Final Exam — Intermediate Macro, Moed B (2023–24) · worked-solution

• #macroeconomics
• #past-paper
• #worked-solution
• #borrowing-constraints
• #public-debt-dynamics
• #production-function
• #investment
• #business-cycles
• #purchasing-power-parity
• #intertemporal-choice
• #labor-supply

1. Q1 — Liquidity constraints across the income distribution and the MPC

   A figure (from McKay and Wolf, 2023, Monetary Policy and Inequality) plots the fraction of US households defined as liquidity (borrowing) constrained on the vertical axis against income quintile on the horizontal axis (1 = bottom 20% of the income distribution, 5 = top 20%). The relationship is negative.

   (a) Briefly explain what can drive the negative relationship between income and being liquidity constrained. What can explain the fact that even among households in the top quintiles some are still liquidity constrained? (3 pts)

   (b) Using the principles of the consumption model, would you expect the marginal propensity to consume (MPC) to be positively or negatively correlated with income quintile? (No formal reasoning required.) (3 pts)

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2. Q2 — Forecasting the change in debt-to-GDP

   You must forecast next year's change in the debt-to-GDP ratio for a country. You collect:

   • current ratio $b = 100\%$;
   • planned primary deficit $d = 4\%$ of GDP;
   • nominal interest rate on government debt $i = 5\%$;
   • expected inflation $\pi^e = 3\%$.

   In addition, 20 years ago real GDP per capita was $10{,}000$, this year it is $18{,}000$, and it is expected to grow in the future at the same rate as the average growth over the last 20 years.

   What is your forecast?

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3. Q3 — Backing out TFP using the capital income share

   An economy has production function

   $$Y = A K^{\alpha} N^{1-\alpha}, \qquad 0 < \alpha < 1.$$

   For year $t$: GDP $= 10{,}000$ units, capital stock $K = 5{,}000$, labor input $N = 10{,}000$. To use capital, producers rent it at real rental price $R = 0.5$. What is total factor productivity $A$ in year $t$?

   (Hint: which parameter is missing from your data, and how can you find it?)

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4. Q4 — Capital tax that keeps the capital stock constant

   Firms invest optimally under the standard investment model:

   • Cobb-Douglas production $Y_t = 10 \times K_t^{1/3} N_t^{2/3}$;
   • current price of capital $p_k = 15$; future price of capital $p_k^{f} = 14$;
   • real interest rate $r = 4\%$; depreciation rate $\delta = 10\%$;
   • current capital stock $K = 700$; labor constant at $N = 1{,}000$.

   The government wants to pick a capital tax rate that keeps the capital stock constant. What tax rate achieves this (5 pts), and what is the implied level of current investment (1 pt)?

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5. Q5 — Pessimism about future productivity in the goods market

   An economy is at its goods-market equilibrium. Consumers and firms become pessimistic about the future level of productivity, believing it will be lower than usual starting next period. Describe the new goods-market equilibrium. In equilibrium, will consumption and investment change in the same direction (both up or both down)?

   (Assume (i) no change in the labor market, and (ii) no government.)

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6. Q6 — PPP GDP per capita growth and catch-up

   Fake data for two countries $A$ and $B$ over two years. Assume every country produces and consumes the same representative basket.

   	Country A	Country B
   Year 1 population	500	25,000
   Year 1 GDP (local)	1,500,000	85,000,000
   Year 1 Price (local)	5	4
   Year 1 Price (US)	3	3
   Year 2 population	550	27,000
   Year 2 GDP (local)	2,000,000	100,000,000
   Year 2 Price (local)	6	5
   Year 2 Price (US)	4	4

   (a) Compute the growth rate of PPP GDP per capita for each country. (4 pts) (b) If these growth rates persist forever, will the poorer country ever catch up with the richer one? (Explain, no calculation needed.) (2 pts)

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7. Q7 — Can lower discount factors cause recessions? Two-period CRRA consumption

   Economists sometimes claim that recessions are triggered by lower discount factors. Evaluate this using the two-period consumption model and goods-market equilibrium.

   Consumers live two periods with disposable income $y_0 > 0$, $y_1 > 0$, no initial assets, and utility from consumption only:

   $$u(c_t) = \frac{c_t^{\,1 - \frac{1}{\sigma}}}{1 - \frac{1}{\sigma}}, \qquad \sigma > 1,$$

   each period $t = 0, 1$. They discount future utility with $0 < \beta < 1$ and take $r > 0$ as given. TFP, labor and capital are constant in the short run.

   1. Write the consumer's optimization problem (objective, budget constraint, choice variables). (6 pts)
   2. Use the Euler equation and budget constraint to solve for the optimal plan $c_0, c_1$ as functions of $y_0, y_1, r, \beta, \sigma$. (7 pts)
   3. Let $s_1$ be current saving (period-0 saving). Express $s_1$ as a function of $y_0, y_1, r, \beta, \sigma$. (3 pts)
   4. Suppose $\beta$ is unexpectedly lower. Effects on current consumption and current saving? Show mathematically and explain. (5 pts)
   5. Describe the new short-run goods-market equilibrium: does the $S$ curve shift? the $I$ curve? what is the resulting equilibrium? (6 pts)
   6. In a typical recession GDP, consumption and investment all fall. (a) Does the model generate this in the short run from a $\beta$ shock? (2 pts) (b) What happens to future capital and output? (3 pts)
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8. Q8 — Labor income tax, the revenue Laffer curve, and productive spending

   Workers have utility over consumption $C$ and labor $N$:

   $$U(C, N) = \ln\!\left(C - \frac{1}{\psi + 1}N^{\psi + 1}\right), \qquad \psi > 0.$$

   They earn wage $w$ per unit of labor, pay a labor income tax $\tau_N > 0$, and may pay a lump-sum tax $T > 0$ or receive a transfer $T < 0$. Budget constraint: $C = (1 - \tau_N)wN - T$.

   1. Show that labor supply is $N = (1 - \tau_N)^{1/\psi}\, w^{1/\psi}$. (You may start from the general static first-order condition.) (8 pts)
   2. What is government tax revenue from the labor income tax, as a function of $\tau_N, w, \psi$? (5 pts)
   3. The government wants to raise labor-tax revenue. One economist says raise $\tau_N$; another says lower it. (a) With the wage fixed at its initial level, can you tell with certainty how revenue changes? Explain mathematically and economically. (5 pts) (b) Now let labor demand be standard so the wage adjusts. Is there a new equilibrium wage? How does it change revenue relative to (a)? Explain all sources of difference. (5 pts) (c) Use this to explain why the revenue-maximizing tax rate is neither 0% nor 100%. (3 pts)
   4. Now suppose new tax revenue is invested in useful projects that immediately raise TFP (instead of being thrown away). Is the revenue-maximizing tax rate lower or higher than before? Explain the labor-market differences. (6 pts)
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