Lindahl equilibrium

In finance , a Lindahl equilibrium is an efficient equilibrium. More precisely, a Lindahl equilibrium is a pair of individual prices and a quantity of the public good, in which the cost shares are composed in such a way that the desired total quantities of the public good are consistent and that the sum of the individual prices of the households equals the marginal costs or the Price for the public good (in units of the private good) are. The allocation in the Lindahl equilibrium is desirable because the Lindahl equilibrium fulfills the Samuelson condition and is therefore Pareto efficient . It thus shows how efficiency can be achieved in an economy if the prices for public goods take account of the principle of equivalence . However, it should be noted that the individual prices of a public good in the Lindahl equilibrium are not understood as market prices, but only represent a concept that can be used to measure the willingness of users to pay .

Problems for which the Lindahl mechanism is intended to be a solution

The Lindahl Mechanism is intended to represent a solution to the fact that the private provision of public goods, without coordination , leads to an inefficient (non-Pareto-efficient) set of public goods.

Formal definition

In the case of two households ( ) with the budget restrictions ${\displaystyle h=1,2}$

${\displaystyle x_{1}+c_{1}\cdot G\leq y_{1}}$

and

${\displaystyle x_{2}+c_{2}\cdot G\leq y_{2}}$

the Lindahl equilibrium must meet the following condition:

${\displaystyle c_{1}+c_{2}=c={\frac {p_{g}}{p_{x}}}}$,

d. H. the sum of the individual prices of household 1 and household two correspond to the marginal costs or the marginal rate of transformation , where the price for the public good and the price for the private good denote. ${\displaystyle c}$ ${\displaystyle p_{g}}$ ${\displaystyle p_{x}}$

On the other hand, there is agreement on the desired total amounts of public goods , so that applies ${\displaystyle G_{L}}$

${\displaystyle G_{1}(c_{1})=G_{2}(c_{2})=G_{L}}$,

With:

1. ${\displaystyle G_{1}}$: Demand for the public good of household 1
2. ${\displaystyle G_{2}}$: Demand for the public good from household 2

Pareto efficiency

It can be shown that the Samuelson condition is fulfilled in the Lindahl equilibrium and the allocation that occurs in the Lindahl equilibrium is therefore Pareto efficient. The sum of the optimality conditions of the individual households results in the Samuelson condition

${\displaystyle \sum \limits _{h=1}^{H}GRS_{h}=\sum \limits _{h=1}^{H}c_{h}}$.

In the case of 2 households

${\displaystyle {\frac {\partial u_{1}\left(x_{1},G\right)/\partial G}{\partial u_{1}\left(x_{1},G\right)/\partial x_{1}}}\;+\;{\frac {\partial u_{2}\left(x_{2},G\right)/\partial G}{\partial u_{2}\left(x_{2},G\right)/\partial x_{2}}}\;=\;{\frac {p_{g}}{p_{x}}}\;=\;c\quad {\text{(Samuelson-Bedingung)}}}$

or.

${\displaystyle GRS_{1}+GRS_{2}=MRT}$,

where denotes the utility function of household 1 and the utility function of household 2. ${\displaystyle u_{1}\left(x_{1},G\right)}$ ${\displaystyle u_{2}\left(x_{2},G\right)}$

criticism

The Lindahl equilibrium is often viewed critically, as households' willingness to pay is usually private information. If the households' willingness to pay is private information, the Lindahl equilibrium cannot be implemented and there is no efficient allocation, but an undersupply of the public good.

Individual evidence

1. ^ Jürgen Eichberger: Fundamentals of microeconomics. 2004.
2. ^ Mark Walker, Lindahl Equilibrium , University of Arizona