This paper concerns empirical measurement of Hicksian consumer welfare under interval-reported income. Bhattacharya has shown that for discrete choice, welfare distributions resulting from a hypothetical price-change can be expressed as closed-form transformations of choice probabilities. However, when income is interval-reported, as is the case in many surveys, the choice probabilities, and hence welfare distributions are not point-identified. We derive bounds on average welfare in such scenarios under the assumption of a normal good. A finding of independent interest is a set of Slutsky-like shape restrictions which are linear in average demand, unlike those for continuous choice. A parametric specification of choice probabilities facilitates imposition of these Slutsky conditions, and leads to computationally simple inference for the partially identified features of welfare. In particular, the estimand is shown to be directionally differentiable, so that recently developed bootstrap methods can be applied for inference. Under mis-specification, our results provide a "best parametric approximation" to demand and welfare. These methods can be used for inference in more general settings where a class of set-identified functions satisfy linear inequality restrictions, and one wishes to conduct inference on functionals thereof. We illustrate our theoretical results using a simulation exercise based on a real dataset where actual income is observed. We artificially introduce interval-censoring of income, calculate bounds for the average welfare effects of a price-subsidy using our methods, and find that they perform favorably in comparison with estimates obtained using actual income.