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

able than to be able to provide a second order approximation (rather than a
first order only as is that case with P^l).

Ad. 4: As there are quite a few equally “superlative” indices it seems
desirable to make an “informed” choice among these index formulas. In this
respect it is often believed that all superlative indices approximate each other
sufficiently close because Diewert has shown that “all the superlative index
formulae . . . approximate each other to the second order around any point
where the two price vectors . .. are equal and where the two quantity vectors
. .. are equal” ([14]Diewert (2000), p. 61). This result is generally understood
as justifying the view that it does not matter much which superlative index
is taken as they will display a similar order of magnitude for whichever data.
The problem, however, is that the “equal (or proportional) price and quantity
point” ([14]Diewert (2000), p.65) where p_t = Ap₀ and q_t = Aq₀ respectively³⁶
is relatively uninteresting. This is so because in that case all reasonable
indices will be identically A anyway. Moreover [20]Hill (2006) found out that
the approximation theorem referring to the quadratic mean of order r (which
is a flexible functional form for all values of r), does not apply to extreme
values of r. It works well in the range 0 ≤ r ≤ 2 but it can yield with a
skewed distribution of price relatives and ∣r∣ → ∞ differences between P^r
and P^rwhich may be much greater than the difference P^l — P^p. So Hill
concluded that “the economic approach does not by itself solve the index
number problem, since it does not tell us which superlative index should be
used”.

In summary we may conclude that the COLI-theory in general and the
SIA (built on this theoretical foundation) in particular need a number of
quite restrictive assumptions so that in “the perennial question of the realism
of the theory’s assumptions” ([27]Triplett (2001), p. 318) in our view the
COLI may well come out at the losing end compared to the COGI. There are
also other aspects we could not deal with here, that may cast doubt on the
general belief of COLI advocates, that the Laspeyres index falls far behind
the Fisher index for example. The superlative quantity index for example
plays a somewhat dubious role. When Qζ_t > 1 is superlative this means that
the standard of living (level of utility) increased because Q^_t approximates
u_t∕u₀ = f (q_t) /f (q₀). On the other hand P^_t measures the minimum cost of
maintaining a constant level of utility.³⁷ The relevance (representativity) of it

³⁶The “equal” point is of course the case λ = 1.

³⁷It is the word “minimum” which Triplett seems to have forgotten when he levies the

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