The multiplier works because one person's spending is another person's income. An initial injection (e.g. $100m of government spending on a new highway) becomes income for construction firms and workers; they spend part of it (the marginal propensity to consume) and withdraw the rest as saving, tax and imports. Each round of spending is smaller than the last, until the total rise in income is a multiple of the initial injection. The bigger the leakages (withdrawals) at each round, the smaller the multiplier.
| Term | Meaning | Formula |
|---|---|---|
| Average propensity to consume (apc) | Share of total income spent on consumption | apc = C ÷ Y |
| Marginal propensity to consume (mpc) | Share of extra income spent on consumption | mpc = ΔC ÷ ΔY |
| Average propensity to save (aps) | Share of total income saved | aps = S ÷ Y; apc + aps = 1 (closed, no tax) |
| Marginal propensity to save (mps) | Share of extra income saved | mps = ΔS ÷ ΔY |
| Average rate of tax (art) | Share of total income paid in tax | art = T ÷ Y |
| Marginal rate of tax (mrt) | Share of extra income paid in tax | mrt = ΔT ÷ ΔY |
| Average propensity to import (apm) | Share of total income spent on imports | apm = M ÷ Y |
| Marginal propensity to import (mpm) | Share of extra income spent on imports | mpm = ΔM ÷ ΔY |
| Economy | Withdrawals | Multiplier formula |
|---|---|---|
| Closed, no government (2-sector) | Saving only | k = 1 ÷ (1 − mpc) = 1 ÷ mps |
| Closed, with government (3-sector) | Saving + tax | k = 1 ÷ (mps + mrt) |
| Open, with government (4-sector) | Saving + tax + imports | k = 1 ÷ (mps + mrt + mpm) = 1 ÷ mpw |
| Open, no government | Saving + imports | k = 1 ÷ (mps + mpm) |
Worked example: calculating the multiplier and the change in income
An economy has mps = 0.1, mrt = 0.2 and mpm = 0.2. The marginal propensity to withdraw (mpw) = 0.1 + 0.2 + 0.2 = 0.5. Multiplier k = 1 ÷ 0.5 = 2. Investment rises by $40bn → ΔY = k × ΔJ = 2 × 40 = $80bn increase in national income. In a closed economy with no government and the same mps (0.1), k = 1 ÷ 0.1 = 10: ΔY would be $400bn. Open economies with high taxes and imports have small multipliers. Check: withdrawals rise by mpw × ΔY = 0.5 × 80 = $40bn = the original injection, so J = W again at the new equilibrium.
Worked example: average and marginal propensities
Income rises from $500bn to $600bn; consumption rises from $400bn to $460bn. apc at Y = 600: 460 ÷ 600 = 0.77; aps = 1 − 0.77 = 0.23. mpc = (460 − 400) ÷ (600 − 500) = 60 ÷ 100 = 0.6; mps = 0.4; in a closed economy with no government k = 1 ÷ 0.4 = 2.5.