# The Samuelson Rule: Simple Derivation

We may also derive the Samuelson rule in a simpler model. Suppose that the utility function is specified as

The production technology function is specified as

The social welfare is given by the sum of the utilities since the two agents are identical. Thus, the government intends to maximize

by choosing Y. The optimal condition is given as

The left-hand side of Eq. (11.12) denotes the marginal cost of public goods and the right-hand side denotes the sum of the marginal benefit of public goods. Thus, Eq. (11.12) denotes the Samuelson rule.

# Numerical Example: A Two-Person Model of Public Goods

**Example 1**

Suppose each person’s utility function is given as and

In addition, the production frontier is simply given as

Then, Vf *=* 1/Y and Vf *=* 2/Y. Hence, the Samuelson rule means that 1/Y + 2/Y = 1. The answer is Y = 3. Figure 11.3 explains this case. MB^{1} = 1/ Y and MB^{2} = 2/Y; hence, MB^{1} + MB^{2} = 3/Y.

Note that we implicitly assume the Bentham criterion as social welfare. Generally, the Samuelson rule alone cannot determine the optimal level of public goods. This rule is a necessary condition of the optimal provision. We need, in addition, the value judgment on social welfare in order to determine the optimal level of public goods.

**Example 2**

Suppose the utility function is given as and the production frontier is simply given as

Fig. 11.3 **The Samuelson rule**

Then, MB^{1} = Ujy = у and MB^{2} = Цу = у. Hence, the Samuelson rule means

Considering the feasibility condition, we have 2Y = M. Thus, the answer is Y = M/2. If agents are identical, social welfare is represented by each agent’s utility. Hence, we can determine the optimal level of public goods explicitly.