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, ...,xn) is called homogeneous of degree r if
g(λχ-ι,..., λχn) = λrg(x1, ...,xn). (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 wi (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, Pot and Qot respectively
vot = ^t^ = po∕.qo∕., (8)
pOqo
called “product test”, or weak factor reversal test24, and in a similar
manner Vot 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).
13
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