Samuelson’s Model of Public Expenditure

With The help of Samuelson’s model, it is shown that government intervention is necessary for the efficient provision of public goods.

Samuelson’s Theory of Public Expenditure

The formal analysis of public goods began with Samuelson (1954), who derived the rule depicting efficient levels of provision of public goods. He tried to explain the efficient provision of pure public good and the role of government in its provision. The rule for efficient provision is typically called the Samuelson rule.

The following analysis will derive the Samuelson rule for a pure public good.

Pure Public Good:

To derive the efficiency rule for a pure public good, it is assumed that there is available a single public good and, initially, that disposal is not possible. The latter assumption implies that all households must consume a quantity of the public good equal to its supply. The extension of too many public goods is simple.

The economy consists of H households, indexed h = 1, …, H. Each household has a utility function:

𝑈^ℎ =   𝑉^ℎ (𝑥^ℎ, 𝐺)

where 𝑥^ℎ is the consumption of household h of private goods, and G is the supply of the public good.

The fact that total supply, G, appears in all households’ utility functions indicates that the public good is pure. It is assumed that the combinations of 𝑥^ℎ, h = 1, …, H, and G that the economy can produce are constrained by production possibilities. The implicit representation of the production set is written:

𝐹 (𝑥, 𝐺) ≤ 0

𝑤ℎ𝑒𝑟𝑒,      𝑥 = ∑^𝐻_ℎ=1 𝑥^ℎ

To characterise the set of Pareto efficient allocations, the government chooses 𝑥ℎ, h = 1, …, H, and G to maximise the utility level of the first household, constrained by the requirement that households 2 to H obtain given utility levels and by production possibilities. Varying the given utility levels for households 2 to H traces out the set of Pareto efficient allocations.

The Lagrangian for this maximisation problem can be written:

where 𝑈^ℎ is the utility level that must be achieved by h = 2, …, H.

Assuming that the specified utility levels can be reached simultaneously, the necessary condition describing the choice of a component 𝑥^ℎ, i from 𝑥^ℎ is:

with 𝜇^ℎ ≡ 1 for h = 1. At an optimum, the above equation holds for all i = 1, …, n.

For the choice of the level of public good, forming the Lagrangian and optimising with respect to G gives:

which is the marginal rate of substitution between the public good and the 𝑖^𝑡ℎ private good for the 𝐻^𝑡ℎ household. The right-hand side of the above equation is the marginal rate of transformation between the public good and private good i.

The above equation can thus be written as:

This equation is the Samuelson rule, which states that Pareto efficient provision of the public good occurs when the marginal rate of transformation between the public good and each private good is equated to the sum, over all households, of the marginal rates of substitution.

The difference between optimal allocations of public goods and private goods occurs due to the fact that an extra unit of public goods increases the utility of all households so that the social benefit of this additional unit is found by summing the marginal benefit, measured by the MRS, to individual households.

At an optimum, this is equated to the marginal cost given by the marginal rate of transformation. In contrast, an extra unit of private good only increases the welfare of its single recipient, and an optimum occurs when marginal benefits are equal across households and to marginal cost.

Two points must be noted in the interpretation of this result:

Firstly, although non-excludability has been adopted as a defining characteristic of pure public good, it played no role in the derivation of the Samuelson rule. In fact, the optimal level of provision is not dependent on the degree of excludability. Instead, excludability is only relevant for determining feasible provision mechanisms.

Secondly, although the Samuelson rule is simple, attention should be given to the fact that it can be easily implemented. In order to derive the rule, it was assumed that the government had complete control over the allocation of resources. Equivalently, the government could employ lump-sum taxation to redistribute income and to finance the provision of the public good.

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