# The Optimal Income Tax

## The Rawls Judgment

First, we investigate this problem based on the Rawls or maximin judgment. In order to find the optimal point on the tax possibility curve, it is useful to draw a social indifference curve associated with Eq. (10.3.2). When the indifference curve is tangent to the tax possibility curve, it corresponds to the optimum. Thus, we investigate the curvature of the indifference curve.

Since the Rawls or maximin criterion is only concerned with the lowest utility person, in the model of Sect. 2 the government only considers the U_{L} of person L. Namely, the relevant indifference curve is a combination of A and t that makes U_{L} constant. An increase in t moves the budget line of L downward, reducing her or his utility. In contrast, an increase in subsidy A enlarges L’s effective income, raising her or his utility. Thus, as shown in Fig. 10.7, the indifference curve is

Fig. 10.7 Optimal income tax: the Rawls criterion upward sloping. If t and A move in opposite directions, U_{L} does not remain constant.

Thus, the optimum is located at E_{R}, on the upward-sloping region of the tax possibility curve. It follows that the optimal tax rate, t_{R}, is always smaller than the revenue-maximizing tax rate, t_{M}. Obviously, it is less than unity. In an endogenous labor supply model, the optimal marginal tax rate is smaller than 100 %, which is the optimal tax rate in the exogenous labor supply model of Sect. 1. Further, it is smaller than the revenue-maximizing tax rate. Thus, we have the following inequality:

Since the Rawls criterion is significantly concerned with inequality, it requires significant redistribution. Nevertheless, it is not desirable to raise the marginal tax rate too high. If this happens, the labor incentive of person H falls significantly, thereby reducing the total tax revenue, which could have been used for redistribution to person L. The reason why we have t_{R} < t_{M} is that even with regard to L, a high t reduces her or his incentive to work.

The tax possibility curve shows that two different tax rates constitute the same tax revenue. From the viewpoint of equity, tax revenue matters. From the viewpoint of efficiency, a smaller tax rate is desirable in order to produce a smaller excess burden. Thus, a smaller tax rate is always more desirable than a larger tax rate to achieve the same revenue. This explains intuitively why the optimal point does not exist in the paradoxical region.