This paper presents a model of consumer demand that aligns with observed expenditure patterns in long-term expenditure surveys and allows for detailed welfare analysis of relative price changes. The authors argue that standard demand models often fail to accurately capture consumer behavior across income groups. They derive a class of integrable quadratic logarithmic expenditure share systems, which are more flexible than popular models like the Translog or Almost Ideal Demand System. These models include quadratic terms in the logarithm of expenditure, allowing for Engel curvature. The authors estimate this model on a large pooled dataset of UK households, finding that models without Engel curvature generate significant distortions in welfare losses associated with tax increases. The paper also discusses the implications of these findings for welfare analysis and indirect tax reform, highlighting the importance of correctly modeling Engel curvature to accurately measure welfare impacts.This paper presents a model of consumer demand that aligns with observed expenditure patterns in long-term expenditure surveys and allows for detailed welfare analysis of relative price changes. The authors argue that standard demand models often fail to accurately capture consumer behavior across income groups. They derive a class of integrable quadratic logarithmic expenditure share systems, which are more flexible than popular models like the Translog or Almost Ideal Demand System. These models include quadratic terms in the logarithm of expenditure, allowing for Engel curvature. The authors estimate this model on a large pooled dataset of UK households, finding that models without Engel curvature generate significant distortions in welfare losses associated with tax increases. The paper also discusses the implications of these findings for welfare analysis and indirect tax reform, highlighting the importance of correctly modeling Engel curvature to accurately measure welfare impacts.