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 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 > be a scalar and N < ∞ be a positive integer Consider a set of prices â ê and any continuous function x : Pε → Pε = p∈RN + : pj /pj ≥ ε for all j and j RN + that satisfies Walras’ Law and is homogeneous of degree 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 220