Macroeconomics guide, Unit 3, step 6 of 23Previous step Next step

Step 6: The Multiplier

By the EconTutorials team. 11-minute read with 6 practice questions.

SyllabusAP Macro 3.2 · Cambridge 9.1 · IB 3.6 (HL)

In 30 seconds

One person’s spending is another person’s income, so an extra $1 of spending can raise real GDP by more than $1.
The spending multiplier is $ \frac{1}{1-\text{MPC}} = \frac{1}{\text{MPS}} $; the tax multiplier is $ -\frac{\text{MPC}}{\text{MPS}} $; the balanced-budget multiplier is 1.
In an open economy with income tax, the multiplier is smaller: $ \frac{1}{\text{MPS}+\text{MPT}+\text{MPM}} $.
Exam favourite: work out the change in G or T needed to close a gap of a given size.

Video lesson coming soonA short animated lesson for this step will appear here.

What the multiplier means

Imagine the council spends $100 on new chairs for a school. The furniture maker now has $100 of extra income. She spends $80 at a local café and saves $20. The café owner spends $64 of his $80 and saves $16. Each new income becomes someone else’s spending, round after round.

Add up all the rounds and the first $100 has created far more than $100 of extra income and output. That knock-on effect is the multiplier effect.

Key termMultiplier effect: an initial change in spending (an injection) leads to a larger final change in real GDP, because each round of spending becomes income for someone else who spends part of it again.

The marginal propensities: MPC + MPS = 1

How big the knock-on effect is depends on how much of each extra dollar people pass on. Economists measure this with two ratios.

The marginal propensity to consume (MPC) is the fraction of each extra dollar of income that is spent: $ \text{MPC} = \frac{\Delta C}{\Delta Y} $.
The marginal propensity to save (MPS) is the fraction of each extra dollar that is saved: $ \text{MPS} = \frac{\Delta S}{\Delta Y} $.

In the simple model (a closed economy with no income tax), every extra dollar is either spent or saved. So the two fractions must add up to one whole dollar:

\[ \text{MPC} + \text{MPS} = 1 \]

If you get an extra $200 and spend $150 of it, your MPC is $ 150 \div 200 = 0.75 $ and your MPS is $ 50 \div 200 = 0.25 $. Notice the word “marginal”: it is about the extra income, not your total income.

Key termMarginal propensity to consume (MPC): the change in consumption divided by the change in disposable income; the share of each extra dollar that is spent.

The spending multiplier, round by round

Take the school chairs again, with MPC = 0.8 and MPS = 0.2. The table follows the first few rounds.

Round	New spending (= someone’s new income)	Saved (MPS = 0.2)	Running total of extra GDP
1	$100.00	$20.00	$100.00
2	$80.00	$16.00	$180.00
3	$64.00	$12.80	$244.00
4	$51.20	$10.24	$295.20
5	$40.96	$8.19	$336.16
All rounds	…	$100.00	$500.00

Each round is 0.8 times the round before, so the rounds keep shrinking. Adding the whole series gives a neat formula:

\[ \text{Spending multiplier} = \frac{1}{1-\text{MPC}} = \frac{1}{\text{MPS}} \qquad \Delta \text{Real GDP} = \text{multiplier} \times \Delta \text{spending} \]

Here the multiplier is $ 1 \div 0.2 = 5 $, so $100 of new spending raises real GDP by $500. Look at the “Saved” column too: when the process ends, total extra saving is exactly $100, equal to the first injection. The process stops when everything injected has leaked out into saving.

The same multiplier works for any change in autonomous spending: government purchases (G), investment (I), consumption that is not caused by income, or net exports. It also works in reverse: a $100 fall in investment cuts real GDP by $500.

Worked example

In an economy, the MPC is 0.75. Firms raise investment by $40 billion. By how much does real GDP change?

Step 1$ \text{MPS} = 1 – 0.75 = 0.25 $

Step 2$ \text{Multiplier} = \frac{1}{0.25} = 4 $

Step 3$ \Delta \text{Real GDP} = 4 \times \$40\text{bn} = \$160\text{bn} $

A $40 billion rise in investment shifts aggregate demand right by $160 billion in total, assuming the price level does not change.

Common mistakeWriting the multiplier as $ \frac{1}{\text{MPC}} $. With MPC = 0.8 that gives 1.25, which is wrong. The bottom of the fraction is always the leakage: $ 1-\text{MPC} $, which is the MPS. A higher MPC means a bigger multiplier.

The tax multiplier and the balanced-budget multiplier

Now suppose the government cuts taxes by $100 instead of buying chairs. Households have $100 more disposable income, but they do not spend all of it in the first round. With MPC = 0.8 they spend only $80 and save $20. So the first round of new spending is $ \text{MPC} \times \$100 = \$80 $, and that $80 is then multiplied by 5.

\[ \text{Tax multiplier} = -\frac{\text{MPC}}{\text{MPS}} \]

With MPC = 0.8: tax multiplier $ = -0.8 \div 0.2 = -4 $. The minus sign tells you the direction: a tax cut (negative $ \Delta T $) raises GDP, and a tax rise lowers it. A $100 tax cut gives $ -4 \times (-\$100) = +\$400 $. That is less than the $500 from $100 of government purchases, because part of the tax cut is saved before it is ever spent. The tax multiplier is always one smaller (in size) than the spending multiplier. The same tax multiplier applies to transfer payments, such as unemployment benefits, with the sign reversed: a rise in transfers acts like a tax cut.

What if the government raises spending and taxes by the same amount, so its budget balance does not change?

G rises by $100: GDP rises by $ 5 \times 100 = \$500 $.
T rises by $100: GDP falls by $ 4 \times 100 = \$400 $.
Net effect: $ +\$500 – \$400 = +\$100 $.

Real GDP rises by exactly the amount of the extra spending. This is the balanced-budget multiplier, and it equals 1 whatever the MPC.

Key termBalanced-budget multiplier: the effect on real GDP of equal increases in government spending and taxes; in the simple model it equals 1, so GDP rises by the amount of the extra spending.

Closing a recessionary or inflationary gap

A favourite exam task gives you the size of an output gap and asks how much the government should change G or T. Work backwards from the formula: divide the gap by the right multiplier.

\[ \Delta G = \frac{\text{gap}}{\text{spending multiplier}} \qquad \Delta T = \frac{\text{gap}}{\text{tax multiplier}} \]

Worked example

Real GDP is $1,700 billion and full-employment (potential) output is $2,000 billion. The MPC is 0.75. How much should government purchases rise to close the gap? How big a tax cut would do the same job?

Step 1$ \text{Gap} = 2000 – 1700 = \$300\text{bn (recessionary gap)} $

Step 2$ \text{Spending multiplier} = \frac{1}{0.25} = 4, \quad \Delta G = \frac{300}{4} = \$75\text{bn} $

Step 3$ \text{Tax multiplier} = -\frac{0.75}{0.25} = -3, \quad \Delta T = \frac{300}{-3} = -\$100\text{bn} $

The government could raise spending by $75 billion or cut taxes by $100 billion. The tax cut must be larger because a quarter of it is saved in the first round. Check: $ 4 \times 75 = 300 $ and $ -3 \times (-100) = 300 $.

An inflationary gap works the same way in reverse. If GDP is $200 billion above potential and MPC = 0.8, the spending multiplier is 5, so G must fall by $ 200 \div 5 = \$40 $ billion. The tax multiplier is −4, so taxes must rise by $ 200 \div 4 = \$50 $ billion.

Common mistakeAnswering “increase G by $300 billion” because the gap is $300 billion. The government does not need to fill the whole gap itself: the multiplier does most of the work. Always divide the gap by the multiplier.

Leakages: the open-economy multiplier

Real economies have more leaks than saving. When your income rises, some of the extra goes in income tax, and some is spent on imported goods, so it becomes income for firms abroad, not at home. Each of these is a leakage (withdrawal) from the circular flow.

MPS: the share of each extra dollar of income that is saved.
MPT (marginal propensity to tax, or marginal tax rate): the share taken in tax.
MPM (marginal propensity to import): the share spent on imports.

If all three are measured as shares of each extra dollar of national income, the multiplier becomes one over the total leakage:

\[ k = \frac{1}{\text{MPS} + \text{MPT} + \text{MPM}} = \frac{1}{\text{MPW}} \]

Here MPW is the marginal propensity to withdraw. The closed-economy formula $ \frac{1}{\text{MPS}} $ is just the special case where MPT and MPM are zero.

Worked example

For example, out of each extra $1 of national income, households save 10 cents, pay 15 cents in tax and spend 25 cents on imports. Exports rise by $60 billion. Find the multiplier and the change in real GDP.

Step 1$ \text{MPW} = 0.10 + 0.15 + 0.25 = 0.50 $

Step 2$ k = \frac{1}{0.50} = 2 $

Step 3$ \Delta \text{Real GDP} = 2 \times \$60\text{bn} = \$120\text{bn} $

Only 50 cents of each extra dollar is spent again on domestic output, so the multiplier is just 2. If saving were the only leakage (MPS = 0.10), the multiplier would be 10. Tax and imports shrink it a lot.

Limitations of the multiplier

The formulas give a neat number, but the real effect is usually smaller and slower. Know these limits for evaluation marks.

Time lags. Each round takes time; by the time the full effect arrives, the economy may have recovered anyway.
Crowding out. If the government borrows to pay for extra spending, it can push up interest rates. Higher rates reduce private investment, which cancels part of the rise in G. Crowding out is strongest near full employment.
Leakages. Higher MPS, MPT or MPM all reduce the multiplier. In an open economy with high taxes, the true multiplier may be well below the simple $ \frac{1}{\text{MPS}} $ figure.
Price level changes. The simple multiplier assumes prices stay fixed. On an upward-sloping SRAS curve, part of the rise in AD becomes higher prices instead of more real output. Close to full capacity, most of it does.
Uncertain MPC. Worried households, or those who see a tax cut as temporary, save more of it, so the multiplier is lower than expected.

DiagramDraw an AD/AS diagram with the price level on the vertical axis and real GDP on the horizontal axis. Start with AD₁ crossing SRAS to the left of the vertical LRAS line: this is a recessionary gap. An increase in G of $75bn (MPC = 0.75) first shifts AD right by $75bn, then the multiplier shifts it further, to AD₂, which is $300bn to the right of AD₁ at each price level. Because SRAS slopes upward, real GDP rises by a little less than the full shift and the price level rises slightly.

Exam tipAP Macro FRQs: show the formula, the numbers and the answer with units (“multiplier = 1/0.25 = 4, so the maximum change in real GDP is $160 billion”). Write “maximum” or “assuming a constant price level” to protect the mark. Say “the tax multiplier is smaller because part of the tax cut is saved”. Cambridge 9708 and IB HL: use $ \frac{1}{\text{MPS}+\text{MPT}+\text{MPM}} $ for an open economy, and earn evaluation marks with crowding out, time lags and the size of leakages.

Practise: 6 questions

Try each one on paper first, then check the worked answer.

EasyQ1. The MPC is 0.6. Find the MPS and the spending multiplier. Show worked answer

MPS = 1 − 0.6 = 0.4.
Spending multiplier = 1 ÷ 0.4 = 2.5.

EasyQ2. The MPS is 0.25. Government purchases rise by $40 billion. What is the maximum change in real GDP? Show worked answer

Multiplier = 1 ÷ 0.25 = 4.
Change in real GDP = 4 × $40bn = $160bn (an increase), assuming the price level does not change.

MediumQ3. The MPC is 0.9. Compare the effect on real GDP of (a) a $20 billion tax cut and (b) a $20 billion rise in government purchases. Show worked answer

MPS = 0.1. Spending multiplier = 1 ÷ 0.1 = 10. Tax multiplier = −0.9 ÷ 0.1 = −9.
(a) Tax cut: −9 × (−$20bn) = +$180bn.
(b) Spending rise: 10 × $20bn = +$200bn.
The spending rise has $20bn more effect, because 10% of the tax cut is saved in the first round. (The $20bn difference is the balanced-budget multiplier of 1 at work.)

MediumQ4. The MPC is 0.5. Investment rises by $1,000. Show the first four rounds of spending, the running total after four rounds, and the final total change in GDP. Show worked answer

Round 1: $1,000. Round 2: 0.5 × 1,000 = $500. Round 3: 0.5 × 500 = $250. Round 4: 0.5 × 250 = $125.
Running total after four rounds = 1,000 + 500 + 250 + 125 = $1,875.
Multiplier = 1 ÷ (1 − 0.5) = 2, so final change = 2 × $1,000 = $2,000. The remaining rounds add the last $125.

MediumQ5. For each extra $1 of national income, 15 cents is saved, 20 cents is taxed and 15 cents is spent on imports. Investment rises by $60 billion. Find the multiplier and the change in real GDP, and explain why the multiplier is smaller than 1 ÷ MPS. Show worked answer

MPW = 0.15 + 0.20 + 0.15 = 0.50.
Multiplier = 1 ÷ 0.50 = 2.
Change in real GDP = 2 × $60bn = $120bn.
1 ÷ MPS would be 1 ÷ 0.15 ≈ 6.67. The real multiplier is smaller because tax and imports are extra leakages: each round, more of the income drains out of domestic spending, so the rounds shrink faster.

ExamQ6. An economy has real GDP of $1,800 billion and full-employment output of $2,000 billion. The MPC is 0.8. (a) Identify the type and size of the output gap. (b) Calculate the spending multiplier. (c) Calculate the minimum change in government purchases needed to close the gap. (d) Calculate the change in taxes that would close the gap instead. (e) Explain why your answer to (d) is larger than your answer to (c). [8 marks] Show worked answer

(a) Recessionary (negative) output gap of 2,000 − 1,800 = $200bn. [1 mark for type, 1 mark for size]
(b) MPS = 1 − 0.8 = 0.2; multiplier = 1 ÷ 0.2 = 5. [1 mark]
(c) ΔG = 200 ÷ 5 = $40bn increase. [2 marks: 1 for method, 1 for answer with direction]
(d) Tax multiplier = −0.8 ÷ 0.2 = −4. ΔT = 200 ÷ (−4) = −$50bn, i.e. a tax cut of $50bn. [2 marks]
(e) A rise in G is spent in full in the first round, but households save part of a tax cut (here 20%), so only $40bn of the $50bn tax cut is spent in the first round. A larger tax cut is needed to create the same first round of spending. [1 mark]

Check yourself

1. If the MPC rises from 0.75 to 0.8, the spending multiplier:

Multiplier = 1/(1 − MPC): 1/0.25 = 4 becomes 1/0.2 = 5. A higher MPC means less leaks out each round.

2. The government raises spending by $50bn and taxes by $50bn. In the simple model, real GDP:

The balanced-budget multiplier is 1, so GDP rises by the amount of the extra spending.

3. MPS = 0.1, MPT = 0.2, MPM = 0.2. The multiplier is:

Total leakage = 0.1 + 0.2 + 0.2 = 0.5, so k = 1/0.5 = 2.

Your score: 0 out of 3

The short version

Spending becomes income, which is partly spent again, so an injection raises real GDP by a multiple of itself.
MPC + MPS = 1; spending multiplier = 1/(1 − MPC) = 1/MPS; tax multiplier = −MPC/MPS; balanced-budget multiplier = 1.
To close a gap, divide the gap by the multiplier: the tax change needed is larger than the spending change.
With tax and imports, the multiplier is 1/(MPS + MPT + MPM), which is smaller.
Time lags, crowding out, leakages and rising prices make the real-world effect smaller than the formula.

Step 6: The Multiplier

What the multiplier means

The marginal propensities: MPC + MPS = 1

The spending multiplier, round by round

The tax multiplier and the balanced-budget multiplier

Closing a recessionary or inflationary gap

Leakages: the open-economy multiplier

Limitations of the multiplier

Practise: 6 questions

Check yourself

The short version

Further reading on EconTutorials

Contact Info

Quick Links

Success!