Lindahl tax

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A Lindahl tax is a form of taxation conceived by Erik Lindahl in which individuals pay for public goods according to their marginal benefits. In other words, they pay according to the amount of satisfaction or utility they derive from the consumption of an additional unit of the public good.

It can be seen as an individual's 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.^[1]

History

Erik Lindahl was deeply influenced by his professor and mentor Knut Wicksell and proposed a method for financing public goods in order 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 person's tax price is set equal to the marginal benefits received at the ideal service level, each person is made better off by provision of the public good and may accordingly agree to have that service level provided.

Lindahl equilibrium

A Lindahl equilibrium is a state of economic equilibrium under a Lindahl tax as well as a method for finding the optimum level for the supply of public goods or services that happens when the total per-unit price paid by each individual equals the total per-unit cost of the public good. It can be shown that an equilibrium exists for different environments.^[2] Therefore the Lindahl equilibrium describes how efficiency can be sustained in an economy with personalised prices. Leif Johansen gave the complete interpretation of the concept of "Lindahl equilibrium", which assumes that household consumption decisions are based on the share of the cost they must provide for the supply of the particular public good.^[3]

The importance of Lindahl equilibrium is that it fulfills the Samuelson condition and is therefore Pareto efficient,^[2] despite the good in question being a public one. 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.

Criticism

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 traditional public goods game.

Preference revelation mechanisms can be used to solve that problem,^[4]^[5] although none of these have been shown to completely and satisfactorily address it. 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, preference revelation is difficult as the size of the population makes it costly both in terms of money and time.^[6]

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).^[7]

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_(else) = 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_(else)= MRS_(person2)

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

MC_(public)/MC_(else)=[P_(public)/P_(else)]=MRT

References

↑ Equity: In Theory and Practice, p. 103.
↑ 2.0 2.1 Mark Walker, "Lindahl Equilibrium", University of Arizona
↑ Leif Johansen (September 1963). "Some Notes on the Lindahl Theory of Determination of Public Expenditures". International Economic Review 4 (3): 346–358.
↑ Revelation, Demand. "Demand Revelation". Economic Theory. Retrieved 30 September 2011.
↑ "Fred Foldvary on Demand Revelation: Better than Voting"
↑ Backhaus, Jürgen Georg;, Wagner, Richard E. (2004). Handbook of Public Finance. ISBN 978-1-4020-7863-7.
↑ http://www.stanford.edu/~jay/micro_class/lecture19.pdf

Foley, Duncan K. (1970), "Lindahl's Solution and the Core of an Economy with Public Goods", Econometrica 38 (1): 66–72, doi:10.2307/1909241 .
"Public Economics" by Gareth D. Myles (October 2001)
Laffont, Jean-Jacques (1988). "2.4 Lindahl equilibrium, 2.5.3 Majority voting or the law of the median voter". Fundamentals of public economics. MIT. pp. 41–43, 51–53. ISBN 978-0-262-12127-9. ISBN 978-0-262-12127-9.
Lindahl, Erik (1958) [1919], "Just taxation—A positive solution", in Musgrave, R. A.; Peacock, A. T., Classics in the Theory of Public Finance, London: Macmillan .
Salanié, Bernard (2000). "5.2.3 The Lindahl equilibrium". Microeconomics of market failures (English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris) ed.). Cambridge, MA: MIT Press. pp. 74–75. ISBN 978-0-262-19443-3. ISBN 0-262-19443-0.
Starrett, David A. (1988). "5 Planning mechanisms (pp. 65–72), 16 Practical methods for large project evaluation (Groves–Clarke mechanism, pp. 270–271)". Foundations of public economics. Cambridge economic handbooks XVI. Cambridge: Cambridge University Press. pp. 65–72, 270–271.

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