Transcript Eco120Int_Lecture3

ECO 120
Macroeconomics
Week 3
AE Model and the Multiplier
Lecturer
Dr. Rod Duncan
Topics
•
•
•
•
Two sector AE model
Three sector AE model
Equilibrium in the AE model
The multiplier in the AE model
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 equation
2. A graph
3. A table
•
For example the consumption function could
be:
–
C = $100bn + 0.5Y
Consumption function
• We can represent this same function with
a graph.
C
C(Y) = $100bn + 0.5Y
$150bn
Slope is 0.5
The MPC is 0.5
$100bn
$100bn
Y
Consumption function
• Or we can
represent
the same
function with
a table.
• Three ways
of
representing the
same
function.
Y
C(Y) = 100 + 0.5Y
C
0
100 + 0.5 (0)
100
100
100 + 0.5 (100)
150
200
300
100 + 0.5 (200)
100 + 0.5 (300)
200
250
400
500
100 + 0.5 (400)
100 + 0.5 (500)
300
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 + 0.5Y
I = 100
AE = C + I = 100 + 0.5Y + 100 = 200 + 0.5Y
Aggregate expenditure function
• This equation is a relationship between
income (Y) and aggregate expenditure
(AE).
AE = 200 + 0.5Y
$250bn
Slope is 0.5
$200bn
$100bn
Y
Aggregate expenditure function
• But we
could
also use
the table
form.
Y
0
100
C
100
150
I
100
100
AE
200
250
200
300
200
250
100
100
300
350
400
500
300
350
100
100
400
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
Y
AE = 200 + 0.5Y
400
400
Y
• The equilibrium
value of Y is
where the 45
degree line and
the AE line
cross. Y* is at
$400bn.
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* = 200
0.5Y* = 200
Y* = 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 Ythe $200bn. This portion is called
“autonomous expenditure”.
– The other part does depend on the value of Ythe 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 + 0.5Y
Y* = AE = 150 + 0.5Y*
Y* – 0.5Y* = 150
0.5Y* = 150
Y* = 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
0
75
I
G
AE
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
400
275
I
G
AE
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 + G
AE = 75 + 0.5Y + 50 + 50
AE = 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 constantsnumbers.
• 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:
1
Y
[a  bT  I  G]
1 b
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:
1
ΔY 
ΔG
1- b
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.
Example of 4 sector model
•
•
Let’s say we start with a linear model with a
consumption function (or “schedule”):
C = 100 + 0.75(Y – T)
Let’s use a 4 sector model of AE with:
I = 80
G = 60
T = 40
NX = 30
Example of 4 sector model
• Firstly graph the
consumption function.
• If we plug in 40 for T,
we get (0.75*40 =
30):
C = 100 + 0.75Y – 30
C = 70 + 0.75Y
• We know the vertical
intercept (Y=0):
C = 70 + 0.75 (0) = 70
C
C=70 + 0.75Y
70
0
Slope is 0.75
Y
Example of a 4 sector model
• From the C function, we know our MPC is
0.75- 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 = 70 + 0.75Y + 80 + 60 + 30
C
I
G
NX
Example of a 4 sector model
• Adding all those
together, we get:
AE = 240 + 0.75 Y
• Just like we did with
the C function, we
can graph this AE
function:
C, AE
240
AE=240 + 0.75Y
C=70 + 0.75Y
70
0
Y
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*.
Example of a 4 sector model
• Using our equation for
AE, we get:
AE = 240 + 0.75Y*
Y* = AE = 240 +
0.75Y*
0.25Y* = 240
Y* = 960
AE
45 deg line
AE = 240
+ 0.75Y
960
240
0
960
Y
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*.