# Analytical Framework

Suppose there are three individuals, person 1, person 2, and person 3, with income inequality. Each person’s identical, additively separable utility function is given as

where c_{i} is the private consumption of person i, Y is the public good, and V(Y) is utility from consuming the public good. For simplicity, we assume that the income effect of private consumption is zero.

The private budget constraint is given as

where M; denotes the income of person i and t is the tax rate. We assume that

Person 1 is poor, person 2 is in the middle-income bracket, and person 3 is rich. The government budget constraint is

The left-hand side of Eq. (12.4) denotes total tax revenue and the right-hand side denotes government spending on public goods.

Let us investigate each person’s desirable tax rate (or the size of public goods). Substituting the private and government budget constraints into the utility function, we have

where M = M_{1} + M_{2} + M_{3} is the total income in this economy.

Then, differentiating this function, Eq. (12.1^{0}), with respect to the tax rate, t, and setting it as zero, we have as the optimality condition,

Thus, the optimal tax rate or optimal public goods of person i is given as

where V_{Y} denotes the marginal utility of public goods, dV/dY.

As shown in Fig. 12.1, the income share of person i, M_{i}/M, is a horizontal line, depending upon each person’s income. The vertical axis denotes the marginal utility of public goods and each person’s income share, and the horizontal axis denotes the level of public goods. Person 1’s income share line is the lowest, while person 3’s line is the highest. The V_{Y} curve is the marginal utility of public goods, which is assumed to be the same among the three persons. Thus, the optimal point for person 1 is given as point L, the intersection of her or his income share line and the V_{Y} curve. The optimal point for person 2 is point M. Finally, the optimal point for person 3 is point R.

Fig. 12.1 **Preferences for small or big government**

As can be easily seen, person 1’s optimal Y, Y_{L}, is the largest among the three; person 2’s optimal Y, Y_{M}, is in the middle; and person 3’s optimal Y, Y_{R}, is the smallest. This simple model explains why rich people like small government with low tax and small spending, while poor people like big government with high tax and large spending. This is because the cost of public spending increases with income, while the benefit of public spending is independent of income.