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 *y _{1}. y_{1}* thus shifts AE

*to AE*

_{1}*.*

_{2}The first increase in AE, *y _{1}*, will reach a point on the AE

*curve which has expenditure greater than income (Y). In order to reach equilibrium, the income of people in the economy thus increases by*

_{2}*x*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

_{1}*y*until the increased level of spending is in accordance with the AE

_{2}*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*

_{2}

Note that as .

To calculate *y _{n+1}*, we look at the gradient of the AE

*function.*

_{2}*y*=

_{n }*x*because the line AE=Y has a gradient of 1, so

_{n }*y*=

_{n+1 }*x*(

_{n}*b*(

*1-t*)) =

*y*(

_{n}*b*(

*1-t*)) where

*b*(

*1-t*) is the gradient of the AE

*function in a closed economy (*

_{2}*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

*y*. Thus, we use the formula for the sum of a geometric series to infinity:

_{1}

We now note that

and *y _{1}* has in effect been multiplied by to reach the total increase in AE. We call the

*multiplier*and denote it as . (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.* 4^{th} edition. Pretoria: Van Schaik Publishers.

Jonathan ShockMarch 16, 2015 at 4:54 pmGreat post Aidan. I’d love to get a definition of some of the terms here if possible. Looking good!

Aidan HornMarch 16, 2015 at 8:10 pmI hope my revision has helped you to understand the terms involved.

Jonathan ShockMarch 17, 2015 at 8:56 amAbsolutely – great, thank you!

Jonathan HornMarch 17, 2015 at 10:09 pmVery impressive Aidan. I would appreciate a discussion on this to get a better grasp and understanding of this multiplier.