# The Tax Possibility Curve

In Fig. 10.6, we draw a curve of the relationship between A and t that satisfies Eq. (10.8). This curve is called the tax possibility curve because it shows how the tax revenue from a proportional income tax used for A varies with the tax rate, t.

Fig. 10.6 The tax possibility curve

As shown in this figure, the tax possibility curve means that if t = 0 and t = 1, then A = 0; and if 0 < t < 1, then A > 0. Namely, if t = 0, the government does not collect taxes. Further, if t = 1, nobody wishes to work; thus, the tax base is zero. LH = Ll = 0 means that A = 0. When the tax rate rises from t = 0, tax revenue first increases. However, when the tax rate is high enough to discourage work incentives, a further increase in the tax rate may reduce the tax base and then tax revenue. This is a paradoxical case. Let us denote the revenue maximizing tax rate by tM. Then, if t > tM, we have a paradoxical case with a tax rate that is too high.

The government must choose A and t in order to maximize social welfare, subject to Eq. (10.8). In other words, it chooses the optimal point on the tax possibility curve. What is the optimal point? It can be determined by considering the social welfare function; namely, optimal income tax is given as the maximization of social welfare on the tax possibility curve.