as for example Pf, Pτ or Pw (and the corresponding quantity indices
Qf, Qτ and Qw of Fisher, Tdrnqvist and Walsh).
2. Hence to be superlative is a rare and most honourable distinction which
most of the hitherto familiar and popular indices like Laspeyres Pl,
Paasche Pp and so do not deserve.2
3. To be “superlative” means that it is distinctly shown that the index
in question is (approximately) equal to the COLI and thus valid for
any utility function whatsoever3 (and any consumer if he behaves ra-
tionally) which dispenses us once and for all from studying behaviour
consumer empirically .
4. It is in particular no longer relevant to study how the average household
responds to changes of relative prices by substituting away from goods
that became dearer. The so called “substitution bias” is correctly mea-
sured as difference between a non-superlative index and a superlative
index, for example Pl — Pf.
5. Although the underlying COLI-Theory had been developed for the case
of one single household only and the compilation of a price index in
one stage only4 the more realistic many-households and multiple-stage
compilation case does not require fundamental changes of the SIA.
In what follows we attempt to show that this interpretation of DiewerFs
SIA is false in all five points and that the approach requires quite restrictive
assumptions which are not likely to be met in real households’ behaviour.
Lacking realism of the theory’s assumptions matters and ensues also that
2In a similar vein it is frequently argued that it is an advantage as such - and therefore
justification - of an index design (like for example the method of chain indices) over other
indices only because it comes closer to a superlative index than the other indices do.
3The notion of a “flexible” function is widely understood as approximating closely
enough an “arbitrary” function (or simply “any” function one might think of) so that
there is no point in dealing any more with specific functions.
4[15]Diewert (2001) is one of the rare occasions where Diewert explicitly discussed
problems (for a COLI) involved in the fact that statistical agencies compile (aggregate)
price indices in two or more stages in practice (using various “component subindices” as
sub-indices on various levels of aggregation). He showed that indices like P f and Pτ satisfy
consistency in aggregation only approximately. He did not, however, refer to problems of
separability of utility functions that is to the microeconomic foundation of the subindices
as opposed to the overall index.