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

• #macroeconomics
• #past-paper
• #worked-solution
• #investment
• #public-debt-dynamics
• #growth-accounting
• #euler-equation
• #unemployment
• #labor-supply
• #fiscal-policy
• #automation

1. Q1 — Backing out future TFP from the investment rule

   Firms make optimal investment decisions according to the standard investment model:

   • Cobb-Douglas production function $Y_t = A_t\,K_t^{1/3}\,N_t^{2/3}$ (all periods)
   • Current price of capital $p_k = 180$; future price of capital $p_k^{f} = 200$
   • Real interest rate $r = 3\%$
   • Depreciation rate $\delta = 8\%$
   • Capital tax $\tau_K = 20\%$
   • Current capital stock $K = 350$; labor constant at $N = 5{,}000$
   • Current investment $I = 53$; current TFP $A = 1$

   An analyst claims: "since we observe positive investment this period, firms must expect a higher future TFP." Is the analyst right? Assuming firms invest optimally, compute the implied future TFP $A^{f}$.

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2. Q2 — Forecasting GDP, debt-to-GDP and total debt

   An economy has:

   • Real GDP in 2000 $= 1{,}000{,}000$; real GDP in 2020 $= 1{,}400{,}000$
   • Future real GDP grows at the average 2000–2020 rate
   • Debt-to-GDP ratio in 2020 $= 80\%$
   • Expected primary deficit $= 2.5\%$ of GDP
   • Real interest rate $r = 4\%$

   Find the predicted real GDP, debt-to-GDP ratio, and total debt in 2021.

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3. Q3 — TFP level, TFP growth and the real wage from data

   You have the following data, with production function $Y_t = A_t K_t^{\alpha}N_t^{1-\alpha}$ and real wage in 2020 equal to $3$:

   Variable	2020	2021
   GDP	100,000	110,000
   Capital stock	10,000	11,000
   Labor input	20,000	22,000

   Calculate the level of TFP in 2020, the growth rate of TFP between 2020 and 2021, and the real wage in 2021.

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4. Q4 — Reading σ and β off a consumption-growth vs. interest-rate figure

   Two consumers share the utility function $u(c) = \dfrac{c^{1-\sigma}}{1-\sigma}$ and the same discount factor $0<\beta<1$. A figure plots the real interest rate $r$ on the horizontal axis against consumption growth $\dfrac{c_{t+1}}{c_t}$ on the vertical axis, with two upward-sloping lines (one solid, one dashed, the dashed being flatter).

   1. Which line corresponds to the consumer with the higher $\sigma$? (You may reason from the Euler equation — no need to solve the full problem.)
   2. From the figure, what is the value of $\beta$? Explain.
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5. Q5 — Unemployment rate, job-finding, separation and steady state from flows

   Labor-market data for two months (assume the entire population is in the labor force):

   Month	Continuing employed	New employed	Continuing unemployed	New unemployed
   1	180,000	9,000	15,000	6,000
   2	180,000	12,000	13,000	5,000

   "Continuing employed" = employed this month and last; "new employed" = employed this month but unemployed last month (similarly for unemployed). Calculate: the unemployment rates in months 1 and 2; the job-finding probability $f$; the separation probability $d$; and the steady-state unemployment rate if these probabilities persist.

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6. Q6 — Optimistic firms, pessimistic consumers: the saving-investment market

   An economy is in goods-market equilibrium, labor is constant, and there is no government. News leads firms to become more optimistic about future TFP while consumers become more pessimistic (about their future income). Describe the new short-run goods-market equilibrium: how do the saving ($S$) and investment ($I$) curves shift, and what happens to saving, investment, and the real interest rate?

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7. Q7 (long) — Three-period cake-eating problem with log utility

   A consumer lives three periods ($0,1,2$), has initial assets $a>0$ and no income ($y_0=y_1=y_2=0$), with per-period utility $u(c_t) = \ln(c_t)$, discount factor $0<\beta<1$, and takes the real rate $r$ as given.

   1. Write the optimization problem, stating the choice variables and the lifetime budget constraint. (5 pts)
   2. Write the Lagrangian and derive the four first-order conditions. (5 pts)
   3. Derive the two Euler equations (periods 0–1 and 1–2); show the general form $\frac{1}{c_t} = \beta(1+r)\frac{1}{c_{t+1}}$. (5 pts)
   4. Solve for the optimal consumption plan $c_0, c_1, c_2$. (6 pts)
   5. Analyze the effect of a higher $\beta$ on $c_0$ (fully) and explain how you would sign $c_1, c_2$; give the intuition for $c_0$. (5 pts)
   6. Analyze the effect of a higher $r$ on $c_0, c_1, c_2$ and give the economic intuition. (6 pts)
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8. Q8 (long) — Inelastic labor supply, TFP and θ shocks, and investment

   Workers have $U(C,N) = \ln(C) - \theta\dfrac{N^{1+\psi}}{1+\psi}$ ($\theta>0,\ \psi>0$), no income other than labor, budget $C = wN$. Firms produce with a standard $AF(K,N)$.

   1. Show the labor supply function is $N = \left(\dfrac{1}{\theta}\right)^{\frac{1}{1+\psi}}$ (you may start from the static FOC — no Lagrangian needed). (7 pts)
   2. What does this labor-supply curve look like? Is it increasing in $w$? What does it imply about income vs. substitution effects? (5 pts)
   3. TFP ($A$) is permanently higher. In the short run: (a) what happens to wages, labor input and output? (b) what happens to optimal future capital and current investment demand? (5+5 pts)
   4. Instead, $\theta$ is permanently lower ($A$ constant). In the short run: (a) wages, labor, output? (b) optimal future capital and investment demand? (5+5 pts)
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9. Miluim — Skill-biased technological change, inequality and polarization

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

   1. Briefly explain what skill-biased technological change (SBTC) is and how it may relate to rising wage inequality and the college wage premium. (6 pts)
   2. Does SBTC provide a good explanation for polarization in the labor market? Briefly explain. (4 pts)
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