increasing (in elements of q), quasi-concave9 (to establish an unequivocal
maximum), and twice continuously differentiable function over all parts of
its domain. These assumptions are known as “regularity conditions” and
they are reconcilable only with a limited number of functional forms so that
not all imaginable functions are eligible for representing f (q).
For the COLI theory it suffices to postulate such quite general proper-
ties of a utility function f (q), however, for the SIA it is necessary to be
more specific regarding the functional form of f (q).10 The problem with
utility maximising is that we cannot be sure whether or not households in
fact perform this optimisation exercise in their decision making.11 The SIA
rationale, however, decisively rests on the assumption that observed expen-
ditures are resulting from utility maximisation. More specific it is assumed
that the scalar product p(qs = ɪbPisQis (^ = 1,. . . , ^ and s = 0, t) is equal
to the value of the cost function12 c (ps, us) in both periods s = 0 and s = t.
Interestingly from the mere definition of c as minimum costs (expenditure)
follows that ptqo ≥ c (pt,uo) and p(1qi ≥ c (po, ut) and therefore
pL = p⅛qo ≥ c(ps,uo) = c(pt,uo)
ot p0qo_ p0qo c (po,uo)
(4)
(5)
and
pp = p⅛ < p⅛q⅛ = c(p⅛1ut)
ot poq⅛ ~ c (po,ui) c (po,ui).
In each equation the rightmost term is a COLI referring to a utility level of
uo and ut respectively.13 It is common practice to refer not only to the same
9This is a curvature condition for f (q). A concave utility function means somewhat
simplified that average quantities of all goods are preferred to extreme or one-sided con-
sumption (0% or 100% of the budget devoted to one specific good).
10We consider these additional assumptions in sec. 2.3.
11It is indeed an advantage the econometric demand-system-approach (DSA) enjoys over
the SIA that empirical results (like ours) indicating a poor goodness of fit of the respective
models to data may be taken as a hint that household possibly are not acting as utility
maximiser so that the theoretic assumptions may not be realistic.
12That is the minimum costs (expenditure) of achieving a utility level us under a pre-
vailing price vector ps (or to put it differently we chose that very vector q of quantities for
which the expenditure q,ps under the price regime of period s is a minimum). A unique
value c of c (ps, us) requires that the above mentioned conditions are met concerning the
utility function us = f (qs) from which the function c(ps,us) is derived.
13This is in the literature sometimes called Laspeyres-KonUs and Paasche-KonUs (price)