Professor Scholz Posted March 1, 2006 Brief Answers for Economics 441, Problem Set #2 Due in class, March 8, 2006

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1 Professor Scholz Posted March 1, 2006 Brief Answers for Economics 441, Problem Set #2 Due in class, March 8, , 10 points) Please do problem 14 from Chapter 8. a) The cost for someone from city A is 10 minutes at $0.50 per minute, for a total cost of $5. The cost for somebody from city B is 20 minutes at $0.50 per minute, for a total cost of $10. b) The individual demand curve goes through the points (10,5) and (5,10). The linear, inverse demand curve that goes through these points is P=15-Q. Consumer surplus for A types is.5*(10)*(10)=50. There are 200,000 inhabitants of city A, for a total surplus of 50*200,000, or $10 million. Consumer surplus for B types is.5*(5)*(5)=12.5. There are 200,000 inhabitants of city B, for a total surplus of 12.5*200,000, or $2.5 million. Total consumer surplus is $12.5 million. c) The park should not be sold. The value of the park is the infinite sum of consumer surplus, which is $12.5/.1, or $125 million, which exceeds the purchase price of $100 million. 2, 5 points each: True, false, or uncertain and explain!) a) In Eden, Eve does the fishing and Adam makes the clothes they then barter clothes for fish. Adam dumps dye into the lake, which lowers Eve s fishing yield. There is no need for a higher power to intervene to address Adam s negative externality on Eve. UNCERTAIN: If the property rights to the lake are clearly assigned either to Adam or to Eve, then there is no need for intervention. We know they have costless bargaining, since they already engage in trade. If the property rights are no assigned (or if you think the higher power would be needed to assign the rights), then there is need for intervention. Note that the fact that dumping raises the price Adam faces for fish does not mean that he internalizes the externality. He internalizes only the effect of dumping on his consumption of fish, not the effect on Eve s consumption of fish. b) There is no reason for state or federal subsidization of education, since people who live in areas with low educational spending prefer low taxes to high education spending. FALSE (or UNCERTAIN). Under the Tiebout mechanism, localities can achieve optimal public good provision if the residents are able to move to communities where other residents have similar preferences. There are numerous reasons why Tiebout might not hold: 1) It is costly to move between towns. 2) There aren t always enough towns to choose from so that people can t find one with their preferred level of spending. 3) Lump-sum taxation is politically undesirable, and towns use other taxing mechanisms (mainly property taxes). With property taxes, not everyone is taxed equally for public goods. This creates problems of free-riding by people who pay lower taxes. 4) Education is not the only public good provided by a town. If towns offer a basket of public goods, it will be a lot harder for someone to find a town that matches her exact preferences. 5) There are numerous spillovers from education across towns. These include reduced crime, more informed voters, and possible production externalities. This last point is especially important with education. 3, 10 points) Your utility function is given by U = ln(4 C), where C is consumption. You make $30,000 per year and enjoy jumping out of perfectly good airplanes. There s a 5 percent chance that in the next year you will break both legs and incur medical costs of $15,000 and $5,000 of lost wages. What is your expected utility without insurance? Suppose you can buy insurance that will cover your medical expenses

2 and foregone salary. How much would such a policy be if it is actuarially fair and what is your expected utility if you buy it? What is the most you would be willing to pay for the policy? Expected utility is 0.95*ln(4*30,000)+.05*ln(4*10,000)= Now there s a 5% chance of paying out $20,000, so an actuarially fair policy would cost $1,000. Your expected utility if you buy is.95*ln(4*(30, ))+.05*ln(4*(30, ))= To calculate the most you would be willing to pay for the policy, calculate 30,000-(exp(11.640)/4)=$1, Of course, the problem was writing so that U = log(4 C), where C is consumption. You could also use the log (rather than natural log) function. If you did this, your answer would be Expected utility is 0.95*log(4*30,000)+.05*log(4*10,000)= Now there s a 5% chance of paying out $20,000, so an actuarially fair policy would cost $1,000. Your expected utility if you buy is.95*log(4*(30, ))+.05*log(4*(30, ))= To calculate the most you would be willing to pay for the policy, calculate 30,000-(inv((log)(5.055))/4)= /4 = $1, , 30 points) In the schools of Beat town students are educated to play music, which they then perform at home for their parents. When more money is spent on the schools, the students learn more songs, and their parents are more entertained. The family utility function for each Beat town family is U=9ln(C)+ln(S) where S is Beat town s per-student expenditure on schooling and C is the amount of money the family has left over for other consumption after paying the school tax. a) Although all the families in Beat town have the same utility function, they have different incomes. 100 families each earn $20,000; 100 families each earn $50,000; and 100 families each earn $80,000. How much does each type of family want to spend on schools, assuming there is a lump-sum tax levied on all families? If the town votes on the level of the lump-sum tax, what level will win? Each family wants to choose consumption C, given income Y: U=9ln(C)+ln(Y-C). The first order conditions imply 9/C=1/(Y-C); and C=0.9Y and S=Y-C=0.1Y. So families earning $20,000 will want $2,000 of schooling, families with $50,000 will want $5,000 of schooling, and families with $80,000 will want $8,000 of schooling. By the median voter theorem, $5,000 will prevail. b) Lump-sum taxes are declared unconstitutional, and Beat town must substitute a flat percentage income tax. If the town votes on the single income tax rate, what rate will win? How will social welfare (the sum of all of the utilities of Beat town residents) change? Explain. Since the tax is flat, the families the earn $50,000 will get $1 of schooling for each $1 they are taxed. This is not true for other families: the poor families get more than $1 of schooling for every $1 they are taxed and the rich families get less than $1 of schooling for every $1 they are taxed. Therefore, the decision for families earning $50,000 is the same as in part a, where the price of schooling was 1. They will want $5,000 of schooling, which can be provided by a 10% tax. Social welfare goes up, but not because everyone is now paying the amount they want to pay the rich families only wanted to pay $8,000 if they got $8,000 in schooling back. Welfare increases because the level of schooling is the same as in part a), but income has been redistributed from the rich families (who now have $72,000 in consumption, compared to $75,000 above), to the poor families (who not have $18,000 in consumption, compared to $15,000 above). Since utility is concave, there are diminishing marginal returns to income, and $3,000 increases the poor families utility by more than the rich families utility. If we weight the families equally, then a transfer from the rich to the poor increases average welfare, even though it decreases the welfare of rich families. 2

3 c) Two new empty towns become available for occupation, for a total of three towns available for residence. How will the residents sort themselves? What will be the per-student spending in each town? How will total social welfare (i.e. efficiency) change? What will happen to the utility of each income group relative to part b)? Explain. The $80,000 families will move to their own town since they don t like paying $8,000 for $5,000 of schooling. Once the highest income families leave, the $50,000 families will also leave to form their own town, since they are now paying $5,000 and only getting $3,500 in education. The $20,000 families will be left on their own in the original town. The $20,000 families will now spend $2,000 and get $2,000 in education. The $50,000 families will now spend $5,000 and get $5,000 in education. The $80,000 families will now spend $8,000 and get $8,000 in education. Everyone is now paying the same amount they paid in b), but the highest income families are getting more schooling, so they are better off, while the poor families are getting less schooling, so they are worse off. Social welfare has decreased because there are diminishing marginal returns to schooling, so the extra $3,000 in schooling that the rich families get does not increase their utility as much as the loss of $3,000 in schooling decreases the poor families utility. d) The state that governs all three towns decides it wants to increase the amount of schooling students get. It considers three proposals: 1) providing matching grants to the towns for each dollar of per-student spending, 2) establishing state-funded schools that spend $6,000 per student, 3) providing unconditional grants to each town. Compare the effects of the proposals on the level of total education spending in each town. Matching grants have both income and substitution effects towards education. They will increase education provided in all three towns. State funded schools increase schooling in the $20,000 town and the $50,000 town, as both towns send their children to the new schools rather than the old $2,000 and $5,000 schools. The effect on the $80,000 town, however, will not be to increase schooling. Either the town will continue to send its children to the $8,000 schools, in which case there is no effect, or they will send their children to the $6,000 state schools, in which case schooling will decrease. Which they decide to do depends on whether they value an increase in consumption from $72,000 to $80,000 more of less than a decrease in schooling from $8,000 to $6,000: 9*ln(80,000)-9*ln(72,000)=0.95 > 0.29 = ln(8000)-ln(6000), so they will decide to send their children to the state schools, decreasing education. Education in each community will rise with unconditional grants given to each town (draw the usual budget sets and indifference curves schooling is a normal good, so consumption increases with income). 5, 60 points) Suppose a passenger aircraft crashes into a remote Pacific island, leaving only three survivors. After days of waiting, it becomes apparent that help may never arrive without some effort by the survivors. Fortunately, this small island is covered with coconut trees. One survivor (Michael) suggests building a raft with the coconut trees to sail off the island and search for help. Another survivor (Shannon) doubts that help would be found with a raft and prefers making a shelter with the wood to make island life more comfortable. The third survivor (Jack) cares equally for building a raft and building a shelter. The castaways must decide how many trees to devote to the raft and how many to shelter. The more trees used for the raft, the more likely it is to be successful. The more trees used on the shelter, the more comfortable the shelter will be. Suppose there exist 100 trees for use on this island. The three castaways have the following utility: 3

4 1 1 Jack: UJ = ln R+ ln S Michael: UJ = ln R+ ln S Shannon: UJ = ln R+ ln S, where R is the number of trees used in the building of the raft, and S is 4 4 the number of trees used to build the shelter. a) How many trees would each person choose to devote to the raft, and how many would each person devote to the shelter if the decision were solely his or hers to make? Each person s utility maximization problem is of the following form: max alnr+blns, such that 100=R+S. Rearranging the budget constraint in terms of S and plugging in, we get: U = alnr + bln(100-r). Maximize U taking the derivative with respect to R and setting it equal to 0 and solve for R, we get R = (100a)/(a+b), but since a+b=1, R=100a. So Jack prefers 50 trees be used for the raft and 50 for shelter (because for Jack, a=0.5 and b=0.5). Michael prefers 75 trees for the raft and only 25 for the shelter, while Shannon favors 25 trees for the raft and 75 for the shelter. b) On a single graph, sketch a plot of each person s utility as a function of the number of trees used for the raft. (Hint: plot the person s optimal number of trees for the raft, and think about what the function would look like if more of fewer trees were used than the optimal amount). Shannon U3 Jack Michael This picture, particularly U3, is not drawn to scale. But all that matters here is that the new utility function (in part f) is not single peaked, since the maximums occur at R=0 and R=100. Remember, with 4

5 utility functions the level of utility is unimportant all that matters is the preference rankings of goods (this is a different way of saying that utility is invariant to monotonic transformations, since these transformations preserve the preference ranking across choices). c) Suppose that the decision on how many trees to devote to a raft is made by majority vote. Based on your results from b, would you expect majority voting to yield a consistent outcome? Explain. Yes, from the graph in b, we see that each person s utility as a function of R has a single peak (the peak for Michael is 75, for Jack is 50, and for Shannon is 25). We know that majority voting yields a consistent outcome (i.e. the majority outcome doesn t cycle and the outcome will not depend on the order in which options are considered) provided preferences are single peaked. d) What final vote would you expect under majority voting? Explain intuitively why a majority vote would settle at this outcome. The median voter theorem states that if everyone s preferences over the decision being voted on are single peaked, then the decision from majority voting will be the same as the choice of the median voter (the person who prefers the median amount of the good). Here it s easy to see graphically from your answer to b. Suppose the three are trying to use 25 trees, more than 25 trees, or fewer than 25 trees on the raft. Jack and Michael would vote for more trees, and win the vote. If they considered 75 trees, Jack and Shannon would vote for fewer and win the vote. They will eventually settle on 50. At 50, a majority will not exist to either increase or decrease the number of trees that are used. Hence, 50 will be the final outcome. Intuitively, this is how the majority settles on the preferred choice of the median voter, in practice, when choosing over a continuum of possibilities for the public good. e) Suppose four more survivors (from the tail section of the plane) are discovered. Three of these 4 1 survivors have utility functions: U1 = ln R+ ln S, and the other has utility function U2 = ln R+ ln S (so three really want to be rescued and one really likes shelter). If these new 5 5 survivors were also given votes, how would you expect the decision under majority voting to change from that in c)? Why? Without carrying out additional calculations, can you determine the exact decision that would result from majority voting? You should now recognize that three of the survivors prefer for more trees to be used for the raft than Michael preferred (the a parameter in utility is greater than 3/4), while one wants fewer trees than Shannon. Hence, the majority should result in a greater number of trees used for the raft than before. Since three life using more than Michael and three like using fewer, Michael is the new median voter. So majority voting will lead to a new optimal choice 75 trees will be used for the raft. f) Suppose instead of those four survivors, only two additional survivors are found. They each have utility functions U3 = max( R, T), so they don t care whether trees are used for a raft or a shelter, as long as resources are only devoted to one use. Add this utility function to your sketch from part b). Will a consistent outcome emerge from majority voting? If so, what is the new outcome? If not, explain or demonstrate why. See the graph in part b. The new preferences are not single-peaked (there is a peak at R=0 and R=100), so the median voter theorem no longer applies. More importantly, majority voting is not longer consistent. The way to see this is to first consider a vote on whether 51 trees should be used, more than 51 or fewer than 51. The voting process will settle at 75, since at 75, exactly half want more and half 5

6 want fewer. If, however, the initial vote was for 49, more than 49, or fewer than 49, the majority vote will settle on 25 trees. Hence, with these preferences, two outcomes are possible under majority voting. This is not consistent, since it depends on how the initial vote was structured. This gives a great deal of power to the agenda setter: if the agenda setter (Michael) started at 51 votes, his preferred outcome would be realized. If instead Shannon were the agenda setter, she could achieve her preferred outcome. 6