We consider how changes in government purchases affect the economy. Since government purchases are one component of expenditure, hgiher government purchases result in higher planned expenditure for any given level of income. If government purchases rises by $\Delta G$, then planned-expenditure schedule $PE$ shifts upward by $\Delta G$:

$\Delta G$

0.00
01

We can see that the equilibrium moves from point A to point B. The increase in government purchases of 0.00 leads to an increase in income of 0.00. That is, $\Delta Y$ is greater than $\Delta G$. The ratio $\frac{\Delta Y}{\Delta G}$ which is here equal to NaN is called the government-purchases multiplier: it tells up how much income increases in response to a \$1 increase in government purchases. The Keynesian cross implies that this multiplier is larger than 1.

Why does fiscal policy have a multiplied effect on income? According to the consumption function $C = C(Y - T)$, consumption is a function of income, so higher income leads to higher consumption. When an increase in government purchases raises income, it also raises consumption, which further raises income, which further raises consumption and so on. Therefore, in this model, an increase in government purchases leads to an even greater increase in income.

To determine the size of the multiplier, we need to trace through each step of the change in income. The process begins when expenditure rises by $\Delta G$, which implies than income rises by $\Delta G$ as well. This increase in income in turn raises consumption by $MPC \times \Delta G$. This increase in consumption raises expenditure and income once again. This second increase in income of $MPC \times \Delta G$ again raises consumption, this time by $MPC \times (MPC \times \Delta G)$, which again raises expenditure and income, and so on. This feedback between consumption and income continues indefinately. We can write the total effect on income as:

$\Delta Y = \Delta G + (MPC \times \Delta G)$
$+ (MPC^2 \times \Delta G) + ...$

We can re-arrange this expression to express the multiplier as an infinate geometric series:

$\frac{\Delta Y}{\Delta G} = 1 + MPC + MPC^2 + ...$

A result from algebra allows us to write the multiplier as:

$\frac{\Delta Y}{\Delta G} = \frac{1}{1 - MPC}$

In this example, the marginal propensity to consume is 0.30 and hence the multiplier is 1/(1 - 0.30) = 1.43. We can explore how the multiplier is affected by changes in the $MPC$:

$MPC$

0.30
01