The multiplier definitions

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,

$Multiplier = \frac{\Delta \; equilimrium\;expenditure} {\Delta \; autonomous\;expenditure}$
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:

1. Equilibrium expenditure is at point $A$


2. The change in autonomous expenditure shifts the $AE$ curve up by 2 trillion


3. The new equilibrium expenditure is at point $B$


4. The change in equilibrium expenditure is 4 trillion


5. $Multiplier = \frac{\Delta \; equilimrium\;expenditure} {\Delta \; autonomous\;expenditure} = \frac{4} {2} =$ 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,

$\Delta Y = \Delta N + \Delta A$
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

Slope of $AE$ Curve = $\frac{\Delta N} {\Delta Y}$ $\Delta Y \, \times$ Slope of $AE$ Curve = $\Delta N$
Substituting this to the other equation gives

$\Delta Y = \Delta Y \, \times$ Slope of $AE$ Curve + $\Delta A$ $\Delta Y - \Delta Y \, \times$ Slope of $AE$ Curve = $\Delta A$ $\Delta Y$ (1 - Slope of $AE$ Curve) = $\Delta A$ $\Delta Y = \frac{\Delta A} {1- Slope\;of\;AE\;Curve}$
Now dividing both sides by $\Delta A$ gives us

$\frac{\Delta Y} {\Delta A} = \frac{1} {1 - Slope\;of\;AE\;Curve} = The \;Multiplier$
Example: If the slope of the $AE$ curve is 0.5, what is the multiplier?

$The \;Multiplier = \frac{1} {1 - 0.5} = 2$
Note: There are other versions of the formula for the multiplier.

$Multiplier = \frac{1} {1 - MPC}, Multiplier = \frac{1} {MPS}$

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:

$Multiplier = \frac{1} {1- Slope\;of\;AE\;Curve}$
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

$Slope\; of\; AE\; Curve = b(1-t)-m$
Substituting this to our multiplier, we have

$Multiplier =$ $\large \frac{1} {1 - [b(1-t)-m]}$
Where:
$b$ = marginal propensity to consume
$t$ = marginal tax rate
$m$ = marginal propensity to import

Therefore, the multiplier can be small if:

1. $b$ is small
2. $t$ (marginal tax rate) is large
3. $m$ (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?

$Multiplier =$ $\large \frac{1} {1 - [b(1-t)-m]} = \frac{1} {1 - [0.5 (1 - 0 ) -0 ] } =\frac{1} {0.5} = 2$
$Multiplier =$ $\large \frac{1} {1 - [b(1-t)-m]} = \frac{1} {1 - [0.5 (1 - 0 ) -0 ] } =\frac{1} {0.75}= 1.\overline{33}$
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.