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ECO 120 Macroeconomics
Week 3
AE Model and the Multiplier LecturerDr. Rod Duncan
News
• Wednesday 4-5pm tutorial has been moved from C2-213 to C2-216 starting tomorrow.
Topics
• Two sector AE model
• Three sector AE model
• Equilibrium in the AE model
• The multiplier in the AE model
What the AE model for?
• Always try to see the forest for the trees!• Always ask “what is this for?” “what is the
purpose of this model?”• The aggregate expenditure (AE) model is a
simple model of GDP:– It can be used to predict what GDP will do? Is
Australian GDP rising? How fast?– It can be used to explain what happened to GDP?
For example GDP might be rising because net exports are rising.
The big picture
P, Wealth, H/h Expectations,H/h Taxes
P, i, Business Expectations, Business Taxes
C
Government policy, and? G
P, and? NX
AE AD
I
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.
1. An mathematical equation2. A graph3. A table
• For example the consumption function could be:
– C = $100bn + 0.5Y
Consumption function
• We can represent this same function with a graph.
Y
C(Y) = $100bn + 0.5Y
$100bn Slope is 0.5
C
$100bn
$150bn
The MPC is 0.5
Consumption function
• Or we can represent the same function with a table.
• Three ways of represent-ing the same function.
Y C(Y) = 100 + 0.5Y C
0 100 + 0.5 (0) 100
100 100 + 0.5 (100) 150
200 100 + 0.5 (200) 200
300 100 + 0.5 (300) 250
400 100 + 0.5 (400) 300
500 100 + 0.5 (500) 350
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 = 100 + 0.5YI = 100AE = C + I = 100 + 0.5Y + 100 = 200 + 0.5Y
Aggregate expenditure function
• This equation is a relationship between income (Y) and aggregate expenditure (AE).
Y
AE = 200 + 0.5Y
$200bn Slope is 0.5
$100bn
$250bn
Aggregate expenditure function
• But we could also use the table form.
Y C I AE
0 100 100 200
100 150 100 250
200 200 100 300
300 250 100 350
400 300 100 400
500 350 100 450
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.
Equilibrium
• The equilibrium value of GDP is $400bn.
Y C I AE
0 100 100 200
100 150 100 250
200 200 100 300
300 250 100 350
400* 300 100 400*
500 350 100 450
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.
Equilibrium
• The equilibrium value of Y is where the 45 degree line and the AE line cross. Y* is at $400bn.
Y
Y
AE = 200 + 0.5Y
400
400
Equilibrium
• Finally, if you are comfortable with the mathematics, you can solve for the equilibrium value of Y using the equations:Y* = AE = 200 + 0.5Y*Y* – 0.5Y* = 2000.5Y* = 200Y* = 400
• You arrive at the same answer no matter which way you use to derive it.
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.
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
0 100 50 150
100 150 50 200
200 200 50 250
300* 250 50 300*
400 300 50 350
500 350 50 400
Scenario
• But you could also find the same answer with some algebra:AE = C + I = 100 + 0.5Y + 50 = 150 + 0.5YY* = AE = 150 + 0.5Y*Y* – 0.5Y* = 1500.5Y* = 150Y* = 300
• Find the answer in the way you feel most comfortable.
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)
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.
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.
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)
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 = 100 + 0.5 (Y – T)
• Let T = G = 50 and let I = 100. We can follow the same steps as before to find our AE function and then to find equilibrium GDP.
Aggregate expenditure function
• Our AE function is:AE = C(Y – T) + I + G
AE = 100 + 0.5(Y – 50) + 100 + 50
AE = 100 + 0.5Y – 25 + 100 + 50
AE = 225 + 0.5Y
• We can also represent this as a table. Our C function with disposable income is:C = 100 + 0.5(Y-50) = 75 + 0.5Y
Table form
Y C = 75 + 0.5Y
I G AE
0 75 100 50 225
100 125 100 50 275
200 175 100 50 325
300 225 100 50 375
400 275 100 50 425
500 325 100 50 475
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.
Equilibrium
Y C = 75 + 0.5Y
I G AE
400 275 100 50 425
425 287.5 100 50 437.5
450* 300 100 50 450*
475 312.5 100 50 462.5
500 325 100 50 475
Equilibrium
• The equilibrium value of Y is 450.
• We could find the answer with our equations:AE = 225 + 0.5Y
Y* = AE = 225 + 0.5Y*
Y* - 0.5Y* = 225
0.5Y* = 225
Y* = 450
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 = 100 + 0.5(Y - T) + I + GAE = 75 + 0.5Y + 50 + 50AE = 175 + 0.5Y
• Solving for Y*, we get:Y* = AE = 175 + 0.5Y*Y* = 350
• Our multiplier = 100/50 = 2 as before.
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)
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.
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:
G]IbT[ab1
1Y
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
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:
ΔGb-1
1 ΔY
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.
Practice exam question
The Howard government plans to spend $600m on new coast guard ships for the Australian armed forces. These ships will be built in Tasmania. Explain (using both a diagram and text based on what you have learned so far in class) how the Australian economy will be affected by:
[Hints: Remember to draw the initial equilibrium before the new spending and new taxes on your diagrams. Treat the new taxes and new spending as separate policies.]
(a) the $600m in new government purchases; and(b) the $600m in new income taxes the government will
have to use to pay for the ships.
