There are quite a lot empirical research based on discrete choice models, in which the consumer selects one of J alternative goods to maximize her indirect utility. The key assumption of these models is that there is an idiosyncratic utility component that follows the extreme value distribution.
The problem I am facing is different. I can observe a large number of individuals' consumption of a single good. The values of these choices are continuous quantities, with a long right tail, and roughly 80% of these values are zero. Even if I can discretize the quantity, I don't think I can assume an indepdent utility term for each discretized choice. So a different modelling approach is needed.
I conjecture the choice problem to consist of two steps. In the first step, the consumer decides whether to consume the good. If yes, the consumer then chooses how much to consume. In the empirical IO literature, is there any established model that is used for such "binary-continuous" choices?