Simple Keynesian Model
National Income Determination
Three-Sector National Income Model
1
Outline
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Three-Sector Model
Tax Function T = f (Y)
Consumption Function C = f (Yd)
Government Expenditure Function G=f(Y)
Aggregate Expenditure Function E = f(Y)
Output-Expenditure Approach: Equilibrium
National Income Ye
2
Outline
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Factors affecting Ye
Expenditure Multipliers k E
Tax Multipliers k T
Balanced-Budget Multipliers k B
Injection-Withdrawal Approach:
Equilibrium National Income Ye
3
Outline
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Fiscal Policy (v.s. Monetary Policy)
Recessionary Gap Yf - Ye
Inflationary Gap Ye - Yf
Financing the Government Budget
Automatic Built-in Stabilizers
4
Three-Sector Model
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With the introduction of the
government sector (i.e. together with
households C, firms I), aggregate
expenditure E consists of one more
component, government expenditure G.
E=C+I+G
Still, the equilibrium condition is
Planned Y = Planned E
5
Three-Sector Model
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Consumption function is positively related
to disposable income Yd [slide 37 of 2sector model],
C = f(Yd)
C= C’
C= cYd
C= C’ + cYd
6
Three-Sector Model
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National Income  Personal Income 
Disposable Personal Income
w/ direct income tax Ta and transfer
payment Tr
Yd  Y
Yd = Y - Ta + Tr
7
Three-Sector Model
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Transfer payment Tr can be treated as
negative tax, T is defined as direct income
tax Ta net of transfer payment Tr
T = Ta - Tr
Yd = Y - (Ta - Tr)
Yd = Y - T
8
Three-Sector Model
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The assumptions for the 2-sector
Keynesian model are still valid for this
3-sector model [slide 24-25 of 2-sector
model]
9
Tax Function
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T = f(Y)
T = T’
T = tY
T = T’ + tY
10
Tax Function
T = T’
T = tY
T = T’ +tY
Y-intercept=T’
Y-intercept=0
Y-intercept=T’
slope of tangent=0 slope of tangent=t slope of tangent=t
11
Tax Function
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Autonomous Tax T’
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Proportional Income Tax tY
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marginal tax rate t is a constant
Progressive Income Tax tY
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this is a lump-sum tax which is
independent of income level Y
marginal tax rate t increases
Regressive Income Tax tY
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marginal tax rate t decreases
12
Consumption Function
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C = f(Yd)
C = C’
C = C’
C = cYd
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C = c(Y - T)
C = C’ + cYd
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C = C’ + c(Y - T)
13
Consumption Function
C = C’ + c(Y - T)
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T = T’
C = C’ + c(Y - T’)  C = C’- cT’ + cY
 slope of tangent = c
 T = tY
C = C’ + c(Y - tY)  C = C’ + (c - ct)Y
slope of tangent = c - ct
 T = T’ + tY
C = C’+c[Y-(T’+tY)]C = C’ - cT’ + (c - ct) Y
slope of tangent = c -14ct
Consumption Function
C = C’ + c (Y - T’)
Y-intercept = C’ - cT’
slope of tangent = c = MPC
slope of ray APC  when Y
15
Consumption Function
C = C’ + c (Y - tY)
Y-intercept = C’
slope of tangent = c - ct = MPC (1-t)
slope of ray APC  when Y
16
Consumption Function
C = C’ + c [Y - (T’ + tY)]
Y-intercept = C’ -cT’
slope of tangent = c - ct = MPC (1-t)
slope of ray APC  when Y
17
Consumption Function
C = C’ - cT’ + (c - ct)Y
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C’ OR T’
 y-intercept C’ - cT’   C shift upward
t
 c(1-t)   C flatter
c
 c(1-t)  C steeper
 y-intercept C’ - cT’ C shift downward
18
Government Expenditure Function
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G only includes the part of government
expenditure spending on goods and
services, i.e. transfer payments Tr are
excluded.
Usually, G is assumed to be an
exogenous / autonomous function
G = G’
19
Government Expenditure Function
Y-intercept = G’
slope of tangent = 0
slope of ray  when Y
20
Aggregate Expenditure Function
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E =C+I+G
given
C
= C’ + cYd
T
= T’ + tY
I
= I’
G
= G’
E = C’ + c[Y - (T’+tY)] + I’ + G’
E = C’ - cT’ + I’+ G’ + (c-ct)Y
E = E’ + c(1-t) Y
21
Aggregate Expenditure Function
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E = C’ - cT’ + I’ + G’ + (c - ct)Y
E = E’ + (c - ct)Y
given E’ = C’ - cT’ + I’ + G’
E’ is the y-intercept of the aggregate
expenditure function E
c - ct is the slope of the aggregate
expenditure function E
22
Aggregate Expenditure Function
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Derive the aggregate expenditure
function E if T = T’
E = C’ - cT’ + I’ + G’ + cY
y-intercept = C’ - cT’ + I’ + G’
slope of tangent = c
23
Aggregate Expenditure Function
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Derive the aggregate expenditure
function E if T = tY
E = C’ + I’ + G’ + (c-ct)Y
y-intercept = C’ + I’ + G’
slope of tangent = (c-ct)
24
Aggregate Expenditure Function
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Derive the aggregate expenditure
function E if T = T’ and I = I’ + iY
E = C’ - cT’ + I’ + G’ + (c + i)Y
y-intercept = C’ - cT’ + I’ + G’
slope of tangent = (c + i)
25
Aggregate Expenditure Function
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Derive the aggregate expenditure
function E if T = tY and I = I’ +iY
E = C’ + I’ + G’ + (c - ct +i )Y
y-intercept = C’ + I’ + G’
slope of tangent = (c - ct +i )
26
Aggregate Expenditure Function
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Derive the aggregate expenditure
function E if T = T’ + tY and I = I’ +iY
E = C’ - cT’ + I’ + G’ + (c - ct +i)Y
y-intercept = C’ - cT’ + I’ + G’
slope of tangent = (c - ct +i)
27
Output-Expenditure Approach
w/ T = T’ + tY
w/ C = C’ + cYd
C
2-Sector
C = C’ + cYd = C’ + cY
Slope of tangent = c = MPC =C/Yd
Slope of tangent = c (1-t) = (1-t)*MPC  MPC
3-Sector
C’
C = C’ - cT’ + c(1-t)Y
C’ -cT’
Y
28
I, G, C, E, Y
Y=E
Y
Planned Y = Planned E
29
Output-Expenditure Approach
I = I’ exogenous function
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E = E’ + (c - ct) Y
[slide 21-22]
In equilibrium, planned Y = planned E
Y = E’ + (c - ct) Y
(1- c + ct) Y = E’
1
Y=
E’
1 - c + ct
E’ = C’ - cT’ + I’ + G’
1
kE=
1 - c + ct
30
Output-Expenditure Approach
I= I’+iY endogenous function
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E = E’ + (c - ct + i) Y
[slide 27]
In equilibrium, planned Y = planned E
Y = E’ + (c - ct + i) Y
(1- c + ct - i) Y = E’
1
Y=
E’
1 - c - i + ct
E’ = C’ - cT’ + I’ + G’
1
k E = 1 - c - i + ct
31
Output-Expenditure Approach
T = T’ exogenous function
I = I’ + iY
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E = E’ + (c + i) Y
[slide 25]
In equilibrium, planned Y = planned E
Y = E’ + (c + i) Y
(1 - c - i) Y = E’
1
Y=
E’
1-c-i
E’ = C’ - cT’ + I’ + G’
1
k E = 1-c-i
32
Factors affecting Ye
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Ye = k E * E’
In the Keynesian model, aggregate
expenditure E is the determinant of Ye
since AS is horizontal and price is rigid.
In equilibrium, planned Y = planned E
E = C’ - cT’ + I’ + G’ + (c - ct + i) Y
Any change to the exogenous variables
will cause the aggregate expenditure
function to change and hence Ye
33
Factors affecting Ye
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Change in E’
If C’ I’ G’  E’  E  Y 
If T’ C’ - c T’ E’ by - c T’E Y
Change in k E / slope of tangent of E
If c  i   E steeper  Y
If c   C’ - c T’  E’  E   Y 
If t   E steeper  Y 
34
I, G, C, E, Y
Y=E
Y
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I, E, Y I’
E’ = I’
 I’
Y
Ye = k E E’
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G, E, Y G’
Y
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C, E, Y C’
Y
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C, E, Y T’
C  by -cT’
Y
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I, E, Y  i
Y
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Digression
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Differentiation
y = c + mx
differentiate y with respect to x
dy/dx = m
41
Expenditure Multiplier k
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Y = k E * E’
1
kE=
kE=
kE=
1 - c + ct
1
1 - c + ct - i
1
1-c-i
E
E’ = C’ - cT’ + I’ + G’
if I=I’ & T=T’+tY
if I=I’+iY & T=T’+tY
if I=I’+iY & T=T’
42
Expenditure Multiplier k
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E
Whenever there is a change in the
autonomous spending C’ I’ or G’ the
national income Ye will change by a
multiple of k E.
It actually measures the ratio of the
change in national income Ye to the
change in the autonomous expenditure E’
Ye/E’ = k E
43
Tax Multiplier k T
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Y = k E * ( C’ - cT’ + I’ + G’)
-c
kT =
if I=I’ & T=T’+tY
kT=
kT=
1 - c + ct
-c
1 - c + ct + i
-c
1-c-i
if I=I’+iY & T=T’+tY
if I=I’+iY & T=T’
44
Tax Multiplier k T
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Any change in the lump-sum tax T’ will
lead to a change in the national income
Ye by a multiple of k T in the opposite
direction since k T takes on a negative
value
Besides, the absolute value of k T is less
than the value of k E.
45
Balanced-Budget Multiplier k
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B
G’  E’   E   Ye  by k E times
T’  E’   E   Ye  by k T times
If G’  = T’  , the change in Ye can be
measured by k B
Y/ G’ = k E
Y/ T’ = k T
kB=kE+kT
k B = 1 + -c
=1
1-c
1-c
46
Balanced-Budget Multiplier k
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B
The balanced-budget multiplier k B = 1
when t=0 & i=0
What is the value of k B if t  0 ?
If k B = 1 an increase in government
expenditure of $1 which is financed by
a $1 increase in the lump-sum income
tax, the national income Ye will also
increase by $1
47
Injection-Withdrawal Approach
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In a 3-sector model, national income is
either consumed, saved or taxed by the
government
Y=C+S+T
Given E = C + I + G
In equilibrium, Y = E
C+S+T=C+I+G
S+T=I+G
48
Injection-Withdrawal Approach
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Since S + T = I + G
SI
TG
I>ST>G
I<ST<G
(Compare with 2-sector model)
In equilibrium S = I
49
Injection-Withdrawal Approach
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T = T’ + tY
S = -C’ + (1-c) Yd
S = -C’ + (1 - c)[Y -_(T’ + tY)]
S = -C’ + (1 - c)[Y - T’ - tY]
S = -C’ + Y - T’ - tY - cY + cT’ + ctY
S = -C’ + cT’ -T’ - tY + Y - cY + ctY
S = -C’ + cT’ - (T’ + tY) + Y - cY + ctY
50
Injection-Withdrawal Approach
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S + T = -C’ + cT’ -(T’+ tY) + Y - cY + ctY +T
S + T = -C’ + cT’ + Y - cY + ctY
In equilibrium, S + T = I + G
-C’ + cT’ + Y - cY + ctY = I’ + G’
(1- c + ct)Y = C’ - cT’ + I’ + G’
Ye = k E * E’
E’ = C’ - cT’ + I’ + G’
[slide 30]
51
Use the Injection-Withdrawal
Approach to solve for Ye if T=T’
52
Fiscal Policy
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The use of government expenditure and
taxation to achieve certain goals, such as
high employment, price stability.
Discretionary Fiscal Policy
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Expansionary Fiscal Policy (when Yf > Ye)
Contractionary Fiscal Policy (when Yf < Ye)
Automatic Built-in Stabilizers
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Proportional / Progressive Tax System
Welfare Schemes
53
Expansionary Fiscal Policy
Recessionary/Deflationary Gap Yf-Ye
Y-line
G’  E’  E  Y
E = E” + (c-ct) Y
E = E’ + (c -ct) Y
G’
Y= k E * E’
Recessionary Gap Ye
Yf
54
Expansionary Fiscal Policy
Recessionary/Deflationary Gap Yf-Ye
T’  E’ by -c T’  E  Y
Y-line
E = E” + (c-ct) Y
E = E’ + (c -ct) Y
-cT’
Y= k E * E’ = k T * T’
Recessionary Gap Ye
Yf
55
Contractionary Fiscal Policy
Inflationary Gap Ye - Yf
Y=E
G’  E’  E  Y
E = E’ + (c-ct) Y
E = E” + (c-ct) Y
G’
Y= k E * E’
Yf
Ye Nominal Y>Yf Inflationary Gap
56
Contractionary Fiscal Policy
Inflationary Gap Ye - Yf
Y=E
T’  E’ by -c T’  E  Y
E = E’ + (c-ct) Y
E = E” + (c-ct) Y
-cT’
Y= k E * E’ = k T * T’
Yf
Ye Nominal Y>Yf Inflationary Gap
57
Automatic Built-in Stabilizers
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Proportional /Progressive Tax System
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Recession: government’s tax revenue 
Boom: government’s tax revenue 
The more progressive the tax system, the
greater is its stabilizing effect. But there will
be greater dis-incentives to earn income
With t, k E  With proportional tax, the
multiplying effect of a discretionary change
in government expenditure G’ reduces
58
Automatic Built-in Stabilizers
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Welfare Schemes
Unemployment benefits, public assistance
allowances, agricultural support schemes
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Recession: government’s expenditure
Boom: government’s expenditure 
Again, if the welfare schemes are
generous, the incentives to work will be
weakened.
59
Discretionary Fiscal Policy v.s.
Automatic Built-in Stabilizers
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If the economy is close to Yf, built-in
stabilizers are useful as they can stabilize the
economy around Yf or potential income level.
However, if the economy is far below Yf,
discretionary fiscal policy is still necessary
(Simple Keynesian model).
Another drawback of the built-in stabilizers is
they may reduce the speed of recovery as
k E   Y = k E * E’
60
Discretionary Fiscal Policy
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Government expenditure G’? Tax T’?
Location of effects
If a recession is localized in a particular
industry  G’
Tax cut will have its impact on the entire
economy
61
Discretionary Fiscal Policy
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Government expenditure G’? Tax T’?
Duration of the time lag
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Decision lag : time involved to assess a situation
& decide what corrective actions should be
taken
Executive lag : time involved to initiate
corrective policies & for their full impact to be
felt
 tax cut has a much shorter executive lag
62
Discretionary Fiscal Policy
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Government expenditure G’? Tax T’?
Reversibility of the fiscal policy
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Government expenditure can easily be increased
but are not so easy to cut as the civil servants
who have vested interests in the present
allocation of government expenditure will resist
Tax is easier to be changed as the civil servants
who administer income tax is independent of the
rate being levied. Of course, voter resistance
should also be considered.
63
Discretionary Fiscal Policy
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Government expenditure G’? Tax T’?
Public reaction to short-term changes
A temporary tax cut raises Yd. Households,
recognizing this situation, may not revise
their current consumption. Instead, they
save a large part of the tax cut.
64
Financing the Government Budget
Increasing Taxes
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By increasing taxes, the government
transfers purchasing power from current
taxpayers to itself
Current taxpayers bear the cost
If the revenue is spent on some investment
project, (current / future) taxpayers may
benefit when the project is completed.
How about the revenue is spent on transfer
payment?
65
Financing the Government Budget
Printing more Money
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This will create inflationary pressure.
Households and firms will be able to buy
less with each unit of money. Fewer
resources are available for private
consumption and investment.
Those whose incomes respond slowly to
changes in price levels will bear most of
the cost of the government activity
66
Financing the Government Budget
Internal Debt
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The government can transfer purchasing
power from any willing lenders to itself in
return for the promise to repay equivalent
purchasing power plus interest in future.
Since, repayment of the debt are made from
tax revenue, future taxpayers will suffer.
However, if the debt raised today is spent on
creating capital assets, the burden on future
generation will be lighter.
67
Financing the Government Budget
External Debt
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Borrowing from abroad transfers
purchasing power from foreigners to
the government.
The burden on future generations will
once again depend on how the debt
raised is used (investment project /
transfer payment)
68
The Problems of the Simple
Keynesian Multiplier k E
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Y = k E * G’
There are several problems with this
method of analysis, i.e., Y may be less
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Sources of financing G’
Effects on private investment I’
Productivity of government projects
69
The Problems of the Simple
Keynesian Multiplier k E
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Sources of financing G’
Increasing Tax
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Increasing Money Supply
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will exert a contractionary effect on the economy
will generate an inflationary pressure
Increasing Debt
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will increase the demand for loanable fund as well
as interest rate  affect private investment
70
The Problems of the Simple
Keynesian Multiplier k E
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Effects on Private Investment I’
Private investment may be crowded out
when government increases its expenditure
It is questionable that the government can
really produce something which is desired
by the consumers
Besides, government investment projects
are usually less productive than private
investment projects
71
The Problems of the Simple
Keynesian Multiplier k E
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Productivity of Government Projects
Government projects may not yield a
rate of return (MEC / MEI) exceeding
the market interest rate.
72