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

By applying Shephard’s Lemma on (41), we receive the QUAIDS budget
share equations

w_i = a, + ∑ 7,j In p_j + B, In (P) +

where we are using P of the form (35) instead of α(p) to ease the com-
parison between AIDS and QUAIDS. To be consistent with the neoclassical
assumptions of consumer theory, the QUAIDS has to fulfill the following
parameter restrictions:

∑o = 1. ∣>, =o,y> = 0. £7_ti = 0¾.      (43)

∑ 7ij = 0 Vi.                        (44)

j

⁷ ij = ⁷ji'                                   ⁽⁴⁵⁾

The restrictions of (43) ensures adding-up, the restriction (44) is required
for homogeneity of degree 0 in p and M of equation (42) and restriction (45) is
a necessary condition for a symmetric Slutsky matrix. Taking the parameter
restricitons (43), (44) and (45) into account, the number of parameters to
estimate of the QUAIDS budget share equation (42) can be determined using
equation N (N — 1) /2 + 3N — 3 (number of parameters of the AIDS plus the
N — 1 additional Λ_i parameters of the QUAIDS). Compared to the AIDS,
the expenditure and price elasticities of the QUAIDS can vary with the level
of total expenditure M, indicating the higher Aexibilitxy of the QUAIDS. In
the QUAIDS it is for example possible, that the same good is identified as
luxury good for a household with low total expenditure and as a necessity
good. A drawback of using the QUAIDS is the danger of obtaining fitted
budget shares for high total expenditure values that fall outside the [0.1]
domain of definition.

By having a look on the budget share equations (34) and (42) of the
AIDS and QUAIDS, it is obvious, that the AIDS is nested in the QUAIDS
(AIDS is the special case of the QUAIDS if Λ_i = 0 for all i). This is a
big advantage because we can easily compare the goodness of fit of the two
models concerning our observed household data described in the next section.

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