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

• #macroeconomics
• #past-paper
• #worked-solution
• #public-debt-dynamics
• #unemployment
• #investment
• #production-function
• #inequality
• #inflation
• #development-accounting
• #consumption-tax
• #intertemporal-choice

1. Q1 — War and the debt-to-GDP dynamics (d, g, r)

   A rating agency (Moody's) forecasts that Israel's government debt ratio will rise to a peak of about 67% of GDP by 2025, up from 60% in 2022 — whereas before the war it had expected the debt burden to decline towards 55% of GDP.

   Using the short model for the evolution of the debt-to-GDP ratio studied in class, explain how each of the three contributing factors — the primary deficit $d$, the growth rate $g$, and the interest rate $r$ — can be affected by the war in a way that is qualitatively consistent with this forecast.

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2. Q2 — Employment next period and the labor-market steady state

   There are 10,000,000 people in the labor force, of whom 600,000 are currently unemployed. The monthly job-finding probability is $f = 35\%$ and the monthly job-separation probability is $d = 3\%$.

   How many workers will be employed next month? Is the economy in its steady state in the current period? In the next period?

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3. Q3 — Investment vs. the current and future price of capital

   A research assistant collects data on investment, the current price of capital $p_K$, and the future price of capital $p_K^{f}$, for firms that behave according to the investment model studied in class. The assistant plots investment against each price but forgets to label the two lines (one solid, one dashed).

   Which line describes investment vs. the current price of capital, and which describes investment vs. the future price of capital? Explain, and describe the economic interpretation of each price's effect on capital accumulation.

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4. Q4 — Permanent TFP rise with an additive production function

   An economy has production function $Y_t = A_t K_t^{\alpha} + N_t^{1-\alpha}$ (note the additive form). A ministry economist claims that encouraging permanent productivity ($A$) improvements will, already in the short run, deliver more output, more investment, and stronger labor demand. A news commentator agrees about output and investment but doubts the labor-demand claim.

   Do you agree about output and investment? What about labor demand? Explain.

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5. Q5 — Inequality ratios (90/10, 90/50, 50/10) and polarization

   Among the measures of inequality, briefly explain what the 90/10, 90/50, and 50/10 ratios are (focus on wages). Then explain briefly what "polarization in the labor market" is, and how some of these ratios can indicate a process of polarization.

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6. Q6 — GDP-deflator vs. CPI inflation with a non-consumption good

   A country produces Pasta, Tomatoes, and Industrial Robots. There are 1,000 identical consumers, who consume only Pasta and Tomatoes (robots are not consumed). Base year 2022.

   Year	Product	Quantity	Price
   2022	Pasta	1,000	5
   2022	Tomatoes	5,000	2
   2022	Industrial Robots	100	500
   2023	Pasta	1,100	5.5
   2023	Tomatoes	4,500	2.2
   2023	Industrial Robots	101	450

   Compute inflation from the GDP deflator and from the CPI (2022 base / fixed 2022 basket). Is there a difference? If so, why?

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7. Q7 — UK growth accounting, temporary TFP shocks, and capital choices (long)

   Data for the United Kingdom, 2007–2009 (Penn World Table 10.1). Assume Cobb-Douglas $Y_t = A_t K_t^{\alpha} N_t^{1-\alpha}$ with a labor share of 0.6 (so $1-\alpha = 0.6$, i.e. $\alpha = 0.4$).

   Year	Real GDP	Employment	Avg Hours	Capital Stock	Total Labor	TFP
   2007	2,635,414	29.32	1,665	11,698	?	?
   2008	2,628,067	29.66	1,660	11,934	?	?
   2009	2,519,950	29.29	1,638	12,080	?	?

   1. Fill in Total Labor (total hours) and implied TFP (10 pts).
   2. Using the standard labor-market model: (a) if TFP shocks are perceived as temporary, what is the short-run effect of a negative TFP shock on labor-market equilibrium (employment, wages) and GDP? (6 pts) (b) Comparing 2008-vs-2007 and 2009-vs-2008, are the model's predictions consistent with the data? (3 pts) (c) Name two other factors (one demand-side, one supply-side) that could reconcile the model with the data (7 pts).
   3. In the investment model, if firms expected these TFP changes, is the observed capital-stock path consistent with the model? (6 pts)
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8. Q8 — Two-period consumption model with a consumption tax (long)

   A two-period consumption model with a consumption tax ($\tau_0, \tau_1$ in periods 0 and 1) that makes consumption more expensive. Budget constraint:

   $$(1+\tau_0)c_0 + \frac{(1+\tau_1)c_1}{1+r} = y_0 + \frac{y_1}{1+r}$$

   Per-period utility is $\ln(c_t)$; lifetime utility is $\ln(c_0) + \beta\ln(c_1)$ with discount factor $0<\beta<1$. Income $y_0, y_1$ is known; $r$ is taken as given.

   1. Write the Lagrangian; state the choice variables; derive the first-order conditions (5 pts).
   2. Show the Euler equation is $\dfrac{1}{c_0} = \beta(1+r)\dfrac{1+\tau_0}{1+\tau_1}\dfrac{1}{c_1}$ (4 pts).
   3. Show $c_0 = \dfrac{1}{1+\tau_0}\dfrac{1}{1+\beta}\left(y_0 + \dfrac{y_1}{1+r}\right)$ (4 pts).
   4. The government raises $\tau_0$, effective immediately. What happens to $c_0$? Explain mathematically and intuitively (2 pts).
   5. Effect on aggregate saving and goods-market equilibrium (assume period-0 production $A,K,N$ unchanged): (a) what happens to consumption expenditure $(1+\tau_0)c_0$ and to private saving? (6 pts) (b) government tax revenue and government saving, if $G$ rises with revenue in period 0 vs. period 1 (6 pts). (c) the new goods-market equilibrium for one of those scenarios (5 pts).
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