Multipliers: Study Guide (Macroeconomics 3.2)

The idea: one person’s spending is another person’s income

In Topic 3.1 you saw that a change in C, I, G or NX shifts the AD curve. This topic answers a different question: by how much?

Suppose a firm spends VD 1 million on a new warehouse. That VD 1 million becomes income for builders and suppliers. They spend part of it on food, transport and clothes, which becomes income for shop owners and drivers. They spend part of that, and so on. A single change in spending sets off rounds of further spending. Total spending, and total output, rise by more than the first change.

The first change is called a change in autonomous expenditure: spending that changes for a reason other than a change in income (new business confidence, a government decision, foreign demand). The rounds that follow are induced spending: spending caused by the new income.

How big the chain becomes depends on one number: the share of each extra valda of income that people spend.

Marginal propensities: MPC and MPS

Households can do two things with an extra unit of disposable income (income after taxes, plus transfers): spend it or save it.

• Marginal propensity to consume (MPC) = change in consumption ÷ change in disposable income.
• Marginal propensity to save (MPS) = change in saving ÷ change in disposable income.

Every extra valda is either spent or saved, so:

MPC + MPS = 1

“Marginal” means “extra”. The MPC uses changes, not totals. A household that spends 90% of its income on average may still spend only 75% of an extra valda.

The two multipliers

Expenditure (spending) multiplier = 1 ÷ (1 − MPC) = 1 ÷ MPS

It tells you how much AD changes for each VD 1 change in autonomous spending by consumers, firms, the government or foreign buyers. It applies to a change in any component: C, I, G or NX.

Tax multiplier = −MPC ÷ (1 − MPC) = −MPC ÷ MPS

It tells you how much AD changes for each VD 1 change in taxes. The sign is negative: higher taxes lower AD, lower taxes raise AD. A change in transfer payments (such as pensions or unemployment benefits) works like a tax change in reverse: a VD 1 rise in transfers raises AD by MPC ÷ MPS.

MPC	MPS	Spending multiplier (1 ÷ MPS)	Tax multiplier (−MPC ÷ MPS)
0.5	0.5	2	−1
0.6	0.4	2.5	−1.5
0.75	0.25	4	−3
0.8	0.2	5	−4
0.9	0.1	10	−9

Two patterns to notice:

• A higher MPC gives larger multipliers. More of each round is passed on.
• The size of the tax multiplier is always exactly 1 less than the spending multiplier.

Change in AD = spending multiplier × change in autonomous spending Change in AD = tax multiplier × change in taxes

The multipliers work in both directions. A fall in spending is multiplied just like a rise. If Valdorian firms cut investment by VD 5 billion and the MPC is 0.8, the first VD 5 billion of lost sales means lost income for workers and suppliers, who then cut their own spending by 0.8 × 5 = VD 4 billion, and so on. AD falls by 5 × 5 = VD 25 billion. Keep the sign of the change: a negative change in spending, or a positive change in taxes, gives a negative change in AD.

Worked example 1: from data to multipliers, round by round

Valdoria is the fictional country used across the Marlbridge macroeconomics pages. Its currency is the valda (VD).

Question. The table shows Valdoria’s disposable income and consumption (VD billion, fictional).

Disposable income	Consumption
400	360
480	420
560	480

(a) Calculate the MPC and the MPS. (b) Calculate the spending multiplier and the tax multiplier. (c) Private firms raise investment by VD 20 billion to build a national cold-storage network. Show the first four rounds of spending and find the total change in AD.

(a) When disposable income rises from 400 to 480 (+80), consumption rises from 360 to 420 (+60).

MPC = 60 ÷ 80 = 0.75. The next row gives the same value: 60 ÷ 80 = 0.75.

Saving = disposable income − consumption: 40, 60, 80. MPS = 20 ÷ 80 = 0.25. Check: 0.75 + 0.25 = 1. ✓

Do not divide 360 by 400 (= 0.9). That is the share of total income spent, not the share of extra income spent.

(b) Spending multiplier = 1 ÷ 0.25 = 4. Tax multiplier = −0.75 ÷ 0.25 = −3.

(c) Each round, households spend 75% of the new income they receive.

Round	New spending (VD billion)	Running total
1 (the investment itself)	20	20
2	20 × 0.75 = 15	35
3	15 × 0.75 = 11.25	46.25
4	11.25 × 0.75 = 8.4375	54.6875
…	each round 0.75 of the one before	approaches 80

Total change in AD = 4 × 20 = +VD 80 billion. AD shifts right by VD 80 billion.

Check. Where does the chain stop? Each round, 25% of new income is saved. The total extra saving is 0.25 × 80 = VD 20 billion, exactly equal to the first injection. Once all of the first VD 20 billion has “leaked” into saving, there is nothing left to pass on.

Why the tax multiplier is smaller

A VD 1 rise in government purchases is spent in full in round 1: the government buys VD 1 of goods.

A VD 1 tax cut is not spending. It raises disposable income by VD 1, and households spend only the MPC of it, say VD 0.75, saving the rest. The chain starts one step later and from a smaller first round. That is why the tax multiplier is smaller in size, by exactly 1:

• Tax cut of VD 1 → first-round spending VD MPC → total = MPC × spending multiplier = MPC ÷ MPS.

So, for the same amount of money, a change in government spending shifts AD by more than a change in taxes.

Worked example 2: spending, taxes and working backwards

Question. Valdoria’s MPC is 0.75 (spending multiplier 4, tax multiplier −3).

(a) The government raises spending on rural clinics by VD 30 billion. Calculate the change in AD. (b) Instead, it cuts taxes by VD 30 billion. Calculate the change in AD. (c) It raises spending by VD 30 billion and raises taxes by VD 30 billion to pay for it. Calculate the change in AD. (d) The government wants AD to rise by VD 60 billion. How large a rise in government spending would do this? How large a tax cut?

(a) ΔAD = 4 × (+30) = +VD 120 billion.

(b) ΔAD = −3 × (−30) = +VD 90 billion. Step by step: households receive VD 30 billion more disposable income, spend 0.75 × 30 = VD 22.5 billion in the first round, and 22.5 × 4 = VD 90 billion. Same answer.

(c) Spending effect +120; tax effect −3 × (+30) = −90. Net ΔAD = 120 − 90 = +VD 30 billion. Even though the budget balance is unchanged, AD still rises, because the spending multiplier is bigger than the tax multiplier.

(d) Divide the target by the multiplier.

• Government spending needed = 60 ÷ 4 = VD 15 billion.
• Tax cut needed = 60 ÷ 3 = VD 20 billion.

Check. 15 × 4 = 60 ✓ and 20 × 3 = 60 ✓. The tax cut must be larger, because part of every valda of tax cut is saved before it reaches round 1.

Interpretation. These numbers give the size of the shift in AD. If the price level stays the same, real GDP changes by the same amount. In the full AD–AS model later in Unit 3, the price level may rise as AD increases, so the change in real GDP can be smaller than the shift.

What the simple multiplier assumes

• The MPC is the same in every round.
• The only leakage from each round is saving. (Background: in real economies, spending on imports and extra taxes paid on new income also leak out, so actual multipliers are smaller than 1 ÷ MPS. You are not asked to calculate this.)
• The price level does not change while the rounds play out.

Common misconceptions

• “Spending multiplier = 1 ÷ MPC.” It is 1 ÷ MPS, or 1 ÷ (1 − MPC). With MPC 0.75: 4, not 1.33.
• Using totals instead of changes for the MPC. Divide the change in consumption by the change in disposable income.
• Using the spending multiplier on a tax change. A VD 30 billion tax cut with MPC 0.75 raises AD by 90, not 120.
• Dropping the sign of the tax multiplier. A tax increase lowers AD. Write the sign so the direction is clear.
• “The multiplier only works for government spending.” It works for any autonomous change in C, I, G or net exports, in either direction. A fall in investment is multiplied too.
• “A balanced-budget change has no effect on AD.” Equal rises in spending and taxes still raise AD, because the two multipliers differ in size.
• “A higher MPS gives a bigger multiplier.” The opposite. More saving each round means a smaller chain.
• “The multiplier is the change in real GDP no matter what.” It gives the shift in AD; the change in real GDP equals it only if the price level is unchanged.

Where this leads

Next, Topic 3.3, Short-Run Aggregate Supply, adds the supply side of the model. You will use the multipliers again for fiscal policy later in Unit 3. Look back at Topic 3.1, Aggregate Demand if shifts of AD are unclear. Try the practice questions now, then use the revision notes and the checklist to consolidate.