Economic growth and economic development 234

Introduction to Modern Economic Growth
Often, we may not want to assume that the economy is indeed inhabited by a
set of identical households, but instead assume that the behavior of the households
can be modeled as if it were generated by the optimization decision of a representative household. Naturally, this would be more realistic than assuming that
all households are identical. Nevertheless, this is not without any costs. First, in
this case, the representative household will have positive meaning, but not always a
normative meaning (see below). Second, it is not in fact true that most models with
heterogeneity lead to a behavior that can be represented as if it were generated by
a representative household.
In fact most models do not admit a representative household. To illustrate this,
let us consider a simple exchange economy with a finite number of commodities and
state an important theorem from general equilibrium theory. In preparation for this
theorem, recall that in an exchange economy, we can think of the object of interest
as the excess demand functions (or correspondences) for different commodities. Let
these be denoted by x (p) when the vector of prices is p. An economy will admit a
representative household if these excess demands, x (p), can be modeled as if they
result from the maximization problem of a single consumer.
Theorem 5.1. (Debreu-Mantel-Sonnenschein) Let ε > 0 be a scalar and
N < ∞ be a positive integer. Consider a set of prices
â
ê
0
and any continuous function x : Pε →
Pε = p∈RN
+ : pj /pj 0 ≥ ε for all j and j

RN
+ that satisfies Walras’ Law and is homogeneous of degree 0. Then there exists an

exchange economy with N commodities and H < ∞ households, where the aggregate
demand is given by x (p) over the set Pε .

Proof. See Debreu (1974) or Mas-Colell, Winston and Green (1995), Proposition 17.E.3.

Ô

This theorem states the following result: the fact that excess demands come
from the optimizing behavior of households puts no restrictions on the form of these
demands. In particular, x (p) does not necessarily possess a negative-semi-definite
Jacobian or satisfy the weak axiom of revealed preference (which are requirements of
demands generated by individual households). This implies that, without imposing
further structure, it is impossible to derive the aggregate excess demand, x (p), from
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