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

Ad 1: It is useful to consider homothetic²¹ or linear homogeneous utility
functions as an interesting and convenient sub-set of utility functions. A
function g(xγ, ...,x_n) is called homogeneous of degree r if

g(λχ-ι,..., λχ_n) = λ^rg(x₁, ...,x_n).                        (6)

The analytic merit of linear (r = 1) homogeneity is that

c (p^, u⁾ = uc(p⁾ = ^{f (}q^)c(p^                      ⁽⁷⁾

where c(p) =c (p, 1) is the “unit cost function” (minimum cost to ac-
quire a utility level of u = 1)²² which provides an enormous simplification.
Homotheticity implies that the COLI is independent of the utility level and
therefore the same for all income classes. Moreover all goods have unitary in-
come elasticities, Engel curves (consumed quantities q as function of income)
are straight lines through the origin²³ and expenditure shares w_i (i = 1,.., N)
are unaffected by changes in income (they are - as desired in the COLI the-
ory - solely reflective of changes in relative prices). However, there is “an
overwhelming amount of empirical evidence contradicting homotheticity of
preferences” ([4]Barnett (1983), p. 218), so that the assumption of homo-
theticity is generally regarded as highly unrealistic. On the other hand the
assumption is extremely convenient. Just as the value index Vot is decom-
posed in a price and quantity index, P_ot and Q_ot respectively

^vot = ^^t^ = ^po∕.^qo∕.,                             ⁽⁸⁾

pOqo

called “product test”, or weak factor reversal test²⁴, and in a similar
manner Vo_t is usually factored in the economic index as follows

²¹It is common to make a distinction between “homothetic” and “non-homothetic”
utility functions. A utility function f is defined (by Shephard) as homothetic if it can
be written as a monotonic transformation of a linearly homogeneous function. However
for all practical purposes it is tolerable to use “homothetic” and “linear homogeneous” as
synonyms (like [16]Diewert did, p.6, footnote 8; this seems to be justified, see also [21]Kats
(1970)).

²²Diewert has shown that c(p) and /(q) satisfy the same regularity conditions.

²³In this respect “quasi-homothetic” preferences are more general in the fact that the
Engel curves “need not be forced through the origin” ([4]Barnett (1983), p.217).

²⁴The (strict) factor reversal test requires the same functional form for P and Q (in-
terchanging prices p and quantities q in the price index P results in the quantity index Q
and vice versa).

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