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

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(q_s = ɪbPisQis (^ = 1,. . . , ^ and s = 0, t) is equal
to the value of the cost function¹² c (p_s, u_s) in both periods s = 0 and s = t.
Interestingly from the mere definition of c as minimum costs (expenditure)
follows that ptq_o ≥ c (p_t,u_o) and p(₁qi ≥ c (p_o, u_t) and therefore

_pL = p⅛qo ≥ ^c(ps^,uo⁾ = ^c(pt^,uo^)
ot p0qo^_ p0qo      ^{c (}po^,uo⁾

and

_pp = p⅛ < p⅛q⅛ = c(p⅛₁^ut)
^ot poq⅛ ~ c (po,u_i)    c (po,u_i).

In each equation the rightmost term is a COLI referring to a utility level of
u_o and u_t 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 u_s 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^,p_s under the price regime of period s is a minimum). A unique
value c of c (p_s, u_s) requires that the above mentioned conditions are met concerning the
utility function u_s = f (q_s) from which the function c(p_s,u_s) is derived.

¹³This is in the literature sometimes called Laspeyres-KonUs and Paasche-KonUs (price)

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