# Optimal Provision of Public Goods: The Samuelson Rule

## The Samuelson Rule: Diagramed Derivation

We now investigate the optimal provision of public goods. In the first best scenario, the government can control resource allocation freely and determine the optimal level of public goods. The necessary condition is called the Samuelson rule.

The Samuelson rule:

The marginal social benefit of public goods, which is given by the sum of each agent’s marginal benefit, should be equal to the marginal cost of public goods.

We can explain this rule with a two-person model. Each person has preferences for private goods x and public goods Y.

Each person’s utility U^{1} increases with the consumption of private goods x_{i} and public goods Y. Production possibilities are given as the production frontier function,

In addition,

denotes the feasibility condition of the private goods.

In Fig. 11.2a, the vertical axis denotes private goods x and the horizontal axis denotes public goods Y. We draw person 1’s indifference curve and production constraint AB. A movement from point B to the right means an expansion of public goods sacrifices private goods. Hence, the production frontier AB is downward sloping and convex to the origin. The slope of this production constraint curve corresponds to the marginal cost of public goods in terms of private goods. The slope of Person 1’s indifference curve means the marginal benefit of public goods of person 1 in terms of private goods.

Then, we fix person 1’s utility as U_{I} and consider the associated indifference curve. With this constraint, the consumption possibility of person 2 may be drawn as curve CD, which is the vertical difference between AB and U_{I}, as shown in Fig. 11.2b. Note that at a given level of public goods F, we have FI = x_{1} and HI = x_{2}. When F changes, we may obtain curve CD as the locus of HI.

The Pareto optimum point, which is the condition of efficient allocation, maximizes person 2’ s utility subject to person 1’s fixed utility. The utility maximizing point of person 2 on curve CD is given by point E, where curve CD and person 2’s indifference curve U_{II} are tangent. At this point, the slope of person

Fig. 11.2 **Optimal provision of public goods. (a) person 1, (b) person 2**

2’s indifference curve or person 2’s marginal rate of substitution, MRS_{2} (= U_{Y}/Ux_{2}), is equal to the gap between the slope of the production curve or the marginal rate of transformation, MRT(= F_{Y}/F_{X}), and the slope of person 1’s indifference curve, MRS_{1}. Note that U_{Y} means the marginal utility of a public good, Uxi means the marginal utility of a private good for person i, F_{Y} means the marginal cost of the public good, and F_{X} means the marginal cost of the private good. Namely, we have

or

This is the Samuelson rule (See Samuelson (1954)).

So far, we have assumed that the government can control resource allocation freely. In a competitive economy, if lump sum taxes are available, the government can attain the first best by realizing the Samuelson condition.