Kuhn-Tucker (KT) demand models characterize consumers’ budget allocation through the Karush-Kuhn-Tucker conditions of utility maximization. Classical specifications, including quadratic utility, are widely used to study pricing, assortment, and competition, but remain difficult to estimate from consumer data. By contrast, models designed for tractable estimation often impose restrictive assumptions on demand behavior. We address this gap with two results. First, for KT models whose utilities are linear in their coefficients, we establish convexity of maximum-likelihood estimation under Gaussian shocks through an equivalent chance-constrained reformulation. The same reformulation reduces likelihood evaluation to at most one-dimensional integration, regardless of the number of goods, enabling scalable estimation with global optimality guarantees. Second, we show that mixtures of these estimable KT models can approximate a broad class of consumption distributions to arbitrary accuracy across varying budgets, prices, and assortments, including behavior generated by nonconcave utilities. Numerical studies demonstrate estimation with 50 goods, compared with at most 10 goods for conventional methods within the same time limit, and improved empirical fit from richer utility specifications.