A model of labour supply is developed in which individuals face restrictions on hours choices. Observed hours reflect both the distribution of preferences and the distribution of offers. In this framework the choice set is limited and observed hours may not satisfy the revealed preference conditions for ‘rational’ choice. We show first that when the offer distribution is known, preferences can be identified. We then show that, where preferences are known, the offer distribution can be fully recovered. We also develop conditions for identification of both preferences and the offer distribution. We illustrate this approach in a labour supply setting with nonlinear budget constraints. The occurrence of nonlinearities in the budget constraint can directly reveal restrictions on choices. This framework is then used to study the labour supply choices of a large sample of working age mothers in the UK, accounting for nonlinearities in the tax and welfare benefit system, fixed costs of work and restrictions on hours choices.