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Article: Lindahl tax
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(Lindahl tax and Pareto optimality: cleanup; article should be clean now)
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A '''Lindahl tax''' is a form of [[taxation]] in which individuals pay for the provision of a [[public good]] according to their [[marginal benefit]]s. So each individual pays according to his/her marginal benefit derived from the public good. 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.
 
  +
Our biggest prize appeared four days later. It was a pocket watch that wouldn’t run, on
 
  +
a chain with an aluminum knife.
Lindahl taxes are sometimes known as '''[[Benefit principle|benefit taxes]]'''. A '''Lindahl equilibrium''' is a state of [[economic equilibrium]] 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.<ref>http://www.saylor.org/site/wp-content/uploads/2011/05/ECON305-2.2.5-Lindahl-Handout-FINAL.pdf</ref>
 
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“You reckon it’s white gold, Jem?”
 
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“Don’t know. I’ll show it to Atticus.”
{{Taxation}}
 
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Atticus said it would probably be worth ten dollars, knife, chain and all, if it were new.
 
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“Did you swap with somebody at school?” he asked.
== Definition ==
 
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“Oh, no sir!” Jem pulled out his grandfather’s watch that Atticus let him carry once a
A Lindahl tax is an individual share of the collective tax burden of an economy.<ref>http://www.answers.com/topic/lindahl-equilibrium</ref>
 
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week if Jem were careful with it. On the days he carried the watch, Jem walked on eggs.
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.<ref name="google1">[http://books.google.co.in/books?id=H0IQ0PKZ4WYC&pg=PA103&lpg=PA103&dq=lindahl+tax&source=bl&ots=CEDhLuQdwh&sig=YN_6zVVOA2PfRc5p_TT1lMP8ZLA&hl=en&ei=nY6BTrKLAYLtrAfBm4X2Dw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEUQ6AEwBA#v=onepage&q=lindahl%20tax&f=false ''Equity: In Theory and Practice''], p. 103.</ref>
 
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“Atticus, if it’s all right with you, I’d rather have this one instead. Maybe I can fix it.”
 
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When the new wore off his grandfather’s watch, and carrying it became a day’s
== Lindahl equilibrium ==
 
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burdensome task, Jem no longer felt the necessity of ascertaining the hour every five
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]] 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.
 
  +
minutes.
The Lindahl equilibrium is [[Pareto efficient]] and it can be proved that an equilibrium exists for different environments.<ref>http://www.u.arizona.edu/~mwalker/11_PublicGoods/LindahlEquilibrium.pdf</ref>
 
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He did a fair job, only one spring and two tiny pieces left over, but the watch would not
 
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run. “Oh-h,” he sighed, “it’ll never go. Scout—?”
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.<ref>http://www.jstor.org/stable/2525312</ref>
 
  +
“Huh?”
 
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“You reckon we oughta write a letter to whoever’s leaving us these things?”
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.<ref>{{cite book|last=Myles|first=Gareth|isbn=978-0-262-12127-9}}</ref>
 
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“That’d be right nice, Jem, we can thank ‘em—what’s wrong?”
 
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Jem was holding his ears, shaking his head from side to side. “I don’t get it, I just don’t
== Background ==
 
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get it—I don’t know why, Scout…” He looked toward the livingroom. “I’ve gotta good
{{Main|Erik Lindahl}}
 
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mind to tell Atticus—no, I reckon not.”
[[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.
 
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“I’ll tell him for you.”
 
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“No, don’t do that, Scout. Scout?”
== Problem with Lindahl taxation ==
 
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“Wha-t?”
{{Public finance}}
 
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He had been on the verge of telling me something all evening; his face would brighten
Lindahl pricing and taxation requires the knowledge of the [[demand function]]s 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]], and people will be inclined to under report their demands for the public goods or service.
 
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and he would lean toward me, then he would change his mind. He changed it again.
 
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“Oh, nothin‘.”
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 auction|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.<ref>http://www.enotes.com/topic/Lindahl_tax</ref> 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.<ref>{{cite book|last=Backhaus, Jürgen Georg;|first=Wagner, Richard E.|title=Handbook of Public Finance|year=2004|isbn=978-1-4020-7863-7}}</ref>
 
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“Here, let’s write a letter.” I pushed a tablet and pencil under his nose.
 
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“Okay. Dear Mister…”
=== Demand revelation ===
 
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“How do you know it’s a man? I bet it’s Miss Maudie—been bettin‘ that for a long
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.<ref>{{cite web|last=Revelation|first=Demand|title=Demand Revelation|url=http://www.amazon.com|work=Economic Theory|accessdate=30 September 2011}}</ref><ref>[http://www.progress.org/fold183.htm Fred Foldvary on Demand Revelation: Better than Voting]</ref>
 
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time.”
 
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“Ar-r, Miss Maudie can’t chew gum—” Jem broke into a grin. “You know, she can talk
=== Revealed preference theory ===
 
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real pretty sometimes. One time I asked her to have a chew and she said no thanks,
{{Main|Revealed preference theory}}
 
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that—chewing gum cleaved to her palate and rendered her speechless,” said Jem
It is an economic theory of consumption, given by Samuelson, it is a method of observing people's preferences and purchasing power.<ref>[http://www.investopedia.com/terms/r/revealed-preference.asp#axzz1Z7zWJG80 Revealed Preference Definition]</ref> 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.
 
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carefully. “Doesn’t that sound nice?”
 
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“Yeah, she can say nice things sometimes. She wouldn’t have a watch and chain
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.'''<ref>{{cite web|last=Theory|first=Revealed Preference|title=Revealed Preference theory|url=http://www.vanderbilt.edu/Econ/sempapers/Pattanaik.pdf|work=Microeconomics|accessdate=30 September 2011}}</ref>
 
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anyway.”
 
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“Dear sir,” said Jem. “We appreciate the—no, we appreciate everything which you
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.
 
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have put into the tree for us. Yours very truly, Jeremy Atticus Finch.”
<ref>http://www.tutorsonnet.com/homework_help/micro_economics/consumption_theory/revealed_preference_theory_of_demand_assignment_help_online_tutoring.htm</ref>
 
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“He won’t know who you are if you sign it like that, Jem.”
<ref>{{cite web|last=Basu|first=Kaushik|title=Revealed Preference of government|url=http://books.google.co.in/books?id=rbU8AAAAIAAJ&pg=PA10&lpg=PA10&dq=revealed+preference+diagram&source=bl&ots=rV6_OQdh88&sig=DUq630j9My5xpCL1vkMOWpbtnVE&hl=en&ei=MLqKTvbSHJDsrQeW1MyNAg&sa=X&oi=book_result&ct=result&resnum=5&sqi=2&ved=0CEEQ6AEwBA#v=onepage&q=revealed%20preference&f=false|work=Microeconomics|accessdate=4 October 2011}}</ref>
 
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Jem erased his name and wrote, “Jem Finch.” I signed, “Jean Louise Finch (Scout),”
 
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beneath it. Jem put the note in an envelope.
== Lindahl tax and Pareto optimality ==
 
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Next morning on the way to school he ran ahead of me and stopped at the tree. Jem
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 rate of substitution|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:
 
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was facing me when he looked up, and I saw him go stark white.
 
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“Scout!”
[[File:Lindahl.jpg|thumb|right|300px|X's demand curve]]
 
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I ran to him.
 
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Someone had filled our knot-hole with cement.
We take a demand curve for a public good.
 
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“Don’t you cry, now, Scout… don’t cry now, don’t you worry-” he muttered at me all the
The less the price of the public good, the more will X want to consume.
 
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way to school.
Let the horizontal line (dashed) be the full price of the public good.
 
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When we went home for dinner Jem bolted his food, ran to the porch and stood on the
Now here, the demand curve implies that X will demand very less.
 
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steps. I followed him. “Hasn’t passed by yet,” he said.
But what if instead of the price decreasing, the percentage of the price X has to pay decreases?
 
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Next day Jem repeated his vigil and was rewarded.
Now X sees the price going down, so his demand for the good increases.
 
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“Hidy do, Mr. Nathan,” he said.
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.<ref name="stanford1">http://www.stanford.edu/~sandersn/130A/Lindahl.pdf</ref>
 
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“Morning Jem, Scout,” said Mr. Radley, as he went by.
 
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“Mr. Radley,” said Jem.
[[File:Lindahl 1.jpg|thumb|right|300px|Y's demand curve]]
 
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Mr. Radley turned around. “To Kill a Mockingbird” By Nelle Harper Lee 34
 
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“Mr. Radley, ah—did you put cement in that hole in that tree down yonder?”
Now as Y observes the price going down it also means that we move further up the vertical axis.
 
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“Yes,” he said. “I filled it up.”
Equilibrium is when both X and Y demand the equal amount of the good.
 
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“Why’d you do it, sir?”
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.
 
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“Tree’s dying. You plug ‘em with cement when they’re sick. You ought to know that,
<ref name="stanford1"/>
 
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Jem.”
 
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Jem said nothing more about it until late afternoon. When we passed our tree he gave
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]].<ref>[http://books.google.co.in/books?id=7pZV5O5n_WwC&pg=PA169&lpg=PA169&dq=lindahl+tax+pareto+optimal&source=bl&ots=JlFv7XLo1I&sig=Qxlbr9VLZJVfU4DUeYdKOvMzlUk&hl=en&ei=sZhxTvu1OozrrQe9_YD3Bg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCoQ6AEwAg#v=onepage&q=lindahl%20tax%20pareto%20optimal&f=false ''Welfare Economics and Social Choice Theory''], p. 169.</ref>
 
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it a meditative pat on its cement, and remained deep in thought. He seemed to be
 
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working himself into a bad humor, so I kept my distance.
[[File:L11.jpg|thumb|left|300px|Lindahl Pricing]]
 
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As usual, we met Atticus coming home from work that evening. When we were at our
 
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steps Jem said, “Atticus, look down yonder at that tree, please sir.”
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.
 
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“What tree, son?”
This point is called the '''Lindahl equilibrium''', and the corresponding prices are called '''Lindahl prices'''.
 
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“The one on the corner of the Radley lot comin‘ from school.”
 
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“Yes?”
== Mathematical representation ==
 
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“Is that tree dyin‘?”
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<sub>public</sub> and the price of everything else can be P<sub>else</sub>.
 
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“Why no, son, I don’t think so. Look at the leaves, they’re all green and full, no brown
* '''α*P<sub>(PUBLIC)</sub>/P<sub>(EVERY)</sub> = MRS<sub>(PERSON1)</sub>'''
 
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patches anywhere—”
 
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“It ain’t even sick?”
This is just the usual price ratio/marginal rate of substitution deal the only change is that we multiply P<sub>public</sub> by α to allow for the price adjustment to the public good.
 
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“That tree’s as healthy as you are, Jem. Why?”
Similarly, Person 2 will choose his bundle such that:
 
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“Mr. Nathan Radley said it was dyin‘.”
* '''(1-ɑ)*P<sub>(PUBLIC)</sub>/P<sub>(EVERY)</sub>= MRS<sub>(PERSON2)</sub>'''
 
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“Well maybe it is. I’m sure Mr. Radley knows more about his trees than we do.”
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
 
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Atticus left us on the porch. Jem leaned on a pillar, rubbing his shoulders against it.
* '''MC<sub>(PUBLIC)</sub>/MC<sub>(EVERY)</sub>=[P<sub>(PUBLIC)</sub>/P<sub>(EVERY)</sub>]=MRT'''
 
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“Do you itch, Jem?” I asked as politely as I could. He did not answer. “Come on in,
 
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Jem,” I said.
== Issues ==
 
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“After while.”
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 Vickrey auctions could be used to overcome this problem).
 
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He stood there until nightfall, and I waited for him. When we went in the house I saw
 
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he had been crying; his face was dirty in the right places, but I thought it odd that I had
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).<ref>http://www.stanford.edu/~jay/micro_class/lecture19.pdf</ref>
 
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not heard him.
 
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Chapter 8
==See also==
 
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For reasons unfathomable to the most experienced prophets in Ma
{{Wikipedia books|Finance}}
 
* [[Erik Lindahl|Lindahl, Erik]]
 
* [[Public choice theory]]
 
* [[Public finance]]
 
* [[Samuelson condition]]
 
* [[Taxation]]
 
* [[Tax choice]]
 
 
== References ==
 
{{Reflist}}
 
* {{Citation |last=Foley |first=Duncan K. |authorlink=Duncan K. Foley |year=1970 |title=Lindahl's Solution and the Core of an Economy with Public Goods |journal=[[Econometrica]] |volume=38 |issue=1 |pages=66–72 |doi=10.2307/1909241 }}.
 
* [http://people.exeter.ac.uk/gdmyles/papers/pdfs/pubec.pdf "Public Economics" by Gareth D. Myles (October 2001)]
 
* {{cite book |last=Laffont |first=Jean-Jacques |authorlink=Jean-Jacques Laffont |year=1988 |chapter=2.4 Lindahl equilibrium, 2.5.3 Majority voting or the law of the median voter|pages=41–43, 51–53|title=Fundamentals of public economics |url=http://books.google.com/books?q=editions:ISBN 0-262-12127-1&id=O5MnAQAAIAAJ |publisher=[http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=7534 MIT]
 
|isbn=978-0-262-12127-9, 978-0-262-12127-9}}
 
* {{Citation |last=Lindahl |first=Erik |year=1958 |origyear=1919 |chapter=Just taxation—A positive solution |title=Classics in the Theory of Public Finance |editor1-last=Musgrave |editor1-first=R. A. |editor2-last=Peacock |editor2-first=A. T. |location=London |publisher=Macmillan }}.
 
* {{cite book |last=Salanié |first=Bernard |chapter=5.2.3 The Lindahl equilibrium|pages=74–75|title=Microeconomics of market failures |edition=English translation of the (1998) French ''Microéconomie: Les défaillances du&nbsp;marché'' (Economica, Paris) |year=2000 |publisher=MIT&nbsp;Press |location=Cambridge,&nbsp;MA |isbn=0-262-19443-0, 978-0-262-19443-3}}
 
* {{cite book|last=Starrett|first=David&nbsp;A.|title=Foundations of public economics|series=Cambridge economic handbooks|volume=XVI|year=1988|number=|chapter=5 Planning mechanisms (pp. 65–72), 16 Practical methods for large project evaluation (Groves–Clarke mechanism, pp. 270–271)|pages=65–72, 270–271|publisher=Cambridge University Press|location=Cambridge|url=http://books.google.com/books?hl=sv&lr=&id=R35yljdyyIkC&oi=fnd&pg=PR11&dq=David+A.+Starrett,+public+economics&ots=66-J_MrQY4&sig=oT54I-xTJ9_fQJ322VyQlOnuJ5s#v=onepage&q=nonconvex%20OR%20nonconvexities&f=false|ref=harv}}
 
 
== External links ==
 
* {{cite web|url=http://www.econ.ucsb.edu/~tedb/Courses/UCSBpf/readings/lindahl.pdf|title=Just taxation—A positive solution}}
 
* http://dept.econ.yorku.ca/~sam/4080/pub_goods/4.pdf
 
 
{{Microeconomics|state=collapsed}}
 
 
[[Category:Economics]]
 
[[Category:Public finance]]
 
[[Category:Economic systems]]
 
[[Category:Economic theories]]
 
[[Category:Economics terminology]]
 
[[Category:General economics]]
 
[[Category:Other economic systems]]
 
[[Category:Taxation]]
 
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Date: Thursday, the 13th of October 2016 at 02:37:58 PM
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Thursday, the 13th of October 2016 at 02:37:58 PM #106409
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