Problem Set 4
Due April 7th before Class Begins
March 26, 2020
You may submit typed or scanned .pdf’s 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 pan-
demic. Or, you can focus on a na
ower 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 you’ve chosen to the concepts we’ve 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 co
ect. 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− h2 .
(Note that we have defined the worker’s 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 pe
day. Thus, she chooses hours worked to solve
max
c,h
c− h2 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
ate 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 worker’s 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 co
espond 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 worker’s
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 co
espond to? How do the income and
substitution effects found here compare to the ones we found in lecture, where a
worker has Co
-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 agent’s 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 agent’s type
and can tell each agent how much to work and how much to consume.
i. Write down the social planner’s problem under the first-best assuming the two
agents have a general utility function u(c,l). Which variables does the so-
cial 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 condi-
tion with respect to each control variable. Remember that you can use the social
planner’s budget constraint to write agent 1’s consumption (c1) in terms of h1,
h2, and c2. So, there should be three FOCs.
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iii. What is the economic interpretation of each of the three first-order conditions?
iv. Using the specific utility function u(c, l) = 12 ln(c) +
1
2 ln(l), solve for the first
est allocation. Denote the first-best allocation by (cFB1 , l
FB
1 ) for agent 1 and
(cFB2 , l
FB
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 com-
pare 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
agent’s type. Furthermore, if an agent works, the social planner cannot observe he
wage rate, only her total income wih. The social planner must now satisfy an incen-
tive 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
2h1
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 2’s incentive
compatibility constraint?
iii. Write down the social planner’s problem under the second-best assuming the
two agents have a general utility function u(c,l). Be sure to include agent
2’s 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 co
ectly 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) = 12 ln(c) +
1
2 ln(l), take the first-orde
condition with respect to each control variable and with respect to the Lagrange
multiplier λ. There should be four conditions – three FOCs with respect to each
control variable, and one condition stating that agent 2’s 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) = 12 ln(c) +
1
2 ln(l), solve for the
second-best allocation. Denote the second-best allocation by (cSB1 , l
SB
1 ) for agent
3
1 and (cSB2 , l
SB
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
Problem Set 4
Due April 7th before Class Begins
March 26, 2020
You may submit typed or scanned .pdf’s 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 pan-
demic. Or, you can focus on a na
ower 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 you’ve chosen to the concepts we’ve 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 co
ect. 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