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

(38) into account, the number of parameters to estimate of the AIDS budget
share equation (34) can be determined using equation N (N — 1) ∕2 + 2N — 2.

[24]Lewbel (1991) extends [19]Gorman^,s (1981) famous Engel curve rank
definition and defines the rank of a demand system to be the maximum
dimension of the function space spanned by the engel curves of the demand
system. Following this definition, [23]Lewbel (1987) shows that the AIDS is
a rank two demand system which implies that it has linear Engel curves (not
necessarily through the origin).

3.2 Quadratic Almost Ideal Demand System (QUAIDS)

Several studies, like for example Lewbel (1991), Blundell et al. (1993) and
Banks et al. (1997), found - at least for some goods - empirical evidence for
non-linear Engel curves. So Banks et al. (1997) extended the AIDS model to
allow budget shares being quadratic in expenditure. The underlying indirect
utility function of this rank three Quadratic Almost Ideal Demand System
(QUAIDS) in its general form

/ -n    f^ⁿ M — In a(p)∖ ¹

"'^{m )}=[(     b(p)     )   ^{+ a} '

is equivalent to a general indirect utility function of a PIGLOG demand
system supplemented by the homogeneous of degree 0 in p function A (p).
Using the already known form 25 from the AIDS for In a(p~) together with
5(p) = ∏ p^βl and A (p) = Y A_i Inp_i we get from equation 39 to the specific
indirect utility function of the QUAIDS

(40)

and to the cost function

ln c (p,u) = «0 + ? α⅛ ln p⅛ + 2 ? ? 7 ⅛j ln p⅛ ln p_j +

25

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