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Index number theory using differences rather than ratios
American Journal of Economics and Sociology, The, Jan, 2005 by W. Erwin Diewert
I
Introduction
When, forty-two years ago, I wrote my doctor's thesis on certain mathematical investigations in the theory of value and prices, I was a student of mathematical physics and, with youthful enthusiasm, dreamed dreams of seeing economics, or one branch of it, grow into a true science by the same methods which had long since built up physics into a true and majestic science.
Irving Fisher (1933: 2)
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Rather than review all of the many contributions of Irving Fisher to the index number literature, we will develop an alternative branch of index number theory that had its origins around the time that Fisher (1911, 1921, 1922) developed his test approach. This alternative branch of index number theory was started by T. L. Bennet (1920) and J. K. Montgomery (1929, 1937), but for various reasons, their approach never prospered and has been mostly forgotten by present-day index number theorists. (1)
In order to explain this alternative approach to index number theory, we first need to explain the traditional approach. Suppose we have collected price and quantity information on N commodities for a base period 0 and a current period 1. Denote the price and quantity of commodity n in period t as [p.sup.t.sub.n] and [q.sup.t.sub.n], respectively, for n = 1, 2, ..., N and t = 0, 1. Define the period t price and quantity vectors as [p.sup.t.sub.n] [equivalent to] ([p.sup.t.sub.1], [p.sup.t.sub.2], ..., [p.sup.t.sub.N]) and [q.sup.t] [equivalent to] ([q.sup.t.sub.1], [q.sup.t.sub.2], ..., [q.sup.t.sub.N]) for t = 0, 1. Then the value of the N commodities in period t is
(1) [p.sup.t] x [q.sup.t] [equivalent to] [[summation].sup.N.sub.n=1][p.sup.t.sub.n][q.sup.t.sub.n] for t = 0,1.
Fisher (1911, 1922) framed the now-traditional test approach to index number theory as follows: find two functions of the 4N price and quantity variables that pertain to the two periods under consideration, say p([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) and Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]), such that the value ratio for the two periods, [p.sup.1] x [q.sup.1]/[p.sup.0] x [q.sup.0], is equal to the product of P and Q, that is,
(2) [p.sup.1] x [q.sup.1]/[p.sup.0] x [q.sup.0] = P([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1])Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]),
and the functions P and Q satisfy certain properties that allow us to identify p([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) as an aggregate measure of relative price change and Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) as an aggregate measure of relative quantity change. Fisher called these properties or axioms tests. The function p([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) is called the price index and the function Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) is called the quantity index. If the number of commodities is one (i.e., N = 1), then the price index P collapses down to the single price ratio [p.sup.1.sub.1]/[p.sup.0.sub.1] and the quantity index Q collapses down to the single quantity ratio [q.sup.1.sub.1]/[q.sup.0.sub.1]. The index number problem (i.e., the problem of determining the functional forms for P and Q) is trivial in this N = 1 case. However, in the multiple commodity case where N > 1, the problem of finding functions P and Q that satisfy Equation (2) and satisfy appropriate tests is far from trivial.
Note that if we have somehow determined the appropriate functional form for P, then Equation (2) can be used to define the quantity index Q that will be consistent with it. Thus we can concentrate on finding a functional form for P that satisfies an appropriate set of tests. This is the traditional test approach to index number theory in a nutshell.
The alternative branch of index number theory that we wish to study is the one that uses a difference counterpart to the ratio Equation (2). Thus we look for two functions of 4N variables, [DELTA]p([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) and [DELTA]Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]), which sum to the value difference between the two periods; in other words, we want [DELTA]P and [DELTA]Q to satisfy the following equation:
(3) [p.sup.1] x [q.sup.1] - [p.sup.0] x [q.sup.0] = [DELTA]P([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) + [DELTA]Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1])
and the functions [DELTA]P and [DELTA]Q are to satisfy certain properties or tests that will allow us to identify [DELTA]P as a measure of aggregate price change and [DELTA]Q as a measure of aggregate quantity or volume change between the two periods.
As was the case with traditional index number theory where P and Q cannot be determined independently if Equation (2) is to hold, then if Equation (3) is to hold, [DELTA]P and [DELTA]Q cannot be determined independently. Thus in what follows, we will postulate axioms or tests for [DELTA]P and once [DELTA]P has been determined, we will define [DELTA]Q using Equation (3).
As the notation [DELTA]P and [DELTA]Q is somewhat awkward, we will use the notation I([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) (for indicator of price change) to replace [DELTA]P([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) and we will use V([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]) (for indicator of volume change) to replace [DELTA]Q([p.sup.0], [p.sup.1], [q.sup.0], [q.sup.1]). Using our new notation, Equation (3) can be rewritten as follows:
