Along each segment of the budget constraint there is a preferred number of hours of work,:
where wi is the net wage on segment i, yi is virtual income for segment i, z represents socioeconomic variables, and α, β, and γ are the parameters to be estimated. For fixed α, β, and -γ, desired hours,, may not be feasible, because may be greater or less than the hours at the end points of the budget segment Hi-1 and Hi. If desired hours are feasible, the indifference curve and the budget segment are tangent. If the budget set is convex, this tangency is unique, and I then use the stochastic specification for the deviation of actual hours from desired hours for person j as
Because observed hours hj ≥ 0, the stochastic term ηj is assumed to be independent truncated normal across individuals in the population. This assumption yields a Tobit specification for the hours-worked variable. However, if= 0, I assume that the individuals choose not to work and set hj = 0. Because the final model has two sources of stochastic variation, the interpretation of ηj differs from that of the error term in standard models. Here the individual chooses, among jobs that differ in normal (long-run) hours worked, the one with working hours closest to his . But observed hj may differ because of unexpected layoffs, short time, overtime, or the worker's poor health together with measurement error. As an empirical matter, the standard deviation of ηj is reasonably small, indicating that people successfully match jobs to their desired hours of work.
If the budget set is nonconvex,is not necessarily unique because multiple tangencies can occur between the indifference curves and the budget set. Then the particular is chosen that leads to maximum utility, which is determined by the use of the corresponding indirect utility function from equation 4. Again I use the stochastic specification of equation 6 to express the deviation of actual hours from desired hours of work. It is interesting to note that, although certain kink points such as H + ̃ in figure 5 in the nonconvex case cannot correspond to desired hours,