The linear expenditure system provides a convenient model for representing consumer response to prices and income. The parameters of the model enter nonlinearly and standard estimation procedures lack desirable properties. This note discusses the problems with existing techniques and derives the maximum-likelihood estimators. The linear expenditure system (LES) was proposed and applied by Stone [7, 8], and it has found further application in [9], [2], and [4]. Aspects of the estimation of the LES are also discussed in [3].