The choices' limited capacities of the discrete choice models must be taken into account. We propose a convex optimisation formulation in which the exponential formulation of the logit model is upheld in the Karush-Kuhn-Tucker (KKT) conditions. The capacities of the choices are then added to the formulation as side constraints. A solution algorithm based on the successive coordinate descent is proposed. For numerical evaluation, we investigate US income tax policies for the years prior to the 2009 crisis using multinomial logit models. The tendency of the states for choice of income tax versus other tax sources is assessed and it is found that: 1) all states show a propensity to levy more income tax; 2) this propensity has a ceiling cap similar to what is already known from the 'Laffer curve'; 3) residents in the states with already high income tax are more likely to be subjected to even heavier income tax within caps.