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^pi^ln Pj ⁺ "’⁰ Π∑^l^,⁽²⁷⁾

i=l i=l j=l

i=l

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

∂ ln c (p,u)
ln Pi

Piqi
^c⁽p^,u⁾

(31)

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⁾

(33)

^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: