This paper presents a model of consumer demand that is consistent with observed expenditure patterns and allows detailed welfare analysis of price changes. The authors argue that Engel curves require quadratic terms in the logarithm of expenditure, unlike popular models like the Translog or Almost Ideal Demand Systems, which assume linear relationships. They derive a class of integrable quadratic logarithmic expenditure share systems and estimate a specification on UK household data. Models that ignore Engel curvature generate significant distortions in welfare loss patterns. The study shows that Engel curves for some goods are nonlinear in log income, while others are linear. This implies that the quadratic logarithmic model allows goods to be luxuries at some income levels and necessities at others. Using UK Family Expenditure Survey data, the authors reject the Working–Leser form for some commodities but find it close to linear for food. The QUAIDS model, a quadratic extension of the Almost Ideal model, is proposed as a practical specification that nests both the AI model and the Translog model. The QUAIDS model is shown to be rank 3, the maximum possible for a demand system linear in income functions. It allows for price-dependent coefficients and includes a quadratic logarithmic term. The model is estimated on pooled FES data and produces a data-coherent description of consumer behavior. The model's coefficients are price-dependent, and regularity conditions like Slutsky symmetry are satisfied. The paper also shows that the QUAIDS model provides a better approximation of Engel curves than linear models, especially for goods with nonlinear Engel curves. The model's results suggest that welfare calculations based on linear Engel curves may be inaccurate. The study highlights the importance of including quadratic terms to account for goods being luxuries or necessities at different income levels. The QUAIDS model is shown to be robust and adequate for empirical analysis, with no need for additional semiparametric terms. The paper concludes that studies based on AI or translog preferences may fail to capture the true distribution of welfare losses due to incorrect modeling of Engel curvature.This paper presents a model of consumer demand that is consistent with observed expenditure patterns and allows detailed welfare analysis of price changes. The authors argue that Engel curves require quadratic terms in the logarithm of expenditure, unlike popular models like the Translog or Almost Ideal Demand Systems, which assume linear relationships. They derive a class of integrable quadratic logarithmic expenditure share systems and estimate a specification on UK household data. Models that ignore Engel curvature generate significant distortions in welfare loss patterns. The study shows that Engel curves for some goods are nonlinear in log income, while others are linear. This implies that the quadratic logarithmic model allows goods to be luxuries at some income levels and necessities at others. Using UK Family Expenditure Survey data, the authors reject the Working–Leser form for some commodities but find it close to linear for food. The QUAIDS model, a quadratic extension of the Almost Ideal model, is proposed as a practical specification that nests both the AI model and the Translog model. The QUAIDS model is shown to be rank 3, the maximum possible for a demand system linear in income functions. It allows for price-dependent coefficients and includes a quadratic logarithmic term. The model is estimated on pooled FES data and produces a data-coherent description of consumer behavior. The model's coefficients are price-dependent, and regularity conditions like Slutsky symmetry are satisfied. The paper also shows that the QUAIDS model provides a better approximation of Engel curves than linear models, especially for goods with nonlinear Engel curves. The model's results suggest that welfare calculations based on linear Engel curves may be inaccurate. The study highlights the importance of including quadratic terms to account for goods being luxuries or necessities at different income levels. The QUAIDS model is shown to be robust and adequate for empirical analysis, with no need for additional semiparametric terms. The paper concludes that studies based on AI or translog preferences may fail to capture the true distribution of welfare losses due to incorrect modeling of Engel curvature.