Samuelson's condition

In the microeconomic theory of economic policy, the Samuelson condition (also: Samuelson-Musgrave condition ) describes a condition for when public goods are efficiently provided in an economy . In this context, public goods are understood to be those goods which, on the one hand, can be consumed by a large number of people without interfering with one another and, on the other hand, nobody can be excluded from their consumption. In the simplest case of an economy with two goods - a private and a public good - the Samuelson condition then states that an allocation of these goods is Pareto-efficient if and only if the marginal rate of the transformation between the two goods is the sum of the household- specific marginal rates corresponds to the substitution between the goods.

The name of the condition goes back to the American economist Paul Samuelson , who first formulated it in 1954 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 foregoes if they receive a unit of the public good in return, so it is a (marginal) willingness to pay. The marginal rate of transformation is the marginal cost of the public good in units of the private good.

Therefore, the condition says that with Pareto efficiency the sum of willingness to pay corresponds to the marginal costs. In the case of private goods, however, each individual willingness to pay corresponds to the marginal costs. The difference is explained by the fact that the provision of the public good benefits several people, but the provision of a private good only benefits one person.

Formal framework (Samuelson model) and derivation

Consider an economy with two goods produced and two households . Let a be a private (rival) consumer good (for example a bicycle) and b a (non-rival) public good (for example national defense). The total amount of the two goods is or . Let further be the amount of k that household i consumes. According to the definition of a private good, it holds for a that , and for b according to the definition of a public good that . ${\displaystyle k=a,b}$ ${\displaystyle i=1,2}$ ${\displaystyle X_{a}}$ ${\displaystyle X_{b}}$ ${\displaystyle x_{k}^{i}}$ ${\displaystyle x_{a}^{1}+x_{a}^{2}=X_{a}}$ ${\displaystyle x_{b}^{1}=x_{b}^{2}=X_{b}}$

The two households each have a continuous and concave (ordinal) utility function that is strictly positive. Let further be a transformation function with for all k and let it hold . All technologically efficient production plans lie on a transformation curve defined in this way - inefficiencies in the production of goods are therefore excluded. ${\displaystyle u^{i}(x_{a}^{i},x_{b}^{i})=u^{i}(x_{a}^{i},X_{b})}$ ${\displaystyle T=T(X_{a},X_{b})}$ ${\displaystyle T:\mathbb {R} ^{2}\rightarrow \mathbb {R} }$ ${\displaystyle \partial T/\partial X_{k}>0}$ ${\displaystyle T=0}$

Samuelson's approach, based on this, consists in finding those allocations from the set of allocations on the transformation curve that maximize the utility of household 1, given a certain utility level of household 2. Since the households are symmetrical, this corresponds exactly the condition for the Pareto optimality of an allocation. The maximization problem is accordingly

${\displaystyle \quad \max u^{1}(x_{a}^{1},X_{b})}$ under the constraints

[1] , ${\displaystyle \quad u^{2}(x_{a}^{2},X_{b})={\overline {u}}^{2}}$

[2] , ${\displaystyle \quad T(X_{a},X_{b})=0}$

[3] and ${\displaystyle \quad x_{a}^{1}+x_{a}^{2}=X_{a}}$

[4] , ${\displaystyle \quad x_{b}^{1}=x_{b}^{2}=X_{b}}$

what about the Lagrange function

${\displaystyle {\mathcal {L}}(X_{a},X_{b},x_{a}^{1},x_{a}^{2};\lambda _{1},\lambda _{2},\lambda _{3})=u^{1}(x_{a}^{1},X_{b})+\lambda _{1}\left[{\overline {u}}^{2}-u^{2}(x_{a}^{2},X_{b})\right]+\lambda _{2}T(X_{a},X_{b})+\lambda _{3}\left[X_{a}-x_{a}^{1}-x_{a}^{2}\right]}$

leads. The important result follows from the corresponding optimality conditions

${\displaystyle \underbrace {\frac {\partial u^{1}\left(x_{a}^{1},X_{b}\right)/\partial X_{b}}{\partial u^{1}\left(x_{a}^{1},X_{b}\right)/\partial x_{a}^{1}}} _{\equiv GRS_{1}(b,a)}\;+\;\underbrace {\frac {\partial u^{2}\left(x_{a}^{2},X_{b}\right)/\partial X_{b}}{\partial u^{2}\left(x_{a}^{2},X_{b}\right)/\partial x_{a}^{2}}} _{\equiv GRS_{2}(b,a)}\;=\;\underbrace {\frac {\partial T(X_{a},X_{b})/\partial X_{b}}{\partial T(X_{a},X_{b})/\partial X_{a}}} _{\equiv GRT(b,a)}\quad \mathrm {\textrm {(Samuelson-Bedingung)}} }$

The efficiency condition for a social planner is that the sum of the household- specific marginal rates of substitution (GRS) - in other words: the sum of the individual marginal willingness to pay - must correspond to the marginal rate of transformation (GRT). This is just the Samuelson condition. If one takes into account the importance of the GRS and the GRT, it can be said in a simplified manner that a Pareto-optimal allocation must be such that the sum of the quantities of the private good that the consumers would be willing to give up for an additional unit of the public good, must be equal to the amount of the private good that is actually needed to produce this additional unit.

If the model is expanded to include additional households, the result does not change in principle, it is then for n households

${\displaystyle GRT=\sum _{i}^{n}GRS_{i}}$,

whereas for private goods, as usual, the efficiency conditions

${\displaystyle GRT=GRS_{1}=GRS_{2}=\dotsc =GRS_{n}}$

be valid. It can be shown that the competitive market solution leads to an inefficiently low provision of the public good, so that the sum of the individual MRS is greater than the MRS (underfunding).

literature

• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .
• Michael Pickhardt: Fifty Years after Samuelson's “The Pure Theory of Public Expenditure”: What are we Left With? In: Journal of the History of Economic Thought. 28, No. 4, 2006, pp. 439-460, doi : 10.1017 / S105383720000941X .
• Agnar Sandmo: Public Goods. In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd ed. Palgrave Macmillan, Internet http://www.dictionaryofeconomics.com/article?id=pde2008_P000245&edition=current#sec1 (online edition).
• Paul Samuelson: The Pure Theory of Public Expenditure. In: The Review of Economics and Statistics. 36, No. 4, 1954, pp. 387-389 ( JSTOR 1925895 ).

Remarks

1. ↑ Samuelson 1954, p. 387 f.
2. ↑ See Pickhardt 2006, p. 440.
3. ↑ In 1954 Samuelson formulated the problem in general for private and public goods. In the following, analogous to Samuelson 1955, only the special case is considered; however, the results can be transferred. For the example here cf. especially Sandmo 2008. ${\displaystyle n}$ ${\displaystyle m+1-n}$ ${\displaystyle n=m=1}$
4. ↑ This equation defines a transformation curve : A point lies on such a curve if and only if . ${\displaystyle (X_{a}^{*},X_{b}^{*})}$ ${\displaystyle T(X_{a}^{*},X_{b}^{*})=0}$
5. ↑ See, for example, Mas-Colell / Whinston / Green 1995, pp. 361-363.