ECON 23950: Economic Policy Analysis Problem Set 2
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ECON 23950: Economic Policy Analysis
Problem Set 2
(Due: October 19, Monday at the beginning of discussion section)
If you type up your solutions nicely, your work will be rewarded with a 10% bonus point.
1 True, False or Uncertain?
Your answers are evaluated on clarity of your argument. Make your assumptions explicit and defend your answer briefly.
1. In a model with capital without labor, a proportional consumption tax does not distort capital accumulation as long as the tax rate is held constant over time.
2. When the government switches from a lump-sum tax to a proportional income tax to raise the same revenue, the fiscal multiplier becomes smaller. Hint: You can think about the first model (with capital but no labor) or the second model (with labor no capital).
3. Lump-sum taxes are non-distortionary.
2 Decentralizing the Crusoe Model
We will recast the model we used to study fiscal multiplier so that it mimics a market-based economy.
Suppose that there are many identical households whose utility function is
We assume that there is no storage technology that allows currently produced goods to be consumed in future. This amounts to assuming that there is no capital stock or that K is nor- malized to unity in the production function. Each agent, however, can borrow and/or lend at a risk free interest rate rt.
Each atomistic agent, taking as given the wage and the real interest rate, maximizes U subject to the following budget constraint:
wtlt+ (1 + rt)bt + πt = ct+ bt+1 + τt
where bt denotes the pre-determined saving that was decided in time t−1 , πt denotes the profit that the household received from the firm, and τt denotes taxes. We assume that the households own the firms so that the profits are rebated back to households. We also assume that the initial holding of assets for all agents are zero: b0 = 0.
Turning to the firm’s problem, we assume that there are many identical firms that have access to the production technology f(l). They take the wage as given and maximizes profit each period:
πt = f(lt) − wtlt
Finally, the government levies lump-sum taxes τt and balances its budget each period so that
τt = gt for all t = 0, 1, 2, 3, ...
As in lecture, we assume the following properties regarding the functional forms of preference and technology: u′ () > 0; u′′ () < 0; v′ () < 0; v′′ () < 0; f′ () > 0; f′′ () < 0.
1. Interpret the budget constraint of the household.
2. Set up the representative household’s maximization problem. Derive the FOCs and opti- mality conditions (both Euler equation and MRS equation).
3. Set up the representative firm’s maximization problem. Derive the FOC.
4. Define the competitive equilibrium of this economy. Clearly state the market clearing conditions.
5. Show that the competitive equilibrium should exhibit the following properties:
6. Will the allocation in the competitive equilibrium be the same as in Robinson Crusoe’s economy we derived in class? Why or why not? What is the role that the Euler Equation play? Explain.
7. Show that the fiscal multiplier 
is positive but less than one.
Suppose now that we relax the government’s budget constraint such that it can borrow and lend:
gt+ (1 + rt)Bt = τt+ Bt+1
where Bt represents government borrowing determined in period t − 1 payable in period t.
8. Rewrite the market clearing conditions in this economy. In particular, what is the credit market clearing condition now?
9. Suppose that the sequence of government spending 
is fixed and the government is free to change the sequence of 
or 
How does the allocation change? Explain intuitively.
3 Dynamic Response to Tax Shock
Let’s analyze how the economy reacts to a tax cut in a simple economy we covered in lecture. The household maximizes the lifetime utility
subject to the resource constraint
Kt(α) = Ct+ Kt+1 − (1 − δ)Kt + Gt
where Gt = τKt(α) and the economy starts with a given initial value of capital K0 .
1. Do the policy experiment numerically on Dynare. Specifically, assume that τ increases from 0.1 to 0.2. You can assume the same parameterization from the previous homework. Print your code and impulse response diagrams and turn them in.
2. Comment on the dynamic response of the economy to the tax increase.
4 Optimal Taxation in Theory and Practice
This question requires reading a paper written by Gregory Mankiw, Matthew Weinzierl, and Danny Yagan (http://www.nber.org/papers/w15071). They summarize main theoretical find- ings of the optimal taxation literature and provide data to show whether governments follow them. What is nice about their work is that they do not use math; they explain things quite simply and intuitively. Read the entire paper to answer the following questions. Do not copy the writing from the article but explain in your own language. Make your answers as brief as possible. Your answer should be no longer than four sentences for each question.
1. Explain why the lump-sum taxation achieves the first best and why it is rarely used in reality.
2. (Lesson 2) Provide two theoretical reasons as to why optimal marginal tax schedule may decline at higher incomes.
3. (Lesson 4) Describe why the optimal extent of redistribution should rise with wage in- equality.
4. (Lesson 4) Does the relevant data across different nations suggest that governments follow the above rule?
5. (Lesson 5) Why does a ”height tax” make sense?
6. (Extra Credit) Can you come up with an optimal tax scheme that minimizes distortion? Describe briefly your policy and its defense.
2024-06-15