CHAPTER 8 - A LIE GROUP APPROACH TO THE INDEX NUMBER PROBLEMS

Abstract

This chapter explains a Lie group approach to the index number problems. It presents an alternative approach to the question of the Fisher-Frisch test criteria for an index number. It deals with the invariant economic index number problem. The Lie group approach has a decided advantage over the standard approach in dealing with invariance properties of economic behavior. The well-known test of an index number is looked at as the conditions on infinitesimal transformations of Lie groups. These tests are actions of Lie groups on the index number. Any index number must satisfy the invariance properties of group transformations so that it may serve as a useful index of measurement of prices and quantities. The chapter also explains the properties of index numbers derived from implicitly holothetic utility functions. Even under the nonhomothetic condition, some of these index numbers satisfy the criterion of factor-reversal test in a broad sense. It describes that the standard quantity index is not completely symmetric with the price index. To overcome this difficulty, a dual quantity index is derived, which is perfectly symmetric with the corresponding price index, satisfying all the requirements of the basic tests.