Lindahl tax

Lindahl and Samuelson defined the Lindahl equilibrium in a general economy, in which there are both public and private goods. Fain, Goel and Mungala[6] present a specialized definition, for the case in which there are only public goods.

• There is a fixed budget B, and k types of divisible public goods.
• The goal is to decide on an allocation x of the budget, such that x₁+...+x_k = B.
• There are n agents; each agent i has a utility function U_i over possible allocations of the budget.

An allocation x is a Lindahl equilibrium if there exist personal price-vectors p₁,...,p_n such that the following two conditions hold:

1. For every agent i, the allocation x maximizes the utility U_i(x) subject to ${\displaystyle \mathbf {p_{i}} \cdot \mathbf {x} \leq B/n}$. That is: x is the best bundle that agent i could buy with his proportional share of the budget and his personal prices. In other words: if each agent chooses the best public bundle he can afford given his personal prices, then all agents unanimously choose the same public bundle.
2. The allocation x maximizes the profit, defined as ${\displaystyle \sum _{i=1}^{n}\mathbf {p_{i}} \cdot \mathbf {x} -|\mathbf {x} |_{1}}$. Intuitively, it is assumed that some producer can produce public goods in cost 1, and he sells them to the public; his profit is the total amount of money he gains from selling the goods in the given prices, minus the total cost of production.

The personalized price-vector p_i can be interpreted as the Lindahl tax on agent i.

Note the difference from a competitive equilibrium in a market of private goods (Fisher market):

• In a Fisher market equilibrium, there is a single price-vector for all agents, but each agent has a different bundle
• In a Lindahl equilibrium, there is a personal price-vector for each agent, but all agents have the same bundle.

A Lindahl equilibrium allocation in a market of public goods has a characterization without the price-vectors.[6]^: Thm.2.1 Specifically, an allocation x is a Lindahl equilibrium if and only if, for every good j, ${\displaystyle \sum _{i=1}^{n}{\frac {{\frac {\partial }{\partial x_{j}}}U_{j}(\mathbf {x} )}{\sum _{m}x_{m}{{\frac {\partial }{\partial x_{m}}}U_{i}(\mathbf {x} )}}}\leq {\frac {n}{B}}}$, where the inequality becomes an equality when ${\displaystyle x_{j}>0}$.

Foley[7] proved that, If the utility functions have continuous derivatives, are strictly increasing, and are strictly concave, then a Lindahl equilibrium exists in the general case of mixed public and private goods, and moreover, it lies in the fractional core. However, his proof is existential and does not provide an efficient algorithm.

The mechanism that, given agents' utilities, computes the Lindahl equilibrium, is not strategyproof, even in the setting with only public goods. This is a special case of the free-rider problem.

Fain, Goel and Mungala[6]^: Thm.2.2 present an algorithm for computing the Lindahl equilibrium using convex programming, in the special case in which agents have scalar-separable non-satiating utilities. Particularly:

• If the agents' utilities are homogeneous of degree 1 and concave, the Lindahl equilibrium allocation can be computed by maxizing the Nash welfare ${\displaystyle \sum _{i=1}^{n}\log U_{i}(\mathbf {x} )}$.[6]^: Cor.2.3 present Moreover, the gradient of this objective function can be computed using quadratic voting.
• For more general non-satiating utility functions, the Lindahl equilibrium can be computed by maximizing a potential function, that can be seen as a regularized version of the Nash welfare.

Peters, PIerczynski, Shah and Skowron[8] present an adaptation of the concept of Lindahl equilibrium to a market with indivisible public goods. In this setting, a Lindahl equilibrium may be not Pareto-efficient.

• Example: there are three goods (a, b1, b2) and two voters, where Alice values a, b1 at 1 and b2 at 0, and George values a, b2 at 1 and b1 at 0. The budget of each agent is 3, and the cost of each good is 2. Producing only {a} is a Lindahl equilibrium, with prices for Alice: 2-0.009, 2-0.006, 0.001 and prices for George: 2-0.009, 0.001, 2-0.006. This is because (1) it maximizes the profit: the producer would not gain from producing b1 or b2; (2) it maximizes the agents' utilities: no agent can afford another desirable good. But {a} is not efficient, since it is dominated by {a,b1,b2}.

If the definition is strengthened to require that the total payment for each public good exactly equals its cost, then the stronger Lindahl equilibrium notion is equivalent to a concept they call strict stable priceability. A stable-priceable committee does not always exist, but extensive simulation experiments show that an "almost" stable-priceable committee almost always exists. It is possible to check in polynomial time whether a given committee is stable-priceable.

When information about marginal benefits is available only from the individuals themselves, they tend to under report their valuation for a particular good. In doing this, an individual can lower his or her tax cost by under reporting the benefits derived from the public good or service. The incentive to lie is associated with the free rider problem; if an individual reports a lower benefit, he or she will pay less taxes, but only see a marginal decrease in the public good. 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.[5]

Preference revelation mechanisms can be used to solve that problem,[9][10] although none of these has 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.[11]

A second drawback to the Lindahl solution is that individuals may be unsure of their own valuation of a public good. Even if individuals are attempting to be honest with their willingness to pay, they may have no idea of their true value. This is especially true for public goods that individuals do not interact with on a day-to-day basis, like fireworks and national defense.[11]

Even if individuals know their marginal willingness to pay, and are honest in their reporting, the government can have extreme difficulties aggregating this into a social value. In situations where few individuals are affected by the public good, like the example below, aggregation can be relatively simple. However, in the case of national defense in the United States, compiling the marginal willingness to pay for this public good of each individual would be nearly impossible.[11]

• Foley, Duncan K. (1970), "Lindahl's Solution and the Core of an Economy with Public Goods" (PDF), Econometrica, 38 (1): 66–72, doi:10.2307/1909241, hdl:1721.1/63828, JSTOR 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. {{cite book}}: External link in |publisher= (help)
• Lindahl, Erik (1958) [1919], "Just taxation – A positive solution", in Musgrave, R. A.; Peacock, A. T. (eds.), 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.
• Starrett, David A. (1988). "5 Planning mechanisms, 16 Practical methods for large project evaluation (Groves–Clarke mechanism)". Foundations of public economics. Cambridge economic handbooks. Vol. XVI. Cambridge: Cambridge University Press. pp. 65–72, 270–71. ISBN 9780521348010.