The multiplier effect

How the process works

1. An injection, such as new government spending or investment, first raises the income of those who receive the new orders.
2. Those earners re-spend a fraction equal to the marginal propensity to consume on further goods and services.
3. That spending becomes income for other households and firms, who in turn re-spend part of it, and so on round after round.
4. Each round is smaller than the last because saving, taxation and imports leak income out of the circular flow.
5. The process stops once the extra leakages together equal the original injection, leaving national income permanently higher by a multiplied amount.

Propensities and tax rates

Average propensity to consume

$$APC=\frac{C}{Y}$$

Marginal propensity to consume

$$MPC=\frac{\Delta C}{\Delta Y}$$

Two-sector identity

$$MPC+MPS=1$$

Propensity to save (average, marginal)

$$APS=\frac{S}{Y}MPS=\frac{\Delta S}{\Delta Y}$$

Propensity to import (average, marginal)

$$APM=\frac{M}{Y}MPM=\frac{\Delta M}{\Delta Y}$$

Rate of tax (average, marginal)

$$ART=\frac{T}{Y}MRT=\frac{\Delta T}{\Delta Y}$$

1. Each average measure divides the total (of saving, imports or tax) by income, while each marginal measure divides the change by the change in income.
2. In a simple two-sector economy the marginal propensities to consume and to save sum to 1, because each extra pound is either spent or saved.

Formula by type of economy

1. In a closed economy with no government, saving is the only leakage, so the multiplier depends only on the MPC (equivalently on the MPS).
2. Adding a government sector introduces taxation as a second leakage, which lowers the multiplier.
3. In an open economy, spending on imports leaks abroad as well, lowering the multiplier further.
4. The combined leakage is the marginal propensity to withdraw (MPW), the sum of the marginal propensities to save, to tax and to import.

Marginal propensity to withdraw

$$MPW=MPS+MPT+MPM$$

Effect on national income

1. Equilibrium national income is where AD equals national output, which is also where planned injections equal planned withdrawals.
2. A change in any injection changes national income by the multiplier times that change.
3. This is why the multiplier is central to judging fiscal policy: the United States' 2020 stimulus cheques were re-spent by households, so the eventual income gain exceeded the initial outlay, though in a small open economy with a high import share much of it would leak abroad.

Change in national income

$$\Delta Y=k\times \Delta injection$$