In this post I will provide a mathematical basis for the multiplier effect which is found when changing an autonomous variable in the aggregate expenditure function (AE). AE is a function which represents the total amount of money that is spent by all consumers in an economy, and is made up of two components: autonomous expenditure (which is exogenous in relation to income) and induced expenditure (endogenous to income). In other words, autonomous expenditure is the y-cut of the AE funtion, and any increases in autonomous expenditure will shift AE vertically. Induced consumption refers to any expenditure that is over and above the level of expenditure at the y-cut, and is directly related to the gradient of the AE function (which includes the marginal propensity to consume¹). Referring to Figure 1 below, let the increase in autonomous expenditure be y1. y1 thus shifts AE1 to AE2.

The first increase in AE, y1, will reach a point on the AE2 curve which has expenditure greater than income (Y). In order to reach equilibrium, the income of people in the economy thus increases by x1 until AE=Y. But because income (Y) has increased, people will be more willing to spend money (according to their marginal propensity to consume¹), so AE will rise by y2 until the increased level of spending is in accordance with the AE2 function. This process repeats over and over again, with smaller and smaller increases in expenditure and income due to the marginal propensity to save. The total increase in AE is represented by $\displaystyle\sum_{n=1}^\infty y_n = \text{E}_2 - \text{E}_1$

Note that $y_n \to 0$ as $n \to \infty$.

To calculate yn+1, we look at the gradient of the AE2 function. yxbecause the line AE=Y has a gradient of 1, so yn+1 = xn(b(1-t)) = yn(b(1-t)) where b(1-t) is the gradient of the AE2 function in a closed economy (b representing the marginal propensity to consume, and t representing the marginal tax rate). But this is a recursive formula of a geometric series, with the ratio between the terms equal to b(1-t) and the first term equal to y1. Thus, we use the formula for the sum of a geometric series to infinity: $\displaystyle S_\infty = y_1 \cdot \frac{1}{1-b(1-t)}$

We now note that $\displaystyle\sum_{n=1}^\infty y_n = y_1 \cdot \frac{1}{1-b(1-t)}$

and y1 has in effect been multiplied by $\frac{1}{1-b(1-t)}$ to reach the total increase in AE. We call $\frac{1}{1-b(1-t)}$ the multiplier and denote it as $\alpha$. (Mohr, Fourie, et al 2009:425.)

¹An increase in consumption caused by an addition to income divided by that increase in income is known as the marginal propensity to consume (MPC) (Encyclopædia Britannica 2009 Deluxe Edition 2009, sv ‘propensity to consume’).

## References

Encyclopædia Britannica 2009 Deluxe Edition. 2009. Sv ‘propensity to consume’. Chicago: Encyclopædia Britannica.

Mohr, P & Fourie, L. 2009. Economics for South African students. 4th edition. Pretoria: Van Schaik Publishers.

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