Macroeconomic Policy — Diagnostic Tests

Unit Tests

UT-1: The Multiplier Effect

Question: In a closed economy with no government, the marginal propensity to consume is $\text{MPC} = 0.75$. An increase in investment of $\pounds 200$~m occurs. Calculate: (a) the multiplier, (b) the total change in national income, (c) the change in consumption. If the government introduces a proportional income tax of $t = 0.2$ and $\text{MPM} = 0.1$, recalculate the multiplier.

Solution:

(a) Simple multiplier: $k = \frac◆LB◆1◆RB◆◆LB◆1 - \text{MPC}◆RB◆ = \frac{1}{1 - 0.75} = \frac{1}{0.25} = 4$.

(b) Change in income: $\Delta Y = k \times \Delta I = 4 \times 200 = \pounds 800$~m.

(c) Change in consumption: $\Delta C = \text{MPC} \times \Delta Y = 0.75 \times 800 = \pounds 600$~m.

With tax and imports, the complex multiplier: $k = \frac◆LB◆1◆RB◆◆LB◆1 - \text{MPC}(1-t) + \text{MPM}◆RB◆$.

$k = \frac{1}{1 - 0.75(1 - 0.2) + 0.1} = \frac{1}{1 - 0.75(0.8) + 0.1} = \frac{1}{1 - 0.60 + 0.1} = \frac{1}{0.50} = 2$.

The tax and imports act as automatic withdrawals, reducing the multiplier from 4 to 2. This means the same $\pounds 200$~m investment increase would now raise income by only $\pounds 400$~m instead of $\pounds 800$~m.

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UT-2: Phillips Curve Analysis

Question: The short-run Phillips curve is given by $\pi = \pi^e - 3(u - 5) + 0.5x$, where $\pi$ is inflation, $\pi^e$ is expected inflation, $u$ is unemployment (%), and $x$ is a supply shock. Initially, $\pi^e = 3\%$ and $x = 0$. (a) If the government reduces unemployment to 3%, what is the inflation rate? (b) In the long run, with adaptive expectations, what happens to the Phillips curve? (c) Calculate the natural rate of unemployment.

Solution:

(a) With $u = 3\%$, $\pi^e = 3\%$, $x = 0$:

$\pi = 3 - 3(3 - 5) + 0 = 3 - 3(-2) = 3 + 6 = 9\%$.

Reducing unemployment by 2 percentage points below the natural rate comes at the cost of 6 percentage points of additional inflation.

(b) With adaptive expectations, $\pi^e$ adjusts to actual inflation with a lag. When inflation rises to 9%, workers and firms eventually revise their expectations upward ($\pi^e$ rises toward 9%). As expectations rise, workers demand higher wages, shifting the SRPC upward. The Phillips curve shifts up until the expected inflation equals actual inflation at the natural rate of unemployment. In the long run, there is no trade-off between inflation and unemployment -- the long-run Phillips curve (LRPC) is vertical at the natural rate.

(c) The natural rate of unemployment ($u^*$) is the rate at which $\pi = \pi^e$ with no supply shock ($x = 0$):

$\pi = \pi^e - 3(u^* - 5) + 0$. For $\pi = \pi^e$: $-3(u^* - 5) = 0$, so $u^* = 5\%$.

The natural rate is 5%.

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UT-3: Fiscal Policy Analysis

Question: The government increases spending by $\pounds 50$~bn, financed entirely by borrowing. The marginal propensity to consume is 0.8, the marginal tax rate is 0.25, and the marginal propensity to import is 0.15. Calculate: (a) the government spending multiplier, (b) the change in the budget deficit (initial and after multiplier effects), (c) the extent of crowding out if the interest sensitivity of investment is such that a 1 percentage point increase in interest rates reduces investment by $\pounds 10$~bn, and the increased borrowing raises interest rates by 0.5 percentage points.

Solution:

(a) Multiplier: $k = \frac◆LB◆1◆RB◆◆LB◆1 - \text{MPC}(1-t) + \text{MPM}◆RB◆$

$k = \frac{1}{1 - 0.8(0.75) + 0.15} = \frac{1}{1 - 0.60 + 0.15} = \frac{1}{0.55} = 1.82$

$\Delta Y = 1.82 \times 50 = \pounds 91$~bn.

(b) Initial increase in deficit: $\pounds 50$~bn (the additional spending).

After multiplier effects: tax revenue increases by $t \times \Delta Y = 0.25 \times 91 = \pounds 22.75$~bn.

New deficit increase $= 50 - 22.75 = \pounds 27.25$~bn (the automatic stabiliser partially offsets the initial deficit increase).

(c) Crowding out: the $\pounds 50$~bn borrowing raises interest rates by 0.5 percentage points.

Investment reduction: $0.5 \times 10 = \pounds 5$~bn.

The crowding out effect reduces the net fiscal expansion. The effective change in aggregate demand is not just the multiplier effect on government spending, but also accounts for the fall in investment:

$\Delta \text{AD} = k_G \times 50 + k_I \times (-5) = 91 + 1.82 \times (-5) = 91 - 9.1 = \pounds 81.9$~bn.

The crowding out reduces the expansionary effect by about 10%. In practice, the extent of crowding out depends on the state of the economy (it is larger when the economy is near full employment and smaller in a recession with a liquidity trap).

Integration Tests

IT-1: Monetary Policy Transmission Mechanism (with The Financial Sector)

Question: The Bank of England raises the base rate from 2% to 3%. Trace the full monetary policy transmission mechanism, explaining each step. Given that the money multiplier is 5 and the banking system holds no excess reserves, calculate: (a) the maximum change in the money supply if the central bank sells $\pounds 2$~bn of government bonds through open market operations simultaneously, (b) the impact on the velocity of money if $\text{MV} = PY$ and $Y$ rises by 2% while $P$ is unchanged.

Solution:

Transmission mechanism:

1. The central bank raises the base rate $\to$ commercial banks raise their lending rates (SVR, mortgage rates, business loan rates).
2. Higher interest rates increase the cost of borrowing $\to$ consumers reduce spending on credit (mortgages, car loans, credit cards) $\to$ consumption falls.
3. Higher rates increase the cost of business borrowing $\to$ firms reduce investment in capital $\to$ investment falls.
4. Higher rates increase the return on saving $\to$ consumers save more and spend less.
5. Higher UK interest rates attract foreign capital inflows $\to$ demand for sterling rises $\to$ the exchange rate appreciates $\to$ exports become more expensive and imports cheaper $\to$ net exports fall.
6. Combined effect: AD shifts left, reducing inflationary pressure but also reducing real GDP and increasing unemployment.

(a) Selling $\pounds 2$~bn of bonds removes $\pounds 2$~bn of reserves from the banking system. With a money multiplier of 5:

Maximum change in money supply $= 5 \times (-2) = -\pounds 10$~bn.

(b) $\text{MV} = PY$. If $Y$ rises by 2% and $P$ is unchanged, then $\text{MV}$ must rise by 2%.

If $M$ decreases (from the open market sale), then $V$ must increase to compensate. The relationship is:

$\frac◆LB◆\Delta(\text{MV})◆RB◆◆LB◆\text{MV}◆RB◆ = \frac◆LB◆\Delta M◆RB◆◆LB◆M◆RB◆ + \frac◆LB◆\Delta V◆RB◆◆LB◆V◆RB◆ + \frac◆LB◆\Delta M◆RB◆◆LB◆M◆RB◆ \times \frac◆LB◆\Delta V◆RB◆◆LB◆V◆RB◆$

For small changes: $\frac◆LB◆\Delta M◆RB◆◆LB◆M◆RB◆ + \frac◆LB◆\Delta V◆RB◆◆LB◆V◆RB◆ \approx 2\%$.

This scenario is contradictory: the interest rate rise and bond sale both reduce the money supply, which should reduce $Y$ or $P$, not increase $Y$ by 2%. If we observe $Y$ rising despite contractionary monetary policy, either: (1) the policy has not yet taken effect (lags), (2) other factors are boosting $Y$ (fiscal expansion, export boom), or (3) velocity has increased significantly as people spend money faster in anticipation of further rate rises.

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IT-2: AD/AS and Supply-Side Policy (with Supply-Side Policy)

Question: An economy is experiencing stagflation: inflation at 8% and unemployment at 8%, with the natural rate at 5%. Using the AD/AS model, explain why demand-side policies alone cannot solve stagflation. A supply-side policy reduces the natural rate of unemployment to 4% and increases potential output by 5%. Assuming initial output is $\pounds 2000$~bn, calculate: (a) the new potential output, (b) the change in the price level if AD is unchanged (assuming unitary elasticity of demand at potential output).

Solution:

Why demand-side policies fail: Stagflation (high inflation + high unemployment) occurs when the SRAS shifts left (adverse supply shock, e.g., oil price increase). If the government uses expansionary fiscal or monetary policy to reduce unemployment, AD shifts right, which further increases inflation without sustainably reducing unemployment (the economy moves along the new, higher SRPC). Conversely, contractionary demand-side policy to reduce inflation would further increase unemployment. There is no single AD shift that can simultaneously reduce both inflation and unemployment -- the economy faces a trade-off.

Supply-side solution: Supply-side policies shift the LRAS (and SRAS) to the right, increasing potential output and reducing the natural rate. This allows lower inflation AND lower unemployment simultaneously.

(a) New potential output $= 2000 \times 1.05 = \pounds 2100$~bn.

(b) With AD unchanged and unitary elasticity (PQ approximately constant at potential):

Initial: $P_0 \times 2000 = \text{AD}$, New: $P_1 \times 2100 = \text{AD}$.

$P_1 \times 2100 = P_0 \times 2000$, so $P_1 = P_0 \times \frac{2000}{2100} = P_0 \times 0.9524$.

The price level falls by approximately 4.76%. The supply-side improvement simultaneously:

• Reduces inflation (price level falls)
• Reduces unemployment (natural rate falls from 5% to 4%)
• Increases economic growth (output rises from $\pounds 2000$~bn to $\pounds 2100$~bn)

This is the key advantage of supply-side policies over demand-side policies in addressing stagflation.

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IT-3: Exchange Rates and Balance of Payments (with The International Economy)

Question: The UK current account is in deficit by $\pounds 50$~bn. The exchange rate is E = \1.25/\pounds$. The Marshall-Lerner condition holds (sum of PEDs for exports and imports$= 1.8$). If the pound depreciates by 20$$1.00/\pounds$: (a) Calculate the J-curve effect in the first 6 months, assuming short-run PEDs sum to 0.6, (b) Calculate the long-run improvement in the current account if import spending was initially$\pounds 500$~bn and export revenue was$\pounds 400$~bn.

Solution:

(a) J-curve effect: In the short run (first 6 months), consumers and firms cannot easily change their trading patterns. The short-run PEDs sum to 0.6 (less than 1), so the Marshall-Lerner condition is NOT satisfied in the short run.

The depreciation makes imports more expensive (more pounds needed per dollar of imports) and exports cheaper (fewer dollars needed per pound of exports). With inelastic demand in the short run, the value effect (higher import prices) dominates the volume effect (quantity changes).

Initial: exports $\pounds 400$~bn, imports $\pounds 500$~bn, deficit $= \pounds 100$~bn (note: deficit is $\pounds 100$~bn by these figures, not $\pounds 50$~bn -- let me use $\pounds 450$~bn imports for a $\pounds 50$~bn deficit).

Let me use: exports $= \pounds 400$~bn, imports $= \pounds 450$~bn, deficit $= \pounds 50$~bn.

With 20% depreciation, import prices rise by approximately 20% (in pound terms) and export prices fall by approximately 20% (in foreign currency terms).

Short-run: $\% \Delta \text{exports} = 0.3 \times 20 = 6\%$ (assuming export PED $= 0.3$). New exports $= 400 \times 1.06 = \pounds 424$~bn.

$\% \Delta \text{imports} = -0.3 \times 20 = -6\%$ (import PED $= 0.3$). New import volume $= 450 \times 0.94 = \pounds 423$~bn at old prices, but at 20% higher prices $= 423 \times 1.20 = \pounds 507.6$~bn.

New deficit $= 507.6 - 424 = \pounds 83.6$~bn. The deficit worsens by $\pounds 33.6$~bn in the short run -- the J-curve effect.

(b) Long-run improvement: PEDs sum to 1.8 (Marshall-Lerner satisfied).

$\% \Delta \text{exports} = 0.9 \times 20 = 18\%$. New exports $= 400 \times 1.18 = \pounds 472$~bn.

$\% \Delta \text{imports} = -0.9 \times 20 = -18\%$. New import volume $= 450 \times 0.82 = 369$ at old prices, but at 20% higher prices $= 369 \times 1.20 = \pounds 442.8$~bn.

New deficit $= 442.8 - 472 = -\pounds 29.2$~bn (surplus of $\pounds 29.2$~bn).

Improvement $= 50 + 29.2 = \pounds 79.2$~bn (from $\pounds 50$~bn deficit to $\pounds 29.2$~bn surplus).

The J-curve illustrates that exchange rate depreciation initially worsens the trade balance before improving it -- the full benefits may take 1--2 years to materialise.

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Section 3: Extended Macroeconomic Policy Practice

UT-4 (Extension). An economy has the following data: $C = 200 + 0.8Y_d$, $I = 150$, $G = 250$, $T = 0.25Y$, $X = 100$, $M = 50 + 0.15Y$. (a) Calculate equilibrium GDP. (b) Calculate the government spending multiplier and the tax multiplier. (c) The government increases spending by 20 while simultaneously increasing the tax rate to 30%. Calculate the new equilibrium. (d) Calculate the change in the government budget balance.

Solution:

(a) $Y = C + I + G + X - M$. $Y = 200 + 0.8(Y - 0.25Y) + 150 + 250 + 100 - (50 + 0.15Y)$. $Y = 200 + 0.6Y + 150 + 250 + 100 - 50 - 0.15Y$. $Y = 650 + 0.45Y$. $0.55Y = 650 \Rightarrow Y = 1181.82$.

(b) $k_G = \frac{1}{1 - MPC(1-t) + MPM} = \frac{1}{1 - 0.8(0.75) + 0.15} = \frac{1}{1 - 0.6 + 0.15} = \frac{1}{0.55} = 1.818$.

Tax multiplier: $k_T = \frac{-MPC}{0.55} = \frac{-0.8}{0.55} = -1.455$.

(c) With $G = 270$ and $t = 0.30$: $Y = 200 + 0.8(Y - 0.3Y) + 150 + 270 + 100 - (50 + 0.15Y)$. $Y = 200 + 0.56Y + 150 + 270 + 100 - 50 - 0.15Y$. $Y = 670 + 0.41Y$. $0.59Y = 670 \Rightarrow Y = 1135.59$.

GDP has FALLEN from 1181.82 to 1135.59 (a decrease of 46.23). Despite the fiscal expansion (higher $G$), the tax increase more than offsets it because it reduces disposable income and consumption.

(d) Original budget: $T = 0.25 \times 1181.82 = 295.46$. $G = 250$. Surplus $= 295.46 - 250 = +45.46$. New budget: $T = 0.30 \times 1135.59 = 340.68$. $G = 270$. Surplus $= 340.68 - 270 = +70.68$. The budget surplus has increased by 25.22, from 45.46 to 70.68. The balanced budget multiplier is NOT 1 here because of proportional taxation and imports (leakages).

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UT-5 (Extension): AD/AS with Simultaneous Shocks. An economy has AD: $Y = 600 - 2P$, SRAS: $Y = 4P - 200$, and LRAS: $Y^* = 500$. (a) Find the current equilibrium and identify any output gap. (b) A positive supply shock shifts SRAS right by 80 units (new SRAS: $Y = 4P - 280$). Calculate the new equilibrium. (c) Simultaneously, consumer confidence falls, shifting AD left by 100 (new AD: $Y = 500 - 2P$). Calculate the combined effect. (d) Is the economy now in a recessionary or inflationary gap?

Solution:

(a) $600 - 2P = 4P - 200 \Rightarrow 800 = 6P \Rightarrow P = 133.33$, $Y = 333.33$. Output gap: $500 - 333.33 = 166.67$ (recessionary gap).

(b) With new SRAS: $600 - 2P = 4P - 280 \Rightarrow 880 = 6P \Rightarrow P = 146.67$, $Y = 306.67$. Output gap: $500 - 306.67 = 193.33$ (larger recessionary gap!). The positive supply shock actually worsened the recessionary gap in the short run because it reduced the price level, which (through the real balance effect) could further reduce AD. But in this model, AD is fixed, so the lower price level simply means a new intersection at lower output. Wait -- this seems wrong.

Let me recheck. New SRAS: $Y = 4P - 280$. At $P = 146.67$: $Y = 586.67 - 280 = 306.67$. Yes, output has fallen. But a rightward shift of SRAS should INCREASE output for a given AD. Let me reconsider.

$600 - 2P = 4P - 280$: $880 = 6P \Rightarrow P = 146.67$. $Y = 600 - 2(146.67) = 306.67$.

Wait, this is LESS output than before (333.33). That cannot be right for a rightward SRAS shift. Let me re-examine.

Original SRAS: $Y = 4P - 200$. At $P = 133.33$: $Y = 533.33 - 200 = 333.33$. Correct.

New SRAS: $Y = 4P - 280$. The intercept has DECREASED (from -200 to -280), which means for any given $P$, output is LOWER. This is a LEFTWARD shift, not a rightward shift. I defined the shift incorrectly.

A rightward shift by 80 means: new SRAS $= 4P - 200 + 80 = 4P - 120$.

Let me redo: $600 - 2P = 4P - 120 \Rightarrow 720 = 6P \Rightarrow P = 120$, $Y = 600 - 240 = 360$. Output has increased from 333.33 to 360 (correct for a rightward SRAS shift). Output gap: $500 - 360 = 140$ (smaller recessionary gap).

(c) Combined shocks: new AD $Y = 500 - 2P$, new SRAS $Y = 4P - 120$. $500 - 2P = 4P - 120 \Rightarrow 620 = 6P \Rightarrow P = 103.33$, $Y = 500 - 206.67 = 293.33$.

(d) Output gap: $500 - 293.33 = 206.67$ (large recessionary gap). The negative AD shock dominates, pushing output well below potential. The positive supply shock partially offset the price level increase but not the output decline.

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IT-4 (Extension): Fiscal Policy and Debt Dynamics. A country has GDP = $\pounds 2000$bn, government debt = $\pounds 1800$bn (90% of GDP), budget deficit = $\pounds 100$bn (5% of GDP), and the interest rate on government debt = 4%. GDP grows at 2% per year and inflation is 3%. (a) Calculate the primary deficit. (b) Calculate the debt-stabilising primary balance. (c) If the government maintains the current primary deficit, calculate the debt-to-GDP ratio after 5 years (using the approximation $b_{t+1} \approx (1 + r - g - \pi)b_t - p$). (d) What primary surplus is needed to reduce the debt ratio to 60% of GDP within 10 years?

Solution:

(a) Interest payments $= 0.04 \times 1800 = \pounds 72\text{bn}$. Total deficit = primary deficit + interest payments. $100 = \text{primary deficit} + 72$. Primary deficit $= \pounds 28\text{bn}$ (1.4% of GDP).

(b) Debt-stabilising condition: the debt ratio is stable when $b_{t+1} = b_t$, i.e., $(r - g - \pi)b = p$. Nominal GDP growth $= g + \pi = 2 + 3 = 5\%$. Real interest rate on debt $= r - \pi = 4 - 3 = 1\%$. Using the approximation: $(r - g_{nominal})b = p$ where $r$ is nominal rate and $g_{nominal} = 5\%$. $(0.04 - 0.05) \times 90 = p \Rightarrow -0.01 \times 90 = p \Rightarrow p = -0.9$.

The debt-stabilising primary balance is a SURPLUS of 0.9% of GDP ($\pounds 18\text{bn}$). Since the current primary balance is a DEFICIT of 1.4%, the debt ratio is rising.

(c) Year-by-year debt ratio evolution ($b_0 = 90\%$, $p = -1.4\%$, $r - g_{nominal} = -1\%$): $b_{t+1} = (1 - 0.01)b_t - (-1.4) = 0.99b_t + 1.4$. $b_1 = 0.99(90) + 1.4 = 89.1 + 1.4 = 90.5$. $b_2 = 0.99(90.5) + 1.4 = 89.6 + 1.4 = 91.0$. $b_3 = 0.99(91.0) + 1.4 = 90.1 + 1.4 = 91.5$. $b_4 = 0.99(91.5) + 1.4 = 90.6 + 1.4 = 92.0$. $b_5 = 0.99(92.0) + 1.4 = 91.1 + 1.4 = 92.5$.

The debt ratio rises from 90% to 92.5% after 5 years. Despite the favourable interest rate-growth differential (nominal GDP growth exceeds the interest rate), the primary deficit pushes the debt ratio up.

(d) To reduce $b$ from 90% to 60% in 10 years: $b_{10} = 60 = (0.99)^{10} \times 90 + p \times \frac{1 - 0.99^{10}}{1 - 0.99}$. $(0.99)^{10} = 0.9044$. $90 \times 0.9044 = 81.4$. $\frac{1 - 0.9044}{0.01} = 9.56$. $60 = 81.4 + 9.56p$. $9.56p = -21.4 \Rightarrow p = -2.24\%$ of GDP.

A primary surplus of 2.24% of GDP ($\pounds 44.8\text{bn}$) sustained for 10 years would reduce the debt ratio from 90% to 60%.

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IT-5 (Extension): Monetary Policy Transmission. The Bank of England raises the Bank Rate from 3% to 4%. (a) If the interest elasticity of investment is $-0.5$ and investment was $\pounds 300$bn, calculate the change in investment. (b) If the multiplier is 2, calculate the change in GDP. (c) If the exchange rate appreciates by 3% as a result, and exports are $\pounds 500$bn with a PED of $-0.8$, calculate the change in export revenue. (d) Calculate the total estimated change in AD.

Solution:

(a) Interest elasticity: $\% \Delta I = \epsilon_I \times \% \Delta r = -0.5 \times \left(\frac{4 - 3}{3} \times 100\right) = -0.5 \times 33.3\% = -16.7\%$. $\Delta I = -16.7\% \times 300 = -\pounds 50\text{bn}$. New investment $= \pounds 250\text{bn}$.

(b) $\Delta Y = k \times \Delta I = 2 \times (-50) = -\pounds 100\text{bn}$.

(c) Exchange rate appreciation of 3% makes UK exports 3% more expensive in foreign currency terms. $\% \Delta X = PED_X \times \% \Delta P = -0.8 \times 3 = -2.4\%$. $\Delta X = -2.4\% \times 500 = -\pounds 12\text{bn}$. New exports $= \pounds 488\text{bn}$.

(d) Total change in AD $= \Delta I + k_{net\_exports} \times \Delta X$. The export change also goes through the multiplier: $\Delta Y_{exports} = 2 \times (-12) = -\pounds 24\text{bn}$. Total $\Delta Y = -100 - 24 = -\pounds 124\text{bn}$.

If GDP was $\pounds 2000$bn, this is a contraction of 6.2%. This is a large effect, illustrating the power of monetary policy. In practice, the effects are spread over 2-3 years and may be partially offset by other factors (e.g., falling inflation boosts real incomes).