Problems of operationalizing the concept of a cost-of-living index

wW   Σ2 ^pit^^QiO^Qit , ~ w   r==l

^POt =        =     = ^POt = ^POt ∙

^pLoPiO у/ QiOQit

So strictly speaking it is the indirect, rather than the direct Walsh index
which is superlative (a quadratic mean of order r = 1 index). In a similar
vein the quadratic mean of order r = 1 superlative quantity index is given
by

so that forr = 1 we get QW = VO_t∕PoW = Qo=¹ saying that the indirect
Walsh quantity index is again a superlative index (now a quantity index).
For pairs of indices satisfying the product test there is no difference between
a direct and an indirect index. Also V_ot = Q∏_tP° = Pot Qo_t so that all four
indices are superlative (be it directly or indirectly).

Ad. 3: For a price (quantity) index P (Q) to be superlative requires
not only that P is exact for c(p) and Q for f (q) but also that f (q), and
c(p) respectively is a flexible functional form. What is meant by “flexible”.
According to Diewert a twice continuously differentiable function f (q) is
flexible if it provides a second order approximation to another function f * (q)
around the point q*, meaning that “the level and all of first and second order
partial derivatives of the two functions coincide at q*.”³⁰ More specific one
may write (in analogy to [5]Barnett and Choi (2008), p.4)³¹:

f (q*) = f * (q), ( lq=q* = ) l_q=_q. and J^fL.1    = AC ∣      (23)

Oq ^{q q}*    оq ^{q q}*     OqOq' ^{q q}*   оqθq' ^{q q}*

Instead of “another” function f * we also find the notion of an “arbitrary

³⁰ [16]Diewert (2008), p. 52. [17]Diewert (2009), p. 13, also stresses the requirement
that q (and q*) need to belong to the “regularity region” (of the utility function). What
is said about f (q) and q* of course also applies mutatis mutandis to the function c(p)
around the point p*. See also [3]Barnett (1983b)for different definitions of “second order
approximation” .

³¹For a graphical interpretation see also [29]von der Lippe (2007), p. 109. It is beyond
the scope of our paper to demonstrate exemplary what has to be done in order to show
that a specific functional form in fact is “flexible” in the definition of Diewert.

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