Revise The Keynesian multiplier for IB Economics (3.9)

The multiplier rests on a single insight: one person's spending is another person's income. When new spending is injected into the economy — say the government spends on building a hospital — the construction workers and suppliers receive it as income. They spend part of that income, which becomes income for shopkeepers and other firms, who spend part again. Each round is smaller than the last because some income leaks away, but adding up every round gives a final rise in national income that is a multiple of the original injection.

A marginal propensity measures how a household splits an ADDITIONAL unit of income. There are four to know. The marginal propensity to consume (MPC = ΔC/ΔY) is the fraction spent on domestic goods and services. The marginal propensity to save (MPS = ΔS/ΔY), the marginal propensity to tax (MPT = ΔT/ΔY) and the marginal propensity to import (MPM = ΔM/ΔY) are the three ways income leaks out. Because every extra unit of income is either consumed domestically or withdrawn, the propensities always sum to one.

$\text{MPC} + \text{MPS} + \text{MPT} + \text{MPM} = 1 \quad\Rightarrow\quad \text{MPC} + \text{MPW} = 1$

The marginal propensity to withdraw collects all three leakages into a single number: MPW = MPS + MPT + MPM. In the simplest model — a closed economy with no government — there is only one leakage, saving, so MPW = MPS and the identity reduces to the familiar MPC + MPS = 1.

The multiplier, k, tells you how many times larger the final change in income is than the injection that caused it. Because 1 − MPC and MPW are the same quantity (both equal the fraction that leaks away), the multiplier can be written in two equivalent forms.

$k = \frac{1}{1 - \text{MPC}} = \frac{1}{\text{MPW}} = \frac{1}{\text{MPS} + \text{MPT} + \text{MPM}}$

Read the formula for its economics. A high MPC means people re-spend most of each extra unit, so 1 − MPC is small and k is large. Bigger leakages — a higher MPS, MPT or MPM — raise the denominator and shrink the multiplier. This is why an open economy with a government (three leakages) has a smaller multiplier than a simple closed economy (one leakage). Once you have k, the final change in real GDP is simply the injection scaled up by it.

$\Delta Y = \text{injection} \times k \quad\text{(e.g. } \Delta Y = \Delta G \times k)$

The multiplier is the reason fiscal policy can move the economy by more than its own headline cost. A government that increases spending or cuts taxes to close a recessionary (deflationary) gap injects demand that is then re-spent, so the final rise in real GDP exceeds the initial injection — a favourable multiplier makes expansionary fiscal policy more powerful. Knowing k also lets a government estimate the injection needed to reach a target level of income: to raise GDP by a desired amount, the required injection is that amount divided by k.