6 Conclusion
The result of the empirical part was that the demand systems estimated had
a poor goodness of fit (table 3) which may possibly indicate the consump-
tion behaviour of households is not well explained by the usual assump-
tions of utility maximisation in microeconomics as they are materialised in
the demand systems. In particular the demand-system and the superlative-
index approach obviously don’t fit together satisfactorily. The Fisher- and
Tbrnqvist- price index index were consistently considerably lower than the
COLI based on a demand system for the same data. We also saw that many
tests for the assumptions of the respective models failed and many results
were such that they did not really make sense. It may be suggested that
these unsatisfactory findings may perhaps be attributed to the specific Ger-
man data. Even if this were the case it is still true, that estimating a demand
system of more than only some few goods (N = 9 in our case) entails so many
insurmountable difficulties that it would by no means be a reasonable option
for official statistics to compile a CPI based on estimated demand systems.
Thus a monthly COLI-type official CPI based on estimated demand systems
comprising a tolerable variety of goods will most probably be impossible.
Even worse, there may be indications that the assumptions underlying the
often praised (alleged) microeconomic foundation of the COLI are unrealistic.
This can be inferred from the unsatisfactory fit of our (demand) regression
equations. In the first place we may conjecture that in reality households
can hardly respond so promptly and rapidly to price signals by substituting
as assumed in theory. Such theory related arguments should be kept in
mind when we consider the superlative-index approach next, because for this
approach to be valid it has to rely on the same assumptions (concerning
consumer behaviour) and to some more in addition (concerning the notion
of “approximating” an “arbitrary” function).
As already mentioned in this situation it appears tempting to avoid all
those econometric difficulties with demand systems by simply calculating a
superlative index combining observable data vectors pt, p0, q0 and qt only
(where perhaps only the timely availability of qt may pose a problem in
practice). This, however, is not that easy. The proof of “superlative-ness”
of an index function requires the restrictive assumptions discussed in section
2 which are unlikely to hold empirically, and together with the assumptions
needed to relate the index to a utility maximum this makes the index no less
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