PS6

Part of: Macro-Economics Submission set: Q1–Q3 (Q4–Q5 are for the TA session — worked here too) Key concepts: Lec_07-Labor Market, Lec_10-Fiscal Policy, Lec_05-Investment, Lec_06-Equilibrium in the Goods Market Builds on: Problem Set 5 Solutions — labor-market equilibrium & income/substitution effects

Setup — the one idea that unlocks the whole set

Every question here is an application of the static labor-supply FOC with taxes from Lec_10-Fiscal Policy:

Two distinctions do almost all the work:

• Lump-sum $T$ vs. distortionary $\tau_N,\tau_C$. A lump-sum tax/transfer moves labor supply only through the income effect (shifts $N^S$ via wealth). A labor-income or consumption tax also drives a substitution effect (changes the after-tax wage), which — under the course assumption that substitution dominates — shifts $N^S$ the opposite way.
• Income effect on/off. Whenever a question says "PVLR is unchanged" (Q1) or you find $T$ has dropped out of a supply function (Q5), the income effect is switched off and only the substitution/distortion channel remains.

Keep the two equilibrium diagrams from Lec_07-Labor Market in your head: labor market ($N^S=N^D$ → $w^*,N^*$) and goods market ($S=I$ → $r^*$). Q1 walks a shock through both.

A specific tax and spending plan [submit]

Assume that workers and firms behave as in the standard model we described in class. The government announces the following policy reform: (i) the labor income tax increases immediately; (ii) all proceeds will be invested in a much-required infrastructure project that will permanently raise Total Factor Productivity ($A$) starting next period; (iii) the tax increase is calculated in a way that will keep workers' PVLR unchanged (i.e. the tax reform does not imply an income effect on labor supply).

1. What is the effect on the labor market in the short run? Explain whether or not the labor supply and labor demand curves shift, in which direction, etc.
2. What is the effect on firms' optimal future capital decision? What does it imply with regards to current investment? (You may assume that the effects of $A^f$ on $MPK^f$ dominate the effects of $N^f$ on $MPK^f$.)
3. Describe the short-run equilibrium in the goods market. Does the $S$ curve shift? The $I$ curve? What is the new level of $r$, and what happens to consumption?
4. Describe the new equilibrium in the labor market in the long run. (You may assume that the demand effects dominate the supply effects.)

A welfare and tax reform [submit]

Assume that workers' preferences can be described using the utility function

where $C$ is consumption, $N$ is the labor input, and $0<\gamma<1$ ('gamma') is a parameter. The budget constraint is $C=(1-\tau_N)wN-T$, where $w$ is the real wage, $0\le\tau_N<1$ is the tax rate on labor income, and $T$ is a lump-sum tax (if $T>0$) or a lump-sum transfer (if $T<0$). In the question for the TA session (Q4) you are asked to show that the optimal labor supply function in this case is:

1. Assume that $T=-0.2$, $w=2$, $\tau_N=0.2$, and $\gamma=0.4$. Calculate the optimal labor supply and the consumption level. Under this tax-and-transfer system, what is the government's budget deficit?
2. You show your calculations to a government official and he responds: "We can't run a deficit and we don't want to run a surplus either, so I know exactly what to do in order to balance the budget: just set the transfers equal to the tax proceeds that you showed me." Is his statement correct? Explain why or why not.
3. Now assume that there are two workers in the economy, $A$ and $B$. Both have the same wage $w=2$ per hour, receive a transfer of $T=-0.1$ each, and face a labor income tax rate $\tau_N=0.2$. They differ, however, in their preferences: $\gamma_A=0.5$ and $\gamma_B=0.3$.
   • (a) Calculate the labor supply and consumption for $A$ and $B$. Who works more? Why?
   • (b) Assume that the government would like to increase aggregate labor supply (i.e. $N_A+N_B$) by lowering the tax rate applied to either type A or type B, but not both. Specifically, the chosen type will pay a labor income tax of 5% instead of 20%, and there are no other changes. Which type should the government pick? Provide a short explanation.
   • (c) What would be the implication of the two alternative policies with regards to consumption inequality in the economy?

Simple public debt dynamics [submit]

1. Go to FRED and search for the latest read of the US debt-to-GDP ratio (Federal Debt: Total Public Debt as Percent of Gross Domestic Product), as well as the latest read on the deficit/surplus as a fraction of GDP (Federal Surplus or Deficit [-] as Percent of Gross Domestic Product).
2. Assume that the annual real interest rate on public debt is 1%. What should the real GDP growth rate be such that the debt-to-GDP ratio does not grow further?
3. Assume that the real interest rate on public debt rises to 1.5%. What should the real GDP growth rate be such that the debt-to-GDP ratio does not grow further?
4. Assume that 2020–2021 were unique years due to the covid crisis and that the deficit is expected to go back to its 2019 level (and the real interest rate is still 1%). What should the real GDP growth rate be such that the debt-to-GDP ratio does not grow further?

Derive the optimal supply function with Cobb–Douglas preferences [TA session]

Assume that workers' preferences can be described using the utility function

where $C$ is consumption, $N$ is the labor input, and $0<\gamma<1$ ('gamma') is a parameter. The budget constraint is $(1+\tau_C)C=(1-\tau_N)wN-T$, where $w$ is the real wage, $0\le\tau_N<1$ is the tax rate on labor income, $\tau_C\ge0$ is the tax rate on consumption, and $T$ is a lump-sum tax (if $T>0$) or a lump-sum transfer (if $T<0$).

1. Show that the utility function satisfies the assumptions we made on workers' preferences: (i) more consumption is better; (ii) marginal utility from consumption is diminishing; (iii) more labor is worse; (iv) the marginal utility from labor becomes more negative as we work more (i.e. it is more 'painful' to work).
2. Solve for the optimal supply function. For this you will need to solve the worker's optimization problem, or use the optimality condition ("static first-order condition") that we discussed in class.
3. Assume that $T<0$ (i.e. workers receive some positive amount of transfer payments). Show that in this case: (a) a further increase in transfer payments reduces labor supply; (b) an increase in the labor income tax reduces labor supply; (c) an increase in the (pre-tax) real wage increases labor supply.
4. Assume that two workers have different preferences such that type $A$ has a lower $\gamma$ than type $B$ ($0<\gamma_A<\gamma_B<1$). Both types face the same $T$, $\tau_N$ and earn the same wage $w$. Which type of worker is more sensitive to a wage change?

Supply function without income effect, heterogeneity, and taxes [TA session]

Assume that workers have the following utility function:

(note that the $\ln$ is applied to the whole term in brackets). Assume that the budget constraint is standard, with labor income tax $0<\tau_N<1$, consumption tax $0<\tau_C$, lump-sum transfers (negative $T$) or lump-sum taxes (positive $T$), and an hourly wage $w$:

1. Start with the standard optimality condition (no need to derive the Lagrangian), $-U_N=\dfrac{1-\tau_N}{1+\tau_C}wU_C$, and derive the optimal labor supply function.
2. Are the lump-sum transfers a part of the supply function? What does that mean with regards to consumer preferences and income effects?
3. Of the factors that may change supply in your function, which will shift the supply curve itself, and which result in movement along the curve?
4. Consider a given wage change (say an increase). Assume that $\theta$ is a larger number. Is the labor supply response stronger or weaker?
5. Assume that there are two types of workers in the economy with $\theta_1>\theta_2$. Suppose that the government would like to cut taxes for one of the two types, and the goal is to maximize the labor supply response. Should the tax cut be applied to type 1 or type 2? Briefly explain.

One-page recap