An analytical solution to determine Equivalence Scales from single equation models

There are basically two ways to estimate Equivalence Scales (ES) from households’ budget data: single equation models, namely Engel and Rothbarth models, and complete demand systems (for example LES, Translog, AIDS, GAIDS). The ES is an index that measures the ratio between the expenditures of the two households at the same level of utility, i.e. at the same level of food share. The aim of this work is to determine an analytical solution for the ES starting from a quadratic single equation model. We will consider only the Engel model, but the methodology we will use can be easily extended to the Rothbarth case.