Lindahl equilibrium

A Lindahl tax is a form of taxation in which individuals pay for the provision of a public good

Public good

In economics, a public good is a good that is non-rival and non-excludable. Non-rivalry means that consumption of the good by one individual does not reduce availability of the good for consumption by others; and non-excludability means that no one can be effectively excluded from using the good...

according to their marginal benefits. So each individual pays according to his/her marginal benefit derived from the public good. eg. If A loves scenic beauty and likes to be close to nature he might be ready to pay 5 dollars per day for sitting in a park, whereas a housewife who does not visit the park very often will not be ready to pay so much, but might agree to pay 1 dollar. So a person who values the good more pays more.

Lindahl taxes are sometimes known as benefit taxes. A Lindahl equilibrium is a state of economic equilibrium

Economic equilibrium

In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...

under such a tax.
Individuals in a society have different preferences based on their nature, personal choice etc. So an individual's willingness to pay for a public good is a function of many factors, like income, preference etc. So a student will want to pay just 1 dollar for entering a museum but a business man will be ready to pay 10 dollars for the same museum. So in such cases the problem of supply of the public good, at optimal levels arises. Lindahl taxation is a solution for this problem.

Definition

A Lindahl tax is an individual share of the collective tax burden of an economy.
The optimal level of a public good is that quantity at which the willingness to pay for one more unit of the good, taken in totality for all the individuals is equal to the marginal cost of supplying that good. Lindahl tax is the optimal quantity times the willingness to pay for one more unit of that good at this quantity.

A Lindahl equilibrium is a method for finding the optimum level for the supply of public goods or services.This idea was given by Erik Lindahl

Erik Lindahl

Erik Robert Lindahl was a Swedish economist and a member of the Stockholm school....

in 1919. The Lindahl equilibrium happens when the total per-unit price paid by each individual equals the total per unit cost of the public good.
The Lindahl equilibrium is Pareto efficient and it can be proved that an equilibrium exists for different environments.

Lindahl equilibrium describes how efficiency can be sustained in an economy with personalised prices. Johansen (1963) gave the complete interpretation of the concept of "Lindahl Equilibrium." The basic assumption of this concept is that every household's consumption decision is based on the share of the cost they must provide for the supply of the particular public good.

The importance of Lindahl Equilibrium is that it fulfills the Samuelson rule and is therefore said to be pareto efficient, despite the existence of public goods. It also demonstrates how efficiency can be reached in an economy with public goods by the use of personalised prices. The personalised prices equate the individual valuation for a public good to the cost of the public good.

Background

Erik Lindahl

Erik Robert Lindahl was a Swedish economist and a member of the Stockholm school....

, was deeply influenced by his professor and mentor Knut Wicksell

Knut Wicksell

Johan Gustaf Knut Wicksell was a leading Swedish economist of the Stockholm school. His economic contributions would influence both the Keynesian and Austrian schools of economic thought....

, proposed a method for financing public goods in order to show that a consensus politics is actually possible. As people are different in nature, their preferences are different, and consensus requires each individual to pay a somewhat different tax for every service, or good that he consumes.. If each person's tax price is set equal to the marginal benefits received at the ideal service level, it turns out that each person will agree to have that service level provided.

Problem with Lindahl taxation

Lindahl pricing and taxation requires the knowledge of the demand functions for each individual for all private and public goods.When information about marginal benefits is available only from the individuals themselves, they tend to under report their valuation for a particular good, this gives rise to a "preference revelation problem." Each individual can lower his tax cost by under reporting his benefits derived from the public good or service. This informational problem shows that survey-based Lindahl taxation is not incentive compatible. Incentives to understate or under report one's true benefits under Lindahl taxation resemble those of a Prisoner's dilemma

Prisoner's dilemma

The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...

, and people will be inclined to under report their demands for the public goods or service.

Preference revelation mechanisms can be used to solve that problem, although none of these have been shown to completely address the problem. Among others the Vickrey–Clarke–Groves mechanism is an example of this, ensuring true values are revealed and that a public good is provided only when it should be. The allocation of cost is taken as given and the consumers will report their net benefits (benefits-cost). The public good will be provided if the sum of the net benefits of all consumers is positive. If the public good is provided side payments will be made reflecting the fact that truth telling is costly. The side payments internalize the net benefit of the public good to other players. The side payments must be financed from outside the mechanism. In reality these preference revelation mechanisms are difficult to implement as the size of the population makes it costly both in terms of money and time.

Demand Revelation

The concept of demand revelation was developed by Edward Clarke, and further by Nicolaus Tideman and Gordon Tullock. It uses economics principles to make efficient social choices. Individuals who manipulate their outcomes, create a social cost:

by creating a situation in which the people have to pay more than what they want, or
by not having the supply of a good that they are willing to pay more for. By paying this social cost, they compensate society for getting their way.

The Demand Revelation theory uses the Clarke tax or pivot mechanism, to ensure that individuals will consider the fact that their choices have a social cost on the social outcome, thus ensuring that they truthfully reveal their preferences,and hence overcoming the free rider problem of public goods provisioning.

Revealed Preference Theory

It is an economic theory of consumption, given by Samuelson, it is a method of observing people's preferences and purchasing power. If a consumer purchases one commodity instead of the other, the consumer reveals his preference for that good. Hence the good is said to be revealed preferred over the other.

The main idea of the theorem is that when an individual spends his entire income on a good, the demand behaviour of the person is in accordance with the Samuelson's Weak Axiom of Revealed Preference.
Now, in the figure, AB is the budget line. All the points below AB are inferior points, hence AOB is the inferior zone. All the points on the left and right of the preferred zone and above the budget line are not ordered by the consumer hence these are the ignorance zones.

Lindahl tax and Pareto optimality

A very important question is that whether a Lindahl tax is a Pareto Optimal equilibrium? A Pareto Optimal allocation happens with public goods when the total of the marginal rates of substitution (MRS) equals the marginal rate of transformation (MRT). So if it can be shown that this holds true in a Lindahl equilibrium, it can be conveniently said that it is Pareto Optimal. This can be shown by following the following steps:
We take a demand curve for a public good.
The less the price of the public good, the more will X want to consume.
Let the horizontal line (dashed) be the full price of the public good.
Now here, the demand curve implies that X will demand very less.
But what if instead of the price decreasing, the percentage of the price X has to pay decreases?
Now X sees the price going down, so his demand for the good increases.
Now lets consider the demand curve of another person, lets say Y. Y sees the vertical axis turned the other way around, with the full price on the bottom and percentage decreasing as you move upwards. Like X, Y will also demand more as his observed price goes down.

Now as Y observes the price going down it also means that we move further up the vertical axis.
Equilibrium is when both X and Y demand the equal amount of the good.
This is possible only when the demand curves of both X and Y intersect each other. If a line is drawn over the price axis from that point of intersection, we get the percentage share for each person that is required to get that price.
In the Lindahl tax scheme it is essential that the system should provide for a pareto optimal output of the public good. The other important condition is that the Lindahl tax scheme should connect the tax paid by an individual to the benefits he derives. This system kind of promotes justice. If the individual's tax payment is equivalent to the benefits received by him, and if this linkage is good enough then it leads to Pareto Optimality.

So it is observed that X is paying P*45% per unit, and Y is paying P*55% per unit, and the economy produces Q* units.
This point is called the Lindahl Equilibrium, and the corresponding prices are called Lindahl prices.

Mathematical representation

We assume that there are two goods in a n economy:the first one is a "public good," and the second is “everything else.” The price of the public good can be assumed to be P_public and the price of everything else can be P_else.

α*P_(PUBLIC)/P_(EVERY) = MRS_(PERSON1)

This is just the usual price ratio/marginal rate of substitution deal the only change is that we multiply P_public by α to allow for the price adjustment to the public good.
Similarly, Person 2 will choose his bundle such that:

(1-ɑ)*P_(PUBLIC)/P_(EVERY)= MRS_(PERSON2)

Now we have both individuals' utility maximizing. We know that in a competitive equilibrium, the marginal cost ratio (price ratio)should be equal to the marginal rate of transformation, or

MC_(PUBLIC)/MC_(EVERY)=[P_(PUBLIC)/P_(EVERY)]=MRT

Issues

Lindahl prices do face some serious drawbacks. First, individual demand curves, and thus individual preferences, are not easily known. Due to the free-rider problem, people have the incentive to hide their true preferences and thus their marginal valuation (However, mechanism design of Groves, which is strongly individually incentive compatible, and Vickery auctions could be used to overcome this problem).

A second drawback to Lindahl prices is that they may be unfair. Consider a television broadcast antenna that is arbitrarily placed in an area. Those living near the antenna will receive a clear signal while those living farther away will receive a less clear signal. Those living close to the antenna will have a relatively low marginal value for additional wattage (thus paying a lower Lindahl price) compared to those living farther away (thus paying a higher Lindahl price).

Some points still remain unanswered, these form the main issues of Lindahl taxation:

How much should the government charge(or tax) each individual to cover the cost of supply of the public good?
In Lindahl pricing everyone is taxed according to the benefit they derive from the public good, this information is difficult to collect, and people also have the incentive to under report.

Benefits Of Lindahl pricing

Lindahl pricing helps in covering the cost of the public sector good, and sometimes also giving a profit if the marginal cost of the good is increasing.
It leads to an efficient allocation of the public goods.
It charges people according to the marginal benefits they derive from the good, so if person 1 has a higher demand for the a particular public good A, as compared to person 2, then person 1 will have to pay more (than person 2).
Also because of Lindahl pricing everyone is left better off than they would have been without the public good. Each person gets a consumer surplus, equal to the area below his/her demand curve, and above the Lindahl price.
Lindahl Pricing is beneficial as it reveals the consumer's marginal benefits form the non excludable goods.
Lindahl tax is a progressing form of taxation. Example: If the willingness to pay for security guards increases with increase in income (which is rational) then Lindahl tax increases with income.

Drawbacks Of Lindahl pricing

There are some drawbacks to Lindahl pricing. Although using Lindahl prices to finance the cost of some public good is often called \benefit pricing", these prices are marginal benefits, not total benefits.

The main problem with Lindahl pricing is the availability of the relevant and complete information.
Every person's individual demand curve is required to be known so that a differential price can be charged. This obviously becomes very difficult as a lot of information is required and knowing each individual's demand curve for each good is almost impossible.
Another problem that arises is that even if we somehow manage to gather all this information, the authenticity of the information is highly questionable, as if people get to know that they will be charged according to their marginal benefits they will tend to under report their demands for the good.

Definition