Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
ECON 23950: Economic Policy Analysis
Problem Set 3
(Due: October 26, 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 A Mini Survey of the Main Results
Are the following statements true or false? Briefly explain your reasoning, based on the models we discussed in class.
1. Proportional taxes are necessarily distortionary.
2. In 2012, Newt Gingrich said ”We want to have zero capital gains tax.” Additionally he said he would cut the corporate income tax to 12.5%. Gingrich claims that this will surely create jobs and increase tax revenues. Your reaction? (Both theoretically and empirically)
3. In theory, higher wage inequality should call for more aggressive redistribution policy.
4. Optimal taxation in the Ramsey framework we studied in lecture calls for setting corporate income tax to zero.
2 Optimal Redistribution with a General SWF
Consider the identical economy as the one we discussed in class. There are two types of agents with equal numbers: type A and type B. The agents are indexed by i and have a common utility function over consumption Ci and labor input Li:
where i = A,B. Also, each agent has a technology transforming labor effort into production: Yi = wiLi; where Yi is output and wi is the wage rate (or labor productivity), which they take as given. We assume that wA > wB , so that agent A is more productive. γ represents the concavity of the utility function and is assumed to be less than unity.
The government collects a labor income tax from agent A (at a constant rate τ) and gives this to agent B as a lump-sum transfer (label it as F). We are going to analyze the effects of this redistribution policy under a more general social welfare function given by
where ψi represents welfare weights for agent i, which we now assume are equal (ψi = 1/I) and ϕ represents the elasticity of substitution between individual utils.
To answer the following questions, you might find this article useful (link here).
1. Show that as ϕ approaches zero, the social welfare function converges to the Leontieff function, namely,
How do you characterize this SWF and why?
2. Show that as ϕ approaches infinity, the social welfare function becomes utilitarian, namely,
3. Setup the maximization problem for the benevolent social planner, using
as the social welfare function.
3 Optimal Taxation
Consider an endowment economy in which income in period t is given by Yt. The government should spend Gt in period t. The values for Yt and Gt for all t = 0, 1, 2, ... are pre-determined. Suppose that the government wants to minimize the present value of deadweight losses subject to the lifetime budget constraint. Let f(Tt/Yt) be a fraction of output lost in DWL. The amount of the deadweight loss is thus f(Tt/Yt)Yt.
1. Assume that f(Y(T)t(t) )′ > 0 and f(Y(T)t(t) )′′ > 0. Interpret these assumptions and explain intu-itively.
2. Derive the lifetime government budget constraint.
3. Set up the government’s optimization problem where the government discounts future DWL at the rate (1 + r)−1 .
4. Show that the average tax rate Tt/Yt should be constant over time.
The optimal tax revenue at each period t = 0, 1, 2, ..., denoted by Tt∗ , should satisfy the following two conditions:
i.e., the lifetime budget constraint of the government should be satisfied, and,
i.e., the overall tax rate is constant over time for all t = 0, 1, 2, ...
Assume Yt = Y , where Y is constant and greater than 2. Also, assume that there is no government debt that should be paid at time 0.
Suppose G0 = 2 and G1 = G2 = ... = 1. That is, the government faces a temporary increase in government spending at time 0 in which the government spending is $2, but after time 0, the spending stays at $1 forever.
5. How much should the government collect as taxes in each period under optimal taxation?
6. Calculate the path of government debt and draw a figure that shows the evolution of debt over time.
7. After time 0, (i.e., in periods 1, 2, ...) does the government collect more taxes than it spends for government consumption (that is, is Tt* > Gt for t = 1, 2, ...)? If so, what does the government do with the “extra” revenue (i.e., Tt* − Gt) in each period?
Now suppose G0 = G2 = G4 = ... = 2 and G1 = G3 = G5 = ... = 1. That is, the government should spend more at even periods than at odd periods.
8. Redo questions 5-7 above with the new expenditure stream.
4 Consumption Tax vs. Capital Income Tax
Suppose we have the infinitely-lived representative agent that maximizes the utility function:
where u′ () > 0, u′′ () < 0, v′ () < 0, v′′ () < 0. The household does production on her own, using her labor input yt = f(lt) where f′ () > 0, f′′ () < 0. The household can borrow and lend at a constant risk-free rate r. Government spending gt is exogenously given. Suppose first that the government raises revenues through consumption tax so that the budget constraint is
(1 + τt(c))ct+ bt+1 = f(lt) + (1 + r)bt
1. Set up the Lagrangian and derive the first-order conditions.
2. Derive the intratemporal MRS condition. Is the leisure-consumption choice distorted? Explain intuitively.
3. Derive the Euler equation. Is the intertemporal consumption choice distorted? Explain intuitively.
Now let’s think about capital income taxation such that the period-by-period budget constraint is:
ct+ bt+1 = f(lt) + (1 − τk )(1 + r)bt
Notice that, with this formulation, both interest and principal payments are taxed.
4. Derive the MRS and Euler equations for this economy.
5. True, False, or Uncertain? Capital income tax works similarly to an increasing consump-
tion tax (τt(c) < τ). Hint: Look only at the Euler equation.
6. Suppose for simplicity that capital income tax is held constant over time. True, False, or Uncertain? Such constant capital income tax distorts intertemporal consumption at a geometrically increasing rate as the time gap widens. In other words, it is similar to a situation where the consumption tax is increasing over time at an increasing rate. Hint: The answer is true. Derive and explain an Euler equation between ct and ct+i.
7. Based on the findings above, comment on the distortionary effects that capital income taxation causes on consumption, labor supply, capital formation (we don’t have K in this model, but you can derive educated guesses), output, and the overall welfare (happiness for the household).
5 John Cochrane on Growth Policy
Read a recent article by Cochrane and summarize key take-aways. You may want to listen to the EconTalk podcast for more discussions.