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

N      ɪ NN            N

In C (p,u) = α₀ + J2"i^ln Pi ⁺ Ô Σ ∑7 *j^{ln p}i^ln Pj ⁺ "’⁰ Π∑^{l ,   (27)}

ù

which is homogeneous of degree one in prices p if the following parameter
restrictions hold:

By multiplying both sides of Shephard’s Lemma ∂c (p,u) ∕∂p_i = q_i with
Pi∕c (p,u) we get

so that we can derive a budget share equation from (31):

N             N

^wi = ai ⁺    7ij^lnPj ⁺ βi^uβо ∏ Pi¹ ∙               ⁽³²⁾

j=l                    i=l

As for a utilitiy maximizing household C (p,u) = M, we can derive the
utiltiy level u via the indirect utility function:

^l ш   ^{ln c (}p^{,u) — ln a (}p⁾

^{v (}p^,M) = 1 м ∖—i---^TΓ = ^u∙

ln о (p) — ln a (p)

Putting the utiltiy level u from equation (33) in (32), we get the common
AIDS budget share equation:

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