http://dept.econ.yorku.ca/~sam/4080_w12/free_rider/1.pdf
http://dept.econ.yorku.ca/~sam/4080_w12/free_rider/2.pdf
http://dept.econ.yorku.ca/~sam/4080_w12/free_rider/3.pdf
Preference Revelation : (a)A Project of Fixed Size When there is a public good,efficiency requires that the sum of people’s marginal rates of substitution ofthe public good for the private good ( their willingness to pay for the publicgood ) must equal the marginal rate of transformation ( the cost of the publicgood ). If some agency — government, private firm, non–profit agency — is totry and achieve this efficient allocation, it must know people’s demand curvesfor public goods. As Samuelson emphasized, the really big problem with publicgoods is not that the efficiency condition is different from the efficiencycondition with private goods, but that efficiency can’t be achieved without alot of information about people’s preferences. So what actually happens in theprovision of public goods such as information, national defense andbroadcasting? Sometimes some of these goods ( some of the ones which areexcludable to some degree ) are privately provided. The allocation here willnot be efficient, since typically these firms exclude some people from somebenefits, because the firms are not able to charge different prices todifferent people. Sometimes these goods are financed voluntarily — which meansthat they will be inefficiently under–provided if people behave in aself–interested manner in deciding their contributions. Sometimes they areprovided by government. Since governments ( at least those in democraticcountries ) are elected, the government officials have some incentive to pleasepeople if they want to get re–elected. But elections do not enable people toconvey exactly their preferences over public goods to politicians. Public good providers could tryand find out exactly what are people’s preferences for public goods by askingthem. This seldom happens in practice. If people were asked their preferencesfor public goods, it seems reasonable to conclude that many people would behavein a self–interested manner in responding. That is, people would not tell thetruth about their demand curves for public goods unless it was in their owninterest to do so. In other words, if people weresurveyed about their demand curves for public goods, we should not assume thatpeople respond truthfully. Rational, selfish people should try and manipulatethe survey in their own interests. That makes the survey a game, that is a situation in people behave strategically. Rational selfish people, responding to a survey,will understand the rules of the game, that is how their responses to thesurvey will affect the taxes they pay and the quantities of public goods thatthey consume. Then they will choose their answers to the survey so as toachieve the outcome most favourable to them. “Simple” questionnaires, in whichpeople simply are asked what are their demands for the public good, would notinduce people to tell the truth ( if people are clever and behave strategically). If people figured out that the government would implement some sort ofbenefit tax system, in which people who expressed a strong demand for thepublic good would pay more in taxes, then people would have an incentive to understate their preferences. Suppose instead people figuredout that public goods would not be finance by benefit taxation, but would befinanced out of general tax revenue. Then they might not understate theirpreference. If a person learned that parks were to befinanced by a cigarette tax, then if the person were a non–smoker she wouldrealize that she would be getting the parks at no cost to herself. Assumingthat cigarette taxes are born entirely by smokers, the added cost of anyexpansion of the park system, if financed by a cigarette tax, would be paidonly by smokers. In this case, the non–smoker does not have an incentive tounderstate her benefits from parks. The value of the benefits which she reportsin the survey don’t affect her tax payments. But she would now have anincentive to overstate herpreferences. If someone else is paying for the parks, then she might as wellexaggerate the benefits she gets. That would make the government agency morelikely to go ahead with the project, or to expand the project. In general, fora person to figure out what would be the best response to a survey about herdemand for the public good might be pretty complicated, and it might depend onher guesses about what other people were responding. But figuring out the beststrategy, although complicated, would be a better option than simply tellingthe truth. However, it turns out that if the taxes people pay arebased on the responses they give, and if the taxes are designed cleverly enough,then people will want to tell the truth, even if they have figured out how thesystem works. That is, there are mechanisms which will induce people to tellthe truth about their tastes for public goods, even if they areself–interested, and even if no–one else knows their preferences — if peoplecorrectly figure out how the tax system works. One feature that is needed for these mechanisms to work is that the government (or whoever is asking thequestions) can commit to themechanism. That is, a mechanism is a set of rules which the governmentannounces. It will be assumed here that the government will actually use therules it announces, and that people believe that the government will use theserules. In the example below, the government proposes rules, for how it will taxpeople, based on responses people give to a questionnaire. In order for thismechanism to work, the government must actually go ahead and use those rules,once the questionnaire has been answered. So it will be taken for granted herethat the government will not try and change the rules once it has someinformation, and that people believe that the rules actually will be used. An Example : A Facility of Fixed Size As an example, consider the issue of whether of not tobuild some public facility of a fixed scale, such as a stadium or theatre. Thesimplification here is that there is no possibility of varying the scale of thefacility : the question is an “all or nothing” issue of whether to build thefacility or not. Suppose that the facility costs $100,000,000 to build. Supposeas well that the government agency is trying to make its decision (whether ornot to build the facility) efficiently : to build the facility if the totaldollar value of all the people’s expressed benefits for the stadium exceed thecost. If people were telling the truth,then that would be a good criterion for the government to use : the project isworth undertaking if the benefits (in dollars) from the project, added up overall people, exceeded the cost of the project. Assume that there are 1 millionpeople in the city, so that the government is going to ask each person tostate, in dollars, what the stadium is worth to him or her. If the sum of theseannounced valuations exceed $100,000,000, then the facility will be built. Further, assume that the cost ofthe stadium, if it is built, will be split equally, $100 per person, among thecity’s residents. Now if the government simply announces that the stadiumwill be built if the sum of the announced valuations exceeds $100,000,000, andalso announces that the cost will be shared equally among all people, people donot have an obvious incentive to tell the truth. [What are the incentives?That’s left to the reader. So think about how you might want to answer a survey,if you knew that you truly valued the project at $25, and if you knew that youwould be assessed a tax of $100 if the project were built. Now think about howyou’d want to answer the survey if you knew that the project was really worth$200 to you, and you still expected to be taxed $100 if the project were built,regardless of your answer.] But in the mechanism I will present here, the governmentputs in an additional tax, a taxwhich applies only if a person’s answer affects the decision. That tax is assessedin addition to the share of the project which a person will pay, if the projectis built. And it might be assessed even if the project is not built. Some notation : for person 1, let V˜1 denote the total of everyone else’s announced valuationfor the facility, leaving out person 1’s own announced valuation. Let v1 be the valuation thatperson 1 announces. So V˜1 ≡ v2 + v3 + ··· + v1000000 where vi is what person i says that the project is worth to her.(Of course what she says it’s worth to her may not be what she really thinksthat it is worth to her : we don’t know her true preferences, just what shetells us her preferences are.) One of the rules the government will stick to is that thefacility will be built if the some of people’s announced values exceeds thecost of the facility, which is $100,000,000. From the definition above of V˜1, as the sum of everyoneelse’s announced value (except for person 1’s), that rule can be written :build the facility if and only if V˜1+ v1 ≥ 100,000,000 If the facilityis built, then everyone will pay a tax of $100 to pay for it. But further,person 1 may pay a special tax, if her response turns out to be pivotal, that is if her responseaffects the decision whether or not to build the facility. Specifically, if V˜1< 99,999,900 and if V˜1+ v1 > 100,000,000 then person 1’s response is pivotal : the facility would not havebeen built without her response being as big as it was. In other words, if theannounced values of all the other people were less than $100 per person, thenthe facility won’t be built, since the cost per person is greater than thebenefit per person (or at least it’s less than the average value of what peoplesay are their benefits). In these circumstances, if person 1 says that herbenefit is really high, then, once her announced benefit is taken into account,the average benefit of all 1,000,000 people will be greater than the cost perperson, so the facility will be built. Of course, this may not happen.But if it does happen that V˜1 < 99,999,900, and that V˜1 +v1 ≥ 100,000,000, then person 1is said to be pivotal, since her response to the survey affects the overalldecision. In that case, she will have to pay aspecial “pivot tax” of 99,999,900 − V˜1 on top of the regular tax of $100. As well,if V˜1≥ 99,999,900 and V˜1+ v1 < 100,000,000 then person 1 would again be pivotal, sincein this case her low announced valuation keeps the facility from being built.In this case, she would also have to pay a pivot tax, this time equal to V˜1− 99,999,900 ( In this case,she is paying a tax, even though the facility is not being built, just becauseshe influenced the decision not to build the facility by announcing such a lowvalue ). Why $99,999,900, and not$100,000,000 in calculating the pivot tax? Because we’re going to build thefacility only if people’s average valuationexceeds the cost per person of $100. I’m pivotal if the average valuation, notincluding mine, is less than $100, and if my announced valuation pulls theaverage above $100 — or if the average valuation, not including mine, weregreater than $100, and if my announced valuation pulls the average below $100. Now, it might seem that thisextra tax would not induce a person to want to tell the truth. It might seemthat it would induce her to want to understate her preferences, to avoid the“pivot” tax. But suppose that she doesunderstate her true valuation. Suppose, for example, that the facility reallyis worth $200 to her. Would it pay her to lie, say to state a valuation of only$100? The effect of her lie depends on what everyone else has stated. If V˜1 is greater than $100,000,000, then the facility gets builtwhatever she says is her valuation. In this case, she is not pivotal, the facility gets built no matter what she says, andshe pays $100 as her share of the cost. Since she won’t be stuck with the pivottax, and since the facility will be built whether she tells the truth or lies,then there is no incentive to lie. Remember : a person pays a pivot tax only ifher answer changes the result. What if V˜1< 99,999,800? Then if shetells the truth ( v1 = 200), the facility will not get built. She can get it built if she exaggerates hervaluation enough — but then she’ll get hit with a pivot tax. For example, if V1 = 99,999,700 and she announces v1 = 400, her lie gets the facility built, since itpushes the sum of announced valuations above $100,000,000. That gets her afacility, a facility which is worth $200 to her — but she would have to paytaxes of $300, the regular $100 tax plus a pivot tax of $99,999,900 − $99,999,700= $200. So it’s not in her interest to overstate her benefit. Overstating onlymakes a difference if she is pivotal. And if she is pivotal, then the extrapivot tax she pays for affecting the decision is greater than the benefit shegets from the facility. If V˜1 isbetween $99,999,800 and $99,999,900, then person 1 would be pivotal if shetells the truth (remember: her true valuation here is assumed to be $200). Shecan avoid the pivot tax by understating her valuation. For instance, if V˜1 = 99,999,850, she couldannounce a valuation of $100, which would mean that the facility would not bebuilt, and she would avoid paying any pivot tax. But in this instance, herpivot tax is only $50 ( 99,999,900−V˜1). She would be better off telling the truth, even though it means payingher regular tax of $100 and herpivot tax of $50, because then she would get to have a facility which is worth$200 to her : telling the truth, and paying a total tax of $150 to get afacility built which is worth $200 is a better strategy than understating herpreferences, paying nothing, and getting nothing.. Finally, if V˜1 is between $99,999,900 and $100,000,000, she couldprevent the facility from being built by understating her true valuationenough. But that would be bad for her in two ways. First of all, she won’t getthe facility. Secondly, in this case her lie would make her pivotal, since herlie, announcing a low valuation was the reason that the facility was not built.So telling the truth gets her a facility worth $200, at a cost of $100, in thiscase. Understating a lot gets her no facility, and a pivot tax as well! So what should she do? The odds are that she won’t haveany influence on the outcome. In that case, it doesn’t matter whether she liesor tells the truth. She is only 1 person among 1,000,000. Her answer onlymatters if the other valuations average out close to $100 per person, so thatshe could be pivotal. And in that case, the analysis above shows that she isbetter off telling the truth than lying. The pivot tax she might end up payingcan never exceed $100 if she tells the truth ( if her true valuation is $200 ),so she would be better off telling the truth, affecting the result, getting thefacility built, and paying the pivot tax, than understating her preference tododge a small pivot tax. Notice that the pivot tax does not depend specifically onwhat she states as her valuation, only on whether she affects the result, andon how that change affects other people. For example, if V˜1= 99,999,850 then her pivot tax will be V˜1− 99,999,900 = 50 whenever she ispivotal in getting the facility built, that is whenever v1 > 150.Understating her valuation slightly ( say stating it’s $180 instead of her true$200 ) won’t reduce her pivot tax : it’s $50 whenever v1 > 150.The only way understating reduces her pivot tax is if she states v1 < 150, in which case she dodges the pivot tax — but also doesn’tget her facility. Now person 1 actually doesn’tknow what everyone else has stated, at the time she is asked her valuation. Shehas to decide her announcement, v1,without knowing what is her V˜1(the sum of everyone else’s vi’s).But the paragraphs above show that she really doesn’t need to know what otherpeople have announced. Either lying about her valuation will have no effect atall, on either whether the facility gets built, or on her taxes ( the mostlikely situation ), or it will have an effect. And if it will have an effect,she is strictly better off telling the truth than lying. In other words, giventhe rules of this tax scheme, telling the truth is a dominant strategy for person 1. Whatever other people’s valuationsare, and regardless of whether they tell the truth or lie, there is no strategyfor person 1 which is any better than simply telling the truth about what thefacility is worth to her. Person 1’s decision process is nodifferent than any other person’s. Person 2 has a similar situation : he willhave to pay a pivot tax if the average of the other 999,999 people’s announcedvaluations is less than $100, and his announcement pushes the average over$100, or if the other people’s average valuation is more than $100, and his lowannounced valuation pushes the overall average below $100. So each person will,if she or he understands the tax rules correctly, want to tell the truth. Eventhough the government has no idea about any person’s true valuation, it candesign a tax scheme which — if understood correctly — will induce everyone toreveal their preferences, because it is in each person’s self–interest to doso. So who’s pivotal? That depends onthe distribution of people’s actual valuations. It might be nobody. For exampleif we had 100 people who valued a facility at $102, 100 people who valued it at$99, and if the facility cost $20,000 to build, then no individual will pay apivot tax if everyone tells the truth.. ( You should check that, in this case, V˜i +vi >19,900 for each person, ifeveryone tells the truth. ) It could be that a lot of people are pivotal : if10 people valued a facility at $151, 10 valued it at $50, and if the facilitycost $2000, then each high–valuation person would face a V˜i of$1859¡$1900, and would pay a pivot tax of $41 if everyone told the truth. What does the government do withthe pivot tax revenue? Notice that this pivot tax revenue ( if there is any )would be money which is over and above the cost of the facility. That may seemlike a nice situation, but the government actually would have to be careful notto return this excess to the taxpayers. Why? If taxpayers realized that theywould be getting a share of the pivot tax revenue, then they might start toadjust their responses, so as to make other people pay more pivot taxes.Figuring this effect out would be very complicated, but if people were reallyclever it would ( very slightly ) offset the incentive to tell the truth.
Preference Revelation : (b)A Project of Variable Size In the example in the previoussection, the project being considered had a fixed size, so that the public goodprovision decision was an “all or nothing” decision : either build the facilityor don’t build the facility. In this section, a somewhat morecomplicated problem is considered : determining the quantity to provide of apure public good. So, unlike the project considered in the previous section,the problem considered here is how much to provide of a pure public good, thatis, how to implement the Samuelson rule when people’s preferences are not known(except by the people themselves). The variable being chosen here is the quantity Z of some pure public good. It isassumed that each person has a downward–sloping demand curve for the publicgood, a demand curve which the person knows, but which no–one else knows. Sonow the preference revelation mechanism must get people to announce theirwillingness to pay for the public good, as a function of the quantity Z providedof the public good. Each person i will be asked to report her willingness to pay vi(Z) for the public good. vi(Z) is the amount, in dollars, that theperson would be willing to pay for a little more of the public good, if aquantity Z is provided. Or it’s whatperson i says is what she is willingto pay : we have no way of verifying whether what she reports is her truewillingness to pay schedule, or not. As in the pivot tax of the previoussection, the preference revelation mechanism here uses a fairly complicated taxrule, which should induce each person to report truthfully herwillingness–to–pay schedule, if she understands the tax rules, and if she wantsto manipulate the system to her own advantage. The notation used here will bequite similar to the notation used in the previous sub–section : the maindifference is that now people have to report a whole demand curve, not just a single number. So vi(Z) will denote the demand schedule which person i reports, some downward–slopingfunction, representing what she says she would be willing to pay for a littlemore of the public good, as a function of the amount provided. This vi(Z) denotes person i’s reported MRS function. As in the previous section, we cannot tell whetherperson i is telling the truth or not— but we’d like to make it worth her while to tell the truth. The marginal cost of eachunit of the public good will be denoted c.That’s just the MRT. The government wants to provide an efficient allocation. Sothe quantity Z∗ which itwill actually provide will depend on the demand schedules reported by thepeople, and it will obey the Samuelson rule, for the reported demand schedules. That is, once all the people havereported their demand schedules for the public good, then the level that willactually get provided is the solution Z∗to the equation v1(Z∗) + v2(Z∗) + ··· + vN(Z∗) = c (1) when the N people report their demand schedules. Equation (1) is theSamuelson condition, except with people’s reportedMRS’s, vi(Z) used,since we don’t actually know their true MRSschedules. As in the simple “pivot tax” mechanismwith a project of fixed size, a person’s tax liabilities here will have twoparts. First of all, part 1 of the tax is simply the person’s share of the costof the public good. So each of the N peoplehas to pay a fraction 1/N of the costof the public good. If T denotes thisfirst part of the tax, then for each person i, file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif (2) since the public good costs c per unit, Z∗ units are being provided, and the cost is beingdivided among all of the N people. The second part of each person’stax is again a “pivot tax”, determined by how the person’s reportedwillingness–to–pay schedule alters the provision of the public good. As in the previous section, consider person 1’s taxes, andher incentives to manipulate the system. As before, let V˜1 denote the sum of everyone else’s announced willingness to pay. V˜1(Z) ≡ v2(Z) + v3(Z) + ··· + vN(Z) So the Samuelson rule (1),which the government has promised to use, can now be written v1(Z∗) + V˜1(Z∗) = c (3) As before,consider what level of the public good would be provided, if person 1’sannounced preferences were not taken into account, but her contribution to thecost were also left out. So define the level of public good provision Z˜1 by theequation file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif (4) In figure 1, the horizontal dotted linerepresents the MRT. But the lowerhorizontal green dashed lines the MRT minusperson 1’s contribution. The downward–sloping green dashed line is the verticalsum of everyone else’s announced demand curves. In the diagram, Z˜1 equals 5, where thevertical sum of everyone else’s announced demand curves crosses a line with aheight (N − 1)/N times the MRT. Notice that person 1 does not get to affect Z˜1 : Z˜1 isdetermined only by the announced demands of other people. The second part of person 1’s tax, the pivot tax, is thearea between the vertical sum of everyone else’s announced demand curves, andtheir share [N − 1)/N]cof the costs, between the level Z˜1of the public good which would be provided without person 1, and thelevel Z∗ which is actuallyprovided. In figure 1, it’s the triangle outlined in green, and labelled PT (for “pivot tax”). Mathematically,the area under a curve is measured using the integral of the function. So file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif (5) In figure 1, Z∗ is to the right of Z˜1 : Z˜1 = 5 and Z∗ = 6 in that figure. That’sbecause, in this example, person 1 seems to have a relatively strong demand forthe public good, so including her valuation pushes up the quantity chosen. Butthat need not be the case : figure 2 illustrates a case in which person 1announces a lower valuation than the average of the other people, so thatincluding her actually reduces the quantity provided. In that case, there isstill a pivot tax PT in the figure :again its the area between Z˜1and Z∗, between thevertical sum of everyone else’s announced willingness to pay, and their sharesof the cost. So formula (5) applies whether or not Z∗ is greater than Z˜1. Person 1’s total taxes are justthe sum of the first part of her taxes, her share of the cost,and her pivottax. That is, in figure 1, her taxes are the triangle labelled PT, plus the red rectangle labelled “taxof person 1 : part 1”. This tax PT really is a generalization of the pivot tax used in the previoussection. When including person 1 affects the level of public good provided, sheis assessed this extra tax. Except with a variable public good, she’ll always(except by extreme coincidence) have some affect on the quantity provided, sothat she’ll aways have some pivot tax to pay. How is person 1’s pivot tax affected by her reported demandschedule? To answer that, consider how PTis affected by the quantity Z∗of the public good which is actually provided. That is, how does thevalue of PT in expression (5) changewith the level Z∗ ofpublic good provision? To answer that,remember thefundamental theorem of calculus, that the derivative of the integral of afunction is just the function itself : file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif Here, equation (5) thensays that file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif) (6) [Why does thatmake sense in the diagram? How much would the green triangle labelled PT grow if Z∗ moved a little to the right? The rate of increasewould be the height of the triangle, the distance between (N − 1)/N and V˜1(Z∗).] There also is a little sense tothis expression (6) for the marginal pivot tax. Suppose that person 1 gets thegovernment to provide a little more of the public good. How does that affectthe other people? The added cost that they would have to pay (since person 1only has to pay for her share 1/N ofthe public good) is [(N − 1)N]cper unit. The added benefit they say that they would get (added up over allthe other people) is V˜1(Z∗). So the marginal pivottax is the net harm person 1 would do to these other people, if she gets thepublic sector to expand, past the point where the cost to these other peopleequals the benefit they get. Suppose now that person 1 were really clever, and couldmanipulate the system to get any level Z∗of the public good that she wanted. What would her total taxes be, from aslight increase in the level of public good provision? From equations (2) and(6), the change in her total tax (part 1 plus the pivot tax) would be file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif) (7) Equation (7) can be simplifiedto file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif) (8) Notice that if Z∗ > Z˜1, person 1 would have to pay moretax if she could somehow get the government to provide more of the public good. But she also does get somebenefit from the public good. She doesn’t want to minimize her taxes : shewants to get the most benefit for the least taxes. So let p1(Z) denote person 1’strue marginal willingness to pay fora little more of the public good. Only she knows that. But it does representwhat she really does think a little more of the public good is worth. If p1(Z) is greater than the marginal taxes she would have to pay, thenshe would like to see the public good provision expanded. p1(Z) representsthe benefit to her of a little more of the public good. Given the complicatedtax rules — and given everyone else’s announced benefits — the right side ofequation (8) represents the marginal cost to her of a little more of the publicgood. So if she could completely manipulate the system, and getany level Z∗ of the publicgood that she wanted, then she would want a level Z∗ such that her true marginal benefit equalled theincrease in her taxes : p1(Z∗) = c − V˜1(Z∗) (9) But the government has announced alreadywhat it will do to determine the public good level that it provides. It’s usingequation (3), which I can re–write as v1(Z∗) = c − V˜1(Z∗) (10) Now look atequations (9) and (10). Equation (9) is the level of public good provision shewants, given this complicated tax rule. Equation (10) is the public goodprovision she will get, given that the government uses the Samuelson ruleapplied to people’s announced demand curves. She can make what she gets intowhat she wants pretty simply : as long as v1(Z∗) = p1(Z∗),then equations (9) and (10) are the same. So she can get the best level of public good provision,from her selfish perspective, simply by telling the truth. Announcing a demandschedule v1(Z) = p1(Z) will guaranteethat the solutions to equations (9) and (10) are the same, whatever the otherpeople do. In the language of game theory, given these tax rules, announcingher true demand schedule is a dominantstrategy to the game played by the taxpayers. So, if people are clever andselfish, the government can get them to reveal voluntarily their willingness topay for the public good, if it uses the Samuelson rule to provide the publicgood, and if it sticks to a policy of making each person’s taxes equal the twoparts, T + PT. Not only does this mechanism getpeople to tell the truth, and provide the efficient quantity of the publicgood, it also ensures that there is enough tax revenue to pay for the publicgood. If we add up the first part of the taxes, over all people, the revenuesums up to cZ∗, exactlythe cost of the public good. Then there are the pivot taxes PT. These must be non–negative : if Z∗ > Z˜1, then [(N − 1)N]c > V˜1(Z) for Z˜1 < Z < Z∗, so the pivot tax triangle has positivearea. But if person 1 announces a low demand, and Z˜1 > Z∗, then V˜1(Z) >[(N − 1)/N]c for Z∗ < Z < Z˜1, so that the pivot taxtriangle defined by expression (5) (and illustrated in figure 2) is againpositive : person 1 is taxed for reducing public good provision, when otherpeople value the marginal units of the public good more highly than the shareof the cost that the have to pay.
Preference Revelation : (c)Complications and Difficulties The preference revelationmechanism described in the previous two sections make it a dominant strategy for a person to tell the truth about the valueshe places on a public good. That is, if she understands the mechanism, and ifthe government commits credibly to obeying its own rules — the level to provideof the public good, the method of dividing up the cost, the rules for theadditional pivot tax — then the person’s own self–interest is best served ifshe announces exactly what her true demand for the public good is. Any attemptto misrepresent her preferences cannot make her better off, and may make herworse off. The fact that this is a dominant strategy for the person means thattelling the truth is the best strategy for her, regardless what other peoplechoose to do when they decide how to report their own preferences. In this section, some extensions of the mechanism arediscussed briefly, and also some potential weaknesses. Sharing the Cost In the mechanism presented in theprevious sections, each person’s tax had two elements. The first quantity washer share of the cost of the public good, and the second was the pivot tax thatshe would have to pay if her answer altered the provided of the public good. It was assumed that her share of the cost was simply afraction 1/N, where N is the total number of people. Thatis, it was assumed that the cost was divided equally among all the people. Thisassumption is unnecessary. Instead, it could be replaced with the followingrule : for each person i, there issome share si of the costof the public good that the person must pay. So (in the variable–quantity publicgood of section (b) above), if thetotal cost of a level Z of the publicgood were cZ, then person i would have to pay sicZ in taxes to cover her share of the cost. The share si does not have to be the same for all people. All that is required is thatthe shares do pay for the public good s1 + s2 + ··· + sN = 1 and that eachperson knows her own share. The rule for determining the quantity of the public good isnot changed. The quantity Z∗ satisfiesthe condition v1(Z∗) + v2(Z∗)+ ··· + vN(Z∗) = c as before. The basic idea of the pivot tax is unchanged. Now the rulefor the pivot tax is changed slightly. The quantity Z˜1 which would be chosen if person 1 were left out ofthe process is now determined as v2(Z˜1) + v3(Z˜1) + ··· + vN(Z˜1)= (1 − s1)c Sothe right hand side of the equation has been modified ; but it still representsthe unit cost of the public good, excluding person 1’s share of the cost. If Z∗ > Z˜1, then the pivot tax paid by person1 would be file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif so that again, the only change is to replaceNN−1c with (1 − s1)c. With this slightly modified mechanism, it will still be thecase that telling the truth is a dominant strategy for person #1 (and for anyother person, if the pivot tax is defined analogously). Of course, people wouldprefer that they be assessed a lower share siof the cost. But given that the shares have already been determined,the best a person can do, when surveyed, is to reveal her preferencestruthfully. General Equilibrium Preference Revelation In the “variable quantity”mechanism (presented in the second part of this section), people are asked tostate their entire inverse demand curve for the public good, some function vi(Z) expressing how much they are willing to pay for a little more ofthe public good, as a function of the quantity provided. Of course, if thedemand curve sloped down, they could just as easily state their regular demandcurve, the quantity of the public good which they would be willing to buy atdifferent prices, were the good excludable and available for sale on privatemarkets. The willingness–to–pay curve vi(Z) is just the inverse of this demandfunction. However, ordinarily quantitydemanded depends not only on price, but on other variables, such as the pricesof other goods and services, and on the person’s income. if the public goodwere a normal good then increases ina person’s income would shift the whole demand curve to the right. So we wouldhave to write the inverse demand curve as vi(Z,Mi) if the good werenormal, where Mi was theperson’s disposable income, and where ∂vi/∂Mi> 0 because the good is normal. So changes in a person’sdisposable income will shift her demand curve for a good, unless the incomeelasticity of demand for the good were exactly zero. But with the mechanism used here,the person may be charged a pivot tax. The amount of the pivot tax depends onthe person’s answers to the survey, but also on everyone else’s answers to thesurvey. Paying a pivot tax will lower the person’s disposable income. So theamount that the person has to pay in pivot tax should affect her demand curve,if the good is normal. The higher the pivot tax, the further the demand curveshifts down and to the left (if the good is normal). But the person does not knoweveryone else’s answers to the survey, at the time that she must figure out herown announced demand curve. So, if the good is normal, she must know the pivottax to correctly figure out her own demand curve, and this is information shewill not have. This problem can be avoided onlyif changes in her disposable income have no effect on the location of herdemand curve for the public good — in other words, if the income elasticity ofher demand for the public good is zero. If the income elasticity of demand is non–zero, then themechanism cannot be so simple. For that reason, the mechanismspresented previously in this section are described as “partial equilibrium”preference revelation mechanisms, in that they ignore income effects. If incomeeffects are significant, then a somewhat different mechanism is needed, a “generalequilibrium” mechanism. Unfortunately, generalequilibrium preference revelation mechanisms do not work as well as partialequilibrium equilibrium mechanisms. It turns out not be possible to design ageneral equilibrium mechanism which i takesinto account these income effects ; ii alwaysguarantees that enough money will be raised to pay for the public good ; iii makes telling the truth a dominantstrategy for all people. It is possible to design general equilibrium preferencerevelation mechanisms which satisfy properties i and ii above, for whichtelling the truth is a Nash equilibrium.That is, telling the truth is best for me, if I know that everyone else istelling the truth. This is a much less powerful property than having tellingthe truth as a dominant strategy. Now people will only be willing to tell thetruth if they think other people are doing the same thing. With a dominantstrategy, as in the partial equilibrium preference revelation mechanism, I ambest telling the truth, even if I suspect that other people may not understandthe rules, and may not themselves reveal their own preferences. Not so withthese general equilibrium mechanisms. Where does the Money Go? The (partial equilibrium)preference revelation mechanism was constructed so that the first part of thetax, the people’s cost shares, exactly paid for the cost of the public good, nomatter what quantity of the public good was chosen. The second part of the tax,the pivot tax, will be positive or zero for each person. It cannot be negative.So the government, if it uses this mechanism, will be guaranteed to collect atleast enough money to pay for the public good, more than enough if the pivottax actually collects positive revenue. What should be done with the excess revenue, the moneycollected through the pivot tax? It cannot be given back topeople. If I knew that I was going to get a share of any pivot tax revenue,then I should take account of this possibility in making my decision. Everyextra dollar I pay in pivot taxes will actually get me back some money. But takinginto account this effect will change my decision–making. It adds another terminto my calculation, and it is no longer going to be true that telling thetruth is a dominant strategy. This will also be the case if Iget any share of the pivot tax revenue collected from other people. If I amperson #2, my answer to the survey affects the pivot tax schedule faced byperson #1. If I, person #2, am going to get some share of that revenue, then Ishould take into account how my answer to the survey affects this tax yield.Again, taking this effect into account may have only a small effect, but itwill change my optimal strategy slightly, away from telling the exact truth. A similar problem will arise ifthe government uses the pivot tax revenue to pay for some other category ofpublic expenditure. As long as part of the pivot tax money is going to fundsome service which is useful to me, or going to reduce my income taxes, then Ishould take into account how my answer to the survey affects the total pivot taxrevenue. And taking this effect into account will alter slightly my incentiveto tell the truth. So the mechanism will work perfectly only if the governmentcan commit not to spend the pivot tax revenue on anything useful to the peoplebeing surveyed. They could just throw the money away. A better approach wouldbe to commit to spend the money on other people. They could, for example, makea deal with another jurisdiction, that the other jurisdiction would get theirpivot tax revenue, and they would get the other jurisdiction’s. Once this dealis completed, I do not care about how my answers to the survey affect pivot taxrevenue. The revenue will be spent in another jurisdiction. I will get somemoney from another jurisdiction’s pivot tax, but my answer to the survey willhave no affect on how much gets collected in the other jurisdiction. Similarly,if the Ontario government used a preference revelation mechanism to findbenefits from subway projects in the Toronto area, they could earmark the pivottax revenue for expenditure in Northern Ontario (and perhaps earmark pivot taxrevenue from a survey of Northern Ontarians’ benefits from highway expansion tobe spent only in the Toronto area). Collusion The (partial equilibrium)preference revelation mechanism is immune to manipulation by individuals. Thatis, if I understand the mechanism, there is no way that I can do better bylying than by telling the truth. Unfortunately, itis not immune too manipulation by groupsof people. As an example, suppose that therewere three people, and an all–or–nothing (fixed quantity) public good was beingconsidered. The public good has a total cost of $1200, so that it will be builtif and only if the sum of the announced valuations of the three people sum to anumber which is greater than or equal to $1200. Suppose that the truevaluations of the three people are $700, $350, and $350. If each person tellsthe truth, then the project will be built, since the sum of the valuations is$1400. But person #1 is pivotal. In this case file:///C:/Users/VT/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif If everyone tellsthe truth, person #1 will be assessed a pivot tax of $100 (= NN−1C − v2 − v3 = 800 − 350 − 350). She isbetter off getting a project built than not : she finds the project is worth$700 to her, and she has to pay only $500 in taxes (her share of $400, plus thepivot tax of $100). So she would not want to lie. But she can benefit from making a deal with person 2 tolie, if she has some idea of other people’s true preferences. She could offerperson #1 a small bribe to change her answer v2 from $350 to (say) $500. If v2 = 500, and if persons #1 and #3 tell the truth, thenno person is pivotal : here v1 +v2 = 1200, v2 +v3 = 850, and v1+v3 = 1050, so that vi +vj > 800 for any two people i and j : the projectwould still be built even if any one person’s share of the costs, and announcedbenefits, were left out. So if person #2 changes hisanswer from $350 to $500, then no–one is pivotal, and no pivot tax is paid.This change saves person #1 $100, since it frees her of the pivot taxliability. The change does not hurt person #2 (or person #3) : the projectwould have been built even if person #2 had told the truth, so having himexaggerate his benefits does not change anything. Thus if person #1 pays person#2 $50 to change his answer from $350 to $500, then both person #1 and person#2 are better off than if person #2 had told the truth. In this example, the lie did notaffect the outcome. But it might. Once people realize that they can colludeprofitably, and that collusion involves lying by one or more of them, then themechanism no longer works. So, if people can easilynegotiate with each other about how they answer the survey, the preferencerevelation mechanisms presented in this section will be susceptible tomanipulation by groups.
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