This paper considers the problem of specifying and estimating demand systems for samples which contain a significant proportion of observation with zero consumption of one or more goods. Our approach uses virtual prices, which are dual to the Kuhn-Tucker conditions, to select the set of goods consumed--the demand regime--and to transform binding nonnegativity constraints into nonbinding constraints. It has the advantage of permitting the use of indirect cost and utility functions such as the translog, and the analytic decomposition of demand effects for goods at the nonnegativity limit.