The spending multiplier, explained with numbers

The test says: "The government spends $100 million more. The MPC is 0.8. By how much does GDP rise?" Half the class writes $100 million, and the other half writes 1 ÷ (1 − 0.8) = 5 without knowing why the answer is five times bigger than the spending. The formula is easy to memorize and easy to misuse. The thing to understand is the ripple: one person's spending is another person's income, and that person spends too.

Spend or save: the MPC

Everything starts with one habit. When a builder on Coralia earns $1,000 more this year, she spends $800 of it (at the bakery, the boatyard, the ferry) and saves $200. Economists call the spent share the marginal propensity to consume, MPC: the share of each extra dollar of income that gets spent, 0.8 here. The saved share is the marginal propensity to save, MPS, 0.2. Every extra dollar is either spent or saved, so MPC + MPS = 1. "Marginal" matters: it is not the share of all income people spend, only of the next dollar they get.

The ripple, round by round

The government pays $100 million to the builders of the harbor. That is $100M of new income: round 1. The builders spend 80% of it, $80M, at the bakery and the boatyard, which is $80M of new income for the bakers and the boatbuilders: round 2. They spend 80% of that, $64M: round 3. Then $51.2M, then $40.96M, and so on. Each round is 80% of the one before, so the rounds shrink but never quite reach zero. Slide through them.

Why does the total stop at $500M and not grow forever? Because at every round 20% leaks out into saving. The $100M of the first round becomes $80M, then $64M: by round 12 a round is under $9M and the total is $465.6M. Keep going and the rounds add less and less; the sum of 100 + 80 + 64 + 51.2 + … gets as close to 500 as you like and never beyond. Saving is the leak that makes the ripple die out, and the size of the leak decides where it stops.

The formula: 1 ÷ (1 − MPC)

There is a shortcut to the end of the ripple. Each round passes on a share MPC of the round before and leaks the rest, 1 − MPC, into saving. The ripple stops when the whole of the first $100M has leaked out, and the leak per round is 0.2 of each round, so the rounds must add up to 100 ÷ 0.2 = 500. In general:

The multiplier is the same thing the rounds told you, in one line. Move the MPC and the whole ripple changes shape: with a lower MPC more leaks at each round, the rounds shrink faster, and the total stops lower.

Notice the direction: the more of each dollar people spend, the bigger the multiplier. At MPC 0.9 only a tenth leaks at each round, the ripple lasts much longer, and the same harbor is worth $1,000M of income. The formula also runs backwards: if $100M of spending ended up as $250M, the multiplier was 2.5, so 1 − MPC = 1 ÷ 2.5 = 0.4 and the MPC was 0.6.

Why a tax cut does less

Plan B: no harbor. The government cuts taxes by $100M instead, so families keep $100M more. Same amount, but the ripple starts differently. The government's $100M for the harbor was spent in full: the builders' income was $100M in round 1. A tax cut goes to families first, and families spend only 80 cents of each extra dollar. Round 1 is 0.8 × $100M = $80M, and the rest of the ripple follows from there: 5 × $80M = $400M, not $500M. The tax multiplier is −MPC ÷ (1 − MPC) = −0.8 ÷ 0.2 = −4; the minus sign means taxes and GDP move in opposite directions. A $100M tax cut raises GDP by $400M, and a $100M tax rise cuts it by $400M.

Plan C is the surprise: a harbor fully paid for by new taxes still raises GDP, by exactly its own cost. The spending adds $500M and the taxes take away $400M, so the balanced-budget multiplier is 1. The $100M that the taxes took out of families' pockets would have been 20% saved; the government spent all of it.

Using the multiplier to close a gap

This is what the formula is for. In the AD-AS lesson, fewer tourists shifted aggregate demand left by $80M and pushed Coralia's real GDP from $1,000M to $960M: a recessionary gap of $40M. The government wants to spend enough to bring AD back. It does not need to spend $80M, because each dollar is multiplied: ΔG = AD shift ÷ multiplier = 80 ÷ 5 = $16M. With a tax cut instead, the shift divided by 4: a cut of $20M.

One thing to watch. On Coralia's graph the price level rises as output recovers, so AD has to shift by the full $80M to close a $40M gap. Many test questions keep the price level fixed; there, the shift needed equals the gap, and the answer would be 40 ÷ 5 = $8M. Read what the question holds constant before you divide.

Why real multipliers are smaller

The formula assumes that the only leak at each round is saving. In a real country every round also leaks into taxes (part of each new dollar of income goes to the government) and imports (part is spent on goods made abroad, income for someone else's country). Less is passed on each round, so the ripple dies out sooner and the multiplier is smaller than 1 ÷ (1 − MPC). Two more things eat into it. Crowding out: to pay for the harbor the government borrows, which pushes interest rates up and makes some firms cancel their own investment. And time lags: seeing the recession, voting the plan and building the harbor can take years, and spending that arrives after the recovery adds to a boom instead of filling a gap.

The real case: in the 2008–09 recession the United States passed a stimulus plan of about $800 billion in spending, tax cuts and aid. The Congressional Budget Office put the multiplier of its federal spending between about 0.5 and 2.5, a long way from 5. The range itself is the lesson: economists still disagree on the size, because taxes, imports, crowding out and timing are different in every recession, and the simple formula knows none of them.

The trap on tests

• The first round is the spending, not the multiplier. $100M of new spending is $100M of income in round 1. The multiplier counts the rounds that follow; the total is multiplier × the first round.
• 1 − MPC is the MPS. The multiplier is 1 ÷ MPS: the smaller the leak, the bigger the ripple. MPC 0.8 gives 5, not 0.8 or 1.25.
• A tax cut starts at MPC × the cut. Families save part before spending, so the tax multiplier is −MPC ÷ (1 − MPC), one less than the spending multiplier in absolute value (4 against 5) and negative.
• Divide the shift, not the gap, unless prices are fixed. ΔG = the AD shift needed ÷ multiplier. If the question holds the price level fixed, the shift equals the gap.
• "Simple" means saving only. Taxes, imports, crowding out and lags make real multipliers smaller; the 2009 estimates were 0.5 to 2.5.

Quick check