A3 Shift of the Social Welfare Function

In this section, we investigate the comparative statics of the weight of the social welfare function. When v changes, the social indifference curve moves, but the tax possibility frontier does not. The optimal point moves on the initial tax possibility frontier. As shown by the movement from W to W’ in Fig. 10.A2, if the absolute slope of the social indifference curve increases with the same values of A and p, the optimal point moves to the right: the optimal level of A decreases and the optimal level of p increases. Thus, it is useful to differentiate dA/dp with respect to v.

We know that w_L < w_H and v_H > v_L. We show that or

Using the envelope theorem, it is straightforward to see that

Fig. 10.A2 Shift of the social welfare function

Since c is an increasing function of w, it follows that the above inequality holds.

Hence, it is easy to see that the sign of Eq. (10.A9) is negative. The absolute value of the slope of the social indifference curve decreases with v. Thus, the optimal value of p decreases with v, and the optimal value of A increases with v. When the social function approaches the Rawls criterion as v ! 1, the optimal tax parameters converge to the Rawls optimal tax parameters. When the social welfare function approaches the Bentham criterion as v ! 0, the optimal tax parameters converge to the Bentham optimal tax parameters. We confirm analytically the conjecture that the optimal marginal tax rate increases with the government’s inequality aversion.