the theory’s relevance is much more limited than what is usually believed.
This is even more a point as the COLI claims precedence over all other
index approaches and index formulas just because it possesses a theoretical
“foundation” while other indices do not.5
A number of steps is needed in order to derive the result that a certain
index function is superlative. In each step some assumptions are implied.
We discuss assumptions in the sequence of these steps, beginning with some
most fundamental assumptions of the COLI theory and proceeding in the
way “superlativity” of distinct index function is proved.
To begin with (sec.2.2) it is assumed that households decide on their
purchases by maximising their utility. For this it is necessary that the vectors
q of quantities of goods and the as great as possible “amount” of “utility”
u are related to one another somehow. This leads to the notion of a utility
function и = f (q). As q is related to и it is maintained that quantities are
endogenous or “explained” in COLI framework and no longer assumed as
given or “exogenous” as they are in other index theories. However, a utility
function in turn requires some quite restrictive assumptions. We are going
to show how these general requirements of a utility function may be inferred
from “rational” consumer behaviour.
In doing so we are dealing so far only with the COLI in general. In order
to move in a further step to the more specific SIA topic we have to consider
specific functional forms for the utility function and related functions, like
the “costs function”. This will be done in sec.2.3 where we introduce two
specific functional forms used by Diewert when he derived some families of
superlative indices. There we also demonstrate how a functional form is
related to its corresponding price and quantity index and what is implied
when the form is called “flexible”.
2.2 Preference formation and utility maximising (COLI
assumptions)
It is generally accepted that the SIA needs to presuppose that households are
engaged in utility maximisation. The COLI theory as basis of the SIA starts
with a single household. Before a utility function exists that can reasonably
5For [27]Triplett (2001) it is clear that the microeconomic theory underlying the COLI
is the only theory we have, and all other indices have no underlying conceptual framework.