Economic growth and economic development 235

Introduction to Modern Economic Growth
the maximization behavior of a single household. This theorem therefore raises a
severe warning against the use of the representative household assumption.
Nevertheless, this result is partly an outcome of very strong income effects.
Special but approximately realistic preference functions, as well as restrictions on the
distribution of income across individuals, enable us to rule out arbitrary aggregate
excess demand functions. To show that the representative household assumption is
not as hopeless as Theorem 5.1 suggests, we will now show a special and relevant case
in which aggregation of individual preferences is possible and enables the modeling
of the economy as if the demand side was generated by a representative household.
To prepare for this theorem, consider an economy with a finite number N of commodities and recall that an indirect utility function for household i, v i (p, y i ), specifies the household’s (ordinal) utility as a function of the price vector p = (p1 , ..., pN )
and the household’s income y i . Naturally, any indirect utility function vi (p, y i ) has
to be homogeneous of degree 0 in p and y.
Theorem 5.2. (Gorman’s Aggregation Theorem) Consider an economy
with a finite number N < ∞ of commodities and a set H of households. Suppose
that the preferences of household i ∈ H can be represented by an indirect utility
function of the form

¡
¢
v i p, y i = ai (p) + b (p) y i ,

(5.3)

then these preferences can be aggregated and represented by those of a representative
household, with indirect utility
v (p, y) =
where y ≡

R

i∈H

Z

ai (p) di + b (p) y,

i∈H

i

y di is aggregate income.

Proof. See Exercise 5.3.

Ô

This theorem implies that when preferences admit this special quasi-linear form,
we can represent aggregate behavior as if it resulted from the maximization of a
single household. This class of preferences are referred to as Gorman preferences
after Terrence Gorman, who was among the first economists studying issues of aggregation and proposed the special class of preferences used in Theorem 5.2. The
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