Economic growth and economic development 236

Introduction to Modern Economic Growth
quasi-linear structure of these preferences limits the extent of income effects and
enables the aggregation of individual behavior. Notice that instead of the summation, this theorem used the integral over the set H to allow for the possibility that

the set of households may be a continuum. The integral should be thought of as
R
the “Lebesgue integral,” so that when H is a finite or countable set, i∈H y i di is
P
indeed equivalent to the summation i∈H y i . Note also that this theorem is stated

for an economy with a finite number of commodities. This is only for simplicity,
and the same result can be generalized to an economy with an infinite or even a
continuum of commodities. However, for most of this chapter, we restrict attention
to economies with either a finite or a countable number of commodities to simplify
notation and avoid technical details.
Note also that for preferences to be represented by an indirect utility function
of the Gorman form does not necessarily mean that this utility function will give
exactly the indirect utility in (5.3). Since in basic consumer theory a monotonic
transformation of the utility function has no effect on behavior (but affects the indirect utility function), all we require is that there exists a monotonic transformation
of the indirect utility function that takes the form given in (5.3).
Another attractive feature of Gorman preferences for our purposes is that they
contain some commonly-used preferences in macroeconomics. To illustrate this, let
us start with the following example:
Example 5.1. (Constant Elasticity of Substitution Preferences) A very common class of preferences used in industrial organization and macroeconomics are the
constant elasticity of substitution (CES) preferences, also referred to as Dixit-Stiglitz
preferences after the two economists who first used these preferences. Suppose that
each household denoted by i ∈ H has total income y i and preferences defined over

j = 1, ..., N goods given by
(5.4)

σ
# σ−1
"N
X¡
σ−1
¡
¢
¢
,
xij − ξ ij σ
U i xi1 , ..., xiN =

j=1

where σ ∈ (0, ∞) and ξ ij ∈ [−¯ξ, ¯ξ] is a household specific term, which parameterizes
whether the particular good is a necessity for the household. For example, ξ ij > 0

may mean that household i needs to consume a certain amount of good j to survive.
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