Problem Set 4 Due April 7th before Class Begins March 26, 2020 You may submit typed or scanned .pdfs on the NYU classes website or, if necessary, photos of your work. All submissions must be strictly prior to the start of class (4:55pm EST). 1. Write a short (less than one page single-spaced) essay that explains how a specific aspect of the covid-19 pandemic can be understood using the ideas we have covered in this course. This assignment is intentionally open-ended; you may pick any aspect of the pandemic that interests you and where you see a clear connection to public economics. You can analyze a general topic for example, in class we discussed how the externalities of disease contagion provide a rationale for governments to regulate economic activity during a pandemic. Or, you can focus on a narrower policy issue for example, a specific bill or policy debate in your home country, such as the economic stimulus bill that is likely to soon be passed by the U.S. Congress. You are free to use news articles, academic articles, or any other reliable information sources to generate ideas and support your arguments. You can also make your argument based on abstract reasoning. The key requirement is that you explicitly connect the topic youve chosen to the concepts weve covered in class. This essay will be graded not only based on effort, but also on whether your writing is clear and your reasoning is precise and correct. Furthermore, you must provide original analysis; it is not sufficient to repeat points about the pandemic that were already made in lecture. I recommend putting significant time and energy into this assignment, especially if you struggled with the short-answer questions on the first midterm. 2. Consider a worker who is paid a wage w per hour. Her preferences over consumption (in dollars) and hours worked are given by u(c, h) = c _ h 2 . (Note that we have defined the workers utility in terms of hours worked rather than hours of leisure). The worker faces a constant marginal tax rate t on her earnings, so her net after-tax wage is w(1 _ t). Assume that the worker cannot work more than 16 hours per day. Thus, she chooses hours worked to solve max c,h c _ h 2 s.t. c _ w(1 _ t)h, 0 _ h _ 16. (a) Solve for the number of hours h(w, t) the worker will choose at wage rate w and tax rate t. (b) Suppose the worker earns a wage of w = $20. How much revenue R(t) will a tax rate t generate? 1 (c) What value of t maximizes revenue from the tax, given the workers wage rate w? You may assume that w _ $32, so that the hours constraint will not bind. Does the answer depend on w? If so, how? For the rest of the problem, assume w = $20. Suppose the government is considering changing its tax policy from t = 0 (no income tax) to a positive tax rate t _ (0, 1). (d) What is the welfare loss to the worker, in terms of equivalent variation, from this policy change? What thought experiment does equivalent variation correspond to? Remember that your answer should be a function of t. (e) Compare the equivalent variation for the worker and the revenue generated by a 50% income tax (t = 0.5, w = 20). (f) Decompose the income and substitution effects of this policy change on the workers chosen hours, h(t). These effects will be functions of the new tax rate t. (g) Interpret (in words) the income and substitution effects in the context of this problem. What thought experiment does each effect correspond to? How do the income and substitution effects found here compare to the ones we found in lecture, where a worker has Cobb-Douglass preferences? You may draw any figures that are helpful in addition to providing a written explanation. 3. Consider a modification of the optimal taxation example we discussed in class. There are two agents, 1 and 2, who have the same utility function over consumption and leisure: u(c, l) = 1 2 ln(c) + 1 2 ln(l) As in lecture, suppose that agent 2 earns a wage of w2 = $40 per hour worked. Agent 1 can also earn income by working, but at a wage of only w1 = $20 per hour. If agent i works h hours, she earns wih dollars and enjoys l = T _ h hours of leisure. (a) Privately Optimal Allocation: Suppose there are no taxes and each agent gets to keep the money she earns from working. i. Write down each agents utility maximization problem. ii. Find the privately optimal amount of hours worked by each agent. iii. Under the private optimum, does agent 1 work more, or agent 2? Which agent is better off? For the rest of the problem, assume the social planner is a utilitarian, i.e. wants to maximize the sum of the two agents utilities: SU = u(c1, l1) + u(c2, l2) (b) First-Best Allocation: Now suppose the social planner knows each agents type and can tell each agent how much to work and how much to consume. i. Write down the social planners problem under the first-best assuming the two agents have a general utility function u(c,l). Which variables does the social planner control? What constraints are there on the set of feasible allocations? ii. Still working with the general utility function u(c, l), take the first-order condition with respect to each control variable. Remember that you can use the social planners budget constraint to write agent 1s consumption (c1) in terms of h1, h2, and c2. So, there should be three FOCs. 2 iii. What is the economic interpretation of each of the three first-order conditions? iv. Using the specific utility function u(c, l) = 1 2 ln(c) + 1 2 ln(l), solve for the first best allocation. Denote the first-best allocation by (c F B 1 , lF B 1 ) for agent 1 and (c F B 2 , lF B 2 ) for agent 2. v. Discuss the first-best allocation. Do agents 1 and 2 work and consume the same amounts, and if not, how do they differ? How does the first-best allocation compare to the privately optimal allocation in this problem? How does it compare to first-best allocation in the example we considered in lecture? (c) Second-Best Allocation: Now suppose the social planner does not know each agents type. Furthermore, if an agent works, the social planner cannot observe her wage rate, only her total income wih. The social planner must now satisfy an incentive compatibility constraint for each agent: each agent must prefer her allocation to mimicking the other agent, which in this problem means earning the same income as the other agent rather than working the same number of hours. You may assume that the only relevant incentive compatibility constraint is for agent 2. i. Write down the incentive compatibility constraint for agent 2. Remember that if agent 1 works h1 hours, agent 2 can mimic agent 1 by working only w1 w2 h1 = 1 2 h1 hours, because then agent 2 will earn the same income as agent 1. ii. Does the first-best allocation you found in part (b) satisfy agent 2s incentive compatibility constraint? iii. Write down the social planners problem under the second-best assuming the two agents have a general utility function u(c,l). Be sure to include agent 2s incentive compatibility constraint. iv. Construct the Lagrangian for this constrained optimization problem. It should look similar to the one we wrote down in lecture, but not identical. v. Explain intuitively how you would expect the second-best allocation to differ from the first-best allocation, and why. Extra Credit: If you have solved the problem set up to this point, you will receive full credit. Solving the questions below correctly will earn you extra credit. This credit will not affect the curve when I determine final grades; instead, it can bump up your grade after the curve. This means your grade will not suffer if you do not solve the remaining parts. vi. Using the specific utility function u(c, l) = 1 2 ln(c) + 1 2 ln(l), take the first-order condition with respect to each control variable and with respect to the Lagrange multiplier _. There should be four conditions three FOCs with respect to each control variable, and one condition stating that agent 2s incentive compatibility constraint holds with equality. vii. What is the economic interpretation of each of the four conditions? viii. Again using the specific utility function u(c, l) = 1 2 ln(c) + 1 2 ln(l), solve for the second-best allocation. Denote the second-best allocation by (c SB 1 , lSB 1 ) for agent 3 1 and (c SB 2 , lSB 2 ) for agent 2. ix. Discuss the second-best allocation. How does it compare to the first-best? Do agents 1 and 2 work and consume the same amounts? If not, how do they differ? 4
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