# Endogenous Labor Supply

So far, we have assumed that labor income Y is exogenous and hence independent of labor income tax, t_{w}. If we relax this assumption, as in Chap. 8, labor income tax is not necessarily more desirable than interest income tax. Then, expenditure tax is not desirable either from the viewpoint of efficiency.

In order to explore this point in an extreme fashion, we now assume that labor income is endogenous but that second-period consumption is somehow exogenously fixed. Namely, we assume that because of some social or institutional constraints, optimal second-period consumption is exogenously fixed, independent of the tax rate, but that labor supply is endogenously determined.

Let us denote first-period leisure by x, first-period available labor supply time by Z, and actual labor supply by L (= Z — x). Then, Eqs. (9.5) and (9.6) are rewritten as

Fig. 9.2 **The optimizing behavior of a household**

Equation (9.5') corresponds to the labor income case and Eq. (9.6') the interest income tax case.

The optimizing behavior of a household with respect to c_{1} and x is described in Fig. 9.2. In this diagram, the vertical axis denotes c_{1} and the horizontal axis denotes x since the agent chooses c_{1} and x subject to her or his budget constraint. AH corresponds to the exogenously given optimal level of second-period consumption. Here the problem is to choose c_{1} and x.

With regard to t_{r} > 0, from Eq. (9.6') the relative price between c_{1} and x is independent of t_{r}. By imposing t_{r}, the budget line moves from AB to DF. However, with regard to t_{w} > 0, from Eq. (9.5') the slope of the budget line becomes flatter in accordance with the tax. The budget line moves from AB to AB^{'}. Hence, as shown in Fig. 9.2, utility at E_{r} is higher than utility at E_{w}. This means that interest income tax is more desirable than labor income tax in order to raise the same tax revenue. This is because the interest income tax is now a lump sum tax on the exogenously given second-period consumption.

In reality, labor supply and second-period consumption are both endogenously determined to some extent. Thus, the issue of which tax is more desirable is generally ambiguous from the efficiency viewpoint. The optimal tax rule that minimizes the excess burden, explained in Sect. 2, suggests that if labor supply is less elastic than second-period consumption, labor income tax becomes more desirable, and vice versa.

# The Negative Incentive Effect and Optimal Taxation

Some argue that it is optimal not to tax interest income or asset income, and that expenditure tax is desirable. However, as explained above, the argument is generally invalid. If saving is less elastic than labor supply, it may be desirable to tax interest income heavier than labor income.

In the real world, labor supply is often institutionally fixed. In this regard, we may justify expenditure tax to some extent. However, labor supply can be volatile in the long run. The negative incentive effect of tax is not generally ignored, even in the case of labor income tax. Thus, it is important to compare various tax bases from the viewpoint of efficiency. The relation between the negative incentive effect or the substitution effect and the tax rate is the main concern of optimal taxation from the viewpoint of efficiency.