The complementary slackness condition (18) has important implications for the shape of the labor supply function. Individuals who face a binding quantity constraint in the credit market react to different labor supply incentives than individuals who are at an interior portfolio optimum.
Consider first the case of an interior solution, when (18) holds as an equality. For individuals for which X > 0, tax arbitrage is driven to the point where the after-tax return on the taxable asset equals p. For these workers, the constraint on short sales is not an issue, and labor supply is determined in a way that parallels our previous analysis. The relevant marginal tax rate is the yield ratio p / r , the same for everyone. It is easy to show that the behavioral functions take the form
i.e., the tax system is linearized. Compared with the analysis in section 3, the only novelty is that (19) and (20) include a wealth effect from the initial land endowment.
To derive the labor supply function for workers who are at a corner solution, we simply set X = 0 in (17). We then have
The labor supply functions of individuals who are at a corner solution will be of the same format as those materializing from the standard labor supply model in the presence of a nonlinear tax system, and with an exogenous income from wealth, rX. In particular, the labor supply function will depend on a vector of parameters, т, describing global properties of the tax system. Although individuals at corners engage in intra-marginal asset trade, what matters for the form of the labor supply function is the extent of tax arbitrage on the margin.
To characterize the determination of equilibrium asset prices we need to derive the selection criterion that allocates individuals between the arbitrage and no-arbitrage regimes. Consider an individual for whom the non-negativity constraint is binding. For this individual we can write (18) as
In the limit the equation ф(т9 w, rX, pir)~ 0 then gives us the surface in (w9 X) space that separates those who are at an interior optimum from those who are at a comer solution. For the particular case of a logarithmic utility function and a tax system with constant residual tax progression, analyzed in the previous section, the separating surface can be given a nice intuitive interpretation. After straightforward but tedious manipulations, one can show that the separating hyperplane can be written as fast cash payday loans