Final Exam — Intermediate Macro, Moed A (2025) · worked-solution

• #macroeconomics
• #past-paper
• #worked-solution
• #development-accounting
• #production-function
• #labor-supply
• #gdp
• #inflation
• #investment
• #intertemporal-choice
• #unemployment
• #business-cycles

1. Q1 — Relative importance of TFP vs. capital

   Two economies, 1 and 2, share the production function $Y = A K^{1/3} N^{2/3}$. Population is $1$ and fully employed ($N_1 = N_2 = 1$):

   Country	TFP ($A$)	Capital ($K$)	Labor ($N$)
   1	4	8,000	1
   2	2	1,000	1

   Two economists in country 2 debate the source of the GDP gap. The first says that because country 2's capital stock is smaller by a factor of $8$, capital differences are mostly responsible for the GDP difference. The second says TFP contributes much more than capital. Calculate the relative importance of TFP and capital, and explain whether either economist is correct.

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2. Q2 — Immigration and the demand for robots

   Firms behave competitively and produce with three factors — capital ($K$), labor ($N$) and robots ($R$) — using $Y = A K^{\alpha}[R + N]^{1-\alpha}$. Immigration raises the number of workers available (new immigrants are identical to residents in all relevant respects) and the price of robots is constant. What is the effect on the demand for robots?

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3. Q3 — Labor supply independent of the labor income tax

   Workers have utility $U(C,N) = \ln(C) - \theta\dfrac{N^{1+\psi}}{1+\psi}$. They earn only labor income and pay a labor income tax; there is no other tax, income source, or transfer. The budget constraint is

   $$C = (1-\tau_N)wN$$

   Show that labor supply does not depend on the tax rate. (You may use the general optimality condition directly — no Lagrangian needed.)

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4. Q4 — Emigration: GDP, GNP and Net Factor Payments

   Because of the Russia–Ukraine war, Ukrainian workers leave Ukraine for a neighboring country and are not replaced. The workers who left find jobs abroad that pay exactly the same wage they used to earn in Ukraine. Explain the effects on Ukrainian GDP, GNP and Net Factor Payments (NFP).

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5. Q5 — EPOP and unemployment rates for young and old

   Data on young and old populations:

   	Young	Old
   Population	500,000	750,000
   Employed	400,000	600,000
   Not employed	100,000	150,000
   Labor-force participation rate	85%	90%

   Calculate the employment-population ratios and the unemployment rates for the young and the old.

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6. Q6 — PCE deflator vs. CPI inflation

   A country produces oranges, bread and machines; 100 identical consumers consume oranges and bread only. Base year 2023.

   Year	Product	Quantity	Price
   2023	Oranges	1,000	5
   2023	Bread	2,000	10
   2023	Machines	120	200
   2024	Oranges	1,100	6
   2024	Bread	1,500	15
   2024	Machines	125	180

   Compute the inflation rate from the PCE deflator (fixed-weight, 2023 base) and from the CPI (2023 representative basket). Is there a difference? If so, why?

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7. Q7 — Long question: optimal mandatory retirement saving (log utility)

   A simple model of a mandatory retirement-saving scheme. Consumers live two periods ($0$ and $1$), have income $y_0 > 0$ in period $0$ and zero income in period $1$, face market rate $r>0$, per-period utility $u(c_t) = \ln(c_t)$, discount factor $0<\beta<1$, and no initial assets or debt.

   1. Write the consumer's optimization problem (choice variables and budget constraint). (5 pts)
   2. Using the Euler equation and the budget constraint, solve for the optimal $c_0$ and $c_1$. (6 pts)
   3. Define the period-0 saving rate as $s_1/y_0$. What is the optimal saving rate? (6 pts) (Call this "voluntary saving" for the rest.)
   4. The government forces retirement saving: a consumer must contribute a rate $\tau$ of period-0 income; each unit saved delivers $1+r$ units of consumption in period 1. (a) If the government maximizes consumer welfare, what should $\tau$ be? (Explain, no proof needed.) (4 pts) (b) If $\tau$ is lower than the optimal level, will voluntary saving be positive / negative / zero? Explain. (3 pts) (c) If $\tau$ is higher than optimal, how does voluntary saving respond? Explain. (3 pts)
   5. Now suppose there are two consumer types, $1$ and $2$, differing only in that $\beta_1 > \beta_2$. Can a single mandatory saving rate be optimal for both? Explain. (5 pts)
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8. Q8 — Long question: TFP, depreciation, investment, wage and government spending

   Data:

   Period	Labor $N$	Capital $K$	GDP $Y$
   $t$	1000	100	100
   $t+1$	1050	105	120

   Assume production $Y = A_t K_t^{0.3} N_t^{0.7}$, real interest rate $r = 2\%$, capital tax $\tau_K = 10\%$, and a constant price of capital $p_k = 10$.

   1. Calculate TFP in both periods. (5 pts)
   2. Explain how you can sign the change in TFP without calculating it. (5 pts)
   3. The choice of $K_{t+1}$ is optimal (investment is decided in period $t$; firms knew $A_{t+1}$). (a) What is the depreciation rate $\delta$? (7 pts) (b) What is investment $I_t$? (3 pts)
   4. For period $t$: (a) If labor is hired optimally, what is the real wage? (4 pts) (b) Firms pay tax at rate $\tau_K$ on output minus labor expenditure (as in the investment model), and this revenue finances government spending $G$. What is $G$? (4 pts) (c) What is consumption? (4 pts)
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9. Miluim question — trend/cyclical components and consumption smoothing

   (Miluim question, 10 pts — only eligible students receive credit.)

   1. Briefly explain the trend and cyclical components of a macroeconomic series (like GDP). (4 pts)
   2. Explain how the properties of the cyclical components of certain consumption categories support the idea of consumption smoothing. (6 pts)
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