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

_v = p⅛q⅛ = c(pt,ut)

°^t    Poqo    c(po,∙⅝) ’

assuming utility maximisation in both periods and using the homothetic-
ity assumption

^ut^{c (}Pt⁾ _ ^{f (}qt)   c⁽pt)

-------:----- —  :----- ^, —:----- .
UoC (Po)    f (qo)  c(po)

where the term (the ratio of utility levels) is said to be the “economic
quantity index”, while the second factor is called “economic price index” or
simply COLI.

In his theory of superlative indices Diewert studied in particular only two
functional forms for f (q) and (derived therefrom) the cost function (we quote
the latter only), viz.

1. the quadratic mean of order r utility and cost function

the coefficient b_ij of B = B/ in the quadratic form

p/Bp have to meet certain restrictions, and

2. the normalised quadratic cost function ([HJDiewert (2009), pp. 18 -
23)

ɪ p/Ap

^cNQ ⁽p⁾ = p^/b + ²---

ɑ/p

where the scalar а/p performs a sort of normalisation and again coeffi-
cients a and b are appropriately restricted. Both models are fairly general.
The first nests the

• “homogeneous quadratic” function (r = 2) for which Fisher’s ideal
index is exact,

• the (homogeneous) translog function (r → 0)

In ^co ⁽p⁾ = ^βo ⁺ ∑ ^β_i In p_i ⁺ 2 ΣΣβi_j In pi In p_j       (13)

i                         i j

for which the Tbrnqvist index is exact,

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