Макроэкономика 1 — МИЭФ, 2025 midterm

Choose one correct answer for each question. Each question is worth 8 marks. There is no penalty for a wrong answer.

1

A Solow economy without technological progress is in the steady state. Profits $\pi$ are equal to zero, so the total income of a representative agent, $y$, is split between capital income, $\alpha y$, and labour income, $(1-\alpha)y$. All capital income is saved and directed to gross investment $i=sy$, while labour income goes to consumption $c=(1-s)y$, where $s$ is the saving rate. Population grows at a rate $n$, while capital depreciates at a rate $\delta$. What is the real interest rate?

• (a) $n$
• (b) $i-\pi$
• (c) $\alpha y^{\frac{\alpha-1}{\alpha}}$
• (d) $\displaystyle \frac{s}{1-s}\left(\frac{s}{n+\delta}\right)^{\frac{1}{s-1}}$

2

In a closed mixed economy with fixed prices and wages, profits constitute a share $\vartheta$ of national income. Households’ income is not taxed, while firms pay lump-sum taxes to finance government purchases $G_0$. All after-tax profits are fully distributed as dividends rather than reinvested. The marginal propensity to consume out of all sources of income is $0<c_1<1$. What will be the economy’s government-purchases balanced-budget multiplier $\partial Y/\partial G_0$ in the short run if the nominal interest rate is fixed?

• (a) $\displaystyle \frac{1}{1-c_1}$
• (b) $\displaystyle \frac{1}{(1-c_1)(1-\vartheta)}$
• (c) $\displaystyle \frac{1}{1-\vartheta}$
• (d) $1$

3

Consider a closed economy at its potential level of output, $Y_n$. Consumption and investment functions are, respectively,

$$C=c_0+c_1\bigl((1-\vartheta)Y-NT\bigr),$$ $$I(r)=I_0-br+\vartheta Y.$$

The central bank’s flexible-inflation-targeting policy follows the rule

$$r=r^*+\alpha(\pi-\pi^*).$$

The aggregate supply function is

$$\pi=\pi^e+\lambda(Y-Y_n),$$

where inflation expectations $\pi^e$ are fully rational and non-anchored. What will be the long-run effect of a balanced-budget fiscal expansion $\Delta G=\Delta NT>0$ on investment if Ricardian equivalence does not hold?

• (a) $\Delta I=0$
• (b) $0<|\Delta I|<\Delta G$
• (c) $|\Delta I|=\Delta G$
• (d) $|\Delta I|>\Delta G$

4

A central bank commits to flexible inflation targeting and allows inflation to deviate from the target $\pi^*=4\%$ by following the nominal-interest-rate rule

$$i=i^*+1.5(\pi-\pi^*).$$

Last year, when CPI rose from $8\%$ to $10\%$ SAAR, the key rate was raised from $14\%$ to $17\%$. This year, when CPI rose from $12\%$ to $14\%$ SAAR, the key rate was raised from $19\%$ to $22\%$. What type of policy was implemented this year compared with the previous one?

• (a) Tight
• (b) Neutral
• (c) Easy
• (d) More aggressive

5

What is the long-run neutral nominal interest rate in Russia according to the most recent estimates?

• (a) $4.0\%$
• (b) $3.5\text{–}4.0\%$
• (c) $6.0\text{–}7.0\%$
• (d) $7.5\text{–}8.5\%$

The supply side of the economy is described by the production function

$$Y=\sqrt{KL},$$

where $K$ is physical capital stock that depreciates at rate $\delta$, and $L$ is the fully employed population, which is not growing. The government considers using lump-sum taxes $T$ to finance non-productive government spending $G$. Private investment $I$ and private consumption $C$ absorb, respectively, constant fractions $s$ and $(1-s)$ of disposable income $Y-T$.

(a) (10 marks) Suppose that the initial steady-state level of GDP per capita $y^*$ was reached by the laissez-faire private economy with no government intervention. Derive the expression for $y^*$ and show it in the Solow diagram.

(b) (10 marks) The government introduces a moderate amount of lump-sum taxes, so each worker pays a fixed amount

$$\tau=\frac{T}{L},$$

which is fully directed to public consumption. Derive the capital-formation equation in per-capita terms. Illustrate the new steady-state levels of capital per capita and private consumption per capita in the new Solow diagram.

(c) (10 marks) Show the immediate effects and transition paths of GDP $Y$ and consumption per capita $c$ from the initial steady states defined in (a) to the new steady states defined in (b).

(d) (20 bonus marks) Find the not-so-moderate level of lump-sum taxes $\hat\tau$ that would ensure only one non-trivial steady state. Will it be stable? Illustrate your answer in the phase diagram $\dot k(k)$.

Consider now the demand-side effects of lump-sum taxes in the medium run. Investment demand is

$$I(r)=I_0-br,$$

where $b>0$. The budget remains balanced. The central bank flexibly targets inflation $\pi^*$ by setting the real interest rate according to

$$r=r^*+\alpha(\pi-\pi^*).$$

The short-run aggregate supply function is

$$\pi=\pi^e+\lambda(Y-Y_n),$$

where $\lambda>0$, $\pi$ and $\pi^e$ are current and expected inflation, and $Y-Y_n$ is the output gap.

(e) (10 marks) Assume that initially, in period 1, people have static inflation expectations:

$$\pi_1^e=\pi_0^e=\pi^*.$$

Illustrate in the four-quadrant $IS-rr-45^\circ-AD-SAS$ diagram the effect of introducing lump-sum taxes on aggregate output, the real interest rate and inflation. Denote the new medium-run equilibrium in $\pi(Y)$ space as $(Y_1,\pi_1)$.

(f) (10 marks) In period 2, all people switch to myopic inflation expectations. Illustrate in the same four-quadrant $IS-rr-45^\circ-AD-SAS$ diagram the effects of further adjustments in aggregate output, the real interest rate and inflation. Denote the new medium-run equilibrium in $\pi(Y)$ space as $(Y_2,\pi_2)$.

Now consider a neoclassical environment in which economic agents have rational forward-looking expectations that are anchored. All prices, including nominal wages, are fully flexible. Assume that the supply-side adjustments discussed in (c) are over and the economy has reached its new steady state.

(g) (10 marks) Illustrate the ultimate equilibrium $(Y_3,\pi_3)$ in a new $IS-rr-45^\circ-AD-LRAS$ diagram. Mind the relationship between the new neutral real interest rate and the monetary-policy rule. Carefully label all curves and explain the adjustment process in three or four sentences.