities. These inequalities are derived from three support functions: the cost function, the revenue function and the profit function, and three gauge (distance) functions: the input distance function, the output distance function and the directional (technology) distance function.