Problem Set 1
This handout includes the questions for Problem Set 1. Answers need to be turned in via bCourses, where you will be prompted to enter your answers. For most questions, you will just enter your answer. Questions marked with an asterisk * here will allow you to enter an explanation, and will be graded for partial credit.
If you have numerical answers that are non-integers, report your answer rounded to the nearest tenth (one decimal). For example, report 1/3 as
0.3. This will help ensure that the computer recognizes all correct answers.
Part I. Economic efficiency.
1. Suppose that a policymaker can choose Policy option A or B. There are five people in the economy whose consumption of a single composite good under A is (2, 5, 7, 13, 15), and under B is (12, 9, 4, 18, 22).
True or False: Moving from allocation A to B is a Pareto improvement. (2 points)
True because option A is better off without making option B worse.
2. True or False: Moving from allocation A to B in the above example is a KaldorHicks improvement. (2 points)
False. Don’t see any cost. Pareto improvement doesn’t mean to be Kaldor Hicks improvement.
3. True or False: Any change in resource allocation that is a Pareto improvement is also a Kaldor-Hicks improvement. (2 points)
No, because Under the Kaldor-Hicks efficiency test, an outcome is efficient if those who are made better off could in theory compensate those who are made worse off and so produce a Pareto efficient outcome. every Pareto improvement is a Kaldor-Hicks improvement, most Kaldor-Hicks improvements are not Pareto improvements.
4. Consider an economy with two people, Donald and Melania, and one good, yachts, denoted y. There are 4 yachts total in the economy. Both people have the same utility function Ui(y) = y1/2. (Assume yachts can be consumed in continuous measure, not just integers.)
5. Now suppose that Donald and Melania live in an economy with two goods, yachts and condos, denoted c. There are 4 yachts and 10 condos in the economy. Both people have the same utility function, Ui(y,c) = y1/2c1/2.
True or False: An allocation where Donald has 3 yachts and 10 condominiums
(and Melania has 1 yacht and 0 condos) is Pareto efficient. (2 points)
Part II. Externalities.
6. There are 10 farms along a river. Each uses fertilizer that causes run off that lowers the profits of a downstream fishery by adding nitrates to the water. The amount of runoff (nitrates that reach the river) per ton of fertilizer used depends on the slope of the land and the proximity of fields to the river. Half of the farms are steeper and closer, so they produce 1 unit of nitrates in the water for every ton of fertilizer. Half are farther and flatter and producer 0.1 units of nitrates in the water for every ton of fertilizer. Suppose that the damage done to the fishery is $4 for every ton of nitrates in the water.
True or False: Because of the externality, the market for fertilizers is not Pareto efficient. Imposing a tax of $4 per ton of fertilizer paid by all farmers would correct this market failure and ensure that the allocation of fertilizers was Pareto efficient. (2 points)
7. Many farmers regularly feed their livestock antibiotics (even while healthy) in order to reduce infections in their populations. The widespread use of antibiotics can accelerate the evolution of bacteria that are resistant to antibiotics, however, which can increase the vulnerability of livestock in the future throughout the country and the world.
True or False: An individual farmer’s use of antibiotics constitutes an externality that implies a market failure. (2 points)
Part III. Pigouvian tax algebra problem. The Pigouvian prescription says to fix an externality by setting a tax rate equal to marginal damages at the optimal quantity. When marginal external damages are constant, the “at the optimal quantity” part is redundant. But, when marginal external damages are changing with the quantity of the good, you have to figure out the right quantity to determine the right tax rate. This problem illustrates this with an algebra example.
Consider a market where total private benefits are equal to TB = 50Q−Q2. Total private costs are TC = 12.5Q + 0.25Q2. Total external damages (costs) are equal to TED = 2.5Q2.
8. Graph this market, showing the supply curve, demand curve, private equilibrium and optimal allocation.* (Upload an image of your graph with your answers. You can draw this by hand and take a picture. Label key values.) (6 points)
9. What is the optimal tax rate on the good? (4 points)
Part IV. Optimal taxes and deadweight loss. The Pigouvian tax does not directly depend on the slope of supply and demand, which means that the tax does not depend on how big the quantity response will be to the imposition of a tax. But, does this mean that the quantity response is irrelevant? This problem provides an illustrative example.
Consider a market where total private benefits are equal to TB = 660Q−Q2. Total private costs are TC = 66Q + 0.5Q2. Total external damages (costs) are equal to TED = 66Q.
10. What is the optimal tax? (2 points)
11. What is the change in quantity that results from introducing the tax? (2 points)
12. Calculate the welfare gain from introducing the optimal tax. (2 points)
Now, suppose that the MB curve was much more vertical (less elastic). Specifically, suppose that TB = 2,244Q − 5Q2.
13. What is the optimal tax now? (Think to yourself: is this different than your answer in question 10? Why or why not?) (2 points)
14. What is the change in quantity that results from introducing the tax? (Think to yourself: is this different than your answer in question 11? Why or why not?) (2 points)
15. Calculate the welfare gain from introducing the optimal tax. (Think to yourself: is this different than your answer in question 12? Why or why not?) (2 points)
Think to yourself: what does this example suggest about the relationship between the welfare gain (elimination of deadweight loss) and the slopes of demand and supply?