Answer Key to Midterm Exam. February

Transcription

1 ECON 525 Farnham/Gugl Answer Key to Midterm Exam February ) Public goods. Suppose there are two identical people in the economy. They privately contribute to a public good. Consumers have the following utility function over a private good x and the public good G u ( x, G) = ln x lng. i i i + The income of each of the consumers is $300, so $600 combined. The prices of both the private good and the public good are equal to 1. a) Find the Nash Equilibrium of the private contribution to the public good game. Given that each person has the same utility function and income, it is sufficient to derive one best response function and then invoke symmetry. max ln such that 300. Rewriting the problem, we get max ln300 ln Taking the derivative with respect to g1 yields. By symmetry,. Solving for the NE, we find 100. b) Show that income transfers from one consumer to the other--as long as both contribute to the public good before the transfer and after the transfer--yield the same utilities for both consumers as in a). The best response function for any amount of income and an interior solution is given by! ". And hence the NE by,. This means that the amount of public good provided is equal to $ %. The amount of private good that each consumer consumes in the Nash equilibrium is equal to &! " %. In the NE both the level of the public good as well as the amount of the private good that each person consumes does not depend on the distribution of the combined income of the two agents. Only the sum of consumers incomes matters. Thus the utility level for each consumer does not change either as long as we have an interior solution and & &

2 c) Is it possible for the government to tax both people to increase the provision of the public good and to make both consumers better off than in a)? Prove your answer. Yes, this is possible. Consider a tax higher than the equilibrium distribution, but no higher than the efficient amount. The efficient level of the public good would be G=300, while the NE level is G=200. So consider a tax t=150 on each person. Given the tax and the resulting level of public good provided by the government of 300, person 1 maximizes max ln ln 300 The best response function is then given by ) * 0. This means, we have a corner solution for person 1. By symmetry the same is true for person 2. Neither consumer contributes to the public good and the total level of the public good is equal to the amount provided by the government, G=300. Finally we need to show that both consumers are indeed better off than in the NE. I use a monotone transformation of the utility function to calculate the utility levels under each regime because this is easier to numerically evaluate than using the function originally given. In NE: + 200, ,000. Under the tax regime, + 150, ,000. d) Consider the Falkinger Mechanism in which each consumer is taxed or subsidized based on the deviation from the average contribution of all the other consumers in the same income group. In the case of only two people this average is just the contribution of the other consumer. Given that the tax/subsidy is calculated by /, find the Nash Equilibrium contributions of each consumer. Given the reward/tax and the level of public good provided by the government, person 1 maximizes max ln ln 300 Taking the derivative with respect to g1 and rearranging, we get 300. We get the same equation for person 2. Note that there is a continuum of NE; but all of them yield a level of the public good of G=300. e) What can you say about the equilibrium level of the public good in d)? The Falkinger Mechanism yields the efficient level of the public good. 2) Consider the Clarke Groves Mechanism. The following table shows the report of 3 people. Assume B1B2. Show that a change in the report of person B would not give person A an incentive to change her report. 2

3 Project 1 Project 2 Person A 7 5 Person B B1 B2 Person C 8 7 If person B makes reports such that project 1 is chosen, person A has no incentive to lie. In this case person A receives 7 and pays a tax of B2+7 (B1+8) or pays no tax. To see this, first note that project 1 is chosen if and only if the sum of reported net benefits is higher for project 1 than for project 2. That is, 7+8+B1 > 5+7+B2. Person A s report in this case may change the group choice and hence a tax of (B2+7 (B1+8)) must be paid or may not change the group choice because B2+7 B1+8. Person A has no incentive to lie if no tax has to be paid because 7>5. Even in the case in which the tax is paid 7-(B2+7 (B1+8)) > 5 Because rearranging this inequality becomes 7+ (B1+8)) > 5+ B2+7 which is true. If person B changes the report so that project 2 is chosen, then person A by reporting truthfully receives 5 and pays no tax. This is the case because for project 2 to be chosen we must have 7+8+B1 5+7+B2. Since both A and C report a higher net benefit with project 1, it must be the case that B2-B1>3 and hence B2+7 > B1+8. If person A lied to get project 1 selected, person A would pay a tax of (B2+7 (B1+8)). This would make person A worse off than receiving 5. To see this, first note we need 7-(B2+7 (B1+8)) 5. Rearranging the inequality we have 7+ (B1+8)) 5+ B2+7 which is true. To conclude, no matter what person B reports, person A has no incentive to lie. 3) Elisabeth likes to set her alarm, so that it goes off just a few minutes before she has to get out of bed. Markus likes to set the alarm so that he has another hour to stay in bed before he gets up. They both have to get up at the same time. Elisabeth s utility function is given by + 2 2,3 where t indicates the minutes before Elisabeth s preferred wake-up time and xe indicates her amount of private consumption. That is, 3 4 0,55. Markus utility function is given by + 5 5,3. Elisabeth s income is 200 and Markus income is 100. One unit of the private consumption good has a cost of $1. a) Briefly explain which kind of economic problem Elisabeth and Markus are facing. The consumption of one person has an unintended negative effect on the other person. We are dealing with a negative consumption externality. 3

4 b) State the efficiency condition and explain briefly what it means. (Note that whatever the value of t is, private consumption in total is always equal to 300. So the MRT of time expressed in units of consumption that have to be sacrificed is 0.) max+ 5 5, ,3 + 2 such that Or equivalently, max+ 5 5, ,3 + 2 Taking the derivative with respect to xm, Taking the derivative with respect t This yields the following condition, The amount of private good Elisabeth needs to receive in order to be compensated for an earlier wake-up time by one minute must equal the amount of private good Markus is willing to give up in order to get an earlier wake-up time by one minute. (Marginal benefit of an earlier wakeup time by one minute needs to equal its marginal cost.) c) Suppose Elisabeth has the right to set the alarm 5 minutes before wakeup time, so t = 0. Can Elisabeth and Markus negotiate an agreement that is efficient? How would this contract look like? Yes, this is efficient. Let the price of permits be q. Elisabeth is the supplier of time permits and Markus is the demander of them. Elisabeth solves the problem max ,3 = 73 = 72 Markus solves the problem max ,3 4

5 = 73 = 75 We can see that this equates the marginal benefit and the marginal cost of t: d) Suppose now Markus has the right to set the alarm 55 minutes before wake-up time, so t = 55. Can Elisabeth and Markus negotiate an agreement that is efficient? How would this contract look like? Yes, this is efficient. Let the price of permits be q. Elisabeth is the demander of quiet time given by (55 t) and Markus is the supplier of it. Elisabeth solves the problem max ,3 = 73 = 72 Markus solves the problem max ,3 = 73 = 75 We can see that this equates the marginal benefit and the marginal cost of t: 5