1. AE Model and the Multiplier
ECO 120 Macroeconomics Week 3
AE Model and the Multiplier
Lecturer
Dr. Rod Duncan

2. Topics Two sector AE model Three sector AE model
Equilibrium in the AE model
The multiplier in the AE model

3. Consumption function
The consumption function relates the level of private household consumption of goods and services (C) to the level of aggregate income (Y).
We can represent the consumption function in three different and equivalent ways.
An mathematical equation
A graph
A table
For example the consumption function could be:
C = $100bn + 0.5Y

4. Consumption function We can represent this same function with a graph.
C(Y) = $100bn + 0.5Y
$150bn
Slope is 0.5
$100bn
The MPC is 0.5 Y
$100bn

5. Consumption function Y C(Y) = 100 + 0.5Y C 100 + 0.5 (0) 100
Or we can represent the same function with a table.
Three ways of represent-ing the same function. Y
C(Y) = Y C
(0)
100
(100)
150
200
(200)
300
(300)
250
400
(400)
500
(500)
350

6. Two sector model
Aggregate expenditure (AE) in the two sector model is composed of consumption (C) and investment (I).
AE = C + I
In this model, we treat I as exogenous, so it is a constant.
Let’s use the same simple linear consumption function:
C = Y
I = 100
AE = C + I = Y = Y

7. Aggregate expenditure function
This equation is a relationship between income (Y) and aggregate expenditure (AE).
AE = Y
$250bn
Slope is 0.5
$200bn Y
$100bn

8. Aggregate expenditure functionY C I AE
100
200
150
250
300
350
400
500
450
But we could also use the table form.

9. Equilibrium in two sector model
Equilibrium in a model is a situation of “balance”. In our AE model, equilibrium requires that demand for goods (AE) is equal to supply of goods (Y).
Y = AE = C + I
For the equilibrium we are looking for the value of GDP, Y*, such that goods demand and goods supply are equal.
In our two sector AE model that means that we can look up our AE table and find where AE = Y.
The equilibrium value of Y will be our prediction of GDP for our AE model.

10. Equilibrium The equilibrium value of GDP is $400bn. Y C I AE 100 200
100
200
150
250
300
350
400*
500
450

11. Equilibrium
We could accomplish the same by using our graph of the AE function.
The AE line shows us the level of goods demand for each value of Y.
The 45 degree line represents the value of Y or supply of goods.
Equilibrium will occur when the 45 degree line and the AE line cross. At the crossing, goods demand is equal to goods supply for that level of Y.

12. Equilibrium
The equilibrium value of Y is where the 45 degree line and the AE line cross. Y* is at $400bn. Y
AE = Y
400
400 Y

13. Equilibrium
Finally, if you are comfortable with the mathematics, you can solve for the equilibrium value of Y using the equations:
Y* = AE = Y*
Y* – 0.5Y* = 200
0.5Y* = 200
Y* = 400
You arrive at the same answer no matter which way you use to derive it.

14. Autonomous expenditure
In our model we have two part of aggregate expenditure:
AE = $200bn + 0.5Y
One part does not depend on the value of Y- the $200bn. This portion is called “autonomous expenditure”.
The other part does depend on the value of Y- the 0.5Y.
In our model part of autonomous expenditure is C and part is I.

15. Scenario: Investment falls
What happens if I drops from 100 to 50 perhaps because of uncertainty due to terrorism scares?
Equilibrium GDP drops to 300. Y C I AE
100 50
150
200
250
300*
400
300
350
500

16. Scenario But you could also find the same answer with some algebra:
AE = C + I = Y + 50 = Y
Y* = AE = Y*
Y* – 0.5Y* = 150
0.5Y* = 150
Y* = 300
Find the answer in the way you feel most comfortable.

17. Multiplier
So a $50bn drop in investment (or autonomous expenditure) leads to a $100bn drop in equilibrium GDP.
The ratio of the change in GDP over the change in autonomous expenditure is called the multiplier:
Multiplier = (Change in GDP)/(Change in I)

18. Multiplier In our scenario the multiplier is:
Multiplier = $100bn / $50bn = 2
This term is given the name “multiplier” because each $1 change in I lead to a 2x$1 change in GDP.
We will come back to the multiplier after we talk about the three sector model.

19. Three sector AE model
Now we make our model slightly more complicated by bringing in the government. The government has two effects on our model:
The government raises tax revenues (T) by taxing household incomes.
The government purchases some goods and services for government consumption (G).
We treat the levels of T and G as exogenous to our AE model. Government policy determines what T and G will be, and policy is not affected by the equilibrium level of GDP.

20. Three sector model
Household consumption depended on household income, Y, in our two sector model.
In the three sector model, the income that households have available to spend or save is now income net of taxes, Y – T. We call this amount “disposable income”, YD.
The consumption function will now depend on disposable income, not income.
C = C(Y – T) = C(YD)

21. Three sector model
Our new aggregate expenditure function includes government purchases of goods and services, so we have:
AE = C + I + G
Let’s assume we have the same linear consumption function as before, but now in disposable income:
C = (Y – T)
Let T = G = 50 and let I = We can follow the same steps as before to find our AE function and then to find equilibrium GDP.

22. Aggregate expenditure function
Our AE function is:
AE = C(Y – T) + I + G
AE = (Y – 50)
AE = Y –
AE = Y
We can also represent this as a table. Our C function with disposable income is:
C = (Y-50) = Y

23. Table form Y
C = Y I G AE 75
100 50
225
125
275
200
175
325
300
375
400
425
500
475

24. Equilibrium
If we want to find equilibrium GDP in our three sector model, we need to find the level of GDP, Y*, for which goods demand (AE) is equal to goods supply (Y).
If we look at our table, we see that for an income level of Y of 400, AE is 425 which exceeds Y. At an income level of Y of 500, AE is 475 which is less than Y.
We would guess that the equilibrium value of Y lies between 400 and 500.
We construct a new table of values of Y between 400 and 500.

25. Equilibrium Y C = 75 + 0.5Y I G AE 400 275 100 50 425 287.5 437.5 450*
300
475
312.5
462.5
500
325

26. Equilibrium The equilibrium value of Y is 450.
We could find the answer with our equations:
AE = Y
Y* = AE = Y*
Y* - 0.5Y* = 225
0.5Y* = 225
Y* = 450

27. Scenario: Investment falls
What happens if we have the same drop in investment in the three sector model? So I drops from 100 to 50?
Using our equations:
AE = (Y - T) + I + G
AE = Y
AE = Y
Solving for Y*, we get:
Y* = AE = Y*
Y* = 350
Our multiplier = 100/50 = 2 as before.

28. Generalizing the AE model
In all our models, we have had Y (and also C) as our endogenous variable.
Exogenous variables are variables in a model that are determined “outside” the model itself, so they appear as constants.
For the aggregate expenditure model, we treat as exogenous:
Investment (I)
Government consumption (G) ( three sector)
Taxes (T) (three sector)
Net Exports (NX) (four sector)

29. Aggregate expenditure
In a two sector model:
AE = C(Y) + I
In a closed (no foreign trade) economy or three sector model:
AE = C(Y - T) + I + G
In an open economy or four sector model:
AE = C(Y - T) + I + G + NX
Changes in the exogenous variables (I, G, T or NX) will shift the AE curve.

30. Equilibrium in the AE model
Exogenous variables are just constants- numbers.
Supply of goods equals demands of goods (in the closed economy):
Y = C + I + G
Y = a + b(Y – T) + I + G
(1 – b)Y = a – bT + I + G
Finally we get:

31. Equilibrium The value 1/(1-b) is the multiplier in our AE model.
So if I or G changes by 1, we know that Y will change by 1/(1-b).
The constant b here is just the MPC.
So we have:
Multiplier = 1 / (1 – MPC)
In our simple linear model, b was 0.5, and we get:
Multiplier = 1 / (1 – 0.5) = 1/0.5 = 2

32. Expenditure multiplier
Imagine the government wishes to affect the economy. One tool available is government consumption, G, or government taxes, T. This is called “fiscal policy”.
Any change in G (∆G) in our AE model will produce:

33. Multiplier
If b=0.75, then the multiplier is (1/0.25) or 4, so $1 of new G will produce $4 of new Y.
Our multiplier is equal to 1/(1-MPC).
Since 0<MPC<1, our multiplier will be greater than 1.
The larger is the MPC, the larger is our multiplier.

34. Example of 4 sector model
Let’s say we start with a linear model with a consumption function (or “schedule”):
C = (Y – T)
Let’s use a 4 sector model of AE with:
I = 80
G = 60
T = 40
NX = 30

35. Example of 4 sector model
Firstly graph the consumption function.
If we plug in 40 for T, we get (0.75*40 = 30):
C = Y – 30
C = Y
We know the vertical intercept (Y=0):
C = (0) = 70 C
C= Y 70
Slope is 0.75 Y

36. Example of a 4 sector model
From the C function, we know our MPC is a fact we will use later.
Now we need to derive our AE curve. We use the 4 sector AE equation:
AE = C + I + G + NX
Plugging in the facts above, we get
AE = Y I G NX C

37. Example of a 4 sector model
Adding all those together, we get:
AE = Y
Just like we did with the C function, we can graph this AE function:
C, AE
AE= Y
240
C= Y 70 Y

38. Example of a 4 sector model
Now we need to find the equilibrium of this model. This occurs where goods demand, AE, is equal to goods supply, Y.
Note that AE is a function of income, Y, and we are also requiring that in equilibrium goods supply, Y, is equal to AE. We are using Y in two different senses, as income and as output. But remember from the first lecture, we can calculate GDP either from income or from production and get the same value.
We are looking for an income level, Y*, that produces an AE level equal to Y*.

39. Example of a 4 sector model
Using our equation for AE, we get:
AE = Y*
Y* = AE = Y*
0.25Y* = 240
Y* = 960 AE
45 deg line
AE = Y
960
240
960 Y

40. Example of a 4 sector model
Since our MPC = 0.75, we know that the multiplier for autonomous expenditure is:
Multiplier = 1/(1-MPC) = 1/(1-0.75) = 4
So a $1 change in I or G or NX would lead to a $4 change in equilibrium GDP, Y*.