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

• the generalised linear utility function (r = 1) f (q) = а/p with the
corresponding general Leontief unit cost function c (p) = b/p and the
CES function with suitable parameter restrictions (a_ij = b_ij = O V
i = j) so that c (p) = a₀ (∑_i a_ip^r_i )^1/r.

A non-homothetic variant of (13) is given by ([17]Diewert (2009), p. 24)

ʃo + Σ βi ^ln Pi + j ΣΣ ^ij ^ln Pi ^ln P^       ⁽¹⁴⁾

+70^{ln (u}) ⁺ ∑ 7i^lnPi^ln («) ⁺ 17oo ^(ln Ы⁾² ∙
i

Obviously with vanishing 7 coefficients (14) specialises to the homothetic
cost function (13). There is not much in the ways of economic theory and
interpretation that can be said about why one functional form should be
preferred over another one.²⁵

Ad. 2: With a given functional form the price and quantity index has to
be found which is exact for it. For example [17]Diewert (2009) has shown
that the function

where p = [pι∙..p,v] coincides with Fisher’s ideal price index FJ_t=² = P]^_t.
Hence F^f is exact for the “homogeneous quadratic” functional form. Corre-
spondingly the Fisher’s quantity index Q^f is equal to

_r=_r=2 _ ^ut _ ^{f (}qt⁾

^Qot =    = fl ∖'

uo    f (qo)

In a similar manner Diewert has shown that the Tbrnqvist price index is
exact for the translog cost function (13).²⁶ The general form of a price index
exact for quadratic mean of order r functional form is given by

²⁵ [28]Turvey (1999), an ingrained opponent of the COLI approach said “Writers on this
[i.e. COLI] theory express no views of which functional form is most realistic”. More
about purely formal criteria in making a choice among functional forms can be found in
[22]Lau (1986).

²⁶He did not consider the Tornqvist quantity index. Note that the Tornqvist index does
not satisfy factor reversibility. The index function even fails the weaker product test.

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