A New Index Number Formula

Abstract

Several years ago (1965, 1967) I started working with price and quantity index numbers, advocated particularly by the Finnish econometrician Törnqvist (1936), whose log-changes are defined as

$$ \sum\limits_{{i = 1}}^n {{{\bar{w}}_{{it}}}\;\log \frac{{{p_{{it}}}}}{{{p_{{i,t - 1}}}}}\;and\;\sum\limits_{{i = 1}}^n {{{\bar{w}}_{{it}}}} \;\log } \frac{{{q_{{it}}}}}{{{q_{{i,t - 1}}}}} $$

(1.1)

where p ₁ ,…,p _n are prices of the n commodities, the q’s are quantities, $m = \sum\nolimits_i \; {p_i}{q_i}$ is total expenditure, w _i , = p _i q _i /m is total expenditure share of the i ^th commodity, while the subscripts t − 1 and t refer to two successive periods and

$$ {\bar{w}_{{it}}} = \frac{1}{2}\left( {{w_{{it}}} + {w_{{i + t - 1}}}} \right) $$

(1.2)

is the arithmetic average expenditure share of the i ^th commodity in the two successive periods. All logarithms are natural logarithms.

This article first appeared in The Review of Economics and Statistics, 55 (1973), 498–502. Reprinted with the permission of Elsevier Science Publishers, B.V. (North Holland). Research supported in part by the National Science Foundation under grant GS-30084. The writer is indebted to Howard Kaikow, programmer-analyst of the Center for Mathematical Studies in Business and Economics of the University of Chicago, for his work on the computations reported in this note.