The Multiplier
In the following table, there are schedules for two different income-expenditure diagrams. The first set of
data for aggregate expenditure, AE0, will serve as our baseline case. AE0 represents the case before we
shock the system with an increase in investment.
The second set of data for aggregate expenditure, AE1, represents the values of aggregate expenditure for
each value of Y following a $500 billion increase in investment. You will notice that each value for AE
has increased by exactly $500 billion when compared to the baseline case.
All numbers are in trillions of dollars.
Real GDP (Y)
5
6
7
8
9
AE0
5.25
6
6.75
7.5
8.25
AE1
5.75
6.5
7.25
8
8.75
We can then graph these income-expenditure data as follows:
Income-Expenditure
10
Real Expenditure
9
8
AE0
AE1
7
6
5
4
4
5
6
7
8
9
10
RGDP (Y)
As you can see from both the table and the graph, the original demand side equilibrium occurs where GDP
and aggregate expenditure both equal $6 trillion.
We then increase investment (I) by $500 billion. The new demand side equilibrium occurs where GDP and
aggregate expenditure both equal $8 trillion.
So, what happened? We increased I by only $500 billion and we ended up with an increase in GDP of $2
trillion. How is this possible?
Some definitions:
Autonomous spending: The components of aggregate expenditure that are not influenced by real GDP. In
English, autonomous spending refers to investment, government purchases, exports (recall that imports are
dependent upon national income) and autonomous consumption. Autonomous consumption is the level of
consumption that would occur in the economy given that real GDP = 0.
Induced spending: Induced spending is that portion of aggregate expenditure that varies with real GDP.
From our earlier discussion, recall that the two main components of induced spending are consumption and
imports.
Back to our story…
When investment is increased by $500 billion, autonomous spending increases by $500 billion. This
increase in autonomous spending moves us from the equilibrium level of aggregate expenditure ($6 trillion)
to an aggregate expenditure of $6.5 trillion. At that point, aggregate expenditure exceeds real GDP.
You are probably getting tired of this portion of the story, but if aggregate expenditure>GDP, then
inventories are falling. Firms will then increase production and, consequently, real GDP.
This increase in real GDP causes an increase in induced spending. This induced spending pushes output up
to the new equilibrium level of $8 trillion.
We are not done yet, but we are getting there. We can now define the multiplier as follows:
Multiplier =  equilibrium expenditure
 autonomous spending
From our example:
Multiplier = $2 trillion = 4
$0.5 trillion
That is, for every dollar we increase autonomous spending,
equilibrium GDP increases by $4.
So, does this induced spending thing work by magic or what? Not really. Here is a very cheesy example to
explain why this works.
If I, the owner of Widgico, manufacturer of quality widgets, decide that I need a new widgetmaking machine, I go to the widget machine factory, Macrosoft and buy a new one. This, of course is an
increase is autonomous spending (investment). So, what happens to that money that I pay to Macrosoft?
To make things simple, let’s take a look at just one of those dollars.
Let’s assume that the one dollar that we are talking about gets paid to a Macrosoft employee (income).
That employee has some marginal propensity to consume (we will assume MPC=.9 for everyone).
Therefore, he will save 10 cents of that dollar and spend the rest of it on shoes. Now the shoe salesman has
90 cents in income. He will save 9 cents and spend 81. What follows on the next page is a list of
hypothetical transactions that occur as a result of this one dollar increase in investment:
Who?
Me
Macrosoft Employee
Shoe salesman
Egg producer
Car salesman
Peanut vendor
Save
$.10
$.09
$.08
$.07
$.066
Spend
$1.00
$.90
$.81
$.73
$.66
$.60
Cum.  in GDP
$1.00
$1.90
$2.71
$3.44
$4.10
$4.70
This process goes on until the entire dollar is saved. But, if we don’t know the total change in GDP
induced by this increase in autonomous spending, then how do we know the value of the multiplier?
We can use the MPC to determine the multiplier:
The total change in GDP is equal to the change in I plus the induced change in C:
Y = C + I
But, as we know, the change in C is determined by the change in Y and the MPC:
C = MPC * Y
Substituting:
Y = MPC * Y + I
Solving for Y:
(1-MPC) * Y = I
Y = I/(1-MPC)
By our previous definition:
Multiplier =  equilibrium expenditure = Y = 1/(1-MPC)
 autonomous spending
I
If you are not bored to tears yet, recall that (1-MPC) = MPS. So,
Multiplier = 1/MPS