Introduction to Modern Economic Growth In other words, if the indirect utility functions of some households 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 ) = (p) + b (p) y i (1) Then any allocation that maximizes the utility of the representative houseP P hold, v (p, y) = i∈H (p) + b (p) y, with y ≡ i∈H y i , is Pareto optimal (2) Moreover, if (p) = 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 , , 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 ≥ for all j, P are nonnegative Pareto weights with i∈H αi = 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