# The multiplier definitions

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##### Intros

###### Lessons

__The Multiplier Definitions__- Change in equilibrium and autonomous expenditure
- Multiplier is always greater than 1
- Why Multiplier> 1?

__Relationship Between the Multiplier & Slope of AE Curve__- Formula for Change in Real GDP
- Formula for Slope of $AE$ curve
- Algebraic Manipulation
- $Multiplier = \frac{1} {1- Slope\;of\;AE\;Curve}$

__The Multiplier Applications__- Import Taxes
- Income Taxes
- The Multiplier becomes smaller

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##### Examples

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###### Topic Notes

__The Multiplier Definitions__The $AE$ curve can be shifted up? How? By the increase of autonomous expenditure.

If the $AE$ curve shifts up, then the equilibrium expenditure changes. How can we find the new equilibrium?

We need to introduce a new concept.

**Multiplier:**shows the magnitude in how much equilibrium expenditure has changed in proportion with the change in autonomous expenditure.

In other words,

The multiplier is important because it helps us see the magnitude of the increase in the $AE$ curve.

Note: The multiplier is always greater than 1. This is because the change in equilibrium expenditure is always bigger than the change in autonomous expenditure.

Suppose we have the following graph

Notice that:

- Equilibrium expenditure is at point $A$
- The change in autonomous expenditure shifts the $AE$ curve up by 2 trillion
- The new equilibrium expenditure is at point $B$
- The change in equilibrium expenditure is 4 trillion
- $Multiplier = \frac{\Delta \; equilimrium\;expenditure} {\Delta \; autonomous\;expenditure} = \frac{4} {2} =$ 2, which is greater than 1

__Relationship Between the Multiplier & Slope of AECurve__If we know the slope of the $AE$ curve, then we can also find the multiplier. How?

Recall that the change in real GDP is due to the changes in both induced expenditure and autonomous expenditure. In other words,

Where:

$\Delta Y$ = change in real GDP

$\Delta N$ = change in induced expenditure

$\Delta A$ = autonomous expenditure

We also know the slope of the $AE$ curve as

Substituting this to the other equation gives

Now dividing both sides by $\Delta A$ gives us

Example: If the slope of the $AE$ curve is 0.5, what is the multiplier?

Note: There are other versions of the formula for the multiplier.

__The Multiplier Applications__Imports and income taxes impacts the size of the multiplier. In fact, they make the multiplier smaller. How?

Recall that the multiplier is:

Notice that the smaller the slope of AE, the smaller the multiplier is. Using mathematical formulas, we can find that the slope of the $AE$ curve is

Substituting this to our multiplier, we have

Where:

$b$ = marginal propensity to consume

$t$ = marginal tax rate

$m$ = marginal propensity to import

Therefore, the multiplier can be small if:

- $b$ is small
- $t$ (marginal tax rate) is large
- $m$ (marginal propensity to import) is large

Example: Find the multiplier if $b=0.5$, $t=0$, $m=0$. Then find the multiplier when $b=0.5$, $t=0.1$, $m=0.2$. What difference do you see?

The multiplier is smaller when there is marginal tax rate and marginal propensity to import.

Note: The smaller the multiplier is from import and income taxes, the less steep the slope of the AE curve is. This reduces the value of the multiplier, which makes the equilibrium expenditure lower.

The slope of the $AE$ curve here is $\frac{1}{2}$, and the multiplier is 2.

The slope of the $AE$ curve here is $\frac{2}{3}$, and the multiplier is 3.

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