The Geometric Structure of Some Systems of Demand Equations
Consider the income compensation function Y = φ(p, y; p0) where Y, y ∈ R are income levels and p, p0 ∈ Rn are prices. By holding p0 fixed we may interpret this as an n-parameter transformation of R to R. With this interpretation we show that a system of demand functions is additively separable in income and prices if and only if φ is a Lie transformation group on R. Sophus Lie's 1888 classification of such groups into three fundamental types provides an alternative derivation of both the rank 3 condition of Gorman and the additively separable functional forms found by him.
Farris, Frank A. with Thomas Russell. “The Geometric Structure of Some Systems of Demand Equations.” Journal of Mathematical Economics 22 (1993): 309–326.