Date: Thu, 23 Sep 1999 12:48:40 -0300
Sender: "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From: "Hector E. Maletta" <hmaletta@OVERNET.COM.AR>
Subject: Re: coding challenge
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Theil's index belongs to the family of so-called log-change indexes.
Think of an ordinary Laspeyres index (such as most price indexes) as a
weighted average of price relatives:
L = sum wi (pit/pi0) where wi is the weight of item i in the
consumer budget. This shows Laspeyres indexes are arithmetic means.
There are also geometric indexes such as:
G = prod (pit/pi0)** wi where ** means "to the power of".
The logarithm of G is often used, thus defining the log-change index:
log G = sum wi log (pit/pi0) = sum wi (log pit - log pi0). A log
change index is given by the artihmetic average of log-changes. Its
anti-log is a geometric index. From the known fact that geometric means
are lower than arithmetic means (except in the trivial case of all items
changing at the same rate), it is G < L.
Theil's index of price variation is a log change index using weights
corresponding to the budget composition at the initial date, t=0.
Another such index is Tornqvist index, sometimes called Tornqvist-Theil
index, where the weights are the simple average of initial and final
budget shares, (wi+wt)/2. The matter is thoroughly discussed in books
about index numbers, beginning with the classical The Making of Index
Numbers by Irving Fisher (1922) where all these formulas are listed,
albeit without the thorough mathematical analysis developed later by
other authors. More recently, see W. Diewert and A.O.Nakamura (eds):
Essays in index number theory (North Holland, 1993); R.A.Pollak: The
theory of the cost of living index, OEP 1989; Fare, Grosskopf & Russell:
Index numbers - Essays in honour of Sten Malmquist, Kluwer Publishers
1997. Of course you may see also Theil's work, for instance his Theory
and measurement of consumer demand, 2 vols, North Holland 1975.
The use of logchange formulas for price and quantity indexes has been
encouraged by their economic implications. For instance, a price index
following Theil or Tornquist formula is able to reflect some degree of
consumer substitution between goods when prices of different goods vary
at different rates, unlike Laspeyres' index which assumes consumers
purchase a fixed physical basket of goods and services irrespective of
changes in their relative prices.
The variation measured by these indexes needs not be over time, of
course: log change indexes may be used to measure, for instance,
relative price-level differences across countries, or relative income
differences across people (the latter is the original problem that
started this thread).
Universidad del Salvador
Buenos Aires, Argentina