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

if r = O and w_it denoting expenditure shares of commodities i = 1,. . . , N
in period t = O,t. This constitutes a family of infinitely many superlative
(price) indices for -∞ < r < +∞. Accordingly if r = O we get

^wiθ+^wit

^p0<=° = ∏ (^t)  ’  = ^p0.               (U)

The corresponding superlative quantity indices are gained by simply sub-
stituting q_it∕q_i0 for p_it∕p_i0 It is important to note that there are an infinite
number of superlative indices on the basis of the quadratic mean of order
r. This functional form, however, requires homothetic preferences. Using
the non-homothetic cost function (14) Diewert could demonstrate that the
Tbrnqvist index is exact for the utility level u* = ^u₀u_t. For other levels
U = u* the (no less) superlative price index c(p_t, U)∕c(p₀,U) will differ from
P^τ. So the SIA involves occasionally some additional assumptions. They
may be without a doubt plausible, however not at all cogent. In this case of
non-homotheticity²⁷ it is an assumption about the absolute level (amount)
of “utility” that had to be introduced. Another situation where such an
assumption is involved is [17]Diewert’s (2009) finding (pp. 21 -23) that

a) P« = | (P⅛⅛ + P<J)                     (18)

£

or

I¹

b) Pf = ɪ '    +1 (Pf Г               (19)

£            £

is exact for the normalised quadratic cost function (12) and therefore
again “superlative”. However, this applies only if we set

a) a = q_t and

b) a = q₀ respectively.

²⁷The result (of Diewert) concerning the index P^τ of Tornqvist is to our knowledge the
only proof of superlativity where the (notoriously unrealistic) assumption of homothetic
preferences is being relaxed.

16

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