Economic growth and economic development 238

Introduction to Modern Economic Growth
In other words, if the indirect utility functions of some households do not take
the Gorman form, there will exist some distribution of income such that aggregate
behavior cannot be represented as if it resulted from the maximization problem of
a single representative household.
In addition to the aggregation result in Theorem 5.2, Gorman preferences also
imply the existence of a normative representative household. Recall that an allocation is Pareto optimal if no household can be made strictly better off without some
other household being made worse off (see Definition 5.2 below). We then have:
Theorem 5.3. (Normative Representative Household) Consider an economy with a finite number N < ∞ of commodities and a set H of households.

Suppose that the preferences of each household i ∈ H take the Gorman form,
vi (p, y i ) = ai (p) + b (p) y i .

(1) Then any allocation that maximizes the utility of the representative houseP
P
hold, v (p, y) = i∈H ai (p) + b (p) y, with y ≡ i∈H y i , is Pareto optimal.

(2) Moreover, if ai (p) = ai for all p and all i ∈ H, then any Pareto optimal
allocation maximizes the utility of the representative household.

Proof. We will prove this result for an exchange economy. Suppose that the
economy has a total endowment vector of ω = (ω 1 , ..., ω N ). Then we can represent
a Pareto optimal allocation as:
X
¡
¢ X i¡ i
¢
max
αi v i p, y i =
α a (p) + b (p) y i
i

{pj }N
j=1 ,{y }i∈H

subject to

−

i∈H

Ã
X ∂ai (p)
i∈H

∂pj

i∈H

∂b (p)
+
y
∂pj
N
X

!

= b (p) ωj for j = 1, ..., N

pj ωj = y,

j=1

where {αi }i∈H

pj ≥ 0 for all j,
P
are nonnegative Pareto weights with i∈H αi = 1. The first set of

constraints use Roy’s identity to express the total demand for good j and set it equal
to the supply of good j, which is the endowment ωj . The second equation makes
224