Samuelson condition

As Samuelson condition (also: Samuelson - Musgrave - condition ) is called in the microeconomic theory of economic policy a condition for which public goods are provided efficiently in an economy. Is understood as public goods, such goods that can be consumed at one of a plurality of persons, without these it interfering with each other, and of their consumption on the other hand nobody can be excluded. The Samuelson condition states then in the simplest case of an economy with two goods - a private and a public good - that an allocation of these goods if and only Pareto efficient if the marginal rate of transformation between the two goods just the sum of the budgetary marginal rates of substitution between the goods corresponds.

The name of the condition comes back to the American economist Paul Samuelson, who in 1954 formulated in an article in the Review of Economics and Statistics.

Intuition

The marginal rate of substitution indicates how many units of the private good a person omitted, if it receives a unit of the public good, so it is a (marginal) willingness to pay. The marginal rate of transformation equals marginal cost of the public good in terms of the private Guts.

Therefore, states the condition that at Pareto efficiency, the sum of the willingness to pay coincides with the marginal cost. For private goods, however, is consistent with each individual 's willingness to pay the marginal cost. The difference is explained by the fact that the provision of the public good benefit several people, the provision of a private good but only one person.

Formal frame ( Samuelson model ) and derivation

Consider whether an economy with two produced goods and two households. It is now a private ( rival ) consumer (eg, a bicycle) and b is a ( non- rival ) public good (such as national defense ). The total existing quantity of the two goods amounts to or. Further, let the set of k that consumed the household i. Now require a according to the definition of a private good that, and for b, by definition of a public good in that.

The two households each have a continuous and concave ( ordinal ) utility function that is strictly positive. Let further a transformation function for all k and suppose. On such a defined transformation curve are all technologically efficient production plans - inefficiencies in the production of goods are thus excluded.

The approach of Samuelson insists building is to find out the amount of lying on the frontier allocations those allocations through which the utility of one household is maximized, given a certain level of utility of household 2 Since the households are symmetric, this is equivalent to just the condition for Pareto optimality of an allocation. The maximization problem is accordingly

What on the Lagrangian

Leads. From the corresponding optimality conditions, the important result follows

The Effizenzbedingung for a social planner is that, in the sum of the budgetary marginal rates of substitution (MRS ) - in other words, the sum of the individual marginal willingness to pay - the marginal rate of technical substitution (GRT ) must comply. This is just the Samuelson condition. Taking into account the importance of the GRS and GRT, can be simplified to say that a Pareto- optimal allocation of just such a character that the sum of the amounts of the private good to abandon the consumer for an additional unit of the public good would be willing to must be equal to the amount of the private good that is actually needed to produce this additional unit.

If we extend the model to more households, changes in principle to the result nothing, is it just for n households

Whereas for private goods as usual, the efficiency conditions

. apply It can be shown that the competitive market solution leads to an inefficient low provision of the public good, so that consequently the sum of individual GRS is greater than the GRT ( underfunding ).

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