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

The corresponding quantity indices are a) harmonic mean Q^_t (corre-
sponding to P_f^t)²⁸ and b) arithmetic mean Qθ_t.

Note that the vector a can also be specialised in a great number of other
ways. We tried for example a = (q_t + q_o)∕2 and got another superlative
price index for which, however, we could not find an obvious (or economically
luminous) interpretation, e.g. in terms of well known index formulas, like P^l,
P^p or so.

It is also remarkable that an index function can be exact for different
functional forms for the cost (or utility) function (and there possibly only
for certain restrictions concerning the parameters) and conversely the same
cost (or utility) function fit to different superlative index formulas. This
may be seen as a challenge for an economic interpretation of the meaning of
“superlative”.

Diewert defined another infinitely large class of superlative indices by
making use of the product test, defined in (8), showing that if P_ot is superla-
tive this applies also to the corresponding “indirect” quantity index. Likewise
if Q_ot is superlative the corresponding indirect price index P_ot = V_ot/Q_ot is
also superlative.²⁹ Direct and indirect indices coincide whenever the index
function is factor reversible which is the case with Fisher’s index.

In this way (via considering indirect indices) also the Walsh index formula
received the honour of generating superlative indices. The quadratic mean
of order r = 1 superlative price index

Por¹ = [∑w (Q)¹'T[∑ - fe f '    <2θ)

_i        ∖JPιO /    J L _i        ∖l^jιt∕

_ ʊ ∕Σ Qit^PioPit _ τr ∕∩W

= ^vot∕v^-----.       = ^Vot/^Qot >

QioQiP y∕PioPit

turns out to be the indirect price index of Walsh /(J^ = P^^r this index
is also superlative. Note that the indirect Walsh price index differs from the
direct Walsh price index

²⁸This index is said to originate from Drobisch in Germany, however Diewert refers to
Bowley and Sidgwick.

²⁹Interestingly Diewert did not make use of such considerations in terms of indirect
indices in a consistent manner. One may for example wonder why the indirect quantity
index of Tornqvist V/P^τ is not “superlative”.

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