# The Neutrality Theorem of Public Goods

## The Neutrality Theorem

Consider the Nash mechanism of public goods provision. This approach assumes that the private sector voluntarily provides public goods instead of the government. As explained in Sect. 3, the Pareto optimum level of public goods cannot be attained and the market fails. In this section, we do not discuss the normative aspect of the Nash mechanism; rather, we explore an interesting outcome of redistribution policy in this formulation.

Warr (1983) showed that in the formulation of the Nash provision of public goods, redistribution policy cannot have any real effect. This policy is perfectly offset by the private reaction to the provision of public goods. This is called the neutrality theorem of public goods. The outcome is consistent irrespective of preferences and the distribution of income.

## The Model of Neutrality Result

We consider a two-person economy as in Sect. 3. Both people provide public goods voluntarily. The government conducts the redistribution policy; namely, the government may tax person 1’s income and give a subsidy to person 2. The redistribution is conducted from person 1 to person 2. Because person 1’s disposable income declines, her or his provision of public goods also declines. However, since person 2’s disposable income rises, her or his provision of public goods also rises.

The neutrality theorem means that at the new equilibrium, a reduction of person 1’s provision is the same as a reduction of her or his disposable income, and an increase in person 2’s provision is the same as an increase in her or his disposable income. As a result, the total provision of public goods does not change. Moreover, the redistribution policy does not change each person’s consumption or welfare.

Consider a two-person economy in which each person’s utility function is given as

where U^{i} is utility, x_{i} is the private good consumption of person i, and Y is a pure public good.

Each person’s budget constraint is given as

where M_{i} is income and y_{i} is the private provision of a public good by person i. The public good is privately provided:

From the optimizing behavior of each person, we have

and

These two equations give the optimal combination of private consumption and public good as a function of each person’s utility. Namely, optimal levels of private good consumption and public good are an increasing function of each person’s utility.

Then, from the above equations as the feasibility condition, we have and

The two Eqs. (11.27) and (11.28), determine the equilibrium values of U^{1} and U^{2}. Since the total income, M_{1} +M_{2}, appears in the first equation, the total income in the economy determines welfare and other economic variables. Since the distribution of income does not affect M_{1} + M_{2}, redistribution does not matter either. In other words, income redistribution does not affect real economic variables. This is the neutrality theorem of public goods.