F Problem Set 6 / Homework 2

Only the first two exercises are required for Homework 2.

F.1 Optimal Taxation: The “Supply Side” (Neoclassical) View (Homework)

We go back to the labor market model, and study redistribution in this neoclassical model. Denoting the (hourly) wage by \(w\), the price of consumption by \(p\), and the number of hours (per year) by \(l\), assume that moreover \(\alpha=0\) so that: \[U(c, l)=c-B\frac{l^{1+\epsilon}}{1+\epsilon}, \qquad f(l)=Al^{1-\alpha} = A l.\]

  1. Derive the Labor Demand curve.

  2. For labor supply, assume that the tax and transfer system is such that \(\tau\) is the marginal tax rate, and \(c_0\) is a real subsistence level of income given by the government: \(pc = (1-\tau)wl + p c_0\). Derive the Labor Supply curve.

  3. Express the number of hours worked per year \(l,\) as well real annual pre-tax income \(y=(w/p) \cdot l,\) as a function of the parameters of the model.

  4. Compute the number of hours worked per year \(\underline{l}\), as well as real pre-tax income \(\underline{y}\) if \(\epsilon = 2,\) \(\underline{\tau}=1/4,\) \(\underline{A}= 1000000/28188\approx 35.5,\) \(\underline{B}= 3000000/491569855488 \approx 6.1 \cdot 10^{-6}.\)

  5. Compute the number of hours worked per year \(\bar{l},\) as well as real pre-tax income \(\bar{y}\) if \(\epsilon = 2,\) \(\bar{\tau}=1/2,\) \(\bar{A}= 1000000/6264\approx 159.6,\) \(\bar{B}=3000000/655426473984 \approx 4.6 \cdot 10^{-6}.\)

  6. Assume that the labor force is \(N = 150\) million, with a fraction \(\lambda = 0.9\) of the type described in question 4, and a fraction \(1-\lambda = 0.1\) of the type described in question 5. What is total output in the economy?

  7. Assume that the tax system has two marginal tax rates: one 25% marginal tax rate above 25K, one 50% marginal tax rate above 200K and that \(c_0=5K\). In other words, if income is \(y\) then taxes are: \[T(y) = -5K + 0.25\cdot \max \{ y-25K, 0 \} + 0.25\cdot \max \{ y-200K, 0 \}.\] Assume a tax reform that lowers the marginal tax rate on the richest by 5 points. How much is the total tax cut? What is the impact on GDP?

  8. Assume a tax reform that lowers the marginal tax rate on the poorest by 5 points (and on the poorest only, even though in reality it would also change how much the rich pay in taxes). How much is the total tax cut? What is the impact on GDP?

F.2 Readings - Paradox of Thrift (Homework)

  1. Paul Krugman, When Consumers Capitulate, New York Times, October 31, 2008; Paul Krugman, Paradox of Thrift, New York Times Blog, February 3, 2009; Paul Krugman, We’re Still In A Paradox Of Thrift World, New York Times Blog, Aug 26, 2010. In what sense have “American consumers long been living beyond their means?” How does Paul Krugman qualify the “paradox of thrift” idea?

  2. Paul Krugman, Crowding In and the Paradox of Thrift, New York Times Blog, April 26, 2015 How did the IMF show empirically the existence of the paradox of thrift?

  3. Neil Irwin, Interest Rates Just Keep Falling. Economic Orthodoxy Is Falling With Them, New York Times, July 4, 2019. Which economic orthodoxes are falling with interest rates, according to Neil Irwin?

F.3 Redistribution between the top 1% and the bottom 99%

During lecture 9, we studied the aggregate demand effects of tax cuts on the bottom 90% financed by tax increases on the top 10%. In this problem, we study another example of these aggregate demand effects, looking at redistribution from the top 1% income share to the bottom 99%.

To illustrate the effects on aggregate demand of redistributive policies between the top 1% and the bottom 99%, we now use the same notations as in lecture 9. There is a share \(\lambda = 99\%\) of population \(N\) who are in the bottom 99%, who earn individual income \(\underline{y}\), pay net taxes \(\underline{t}=\underline{t}_0 + t_1 \underline{y}\), have an MPC \(\underline{c}_1\), baseline consumption \(\underline{c}_0\). Notations for high income are similar, but with bars: \(\bar{y}\), \(\bar{t}=\bar{t}_0+t_1 \bar{y}\), \(\bar{c}_1\), \(\bar{c}_0\). Total GDP is \(Y\), investment is \(I=b_0 +b_1 Y\), government spending is exogenous and equal to \(G\).

  1. Use Google to find out how much income would put you in the top 1%.

  2. The World Income Database suggests that the top 1% captures approximately 20% of total U.S. income in 2017, while it was approximately 10% in 1980. Using the notations of the class, what is \(\gamma=\bar{y}/\underline{y}\)?

  3. Compute aggregate consumption \(C= \underline{C} + \bar{C}\), as in lecture 9, using the following notations: \[ \begin{aligned} c_{1}&\equiv\frac{\lambda\underline{c}_{1}+\left(1-\lambda\right)\gamma\bar{c}_{1}}{\lambda+(1-\lambda)\gamma}\\ C_{0}& \equiv \lambda N \underline{c}_0 + (1-\lambda) N \bar{c}_0\\ \underline{T}_{0}& \equiv \lambda N \underline{t}_0\\ \bar{T}_0 & \equiv (1-\lambda) N \bar{t}_0. \end{aligned} \]

  4. What is an economic interpretation for \(c_1\) ? Calculate \(c_1\) if \(\underline{c}_1=1\) and \(\bar{c}_1=1/4\).

  5. Using the expression for \(I\), and for aggregate consumption \(C\), compute \(Y\).

  6. Assume that \(t_1=1/4\) and \(b_1=1/6\). Compute the impact on GDP of a 100 billion dollars tax cut on the top 1%. What is the impact on the government deficit of such a cut?

  7. Assume that \(t_1=1/4\) and \(b_1=1/6\). Compute the impact on GDP of a 100 billion dollars tax cut on the bottom 99%. What is the impact on the government deficit of such a cut?

  8. Assume that \(t_1=1/4\) and \(b_1=1/6\). Compute the impact on GDP of a transfer of 100 billion dollars from the top 1% to the bottom 99%. What is the impact on the government deficit of such a transfer?

  9. What happens if there are no automatic stabilizers in this economy (\(t_1 = 0\))? Explain.

F.4 Readings - Redistributive Policies

  1. Laura D’Andrea Tyson, Owen Zidar. “Tax Cuts for Job Creators.” New York Times Blog Post, October 19, 2012. How can we test “trickle-down economics” using state-level data? What additional arguments are given in favor of the main thesis in this article?

  2. “Inequality Will Eventually Hurt the Rich, Too”, Michael Pettis, New York Times, April 18, 2019. What is the link between debt accumulation and consumer demand? When is inequality potentially beneficial?

F.5 Paul Krugman VS Robert Barro on the Bush tax cuts

Watch this debate between Paul Krugman and Robert Barro, moderated by Charlie Rose.

  1. Explain Robert Barro’s view on taxes, using the supply-side view of redistribution.

  2. What is Paul Krugman’s take on taxes? Compare it to the Keynesian view on redistribution.

  3. What is the discussion on public debt, and the deficit? Explain.

  4. From 11:26 Robert Barro suggests to move from pay-as-you-go to “personal accounts”. Explain their discussion in terms of the overlapping-generations model.