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

from which it is derived) is not observable. Given that we can’t know how
the “amount” of “utility” is related to the quantity of goods consumed we
have to find ways to make the notion of a “constant-utility-index” or COLI
nonetheless “operational” or “measurable”. There are in principle three ways
proposed in order to accomplish this task

1. historically the first approach was to define upper and lower bounds
for a COLI (it is for example well known that under fairly general
conditions POf < P⅛ < POf holds),

Of       Of        Of

2. then some attempts were undertaken in order to estimate (economet-
rically) “demand systems”, that is systems of N demand equations
for N goods from which the theoretical cost-functions as numerator
and denominator of the COLI can be derived (and also the shape of
the Engel-curves and the estimates of some parameters such as various
“elasticities”. Those may be interesting regarding the economic inter-
pretation of the empirical results). However, as this approach turns out
to be extraordinarily difficult to carry out in practice, it became more
and more popular, to

3. make use of the theory of “superlative indices” developed by [12][13]W.
Erwin Diewert (1976, 1978) according to which certain observable price
indices¹ - each “using the quantities in the base period as well as in
the current reference period as weights in a symmetric fashion” - are
capable of approximating a COLI derived from a fairly general (or
“flexible”) demand function.

The first approach certainly is less promising from a practical point of
view of compiling an official CPI on a monthly basis, because it can at best
provide intervals only rather than an exact numerical value. The second and
in particular the third approach may appear more pertinent and successful.

The focus of our paper therefore is on the second and third approach.
For this purpose we have undertaken both, a theoretical analysis of the as-
sumptions explicitly (or implicitly) underlying Diewert’s theory of superlative
indices, and an empirical study of demand systems (using data taken from

¹Such as Pq) = ʌ/P(⅜Pq) (Fisher price index) or Po) = ∏()^Si (Tornqvist price
index) where Si = l∕2(⅛io + зц) is the arithmetic mean of the weights of the Laspeyres
and Paasche price index formulas.

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