linear homogenous function” .32 It is important to note that the approxima-
tion is evaluated at the point q* of the two function in question, so that the
situation may well differ substantially at other points. Furthermore flexibil-
ity is not the only desirable property of a functional form. There are other
equally important criteria which may not be reconcilable with flexibility.33
Hence flexibility which makes an exact index superlative seems to be a
purely formal criterion for which a “luminous” economic interpretation is
lacking. Moreover the criterion should also be set against other more restric-
tive and unrealistic assumptions made in the course of proving the superla-
tivity of an index formula. One may for example argue, that homothetic
preferences are assumed in most of the SIA results (for example concerning
Fisher’s index Pf) such that the income level is irrelevant for the choice of a
vector qt as opposed to q0 which is undisputedly unrealistic.34 This implies
to focus exclusively on the substitution bias (of a non-superlative index as by
contrast to a superlative one) while in the case of non-homothetic preferences
the combined income and substitution bias may well offset the substitution
bias.35 So to be adequate for non-homothetic preferences may be more valu-
32 [16]Diewert (2008), p. 15. [22]Lau (1986), p. 1539) emphasised explicitly “the ability
... to approximate arbitrary but theoretically consistent behaviour through an appro-
priate choice of the parameters”. Aside from sounding a bit contradictory (theoretical
consistency or regularity is anything but “arbitrary11) this definition of “flexibility” un-
derscores the idea of having left a sufficient number of free parameters to reflect different
decisions of consumers and to generate for example different values for certain elasticities,
expenditure shares, rates of substitution etc. Accordingly he wrote “so that . .. their
own and cross price elasticities are capable of assuming arbitrary values .. . subject only
to the requirements of theoretical consistency” (p. 1540). Arguing that the normalised
quadratic function is flexible Diewert also pointed out that it has the “minimal number of
free parameters that is required ... to be flexible.” ([17]Diewert (2009), p. 20)
33[22]Lau (1986) lists five criteria. These are in addition to flexibility theoretical con-
sistency, a (wide) domain of applicability (whether consistency is globally, over the whole
domain, or only locally valid), computational facility and factual conformity (according to
which he rejects for example all those functional forms which generate linear Engel curves,
as this appears to him highly unrealistic).
34Constant returns to scale is the production theory counterpart to homotheticity in
consumption theory.
35This is in principle an argument of [18]Dumagan and Mount (1997). Their theoretical
and empirical work amounts in no small measure to a vindication of the Laspeyres formula.
They also criticise from the point of view of microeconomic theory the usage of both weights
qo as well as qt (or expenditure shares w,∙o as well as w,t ) in a symmetrical fashion which
interestingly - at least to our knowledge - takes place in all superlative index formulas.
There is no superlative index which refers to the weights of one of the two periods only.
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