Additional Questions. Externalities and Public Goods.

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1 Additional Questions. Externalities and Public Goods. Problem 1. The purpose of this problem is to help you understand the difference in market demand for purely private and purely public good. For each of the following situations suppose that private marginal benefits are given by MP B = 10 Q, where Q is the quantity of a good, keep in mind that MP B measures the maximum price a person will be willing to pay for an additional unit of the good. a) Suppose the good is purely private, for example, apples. On a diagram plot individual demand for apples and construct market demand for apples if there are two consumers in the market. Market demand for private good is obtained by adding quantities demanded at each price. You can construct market demand by points: at price P = 10 none of the consumers is willing to purchase any quantity, then total quantity demanded is Q = 0, this is the first point on your diagram (0,10); at P = 9 each person will demand 1 apple and the total quantity demanded in the market is 2 1 = 2, the second point is (2; 9) and so on. You can also use algebra: from the MPB you can find individual quantity demanded as a function of price Q 1 = 10 P is demand by the first person and Q 2 = 10 P is quantity purchased by the second person at each price; market demand is the sum of individual quantities Q M = Q 1 +Q 2, substitute the expressions for the individual demands (plug 10-P for Q 1 and Q 2 ) which will give you Q M = 10 P + 10 P, simplify to Q M = 20 2P and single out price as a function of Q M. Both methods give you exactly the came market demand curve: P = 10.5Q M where Q M is the total quantity demanded at each price. Notice that the market demand curve has the same vertical intercept as individual demands, has half of the slope and twice the horizontal intercept. b) Suppose the good is purely public, for example a street lights installed in the neighborhood. On a diagram plot private marginal benefits and construct market demand for the public good if there are two consumers. Pure public good is nonexcludable and nonrival in consumption. Nonexcludable means that if one of the consumers purchases one unit of the good, the other consumer will be able to consume that unit as well and there is no way to preclude the second consumer from enjoying the benefits. Nonrival means that the fact that the second person is consuming the good does not diminish the benefits to the first person. When you construct the market demand for public good use the notion of maximum amount all consumers will be willing to pay in order to purchase an additional unit. You can construct the market demand curve by points: for example, for the first unit Q = 1 of the good each of the consumers is willing to pay P=9 dollars, it means that the maximum that both will be willing to pay in order to install the first unit of the public good is 18, so one of the points on the market demand for the public good is (1, 18); for the second unit one person will pay up to 8 dollars, max. that two will pay is 16, the second point is (2, 16) etc. You can also notice that when you construct market demand for public good 1

2 you are summing up the individual marginal benefits for each unit, therefore D M = 2 MP B = 2(10 Q) = 20 2Q. The market demand curve has vertical intercept twice higher compared to the individual demand, as twice the slope and the same horizontal intercept. c) Compare the two diagrams. What will happen to the shape of market demand curves if we add more consumers in each case? The most important thing to notice about diagrams is the following. When we aggregate the individual demands for a private good, the resulting market demand curve is a horizontal summation of individual demands. When we find market demand for the public good we use a vertical summation of the individual demands. I think that in this case demand for public good might seem like a misnomer, because the curve literally tells you the maximum amount that consumers would collectively pay for an additional unit, and therefore rather represents the marginal social benefits, as we will see from the next problem it has little to do with the quantities that will be privately purchased in the market if consumers do not share the costs of purchasing each unit. In this simplified case with identical consumers adding more people will look as follows. In case of private good the vertical intercept will always be at 10, but as we add more consumers the horizontal intercept will move proportionately further from the origin, therefore the slope will always be equal to the slope of individual demands divided by the number of consumers. In case of public good adding more people to the society will move vertical intercept proportionately higher while leaving the horizontal intercept in the same spot, this means that the slope will be getting steeper. Problem 2. Marginal cost of planting a tree is constant and is equal to MC = 25 dollars. There are 15 people living in a small town who have different preferences over the trees planted on the streets (trees are obviously a public good). Five people have high marginal benefits given by MB H = 100.5Q and the remaining ten people have low marginal benefits MB L = 50.25Q, where Q is the number of trees planted on city streets. a) Plot individual MBs and the MC on a diagram. What is the maximum number of trees that can be potentially privately provided in this town if people do not share costs? Do you think it is possible that no trees will be privately provided? MC intersects MB L at Q = 100 and intersects MB H at Q = 150. If people do not share cost, then each person who decides to plant a tree has to incur private cost of 25 dollars per tree. Recall the idea behind individual choice whether to buy a unit or not: each consumer will be willing to pay for an additional tree as long as he gets marginal benefits from the tree at least as high as the cost. This means any person with low marginal benefits will potentially buy 100 trees and each person with high marginal benefits will be willing to buy up to 150 trees. Notice the following. Since trees are a public good, once a tree is planted everybody will be able to enjoy the benefits of it. This means that the maximum number of trees that will be privately planted cannot exceed 150. If there is one person in the city who unwittingly plants 150 trees, that will be the end of story. At Q = 150 MB L = 12.5, 2

3 clearly none of the guys with low marginal benefits will be buying any more trees. How about the guys with high value of the trees? A person looks at the streets and sees 150 trees. If he buys one more tree, it will be 151 st tree with MB H = = 24.5, which is lower than the cost of the tree - clearly this 151 st tree will never see the light of the day. Conclusion: we cannot tell who exactly will decide to plant how many trees, but the possible maximum number of trees privately provided is 150. It is highly unlikely that no trees will be planted in the city, the number of trees actually planted will depend on how well people can strategically assess the possible scenarios and what are their beliefs about actions of the fellow citizens. Let s consider the following case. A person walks out on a street and sees no trees, at this moment marginal benefits of having at least one tree are higher than the cost of getting it. The person then has a question: should I purchase the tree? It is of course in his private interest to do so, because MB > MC. But the smart person will actually not buy the tree: it is in everybody s private interest to get that 1st tree, so why bother? Let somebody else buy it and then enjoy the benefits without paying the cost. But then if everybody is as smart as that, then there will be no trees. This might seem like a paradox: if nobody buys the trees, it is individually rational to purchase the trees, but then if you are going to get the tree, isn t there somebody else who is thinking the same, so it is better just to wait until the the trees will be provided by other people? I think that in this case since private MB are higher than the MC, there will be some trees privately provided, but if MC are higher than individual MBs and purchasing the public good requires coordinated effort, the public good will definitely not be provided by individually rational people. b) Find and plot on a diagram social marginal benefit curve. Calculate socially optimal number of trees planted in town. MSB = 5 MB H + 10 MB L = 5(100.5Q) + 10(50.25Q) = 1, 000 5Q. Efficiency requires MSC = MSB, 25 = 1, 000 5Q, Q = 195 trees. c) How will your answer to parts (a) and (b) change if the number of people in town increases? In part (a), if anything will change at all, it will be to the worse: the more people there are the more likely they are to expect somebody else to plant the trees. In part (b) the more people there are the higher will be the MSB of each tree and therefore the higher the optimal number of trees. d) Recall Cause theorem: under certain conditions private negotiations will achieve efficiency in presence of externalities. Using the logic of Cause theorem discuss whether it is possible that people in town can reach efficient outcome. If it is indeed only 15 people who know each other relatively well, can communicate easily and also, what is important, know each other s MBs of the trees 1, they can collectively decide to share the costs and achieve efficient number of trees. If costs are shared then the price that is paid by each person will only be a fraction of the 25 dollars so people will be willing to buy more than 150 trees. Notice that the costs should not be shared 1 Which brings us to the conditions necessary for efficiency in unregulated outcome according to Coase theorem: low transaction costs and no asymmetric information. 3

4 proportionally: people with higher MB of trees should be paying more. This potentially creates a problem: if people know that their contributions will be proportional to their MBs, they have an incentive to lie and pretend they do not care about the trees at all so that they will not contribute, but will still enjoy the benefits. Problem 3. Externalities, Efficiency, and Property Rights. Alfred and Ben share an apartment. Alfred likes loud music and Ben does not. Suppose that music imposes an increasing marginal cost on Ben MEC = Q where MEC is dollars and Q is hours of loud music. Alfred s marginal value of each hour of loud music measured in dollars is MP B = 10 Q. (a) Represent this situation on diagram. Show efficient consumption of loud music and situation when Alfred listens to loud music as much as he wants. Since you are not given any info about private costs, assume MPC=0; On his own Alfred will consume where his MPB=MPC, Q=10. For efficiency we need MSC=MSB. MSC = MEC and MSB = MP B, efficient consumption of loud music is 5 hours, at Q=5 MSB=10-5=5=MSC. Welfare in unregulated outcome: DWL=25 Alfred s total benefits from the music=total value of 10 hours=50, denote U A = 50, Ben s total costs are summation of MEC from each hour=area under MEC, which is 50, denote U B = 50. (b) Suppose that there are no legal restrictions on volume of music and Alfred has a right to listen to music as loud as he wants and as much as he wants. Ben approaches Alfred with an offer to pay money for reducing the number of hours of the loud music. Will negotiations achieve efficient outcome? For simplicity suppose both guys know that if there is no agreement Alfred will listen to music for 10 hours. Negotiations will work as follows. For each hour when Alfred does not turn up the volume Ben will be willing to offer payment up to the amount of the marginal cost imposed on Ben. For example 10 th hour causes Ben MEC=10, meaning that it makes Ben so sick he will pay up to 10 dollars to avoid it. Alfred s marginal value of 10 th hour is MP B = = 0, meaning that he will accept any payment in order to give up the last hour of music. Clearly they can agree on some payment that will make both better off. They will continue negotiations until price that Ben is willing to pay is equal to price that Alfred is willing to accept, which happens at 5 hours of music, which is the efficient level of consumption. Suppose that Ben paid 5 dollars for each hour of music reduction. Let s calculate guys welfare in this case: U A = = 62.5, which is Alfred s total value of 5 hours consumed 2 + payment from Ben. Ben s utility U B = = 37.5, which is total damage from 5 hours of music and the payment. Notice what happened: both are better off compared to situation when Alfred listened to music for 10 hours, and sum of each guys increase in welfare = DWL, this means that during negotiations they managed to capture the efficiency loss brought about by the externality. Actual price for which the externality is traded does not have to be $5/hour and is indeterminate - there are many prices that will be acceptable for both guys, but regardless of the price at least one of them will benefit from reducing level of music while the other one will not be hurt. 2 Total value = area under D curve for the quantity consumed. 4

5 Internalizing the externality in this case works as follows. although Alfred does not have to pay any price in order to listen to music, possibility of getting paid for not listening creates the opportunity cost, which as you remember is a part of economic costs: every time Alfred listens to 1 hour of music he loses an opportunity to get some money from Ben. (c) Suppose Ben has a right for quiet environment. This means that loud music is legal, but Ben has a right to prohibit Alfred from playing music loudly any time. Suppose Alfred approaches Ben with an offer to compensate Ben for each hour of loud music. Will negotiations reach efficient outcome? In order for Ben to accept an offer the compensation per hour should be at least as high as MEC, for example the minimum he will accept for the first hour is 1 dollar etc. Alfred will only pay up to his marginal value of the music, for the first hour he will pay about 9 dollars. Clearly it is possible for both to benefit from trading on the first hour. Alfred will purchase 5 hours of music in total: all gains are exploited when price that Alfred is willing to pay for one more hour is equal to price that Ben is willing to accept. Let s again calculate welfare assuming that Alfred paid 5 dollars per each hour. U A = 12.5, which is his consumer surplus = TV-payment, Ben s welfare U B = 12.5, which is the payment he received minus the cost of the music. (d) Compare parts (b) and (c) and make a conclusion. If you did everything correctly you already know that in both cases the level of music is the same and it is efficient. The difference is apparently in who pays. Notice that whoever has property rights is in advantageous position and will be able to gain from selling the rights. (e) Discussion. This problem is a demonstration of Coase theorem. Explain how does Coase theorem apply to the situation described in this problem. Can you think of any reasons why negotiations may fail between Ben and Alfred? Coase theorem: when there is small number of parties involved, transaction costs are zero, there is no hidden information and property rights are clearly defined, private parties will achieve efficient solution even if externality is present. More or less obvious reason for unsuccessful negotiations is asymmetric information: if guys do not know each other s costs and benefits they might not reach an agreement. Transaction costs/contract enforcement - what if Alfred takes money from Ben and then cheats and listens to music anyways? Notice that if guys lived in a residence building where half of people loved loud music and the other half did not, transaction costs would be extremely high and even if property rights were defined, it would be very hard to reach efficient solution. Discussion Questions 1. Summarize the main differences between pure private and pure public goods. Explain why in case of pure public goods private markets are likely to fail to achieve efficiency. 2. Is pollution abatement a public good? Explain your answer. 5

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