Introduction to Economic Analysis Chapter 8 Public Goods

Chapter Public Goods Consider a company offering a fireworks display . Pretty much anyone nearby can watch the fireworks , and people with houses in the right place have a great View of them . The company that creates the can compel those with nearby homes to pay for the fireworks , and so a lot of people get to watch them without paying . This will make it or impossible for the company to make a . Free Riders LEARNING OBJECTIVE . What are people who just use public goods without paying called , and what is their effect on economic performance ?

A public good is a good that has two attributes , which means the producer can prevent the use of the good by others , and , which means that many people can use the good simultaneously . The classic example of a public good is national defense . National defense is clearly because , if we spend the resources necessary to defend our national borders , it isn going to be possible to defend everything except one apartment on the second of a apartment building on East Maple Street . Once we have kept our enemies out of our borders , we ve protected everyone within the borders . Similarly , the defense of the national borders exhibits a fair degree of , especially insofar as the strategy of defense is to deter an attack in the place . That is , the same expenditure of resources protects all . It is theoretically possible to exclude some people from the use of a poem or a mathematical theorem , but exclusion is generally quite difficult . Both poems and theorems are URL books 178

. Similarly , technological and software inventions are , even though a patent grants the right to exclude the use by others . Another good that permits exclusion at a cost is a highway . A toll highway shows that exclusion is possible on the highways . Exclusion is quite expensive , partly because the require , but mainly because of the delays imposed on drivers associated with paying the time costs of toll roads are high . Highways are an intermediate case where exclusion is possible only at a cost , and thus should be avoided if possible . Highways are also rivalrous at levels , but at levels . That is , the marginal cost of an additional user is essentially zero for a sizeable number of users , but then marginal cost grows rapidly in the number of users . With fewer than 700 cars per lane per hour on a highway , generally the of traffic is unimpeded . As congestion grows beyond this level , traffic slows down and congestion sets in . Thus , west Texas interstate highways are usually , while Los Angeles freeways are usually very rivalrous . Like highways , recreational parks are at levels , becoming rivalrous as they become sufficiently crowded . Also like highways , it is possible , but expensive , to exclude potential users , since exclusion requires fences and a means for admitting some but not others . Some exclusive parks provide keys to legitimate users , while others use gatekeepers to charge admission . Take the example of a neighborhood association that is considering buying land and building a park in the neighborhood . The value of the park is going to depend on the size of the park , and we suppose for simplicity that the value in dollars of the park to each household in the neighborhood is , where is the number of park users , is the size of the park , and a and are parameters satisfying a . This functional form builds in the property that larger parks provide more value at a diminishing rate , but there is an effect from URL books 179

congestion . The functional form gives a reason for parks to be is more efficient for a group of people to share a large park than for each individual to possess a small park , at least if a , because the gains from a large park exceed the congestion effects . That is , there is a scale doubling the number of people and the size of the park increases each individual enjoyment . How much will selfish individuals voluntarily contribute to the building of the park ?

That of course depends on what they think others will contribute . Consider a single household , and suppose that each household , i , thinks the others will contribute to the building of the park . Given this expectation , how much should each household , i , contribute ?

If the household contributes , the park will have size , which the household values at ( Thus , the net gain to a household that contributes the others contribute is ( Exercise shows that individual residents gain from their marginal contribution if and only if the park is smaller than ( Consequently , under voluntary contributions , the only equilibrium park size is So . That is , for any park size smaller than So , citizens will voluntarily contribute to make the park larger . For any larger size , no one is willing to contribute . Under voluntary contributions , as the neighborhood grows in number , the size of the park shrinks . This makes benefits of individual contributions to the park mostly accrue to others , which reduces the payoff to any one contributor . How large should the park be ?

The total value of the park of size to the residents together is times the individual value , which gives a collective value of and the park costs , so from a social perspective the park should be sized to maximize , which yields an optimal park of size ( Thus , as the neighborhood grows , the park should grow , URL books 180

but as we saw , the park would shrink if the neighborhood has to rely on voluntary contributions . This is because people contribute individually , as if they were building the park for themselves , and don account for the value they provide to their neighbors when they contribute . Under individual contributions , the hope that others contribute leads individuals not to contribute . Moreover , use of the park by others reduces the value of the park to each individual , so that the size of the park shrinks as the population grows under individual contributions . In contrast , the park ought to grow faster than the number of residents grows , as the per capita park size is , which is an increasing function of The lack of incentive for individuals to contribute to a social good is known as a problem . The term refers to the individuals who don contribute to the provision of a public good , who are said to be free riders , that is , they ride freely on the contributions of others . There are two aspects of the rider problem apparent in this simple mathematical model . First , the individual incentive to contribute to a public good is reduced by the contributions of others , and thus individual contributions tend to be smaller when the group is larger . Put another way , the size of the problem grows as the community grows larger . Second , as the community grows larger , the optimal size of the public good grows . The market failure under voluntary contributions is greater as the community is larger . In the theory presented , the optimal size of the public good is ( and the actual size under voluntary contributions is ( a gap that gets very large as the number of people grows . The upshot is that people will voluntarily contribute too little from a social perspective , by free riding on the contributions of others . A good example of the provision of public goods is a term paper . This is a public good because the grade given to the paper is the same for each author , and the quality of the paper depends on the sum of the efforts of the individual URL books 181

authors . Generally , with two authors , both work pretty hard on the manuscript in order to get a good grade . Add a third author , and it is a virtual certainty that two of the authors will think the third didn work as hard and is a free rider on the project . The term paper example also points to the limitations of the theory . Many people are not as as the theory assumes and will contribute more than would be privately optimal . Moreover , with small numbers , bargaining between the contributors and the division of labor ( each works on a section ) may help to reduce the problem . Nevertheless , even with these limitations , the problem is very real and it gets worse the more people are involved . The theory shows that if some individuals contribute more than their share in an altruistic way , the more selfish individuals contribute even less , undoing some of the good done by the . KEY TAKEAWAYS A public good has two attributes , which means the producer ca prevent the use of the good by others and , which means that many people can use the good simultaneously . Examples of public goods include national defense , fireworks displays , and mathematical theorems . implies that people do have to pay for the good means that the efficient price is zero . A free rider is someone who does pay for a public good . Generally voluntary contributions lead to too little provision of public goods . In spite of some altruism , the problem is very real , and it gets worse the more people are involved . Verify that individual residents gain from contributing to the park ( gain from reducing their contributions if ( URL books 182

. For the model presented in this section , compute the elasticity of the optimal park size with respect to the number of is , the percentage change in for a small percentage change in Use the linear approximation trick ( for A near zero . the model presented in this section , show that an utility when the park is sized and the expenses are shared equally among the individuals is ( Does this model predict an increase in utility from larger communities ?

Suppose two people , Person and Person , want to produce a playground to share between them . The value of the playground of size to each person is , where is the number of dollars spent to build it . Show that , under voluntary contributions , the size of the playground is and that the efficient size is . For the previous exercise , now suppose Person offers matching funds is , offers to contribute an equal amount to the contributions of Person . How large a playground will Person choose ?

Provision With Taxation LEARNING OBJECTIVE . If people wo pay for public goods , can society tax them instead ?

Faced with the fact that voluntary contributions produce an inadequate park , the neighborhood turns to taxes . Many neighborhood associations or condominium associations have taxing authority and can compel individuals to contribute . One solution is to require each resident to contribute the amount , resulting in a park that is sized at , as clearly shown in URL books 183

the example from the previous section . Generally it is possible to provide the correct size of the public good using taxes to fund it . However , this is challenging in practice , as we illustrate in this slight of the previous example . Let individuals have different strengths of preferences , so that individual the public good of size at an amount that is expressed in dollars . It is useful to assume that all people have different values to simplify arguments . The optimal size of the park for the neighborhood is ( where is the average value . Again , taxes can be assessed to pay for an sized park , but some people ( those with small values ) will View that as a bad deal , while others ( with large ) will view it as a good deal . What will the neighborhood choose to do ?

If there are an odd number of voters in the neighborhood , we predict that the park size will appeal most to the median voter . This is the voter whose preferences fall in the middle of the range . With equal taxes , an individual obtains . If there are an odd number of people , can be written as . The median voter is the person for whom there are values vi larger than hers and values smaller than hers . Consider increasing If the median voter likes it , then so do all the people with , and the proposition to increase passes . Similarly , a proposal to decrease will get a majority if the median voter likes it . If the median voter likes reducing , all the individuals with smaller vi will vote for it as well . Thus , we can see the preferences of the median voter are maximized by the vote , and simple calculus shows that this entails ( Unfortunately , voting does not result in an efficient outcome generally and only does so when the average value equals the median value . On the other hand , voting generally performs much better than voluntary contributions . URL books 184

The park size can either be larger or smaller under median voting than is efficient . KEY TAKEAWAYS a solution to the problem . An optimal tax rate is the average marginal value of the public good . Voting leads to a tax rate equal to the median marginal value , and hence does not generally lead to efficiency , although it voluntary contributions . Show for the model of this section that , under voluntary contributions , only one person contributes , and that person is the person with the largest vi . How much do they contribute ?

Which individual i is willing to contribute at the largest park size ?

Given the park that this individual desires , can anyone else benefit from contributing at all ?

Show that , if all individuals value the public good equally , voting on the size of the good results in the efficient provision of the public good . Local Public Goods LEARNING OBJECTIVE . What can we do if we disagree about the optimal level of public goods ?

The example in the previous section showed the challenges to a neighborhood provision of public goods created by differences in the preferences . Voting does not generally lead to the efficient provision of the public good and does so only rarely when all individuals have the same preferences . URL books 135

A different solution was proposed by in 1956 , which works only when the public goods are local . People living nearby may or may not be excludable , but people living farther away can be excluded . Such goods that are produced and consumed in a limited geographical area are local public goods . Schools are distant people can readily be excluded . With parks it is more to exclude people from using the good nonetheless , they are still local public goods because few people will drive 30 miles to use a park . Suppose that there are a variety of neighborhoods , some with high taxes , better schools , big parks , beautifully maintained trees on the streets , frequent garbage pickup , a fire department , extensive police protection , and spectacular fireworks displays , and others with lower taxes and more modest provision of public goods . People will move to the neighborhood that fits their preferences . As a result , neighborhoods will evolve with inhabitants that have similar preferences for public goods . Similarity among neighbors makes voting more efficient , in turn . Consequently , the ability of people to choose their neighborhoods to suit their preferences over taxes and public goods will make the neighborhood provision of public goods more . The theory shows that local public goods tend to be provided . In addition , even private goods such as garbage collection and schools can be publicly provided when they are local goods , and there are enough distinct to offer a broad range of services . KEY TAKEAWAYS When public goods are living nearby may or may not be excludable , whereas people living farther away may be goods are local public URL books 186

Specialization of neighborhoods providing in distinct levels of public goods , when combined with households selecting their preferred neighborhood , can lead to efficient provision of public goods . a babysitting cooperative , where parents rotate supervision of the children of several families . Suppose that , if the sitting service is available with frequency , a person i value is and the costs of contribution is , where is the sum of the individual contributions and is the number of families . Rank , a . What is the size of the service under voluntary contributions ?

Hint Let yi be the contribution of family i . Identify the payoff of family as ( What value of maximizes this expression ?

What rib ions maxi i total ( Let and ( Conclude that , under voluntary contributions , the total value generated by the cooperative is ( Hint It helps to know that ' URL books 187