meet the quite restrictive “regularity conditions” at least in relevant parts of
its domain. It therefore cannot be any arbitrary function.
Before embarking on a discussion of suitable functional forms for f (q) it
may be interesting to note that [5]Barnett and Choi (2008) called in question
Diewerfs way to concentrate only on functions that exist in an algebraically
closed form. They contend that there are many functions that can “only
be tabulated”18 and evaluated by “using partial sums of series of (Taylor)
expansions” his method “spans only a strict subset of the general class” (of
superlative indices) in question.19 They also showed that all indices of the
class of log-change indices20 (of which Pτ is a member) are “superlative”.
Note that even if we confine ourselves to those functions for which a closed
and parameterised algebraic form exists we may find an infinite number of
superlative index formulas of which only a small subset is known and capable
of a plausible interpretation.
To develop a theory of superlative indices and make use of the SIA four
steps are needed:
1. A functional form has to be found which complies with all assumptions
needed to reflect rational consumer behaviour, and
2. the price and quantity index corresponding to (“exact” for) this func-
tional form have to be derived, and
3. it has to be demonstrated that the functional form in question is “flex-
ible” in the definition of Diewert, and finally
4. as there is an abundance of superlative indices it is desirable to have
some guidance in making a choice among these formulas.
18Examples they referred to are trigonometric or hyperbolic functions.
19[6]Barnett et al. (2003) therefore also wondered why only a small number of index
numbers (of a potentially infinite number) have so far been found and maintained that
the search process of an index number which is exact for a given functional form or - the
other way round - for a functional form which is exact for a given index function is not
yet formalised (“No simple procedure has been found for either direction. For example
the miniflex Laurent aggregator function, originated by [7]Barnett and Lee (1985) ..., is
known to be second order, but no one has succeeded in finding the index number, that
can track it exactly” ).
20See [29]von der Lippe (2007), pp. 226 - 254. These indices are also called (e.g. in
[5]Barnett and Choi (2008)) “Theil Sato Indices”.
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