Lindahl Model: Concept, Equilibrium & Limitations

6 months ago

Atiqur Rahman

11 minutes

Erik Lindahl (21 November 1891 – 6 January 1960) was a Swedish economist and a professor of economics at Uppsala University. He was also an advisor to the Government of Sweden and the central bank.

Lindahl modelled the question of financing public goods in harmony with individual benefits. The quantity of the public good satisfies the requirement that the total marginal benefit equals the marginal cost of providing the good.

The necessary and sufficient conditions for such an equilibrium are:

(i) the sum of the declared willingness be greater than the cost of provision, and
(ii) the minimum willingness to pay is positive & non-zero.

Erik Lindahl was deeply influenced by his professor and mentor Knut Wicksell and proposed a method for financing public goods permissible to show that consensus politics is 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 individual’s tax price is set equivalent to the marginal benefits received, each individual is made better off by the provision of the public good and may accordingly agree to have that service level provided.

Concept of Lindahl’s Model

A Lindahl tax is a system of taxation in which individuals pay for the provision of a public good in accordance with their marginal benefits. So, each individual pays according to his/her marginal benefit derived from the public good.

For e.g., 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 college student 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. In such cases, the problem of the supply of the public good at optimal levels arises. Lindahl taxation is a solution to this problem.

Lindahl tries to solve the following three problems:

Extent of state activity
Allocation of the total expenditure among various goods & services
Allocation of tax burden

Consider the following Figure 1:

Supply and Demand for State Services/ Public Goods

In the Lindahl model, if SS is the supply curve of public goods, it is assumed that the production function of public/social goods is linear and homogenous. DDa is the demand curve of taxpayer A, and DDb is the demand curve of taxpayer B. The vertical summation of the two demand curves results in the community’s total demand schedule for public goods. A and B pay different proportions of the cost of the services.

When QN is the amount of public goods produced, A contributes NE, and B contributes NF; the cost of supply is NG. Since the state is non-profit, it increases its supply to QM. At this level, A contributes MJ and B contributes MR (the total cost of supply). Equilibrium is reached at point P on an Voluntary-exchange basis.

Lindahl Equilibrium

Lindahl taxes are also known as Benefit taxes. Lindahl equilibrium is a sort of economic equilibrium under such a tax. It is a method of finding the optimal level for the supply of public goods or services. The Lindahl equilibrium happens when the total per unit price paid by each individual equals the total per unit cost of the public good.

Lindahl equilibrium shows how efficiency can be sustained in an economy with personalized prices. Johansen (1963) gave the complete interpretation of the concept of “Lindahl equilibrium.” The simple 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 fulfils the Samuelson rule and is therefore said to be Pareto efficient despite the existence of public goods. It also establishes how efficiency can be reached in an economy with public goods by the use of personalized prices. The personalized prices equate the individual estimate for a public good to the cost of the public good.

Lindahl pricing and taxation require 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 underreport their valuation for a particular good, and this gives rise to a “preference revelation problem.”

Each individual can lower his tax cost by underreporting 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 underreport one’s true benefits under Lindahl taxation resemble those of a Prisoner’s dilemma, and people will be inclined to underreport their demands for public goods or services.

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 Lindahl equilibrium, it can be conveniently said that it is Pareto Optimal. This can be shown by following the following steps.

Consider figure 2:

We take a demand curve for a public good. X will want more goods to consume when the price of the public good is less. 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, let’s consider the demand curve of another person; let’s say Y. Y sees the vertical axis turned the other way around, with the full price on the bottom and the percentage decreasing as you move upwards. Like X, Y will also demand more as his observed price goes down.

Consider figure 3:

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 an 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 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.

Consider the figure 4:

Lindahl Pricing

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 of Lindahl

We assume that there are two goods in an 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_(ELSE) = MRS_(PERSON1)

This is just the usual price ratio/marginal rate of substitution deal; the only change is that we multiply Ppublic 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_(ELSE) = MRS_(PERSON2)

Now, we have both individuals’ utility maximizing. Moreover, in a competitive equilibrium, the marginal cost ratio (price ratio) should be equal to the marginal rate of transformation, i.e.:

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

Limitations of Lindhal’s Model

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, the mechanism design of Groves, which is strongly individually incentive compatible, and Vickrey auctions could be used to overcome this problem).

Another drawback is that Lindahl’s prices may be unfair. Consider a television broadcast antenna that is discretionarily placed in an area. Those living near the antenna will receive a clear signal, while those living farther away will not receive a 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).

Lindahl Model: Concept, Equilibrium & Limitations

Concept of Lindahl’s Model

Lindahl Equilibrium

Lindahl Tax and Pareto Optimality

Lindahl Pricing

Mathematical Representation of Lindahl

Limitations of Lindhal’s Model

Read More in: Theory of Public Finance

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Concept of Lindahl’s Model

Lindahl Equilibrium

Lindahl Tax and Pareto Optimality

Lindahl Pricing

Mathematical Representation of Lindahl

Limitations of Lindhal’s Model

Read More in: Theory of Public Finance

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