One of the most important current questions in economic analysis is whether or not labor markets clear in the short run. To answer this, it is necessary to be able to distinguish between restricted and unrestricted behavior by consumers supplying labor. This paper investigates the forms of preferences which lie behind linear models of labor supply, and derives the functional forms for commodity demands which accompany them, both with and without quantity restrictions in the labor market. Simple linkages between restricted and unrestricted demands are also considered as is the question of perfect aggregation over consumers in the presence of quantity restrictions.