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

W_i

Σⁿ , n Л¹¹^c ⅛P∙") - '^π^a (p)ʌ п π ,
_J=1 ^7ij^1П^P^j⁺^βⁱ к ln Ъ (р) - ln a (р) )^β⁰ П ^P

^ai + ∑ 7ij lnPj + Pi (ln ^М - ln a (p))
j=l

^ai ⁺ Σ 7ij ^ln Pj ⁺ Pi ^lnj=l

(34)

where P is a price aggregator

N ɪ NN

ln P = <>0 + Vv ln Pi +2∑ ∑7 ij ln Pi ln Pj (35)

i=l i i=l j=l

and where

7ij = ɪ (Pi_j ⁺ 7ji) ∙

(36)

The parameter restrictions (28) - (30) of the cost function (27) should in
general also hold for the budget share equation (34). Only the restrictions
(29) and (30) have to be changed slightly due to equation (36) to:

∑ 7ij = 0 ∀j,

(37)

i=l

and

7 i_j = 7ji∙ (³⁸)

As [11]Deaton and Muellbauer (1980) show, adding-up (ɪ) W_i = ɪ) re-
quires the parameter restrictions (28) und (37) to be satisfied, the restriction
(37) ensures the homogeneity of degree 0 in p und M of the budget share
equation (34) and restriction (38) is the requirement for a symmetric Slutsky
matrix. The AIDS can either be estimated by directly imposing the re-
strictions (28) - (30) or by testing the restrictions after having estimated the
model without restricitions. Taking the parameter restricitons (28), (37) and

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

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