Hierarchic Preferences.pdf - Hierarchic Preferences 2.1...

Hierarchic Preferences
2.1 Basic Set-up
In this chapter we propose a general formulation of non-homothetic hierarchic
preferences. The purpose is to develop functional forms which are still tractable
for macro models but which are general enough to match the empirical facts
about the income dependent structure of demand. To start from general notion of
preferences has the advantage that it allows us to identify the critical assumptions
on the curvature of the utility function needed to match the empirical facts. In the
first part of the chapter we focus on a static set-up which allows us to remain fairly
general in our choice of the utility function. The second part of the chapter shows
which additional assumptions on the utility function and on the price structure are
needed to deal with dynamic problems. When the hierarchy function is a power
function and the supply side satisfies certain symmetry conditions, we obtain a
result wherein the indirect utility function takes a constant relative risk aversion
(CRRA) form which is compatible with steady states. This result is very helpful.
Although the composition of the consumption basket changes over time we can
find a consumption aggregator that has the same properties (i.e. CRRA) as the
composite commodity in standard monopolistic competition models. We will make
use of this result in the following chapters 3 and 4. At the end of this chapter, we
compare our hierarchic utility function to different functional forms suggested in
the previous literature.

10 2. Hierarchic Preferences
Consider an economy with an infinite number of potentially producible goods
ranked by an index j . A certain need j can be satisfied by consuming the cor-
responding good j . Put in other terms, a good represents a "technology" which
satisfies a given need. A meaningful specification of hierarchic preferences then
has to take account of three facts.
(i) Needs are ordered.
(ii) Some goods may not be consumed, i.e. some needs remain unsatisfied,
because the consumers cannot afford it. Technically speaking, marginal utility at
zero must be finite, at least for goods of lower priority.
(iii) If a consumer has additional income, he should spend it primarily on goods
that have lower priority because the needs of higher priority are already saturated
(at least in relative terms).
Therefore, we study the structure of consumption that is generated by prefer-
ences of the form^
/•OO
"(Mi)})= / aJHc{j))dj (2.1)
Jo
where v{c{j)) is an indicator for the utility derived from consuming good j in
quantity c. The 'baseline' utility v{c{j)) satisfies the usual assumptions v' > 0
and v'^ < 0; and the 'hierarchy^ function ^(j) is monotonically decreasing in j ,
C\j) < 0, hence low-j goods get a higher weight than high-j goods. It is important
to note that we make three important assumptions or restrictions, respectively,
at this stage. First, the marginal utility of good j only depends on c{j) but does
not depend on the consumption level of other goods. Thus, utility is assumed
to be additively separable.