Assignment:
Public Economics
David Benjamin
Q 1. Let Joe and Mojo have preferences ai0.25bi0.75 . Assume each agent has an equal endowment of ale and bread of five units.
(a) Calculate Joe’s excess demand as a function of the price of ale.
(b) Compute the equilibrium by setting setting the excess demand functions equal to zero. Could you have taken a short cut?
(c) Compute Joe’s equilibrium Marginal Rate of Substitution.
(d) What happens to the equilibrium if Joe has instead 15 units of ale?
Q 2. Assume a firm can generate 5q1 units of profit for each unit of output q1. Assume that each unit of production produces q12/5
units of pollution. Suppose the firm is endowed with one emission permit and can purchase additional permits at price p.
(a) Find the demand for permits as a function of p. What is the corresponding output level?
(b) Assume a second firm whose permit excess demand is 1−p. What is the equilibrium permit price?
Q 3. Describe an environment in which permit trading can contribute to economic efficiency. How do permits make such a contribution? Why in a traditional model are permits and Pigouvian taxes considered equivalent in implementation? Give a reason that in practice they may not be.
Q 4. Suppose the government does not know the level of the marginal damages, but that the neighbors themselves can effectively petition the government to impose a Pigouvian tax of any size after a failed negotiation.
Lets start from the assumption that the neighbors can pocket the tax revenues and will ask for the tax rate that maximizes their revenues? What will the set of outcomes of these negotiations include?
Lets say the government wanted to implement the efficient quantity but with this feedback mechanism, how might the government implement a reporting scheme (with possible fines on both sides) to insure an effi- cient outcome if the original model does not. What if the government has access to a costly auditing technology?
Q 5. Assume a Total damage function of D = 3q2, a demand function constantwith price 200 and a supply function of T C = 4q3 + q +4.
Find the equilibrium quantity and set of possible transfers both with and without negotiations and with property rights that either belong to the firm or the neighbors. What is the Pigouvian tax that would have produced the same outcome in terms of quantity?
Q 6. Assume UG = z0.333xg0.667. Assume George is endowed with 10 units of the private good which he can convert to public goods at a one for one ratio.
(a) Take the other agents donations of private goods to be converted to public goods as given and equal to zero. Write down George’s utility as a function of his donation.
(b) How much of the private good will George donate in equilibrium.
Q 7. Derive the Samuelson condition graphically.
Q 8. What does it mean for a good to be non-excludable? In theory one may worry that such a problem could completely shut down private markets. How did we deal with that in designing a private environment for the public goods case so that this doesn’t happen? How and why did we use game theory in this design?
Q 9. Traditionally we look at the distribution of consumption as a fairness concern, not an efficiency concern. Why is that the case? Is there a perspective from which income distribution looks like a public good (or the result of a public good)? [Hint: the book offers one in a later chapter and public discourse may offer additional.] Why can education or knowledge be considered a public good? Note a good that is partially rival, excludable and reproducible can be efficiently provided for as a club good in which there is a charge for both membership and usage. Using these two results, Milton Friedman once argued that there was a strong case for subsidized education, but not necessarily publicly provided education. Defend his claim. If there is a limit to his logic
what is it?
Q 10. Let’s implement a public good model with a little bit of a twist. Suppose we have two individual each with 10 units of land that each without conversion can produce 10 units of a private good. But Joe’s land can be converted at a rate of 5 to 1 into a public good whereas Dinah’s land can be converted into the same public good at a rate of 1 to 1. Suppose they have preferences given by log(x) + log(z) where both are public goods and the derivative of log(x) is 1/x .
(a) Solve for the equilibrium with private contributions. Suppose Joe can give Dinah land under the condition that Dinah donate the land to be developed into the public good. What is the outcome here? Suppose Joe can unconditionally give Dinah land and Dinah
can plan to regift a constant proportion of Joe’s to her for development into public land (any fraction), what would be the optimal amount of a donation.
(b) Solve for the optimal level of public provision with uniform contributions from both Dinah and Joe. Compared to the above benchmarks is it efficient?
(c) Suppose that Dinah and Joe’s donation will produce two parks. Joe’s consumption of parkland will equal zj0.8zd0.2 and Dinah’s will equal the opposite. Repeat the above two parts. Frequently one issue with public provision of private goods that has been pointed out in the literature is that prices of publicly provided goods may not capture the cost of providing the good and thus may be costly. Comment on that with regard to these results.
Q 11. One area in which externalities arise is in congestion which is at the heart of the common’s problem but we can capture with a simple example. Suppose an inelastic demand for good x at price 1 which doesn’t change as the quantity changes. Suppose that there are a supply of N workers with one hour to supply and that the aggregate supply of the good will be y(n) where n is the number of workers who attempt to supply the good while sacrificing a wage of w producing other good. In a traditional model without externalities y(n) will typically be linear until it hits its maximum of N.
(a) Suppose that instead y(n) = 2∗n0.5 . Plot y. Define Social Surplus for this economy. Show both the efficient quantity of work on your figure and the ”market outcome” if all workers who produce the good corner an even share of the market. Assume the market wage for outside labor is 1/16 . Also assume that the workers start the day in a line in front of two doors, one for production of the congested good and one for traditional goods. They pick the one that will them higher profits or wages given the number of people
who have entered each door and the number of people behind them they expect to rationally pick each door. Explain the nature of the externality in this problem. Suppose the government wanted to fix the inefficiency with a Pigouvian tax, how big should it be?
(b) Suppose that instead we have a production function given by y(n) = (1/32)n4 . Suppose there are a sufficiently large number of
potential workers (N=3000). How many equilibria are there? Are either of them efficient? Can public policy eliminate the bad equilibrium? How?
Q 12. Assume a constant demand function of 5 and a constant MD of 1. Assume also a linear MC function of 2Q. Assume the government does not MD but wants to use a Pigouvian tax. Lets say both the neighbors and the firm know MD but cannot prove it. Lets also suppose that this a repeated economy where both sides can overcome coordination issues.
(a) Lets say the possible taxes are bounded above by 5 and below by 0 and that in the default the government will tax 2.5 per unit. Suppose the government auctions off the right to set the tax. What will the outcome likely be?
[Hint: Assume both sides are willing to pay at most the surplus from changing the tax and are willing to bid one penny more than what they perceive the other side’s maximum bid.]
(b) Suppose instead the government begins in period 2 by bidding the right to change the tax from its starting point by 50 cents per
unit. Suppose a tie bid keeps the tax where it is set. What will the outcome be of this process?
(c) What happens if only neighbors who are responsible for half of the damages are willing to pay their part of the bid?
13. What is meant by least cost pricing in carbon models? What is the range for the mean of GDP falls these models predict by 2050 in percentage terms? What is the main deviation from least cost pricing Edmonds attempts to capture? How does he model it? How does this affect results on the costs of eliminating carbon?
14. How do researchers estimate the cost of climate change in terms of agriculture and sea level? What results do they find?
15. How do researchers typically construct a marginal damages function for climate change from individual estimates of total damages? What are some of the limitations of this approach?
Attachment:- cpemanual.zip