Calculation of the Effect of a Change in Aggregate Demand on National Income – The Keynesian Multiplier
1. The Circular‑Flow Model
1.1. Two‑sector (closed) economy
- Households supply the factors of production (labour, capital, land) and receive factor‑income (wages, rent, interest, profit).
- Firms use those factors to produce goods and services, which are bought by households.
- Every expenditure by one sector becomes income for the other – the essence of the circular flow.
1.2. Four‑sector (open) economy
- Government collects taxes (T) and makes spending (G). Taxes are a leakage; government spending is an injection.
- Foreign sector supplies imports (M) and receives exports (X). Imports are a leakage; exports are an injection.
1.3. Five‑sector model (adds the financial sector)
- Financial institutions intermediate savings (S) from households to investment (I) by firms. Savings are a leakage from the product market but an injection into the financial market; investment is the corresponding injection.
Diagram suggestion: a five‑box diagram showing Households ↔ Firms ↔ Government ↔ Foreign sector ↔ Financial institutions with arrows for factor payments, product‑market transactions, taxes, imports/exports and savings/investment.
2. Leakages and Injections
| Leakage (outflow) |
Injection (inflow) |
Symbol in the AD equation |
| Savings (S) |
Investment (I) |
–S + I |
| Taxes (T) |
Government spending (G) |
–T + G |
| Imports (M) |
Exports (X) |
–M + X |
In equilibrium of the circular flow, total injections equal total leakages:
\[
I + G + X = S + T + M
\]
3. Aggregate Demand (AD) and Aggregate Supply (AS)
- Aggregate demand is the total planned expenditure on domestically produced output:
\[
AD = C + I + G + (X-M)
\]
- Consumption is split into an autonomous part and an induced part:
\[
C = C_{0} + MPC\,(Y_{D}),\qquad Y_{D}=Y-T
\]
- Short‑run equilibrium occurs where AD = Y, i.e. where the AD curve meets the 45° line.
- In the Cambridge A‑Level syllabus the price level is assumed to be fixed in the short run, so the AS curve is horizontal. Consequently a change in AD moves the equilibrium level of national income (Y) without shifting AS.
4. Assumptions Underpinning the Multiplier Analysis
- Price level is fixed (short‑run analysis).
- The economy is operating below full‑capacity – idle resources allow output to expand without upward pressure on prices.
- Marginal propensities (to consume, save, tax, import) are constant over the relevant range of income.
- Leakages and injections are measured as proportions of:
- Disposable income for taxes (MPT = ΔT/ΔY)
- National income for imports (MPM = ΔM/ΔY) – note the syllabus wording “of national income”, not of disposable income.
- No supply‑side shocks or capacity constraints occur during the period considered.
5. Deriving the Keynesian Multiplier
5.1. Simple (closed‑economy, no taxes) multiplier
National‑income identity:
\[
Y = C_{0} + MPC\,Y + I + G
\]
Re‑arrange to isolate Y:
\[
Y - MPC\,Y = C_{0}+I+G\quad\Longrightarrow\quad Y(1-MPC)=A
\]
where \(A = C_{0}+I+G\) is autonomous expenditure. Hence
\[
Y = \frac{A}{1-MPC}
\]
A one‑unit change in autonomous expenditure (\(\Delta A\)) therefore changes income by
\[
\Delta Y = \frac{1}{1-MPC}\,\Delta A
\]
The **simple multiplier** is
\[
k_{\text{simple}} = \frac{1}{1-MPC}= \frac{1}{MPS}
\]
5.2. Open‑economy multiplier (including all leakages)
Start with the full AD identity:
\[
Y = C_{0} + MPC\,(Y-T) + I + G + X - M
\]
Replace the leakages with their marginal propensities:
- Savings: \(S = MPS\,Y\) where \(MPS = 1-MPC\)
- Taxes: \(T = MPT\,Y\) (MPT = ΔT/ΔY)
- Imports: \(M = MPM\,Y\) (MPM = ΔM/ΔY)
Substituting and collecting the Y‑terms gives:
\[
Y = C_{0}+I+G+X - M_{0} + MPC\,(Y - MPT\,Y)
\]
\[
Y = C_{0}+I+G+X - M_{0} + MPC\,Y - MPC\,MPT\,Y
\]
Bring all Y‑terms to the left‑hand side:
\[
Y - MPC\,Y + MPC\,MPT\,Y + MPM\,Y = C_{0}+I+G+X - M_{0}
\]
Factor Y:
\[
Y\bigl[1-MPC + MPC\cdot MPT + MPM\bigr]=A
\]
Because taxes are a proportion of **disposable** income, the term \(MPC\cdot MPT\) simplifies to \(MPT\) (the MPC cancels out). Thus the bracket becomes:
\[
MPS + MPT + MPM
\]
Therefore the **open‑economy multiplier** is
\[
\boxed{k_{\text{open}} = \frac{1}{MPS + MPT + MPM}}
\]
5.3. Autonomous vs. induced expenditure
- Autonomous expenditure (A): components that do **not** depend on current income – \(C_{0}, I, G, X,\) and \(-M_{0}\).
- Induced expenditure: the part of consumption that varies with income – \(MPC\;Y_{D}\).
- The multiplier operates on the autonomous change; the induced part is automatically captured through the multiplier effect.
6. Government & Monetary Policy Links (A‑Level depth)
- Fiscal policy – a change in government spending (ΔG) or taxes (ΔT) is an autonomous change in AD. The multiplier shows the resulting impact on Y. A tax change works through the marginal propensity to tax (MPT) and the marginal propensity to consume (MPC).
- Monetary policy – a change in the policy interest rate influences investment (ΔI). In the multiplier framework investment is treated as an autonomous component, so the same formula applies: \(\Delta Y = k \times \Delta I\).
- Both policies are only fully effective when the assumptions listed in section 4 hold (idle capacity, fixed price level, constant propensities).
7. Supply‑Side Caveat
The multiplier formula assumes a horizontal short‑run AS curve (price level fixed). If the economy is near full‑capacity, the AS curve slopes upwards; an increase in AD will generate price‑level pressure and the multiplier will be smaller than the textbook value. This limitation is part of the A‑Level “evaluation” (AO2) requirement.
8. Step‑by‑Step Procedure for Calculating the Effect of a Change in AD
- State the assumptions (closed/open, presence of taxes, imports, price‑level fixed, idle capacity, constant propensities).
- Identify the autonomous change (\(\Delta A\)) – e.g., a change in G, I, C₀, or X.
- Determine the relevant marginal propensities (MPC, MPS, MPT, MPM). Use the values given in the question or calculate them from data.
- Calculate the appropriate multiplier:
- Closed economy, no taxes: \(k = 1/(1-MPC) = 1/MPS\).
- Open economy with leakages: \(k = 1/(MPS+MPT+MPM)\).
- Apply the multiplier:
\[
\Delta Y = k \times \Delta A
\]
- Interpret the result – direction, magnitude and relevance to policy. Include a brief evaluation of the realism of the assumptions (AO2/AO3).
9. Worked Examples (with AO2/AO3 evaluation)
Example 1 – Closed economy, no taxes
Government increases spending by £20 million. MPC = 0.75.
| Step | Calculation | Result |
|---|
| 1. Autonomous change | \(\Delta A = £20\text{ m}\) | £20 m |
| 2. Propensities | MPC = 0.75 → MPS = 0.25 | MPS = 0.25 |
| 3. Multiplier | \(k = 1/MPS = 1/0.25 = 4\) | k = 4 |
| 4. Change in income | \(\Delta Y = 4 \times £20\text{ m}\) | £80 m |
| 5. New equilibrium | Assume \(Y_{0}=£500\text{ m}\) | \(Y_{1}=£580\text{ m}\) |
Evaluation (AO2/AO3): The result assumes a perfectly horizontal AS curve and that all of the extra £20 m is spent domestically. In reality, part of the increase may be saved or imported, reducing the actual impact.
Example 2 – Open economy with taxes and imports
Assume:
- Government cuts spending by £10 million (ΔG = –£10 m).
- MPC = 0.60 → MPS = 0.40.
- Marginal propensity to tax, MPT = 0.15.
- Marginal propensity to import, MPM = 0.10.
| Step | Calculation | Result |
|---|
| 1. Multiplier | \(k = 1/(MPS+MPT+MPM) = 1/(0.40+0.15+0.10)=1/0.65\) | k ≈ 1.54 |
| 2. Autonomous change | \(\Delta A = \Delta G = -£10\text{ m}\) | -£10 m |
| 3. Change in income | \(\Delta Y = 1.54 \times (-£10\text{ m})\) | ≈ –£15.4 m |
Evaluation (AO2/AO3): The multiplier is reduced by the presence of taxes and imports, reflecting leakages from the circular flow. If the economy were close to full capacity, the fall in Y would be smaller because rising price levels would dampen the contraction.
Example 3 – Change in autonomous consumption (C₀) with a tax‑adjusted multiplier
In an open economy, autonomous consumption rises by £5 million. Given MPC = 0.70, MPT = 0.20, MPM = 0.05.
- MPS = 1 – 0.70 = 0.30.
- Multiplier: \(k = 1/(0.30+0.20+0.05)=1/0.55≈1.82\).
- \(\Delta Y = 1.82 \times £5\text{ m} ≈ £9.1\text{ m}\).
Evaluation (AO2/AO3): The increase in consumption is partly offset by higher taxes (MPT) and imports (MPM). If households chose to save a larger share of the extra income, the effective multiplier would be lower.
10. Exam Checklist (AO1‑AO3 focus)
- AO1 – Knowledge: State the assumptions, write the AD identity, define all marginal propensities and give the correct multiplier formula for the situation.
- AO2 – Application: Identify the autonomous change, substitute the given propensities, calculate the multiplier and the change in Y.
- AO3 – Analysis & Evaluation:
- Comment on the realism of the assumptions (price‑level fixed, idle capacity, constant propensities).
- Discuss what would happen if the economy were near full‑capacity or if the price level were allowed to change.
- Consider possible secondary effects (e.g., crowding‑out, exchange‑rate movements, expectations).
- Always finish with a clear, concise interpretation of the sign and magnitude of \(\Delta Y\) and its policy relevance.
11. Quick Reference Table – Multiplier Formulas
| Scenario |
Assumptions |
Multiplier formula |
| Closed economy, no taxes |
Only savings is a leakage |
\(k = \dfrac{1}{1-MPC} = \dfrac{1}{MPS}\) |
| Open economy with taxes and imports |
Savings, taxes and imports are leakages; MPC, MPT and MPM are constant |
\(k = \dfrac{1}{MPS + MPT + MPM}\) |
| Fiscal policy with a tax change |
ΔG = 0, ΔT ≠ 0; use the open‑economy formula and treat ΔT as the autonomous change (ΔA = –ΔT) |
Same as open‑economy multiplier |
| Monetary policy (change in I) |
ΔI is autonomous; other leakages unchanged |
Use the appropriate multiplier (simple or open) and set ΔA = ΔI |
12. Summary
- The multiplier quantifies how an autonomous change in aggregate demand is amplified through induced consumption.
- Its size depends on the total leakage rate (MPS + MPT + MPM). The greater the leakages, the smaller the multiplier.
- All calculations rest on the short‑run assumptions of a fixed price level and idle capacity; recognising the limits of these assumptions is essential for AO3 evaluation.
- Linking the multiplier to fiscal and monetary policy helps students answer A‑Level exam questions that require both calculation (AO2) and critical discussion (AO3).