Introduction to Modern Economic Growth The utility function (5.4) is referred to as constant elasticity of substitution (CES), since if we define the level of consumption of each good as xˆij = xij −ξ ij , the elasticity of substitution between any two xˆij and xˆij would be equal to σ Each consumer faces a vector of prices p= (p1 , , pN ), and we assume that for all i, N X pj ¯ξ < y i , j=1 so that the household can afford a bundle such that xˆij ≥ for all j In Exercise 5.6, you will be asked to derive the optimal consumption levels for each household and show that their indirect utility function is given by i h P N i i p ξ + y − j ¡ ¢ j j=1 , (5.5) vi p,y i = h PN 1−σ i 1−σ j=1 pj which satisfies the Gorman form (and is also homogeneous of degree in p and y) Therefore, this economy admits a representative household with indirect utility: h P i − N p ξ + y j=1 j j v (p,y) = h PN 1−σ i 1−σ j=1 pj R R where y is aggregate income given by y ≡ i∈H y i di and ξ j ≡ i∈H ξ ij di It is also straightforward to verify that the utility function leading to this indirect utility function is (5.6) σ # σ−1 "N X¡ ¢ σ−1 xj − ξ j σ U (x1 , , xN ) = j=1 We will see below that preferences closely related to the CES preferences will play a special role not only in aggregation but also in ensuring balanced growth in neoclassical growth models It is also possible to prove the converse to Theorem 5.2 Since this is not central to our focus, we state this result in the text rather than stating and proving it formally The essence of this converse is that unless we put some restrictions on the distribution of income across households, Gorman preferences are not only sufficient for the economy to admit a representative household, but they are also necessary 223