Simple Keynesian Model
National Income Determination
Two-Sector National Income Model
1
Outline
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Macroeconomics [2.1]
Exogenous & Endogenous Variables
[2.3]
Linear Functions [2.6]
Aggregate Demand & Supply [3.2]
National Income Determination Model
OR Simple Keynesian Model [3.3]
2
Outline
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National Income Identities [3.4]
Equilibrium Income [3.5 & 3.11]
Consumption Function [3.6]
Investment Function[3.7]
Aggregate Demand Function [3.8]
3
Outline
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Output-Expenditure Approach to
Income Determination[3.9 ]
Expenditure Multiplier [3.9]
Saving Function [3.10]
Injection-Withdrawal Approach to
Income Determination [3.10]
Paradox of Thrift [3.13]
4
Macroeconomics

National income, general price level,
inflation rate, unemployment rate,
interest rate and the exchange rate are
the economic measures to be explained
in the macroeconomic models / theories
5
Exogenous & Endogenous Variables
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Exogenous Variable
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the value is determined by forces outside
the model
any change is regarded as autonomous
I, G, X ( Micro: Income/Population)
Endogenous Variable
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the value is determined inside the model
factor to be explained in the model
Y, C, M ( Micro: Price/Quantity)
6
Linear Functions
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A function specifies the relationship
between variables
y is the dependent variable
x is the independent variable
y=f(x)
7
Linear Functions
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y=f(x)
y= c
y=mx
y=c+mx
m, c are exogenous variables
y, x are endogenous variables
8
Linear Functions
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Consumption Functions
C= f(Y)
C= C’
C= cY
C= C’ + cY
9
Linear Functions
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C’, c are exogenous variables
C, Y are endogenous variables
Y is independent variables
C is dependent variables
10
Linear Functions
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Can you express the 3 consumption
functions graphically?
11
Linear Functions
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The parameter C’ is autonomous
consumption
It summarizes the effects of all factors on
consumption other than national income.
What is the difference between a change in
exogenous variable (autonomous change)
and a change in endogenous variable
(induced change)?
12
Linear Functions
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C= f(Y, W)
If wealth is deemed as a relevant factor
but is not explicitly included in the
consumption function C=C’+ cY
a rise in wealth W  will lead to a rise in
the exogenous variable C’
graphically, the consumption function C
will shift upwards
13
Linear Functions
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What happens if c  ?
What happens if Y  ?
14
Linear Functions
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Consumption function can also be a
relationship between consumption C
and interest rate r.
What do you think of the relationship
between the variables, i.e.,
consumption C and interest rate r?
Are they positively correlated or
negatively correlated?
15
Aggregate Demand & Supply
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Aggregate Demand
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the relationship between the total amount
of planned expenditure and general price
level (v.s. aggregate expenditure E)
Aggregate Supply

the relationship between the total amount
of planned output and the general price
level
16
Aggregate Demand & Supply
Price Level
Aggregate Supply
Equilibrium: no tendency to change and
the values of the endogenous variables
will remain unchanged in the absence of
external disturbances
Aggregate Demand
National Output
17
Aggregate Demand & Supply
P
AS When AS is vertical
A shift of AD will cause a change
In P only but have no effect on Y
AD2
AD1
Yf
Y
18
Aggregate Demand & Supply
P
AD1
AD2
When AS is horizontal
A shift of AD will cause a change
in Y only but have no effect on P
AS
Y
19
Aggregate Demand & Supply
AS
AD
Ye
Yf
20
Aggregate Demand & Supply
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The Upward Sloping AS
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When the economy is close to but below
full employment level Y < Yf, the attempt
to raise output by increasing aggregate
demand will face supply side limitations
both price and output will increase
21
Aggregate Demand & Supply
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The Vertical AS (slide 18)
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When full employment is attained Y = Yf,
an increase in aggregate demand can only
cause prices to rise
22
Aggregate Demand & Supply
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The Horizontal AS (slide 19)
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When output is far below Yf, the
equilibrium output is determined by AD
The supply side has no effect on income
level as firms could supply any amount of
output at the prevailing price level
The Keynesian Model analyses the situation
of an economy with fixed prices and high
unemployment Y < Yf
23
National Income
Determination Model
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Assumptions:
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National income Y is defined as the total
real output Q
A constant level of full national income Yf
Serious unemployment, i.e., there are
many idle or unemployed factors of
production
24
National Income
Determination Model (cont’d)
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Income / output can be raised by using
currently idle factors without biding up
prices
Price rigidity or constant price level
There are only households and firms (2sector). No government and foreign trade
25
National Income Identities
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An identity is true for all values of the
variables
In a 2-sector economy, expenditure
consists of spending either on consumption
goods C OR investment goods I.
Aggregate expenditure (AE OR E) is ,by
definition, equal to C plus I
EC+I
26
National Income Identities
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National income Y received by
households, by definition, is either
saved S OR consumed C.
YC+S
27
National Income Identities
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Aggregate expenditure E is, by
definition, equal to national income Y
YE
C+SC+I
SI
28
Equilibrium Income
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Equilibrium is a state in which there is no
internal tendency to change.
It happens when
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firms and households are just willing to
purchase everything produced Y = E (v.s.
Micro: Qs = Qd) [slide 30-36]
Income-Expenditure Approach [slide 37-60]
planned saving is equal to planned
investment S = I
Injection-Withdrawal Approach [slide 61-74]
29
Equilibrium Income
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
What is the definition of GNP (/ GDP) in
national income accounting?
The total market value of all final goods
and services currently produced by the
citizens (/within the domestic boundary)
of a country in a specified period
30
Equilibrium Income

Ex-ante Y > E  Excess supply
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planned output > planned expenditure 
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unexpected accumulation of stocks OR
unintended inventory investment OR
involuntary increase in inventories
In national income accounting, this amount Y-E is
treated as (unplanned) investment by firms
31
Equilibrium Income
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Ex-post Y= E
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Actual (Realised)=Planned
+
Expenditure
Expenditure
Unplanned
Investment
Actual (Realised) Output = Actual Expenditure
Firms will reduce output
32
Equilibrium Income

Ex-ante Y < E  Excess Demand
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planned output < planned expenditure 
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unexpected fall in stocks OR
unintended inventory dis-investment OR
involuntary decrease in inventories
However, in national income accounting,
this amount E - Y consumed is not
currently produced
33
Equilibrium Income
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Ex-post Y= E
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Actual (Realised)=Planned
Expenditure
Expenditure
Unplanned
Dis-investment
Actual (Realised) Output = Actual Expenditure
Firms will increase output
34
Equilibrium Income
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Ex-ante Y= E  Equilibrium
There is no unintended inventory
investment OR dis-investment
Ex-post Y=E
35
Equilibrium Income
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When there is excess supply, i.e., planned
output > planned expenditure, firms will
reduce output to restore equilibrium
When there is excess demand, i.e., planned
expenditure > planned output, firms will
increase output to restore equilibrium
In the Keynesian model, it is aggregate
demand that determines equilibrium output.
Remember the horizontal AS [slide 19]
36
Consumption Function
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Now, we will look at the 1st component
of the aggregate expenditure E  C + I
i.e. C
Empirical evidence shows that
consumption C is positively related to
disposable income Yd
Yd = Y since it is a 2-sector model
Remember the 3 consumption functions
[slide 9 & 11]
37
Consumption Function
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Autonomous Consumption C’
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It exists even if there is no income. This
can be done by dis-saving, i.e., using the
past saving
Then, saving will be negative when income
is zero.
It is totally determined by forces outside
the model
What happens to the 3 consumption
functions if C’  ? Or C’  ?
38
Consumption Function
C’ = y-intercept
 In C’
C = C’
C = cY
C = C’ + cY
39
Consumption Function
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Marginal Propensity to Consume MPC = c
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It is defined as the change in consumption per unit
change in income
MPC = C / Y
It is the slope of the tangent of the consumption
function
For a linear function, MPC is a constant
What does the consumption function C look like if
MPC is increasing? Decreasing?
It is assumed that 0 < MPC < 1
What happens to the 3 consumption functions if c ?
or c  ?
40
Consumption Function
MPC = slope of tangent
 in MPC or  in c
C = C’
C = cY
C = C’ + cY
41
Consumption Function
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Average Propensity to Consume APC
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It is defined as the ratio of total
consumption C to total income Y
APC = C / Y
It is the slope of ray of the consumption
function
When C = C’ OR C = C’ + cY, APC
decreases when Y increases.
When C = cY, APC = MPC = c = constant
42
Consumption Function
APC = slope of ray
C = C’
C = cY
C = C’ + cY
43
Consumption Function
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Relationship between APC and MPC
C = C’
Divide by Y
C/Y = C’/Y
APC = C’/Y
APC  when Y 
Slope of ray flatter when Y 
Slope of tangent = MPC = c = 0
44
Consumption Function
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Relationship between APC and MPC
C = cY
Divide by Y
C/Y = c
APC = MPC = c
Slope of ray=Slope of tangent=constant=c
45
Consumption Function
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Relationship between APC and MPC
C = C’ + cY
Divide by Y
C/Y = C’/Y + c
APC = C’/Y + MPC
C’ +ve
APC > MPC
Slope of ray steeper than slope of tangent
Slope of tangent constant
Slope of ray flatter when Y 
APC  when Y 
46
Investment Function
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Let’s look at the 2nd component of the
aggregate expenditure E  C + I
An investment function shows the
relationship between planned
investment I and national income Y
It can be a linear function or a nonlinear function
47
Investment Function
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Again, there can be 3 investment functions
I = I’
I = iY
I = I’ + iY
Economists usually use the first one, i.e., I= I’
as investment is thought to be correlated with
interest rate r, instead of Y
I’ , i are exogenous variables
I , Y are endogenous variables
48
Investment Function
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Autonomous Investment I’
It is independent of the income level
and is determined by forces outside the
model, like interest rate.
I’ is the y-intercept of the investment
function
49
Investment Function
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Marginal Propensity to Invest i
It is defined as the change in investment I
per unit change in income Y
MPI = I / Y
MPI would not correlate with Yd
It is the slope of tangent of I
It is also determined by forces outside the
model
50
Investment Function
MPI = i =slope of tangent
I’ = y-intercept
API  when Y
MPI =0
I = I’
I = iY
I = I’ + iY
51
Aggregate Expenditure Function
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Given E = C + I
C = C’ + cY
I = I’
 E = I’ + C’ + cY
 E = E’ + cY
52
Aggregate Expenditure Function
I
C, I, E
C
Slope of tangent = c
Slope of tangent=0
Y
I = I’
C = C’+cY
E = I’ + C’+ cY
53
Aggregate Expenditure Function
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Autonomous Change
When C’ or I’   E’   shift upward
When c   slope of E steeper  rotate
Induced Change
When Y   E   move along the curve
54
Output-Expenditure Approach
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National income is in equilibrium when
planned output = planned expenditure
We have planned expenditure E=C+I
Equilibrium income is Ye=planned E
A 45°-line is the locus of all possible
points where Y = E
When E = planned E, Y = Ye
55
Output-Expenditure Approach
Y=E
C, I, E
Planned E=C +I
Planned E < Y
Y=planned E
Planned E>Y
Unintended
inventory disinvestment
Actual E =Y
Unintended
inventory
investment
Actual E = Y
Y
Y
Ye
Y
56
Output-Expenditure Approach
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Y = planned E
Y = I’ + C’ + cY
Y = E’ + cY
(1-c)Y = E’
Equilibrium condition
1
Y=
E’
1-c
57
Output-Expenditure Approach
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If C’ or I’   E’  E   Ye 
If c   E steeper  Ye 
If we differentiate the equilibrium
condition,
Y/E’ = 1/(1-c)
Given 0 < c < 1 1/(1-c) > 1
E’   Ye by a multiple 1/(1-c) of E’
58
Expenditure Multiplier 1/(1-c)
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Assume c=0.8, E’ = 100
The one who receive the $100 as
income will spend 0.8($100)
then the one who receives 0.8($100) as
income will spend 0.8*0.8($100)
The process continues and the total
increase in income is
$100+0.8($100) +0.8*0.8($100) +…
59
Expenditure Multiplier 1/(1-c)
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The total increase in income is actually
the sum of an infinite geometric
progression which can be calculated by
the first term divided by (1- common
ratio)
The first term here is E’ = $100 and
the common ratio is c =0.8
The sum of GP is E’ * multiplier
60
Saving Function
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We have Y  C + S [slide 27]
Saving function can simply be derived from
the consumption function
S=Y–C
if C = C’ + cY
S = Y – C’ – cY
S = -C’ + (1-c) Y
S = S’ + sY
S’= -C’
s=1-c
S’ < 0 if C’ >0
S’ = 0 if C’ = 0
61
Saving Function
S
S
S = sY
S = S’+ sY
S = (1-c)Y
S =-C’+(1-c)Y
Slope of tangent = s =1- c
Y
Y > Y*  S+ve
Y*
Y
S’
Slope of ray = slope of tangent Slope of ray < slope of tangent
62
Saving Function
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Autonomous Saving S’
Since S= -C’ + (1-c)Y
If C’= 0 when C= cY
 S = (1-c)Y  S’ = 0

If C’ +ve when C = C’ + cY
 S = -C’ + (1-c)Y  S’ –ve

If Y= 0  S’ = -C’  Dis-saving

63
Saving Function
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Marginal Propensity to Save MPS = s
It is defined as the change in saving per unit
change in disposable income Yd OR income Y
(in a 2-sector model)
MPS = S/ Y
It is the slope of tangent of the saving
function
MPS is a constant if the consumption / saving
function is linear
64
Saving Function
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Average Propensity to Save APS
It is defined as the total saving divided
by total income
APS = S/Y
It is the slope of ray of the saving
function
65
Saving Function
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Average Propensity to Save APS (cont’d)
When S= sY
 APS = MPS = s = constant
When S=S’+ sY
 APS < MPS as S’ –ve
 APS –ve when Y < Y* [slide 62]
 APS = 0 when Y = Y*
 APS +ve when Y > Y*
 APS  when Y 
66
Saving Function
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Y=C+S
Differentiate wrt. Y
Y/Y=C/Y + S/Y
 1= MPC + MPS
1=c+s
[slide 61]
S = S’ + sY
S/Y = S’/Y + s
APS=S’/Y+MPS
Divided by Y
[slide 66]
67
How to determine Ye?
Y=E
C, S, I, E
Planned Y = planned E
+ve S
Planned I
-ve S
C
Planned C
Y<C Y*=C
Ye
Y>C
Y
68
Y=E
Y = C  S =0
 No Dis-saving
Y<E
 Unintended Inventory Dis-investment
 Actual I =Planned I – Unintended I
E=C+I
Y<C
Y<E
Planned I
C
Planned C
Planned Y < Planned E
Ye
69
Y=E
Y > C  S +ve
 Saving
Y>E
 Unintended Inventory Investment
 Actual I =Planned I + Unintended I
Planned I
E=C+I
C
Planned C
How about Y*<Y<Ye?
Ye
Planned Y > Planned E
70
Injection-Withdrawal Approach
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
Remember the national income identity
S  I [slide 28]
The equilibrium income happens when
planned Y= planned E as well as
planned S = planned I [slide 29]
71
Injection-Withdrawal Approach
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S’+ sY = I’
sY = I’ – S’
S’=-C’
s=1-c
(1-c)Y = I’ + C’ = E’
Equilibrium condition [slide 57]
1
Y=
E’
1-c
72
Equilibrium Income


No matter which approach you use, you
will get the same equilibrium condition.
Can you derive the equilibrium
condition if investment I is an induced
function of national income Y, using the
2 approaches?
73
Equilibrium Income


Write down the investment function I first.
Then write down the saving function S.
Remember planned S = planned I when Y is
in equilibrium {Injection-Withdrawal}
Write down the investment function I as well
as the consumption function C. Together they
are the aggregate expenditure function E.
Remember planned Y = planned E when Y
is in equilibrium {Output-Expenditure}
74
Injection-Withdrawal Approach
75
Output-Expenditure Approach
76
Y<C
Y=C
Y>C
Y=E
E=C+I
C=C’+cY
E’=C’+I’
C’
I’
S’=- C’
S = S’ + sY
I=I’
77
Planned Y=Planned E
Unintended
Inventory
Investment
E=C+I
S=S’+sY
E’=C’+I’
I’
Unintended
Inventory
Disinvestment
Unintended
Inventory
Disinvestment
Planned S=Planned I
Unintended
Inventory
Investment
I=I’
78
Paradox of Thrift
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
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


This is an example of the “fallacy of
composition”
“Thriftiness, while a virtue for the
individual, is disastrous for an economy”
Given I = I’
Given S = S’ + sY OR S = -C’ + (1-c)Y
Now, suppose S’ 
Will Ye increase as well?
79
A rise in thriftiness causes a decrease
in national income but no increase in
realised saving.
S=S” +sY
S= S’+ sY
Excess Supply
I=I’
Ye
80
Paradox of Thrift




If a rise in saving leads to a reduction in interest
rate and hence an increase in investment (Think of
the loanable fund market), national income may not
decrease
Ye will increase if I’ increase more than S’
Ye will remain the same if I’ increase as much as S’
Ye will decrease if I’ increase less than S’
81
I  > S
S=S” +sY
S= S’+ sY
I=I”
I=I’
Ye
82
I  = S
S=S” +sY
S= S’+ sY
I=I”
I=I’
Ye =Ye
83
I  < S
S=S” +sY
The reduction in Ye is
less than the case when
I does not increase
S= S’+ sY
I=I”
I=I’
Ye
What about the case if I is an induced function of Y?
84