# A Two-Person Model with Income Inequality

First, we explain the classical argument for extreme income redistribution. The purpose of this argument is to justify an extremely progressive income tax. Imagine a simple two-person model with income inequality. There are two different persons with respect to income, H and L. Let us denote each person’s income by Y_{H}, Y_{L}, and Y_{H} > Y_{l} respectively. H is a rich person and L is a poor person.

**© Springer Science+Business Media Singapore 2017 **267

**T. Ihori, ***Principles of Public Finance,*** Springer Texts in Business and Economics,**

**DOI 10.1007/978-981-10-2389-7_10**

Utility from income may be represented by the common utility function,

Utility increases with income and marginal utility decreases with income. Without any redistribution, we have

The utility of person H is higher than the utility of person L. If this outcome is regarded as inequitable, redistribution is needed to some extent.

# The Social Welfare Function

The government conducts a redistribution policy so that it imposes a tax on person H and transfers it to person L. How much should the government redistribute between them? This may depend on various factors including a value judgment on inequality. First, we formulate this social judgment by using the social welfare function:

where W is social welfare and U_{H} and U_{L} are the utility levels for H and L respectively. Normally, W increases with U_{H} and U_{L}.

There are two special functional forms of the social welfare function:

Equation (10.3.1) refers to the Bentham judgment in the sense that social welfare is given as the sum of utilities. This judgment is also called the utilitarian criterion since it concerns the utilities of all agents. Equation (10.3.2) refers to the Rawls judgment in the sense that social welfare is given as the worst person’s utility. This judgment is also called the maximin criterion since it intends to maximize the minimum utility.

In Fig. 10.1, the vertical and horizontal axes denote utility for person H and person L respectively. Let us draw the social indifference curve, the combination of U_{H} and U_{L}, in order to maintain the same social welfare. The social indifference curve associated with the Bentham judgment, Eq. (10.3.1), is curve I_{B}, which is the line with the slope of 45 degree. The social indifference curve associated with the Rawls judgment, Eq. (10.3.2), is curve I_{R}, the line for which is at right angles on the 45-degree line.

Let us explain the property of social indifference curves intuitively. For the Bentham judgment, Eq. (10.3.1), we have as an indifference curve

Fig. 10.1 **Socially optimal point**

where W_{0} means a fixed level of welfare. The slope of this curve is given by —1, as shown in Fig. 10.1. When W_{0} is high, the associated indifference curve is located in the upward region in Fig. 10.1.

For the Rawls judgment, Eq. (10.3.2), we have as an indifference curve

Thus, the indifference curve is a vertical line if U_{L} < U_{H}, and is a horizontal line if U_{H} < U_{L} in Fig. 10.1.

These two formulations are rather extreme. Generally, normal formulations on social welfare imply that the associated indifference curve lies between I_{B} and I_{R}. If the judgment is more concerned with inequality, the slope of the associated indifference curve is steeper and approaches I_{R}.