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 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 = (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 (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 221