The Socially Optimal Point

The utility frontier, which is the combination of utilities associated with different redistributions between Y_H and Y_L, may be drawn as curve AB in Fig. 10.1. Curve AB is convex toward the origin. Since the marginal utility of income decreases with income, social welfare increases when redistribution is more equitable and approaches the 45-degree line. The socially optimal point maximizes social welfare on the utility frontier.

The socially optimal point is given by the point where the indifference curve is tangent to the utility frontier. With regard to the Bentham judgment, it is easy to see that the optimal point E is on the 45-degree line since at this point the slope of the indifference curve is equal to — 1. With regard to the Rawls judgment, the slope of the indifference curve at point E is kinked; however, it is also easy to see that E is on the 45-degree line.

Thus, as shown in Fig. 10.1, the optimal point E is on the 45-degree line, irrespective of the social value judgment. In other words, in either the Bentham judgment or the Rawls judgment, it is optimal to redistribute income so as to attain perfect equality. It follows that point E is also optimal at a wide range of reasonable judgments on equity. At point E, the after-tax income of each person is equal to the average income, (Y_H + Y_L)/2.