Where a multiplier comes from
New spending becomes someone's income. They spend the marginal-propensity-to-consume share of it, which becomes someone else's income, and so on. With an MPC of 0.75, $100 of new spending generates rounds of $100, $75, $56.25 and so on — a geometric series that sums to 100 ÷ (1 − 0.75) = $400.
The spending multiplier is therefore 1 ÷ (1 − MPC), or 1 ÷ MPS: the smaller the share of each dollar that leaks into saving, the longer the chain of re-spending and the larger the total.
Worked example
MPC = 0.75
Rounds: 100 + 75 + 56.25 + …
k = 1 ÷ 0.25 = 4
ΔY = 4 × 100 = 400
A tax cut starts one step later
A $100 tax cut does not enter spending directly. It raises disposable income by $100, and households spend only the MPC share of that — $75 — in the first round. From there the chain is the same as before, so the tax multiplier is the spending multiplier applied to the MPC share: −MPC ÷ (1 − MPC) = −0.75 ÷ 0.25 = −3.
The minus sign records that taxes and output move in opposite directions. A $100 cut raises output by $300; a $100 increase lowers it by $300. And because the first round is partly saved, the tax multiplier is always exactly one smaller in size than the spending multiplier in this model.
Worked example
k_T = −0.75 ÷ 0.25 = −3
ΔT = −100 (a cut)
ΔY = −3 × (−100) = +300
The balanced-budget multiplier is 1
Raise government purchases by $100 and pay for them with a $100 lump-sum tax. The budget balance does not move, but output does: the spending adds 4 × 100 = $400 and the tax subtracts 3 × 100 = $300, leaving output $100 higher. The balanced-budget multiplier, usually credited to Trygve Haavelmo, is 4 + (−3) = 1.
The result holds for any MPC in the simple model, because the two multipliers always differ by one. It is the tidiest illustration of the asymmetry between the two levers — and it disappears once taxes vary with income, as the next section shows.
Worked example
ΔG = +100 → +400
ΔT = +100 → −300
ΔY = +100
Income taxes and imports shrink both multipliers
Real economies leak spending in more places than saving. A proportional income tax takes a share t of each extra dollar before it can be spent, and a share m of spending goes on imports, which raise output abroad. Collecting the leaks, the denominator becomes 1 − MPC(1 − t) + m.
With MPC 0.75, t = 0.2 and m = 0.1, the denominator is 1 − 0.6 + 0.1 = 0.5, so the spending multiplier falls from 4 to 2 and the tax multiplier from −3 to −1.5. The OpenStax textbook's own example — MPC 0.7, a 10% tax and imports of 0.1 — gives 1 ÷ 0.47 = 2.13. Imports enter the denominator with a minus sign. They are a leak, so they add to it; only re-spending at home (consumption after tax, and in fuller models induced investment) subtracts.
Worked example
MPC = 0.75, t = 0.2, m = 0.1
den = 1 − 0.75 × 0.8 + 0.1 = 0.5
k = 2, k_T = −1.5
What the evidence says about real tax multipliers
Estimating the real effect of tax changes is hard because governments change taxes in response to the economy. Christina and David Romer sidestepped that by reading the legislative record and keeping only tax changes made for reasons unrelated to current conditions. They found those changes strongly contractionary: a tax increase of 1% of GDP lowered real output by almost 3% over roughly the following three years.
That is larger than many estimates of spending multipliers — the opposite of the ranking in the simple model. The model captures one channel, spending out of disposable income; real tax changes also change incentives to work and invest, and are shaped by what people expect about future taxes and by how the central bank responds.
Why real multipliers vary so much
Two conditions matter more than the formula. First, spare capacity: with idle workers and machines, extra demand raises output; near full employment it mostly raises prices. Second, the monetary response: if a central bank raises interest rates to offset a fiscal expansion, some private spending is crowded out; at interest rates near zero it is not.
Composition matters too. Transfers and tax cuts aimed at households with little savings have larger effects, because those households have higher MPCs; a temporary cut is spent less than a permanent one. The tax multiplier calculator handles the simple and the fuller textbook forms; the evidence is the reminder that both are upper-bound teaching tools.
Common mistakes
Using the spending multiplier for a tax change.The first round of a tax cut is only the MPC share of it, so the multiplier is −MPC ÷ (1 − MPC).Dropping the sign on the tax multiplier.A tax increase lowers output; the negative sign is the answer, not a formatting detail.Adding induced investment as a leak.Investment that rises with income is re-spending and makes the multiplier larger; imports, saving and taxes are the leaks.Reading a textbook multiplier as a forecast.Real multipliers depend on spare capacity, monetary policy and who receives the money, and are usually smaller.