# The Three-Person Model of Regional Redistribution

Regional redistribution may not produce a desirable outcome if some agents can choose the regions. Let us explain this paradoxical case using a simple numerical example.

Imagine that initially one person lives in region A and two persons live in region B (see Table 13.1a). In region A, an agent earns an income of 10 and in region B an agent earns an income of 40. A is a poor rural area and B is a rich urban area; thus, an agent living in region B can earn a higher income than in region A. Because earning an income requires a cost (or excess burden) to some extent, an agent’s welfare is assumed to be half of his or her earned income.

Table 13.1a Outcome before move

Table 13.1b Outcome after move

 Region A B Earned income 10, 10 40 Cost of earning income 5,5 20 Initial welfare 5,5 20 Income after redistribution 20, 20 20 Welfare after redistribution 15, 15 0 Initial social welfare 5 Social welfare after redistribution 0
 Region A B Earned income 10 40, 40 Cost of earning income 5 20, 20 Initial welfare 5 20, 20 Income after redistribution 30 30, 30 Welfare after redistribution 25 10, 10 Initial social welfare 5 Social welfare after redistribution 10

Hence, in region A an agent’s welfare is 5 and in region B it is 20. Because welfare is still higher in region B than in region A, people prefer to live in region B if they can move. In this three-person model, we assume that only one person can move and that other persons cannot move but live in their original regions A and B. Thus, the person who can move lives in region B.

Now let us introduce an extreme redistribution policy managed by central government. Namely, central government imposes a tax of 10 on a person in region B and transfers 10+10 = 20 to a person in region A; thus, after-tax income is equalized as 30 for all persons. Then, welfare in region A is 25 (=5 + 20) and welfare in region B becomes 10 (=20 — 10). From the viewpoint of the Rawls criterion, social welfare increases from 5 to 10. In this sense, the redistribution policy may be desirable.

For the person who can move, it is now better to move to region A from region B because welfare in region B is 10, which is lower than welfare in region A, which is now 25. Thus, he or she moves to region B. Table 13.1b presents the outcome after the person moves to region A.

In region A, two persons earn 10 and in region B one person earns 40. Because of the extreme redistribution policy, a person in region B now pays a tax of 20 and two persons in region A receive a subsidy of 10. Thus, after-tax income is equalized as 20 for all persons. Welfare after redistribution is now 15 (=5 + 10) in region A and 0 (=20 — 20) in region B. The person who can move obtains higher welfare compared with the person who is unable to move in region B. Thus, the person who can move prefers to move to region A.

Social welfare under the Rawls criterion is now 0, which is the welfare of the person who is unable to move in region B. In other words, social welfare declines from 5 to 0. Hence, considering the possibility of movement between the two regions, a redistribution policy may not be desirable. We may derive a similar policy implication if we adopt the Bentham criterion instead of the Rawls criterion.