Chapter 4
Income Determination I --Income-expenditure Model
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Contents:
•
•
•
•
•
•
•
•
•
Assumptions of income-expenditure Model
Two-sector economy
Three-sector economy
Four-sector economy
Different kinds of multipliers in different economies
Other points to be noticed
Paradox of thrift
Implications of private saving, public saving & national saving
Advanced Material 4.1 : Equality between investment and
saving in a two-sector economy
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Assumptions of Incomeexpenditure Model
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Assumptions of income-expenditure model
(or elementary Keynesian model)
The amount of resources and the state of technology
remain unchanged, i.e., Yf is a constant.
There exists an unemployment of resources. The model
is to find out the determinants of equilibrium GNP and
the ways to eliminate unemployment.
No indirect taxes, subsidies, depreciation or net factor
income from abroad, i.e., Y = GDP = GNP.
The interest rate and the price level are fixed. So
nominal variables = real variables and nom. r = real r.
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Two-sector Economy
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Two-sector economy
Households
Firms
 Composed of households and firms only
 No government sector
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Features of households
Households provide factor services for income.
Factor Services
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Income
Features of households
Disposable income is either consumed or saved.
Disposable Income
Consumption
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Saving
Consumption function
Marginal Propensity to
Consume (MPC)
C = cYd + C*
Disposable Income
Autonomous Consumption
 Marginal propensity to consume (MPC or c) is the change in
consumption resulting from a unit change in disposable income. c < 1.
Autonomous consumption (C*) is the consumption at zero
disposable income (the minimum amount for subsistence). C* > 0.
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Consumption function
C
Graphical Illustration
C = cYd + C*
c
C*
+1
0
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Yd
Saving function
Marginal Propensity to
Save (MPS)
S = sYd + S*
Disposable Income
Autonomous Saving
Marginal propensity to save (MPS or s) is the change in saving
resulting from a unit change in disposable income. s = 1 – c. Why?
Autonomous saving is the saving at zero disposable income.
S* = -C*. Why?
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Saving function
S
Graphical Illustration
S = sYd + S*
s = (1-c)
+1
0
S* = -C*
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Y
Features of firms
1. Firms employ factor services to produce goods.
Factor Services
Products
Firms
2. Firms also consume final products
(fixed investment & inventories)
to help production.
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Investment function
Marginal Propensity to
Invest (MPI)
I = i Y + I*
National Income
Autonomous Investment
Marginal propensity to invest (MPI or i) is the change in
investment resulting from a unit change in income.
i > 0. Why?
Autonomous investment is the investment at zero income.
I* > 0. Why?
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Investment function
r
Graphical illustration
I = iY + I*
I*
i
+1
0
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Y
Equilibrium condition (2-sector economy)
Aggregate supply (AS) of final products is GNP or Y.
Without government or taxation
 AS = Y = Yd = C + S
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Equilibrium condition (2-sector economy)
Aggregate demand (AD) for final products
is aggregate expenditure (E).
AD = E = C + I
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Equilibrium income (or equilibrium GNP)
is reached when AS = AD
AS = AD
Y=E
C+S=C+I
S = I
Withdrawal =
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Injection
 Withdrawal (撤出 , W)
 is the amount of income withdrawn from the
circular flow, not being spent on final products.
 In a 2 sector economy, saving is the withdrawal.
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Injection (注入, J)
 is the amount of expenditure on final products
injected into the circular flow
 not financed by income earned from production.
 In a 2 sector economy, investment is the injection.
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Circular flow of a 2-sector economy
Injection
Withdrawal
Y
ABC Ltd.
Firms
Households
E
Investment
Yd
Consumption
Saving
When I = S, an equilibrium is achieved.
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Meaning of a 45° line
E
V (Y < E)
Meaning of a
45o line
U (Y = E)
0
45o
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Y=E
Z (Y > E)
Y
Aggregate expenditure function
Add consumption and investment functions vertically
Y=E
E
E= C+I
C
C*
I*
0
I
45o
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Y
Adjustment mechanism
2 approaches:
Income-expenditure approach
Injection-withdrawal approach
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E
Income-expenditure approach
 production
Y=E
At Y2, Y < E
At Y1, Y > E
}
Unplanned
decrease in
inventories
E=C+I
 Unplanned
increase in
inventories
 production
{
0
Y2
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Ye
Y1
Y
Injection-withdrawal approach
E
I*
0
-C*
 production
At Y2, J > W 
Unintended
inventory
disinvestment
{
Y2
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S At Y1, J < W
}I
 Unintended
inventory
investment
 production
Y
Ye
Y1
Multiplier effect
E:
ΔEi
ΔEa
E
ΔEi’
•••
Y:
Y
Initial equilibrium
: Change in Y brought by ΔEa
Total change in Y = ΔE + c•ΔE + c•c• ΔE + c•c•c• ΔE + …
= (1 + c +
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c2 +
c3 +
…) • ΔE =
1
•ΔE
1 c
Multiplier effect
Multiplier (k)
 is the ratio of the total change in equilibrium income
to the initial change in autonomous expenditure
(or autonomous withdrawal) that brought it about.
 Mathematical expression:
k 
Y
1

E
1 c
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Mathematical derivation of equil. income & multiplier
In a 2-sector economy
 E = C + I, where C = cYd + C* and I = I*
In equilibrium, Y = E = C + I
= cYd + C* + I*



= cY + C* + I*
Y – cY = C* + I*
(1 – c)Y= C* + I*
1
Equilibrium Y =
(C* + I*)
1 c
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.
Mathematical derivation of equil. income & multiplier
When C* or I* changes by ΔEa
 ΔY =
1
. ΔEa
1 c
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
1
Y

Ea 1 c
Q4.1:
Calculate the value of multiplier and explain its
meaning in each of the following cases.
(a) MPC = 1
(b) MPS = 1
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Q4.2:
If the autonomous expenditure decreases by ΔE,
what will be the change in equilibrium income?
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Three-sector Economy
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Three-sector economy
Firms
Households
Government
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Government
 Government’s expenditure
is mainly financed by taxation
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Government expenditure function (G)
 is fixed by the budget at the beginning of a
fiscal year.
 G is a constant (G*) independent of any variables.
G = G*, where G > 0
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Tax function
No indirect taxes is assumed.
Only direct taxes are concerned. There exist
three kinds of direct tax systems.
Tax System
Formula
Example
T = T’
Poll tax
Proportional tax system
T = tY + T*
Profits tax
Progressive tax system
T = t*Y + T^ Salaries tax
Lump-sum tax system
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Equilibrium condition (3-sector economy)
 Aggregate supply (AS) of final products is GNP or Y.
 Yd = Y – T = C + S
 Y = Yd + T = C + S + T
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Equilibrium condition (3-sector economy)
 Aggregate demand (AD) for final products is E.
E=C+I+G
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Equilibrium income (or equilibrium GNP)
is reached when AS = AD
 AS = AD
 Y=E
C+S+T=C+I+G
S+T=I+G
Withdrawal
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=
Injection
Circular flow of a 3-sector economy
Injection
Withdrawal
Y
ABC Ltd.
Firms
Households
E
Consumption
Investment
Government expenditure
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Saving
Tax
When I + G = S + T,
equilibrium is achieved.
Mathematical derivation of equil. income & multiplier
In a 3-sector economy with a lump-sum tax system
 E=C+I+G, where C = cYd + C*; Yd = Y – T’ and
I = I* and G = G*
In equilibrium, Y = E = C + I + G
= cYd + C* + I* + G*
Y = c(Y-T’) + C* + I* + G*
= cY- cT’+ C* + I* + G*

(1–c)Y = C* + I*+ G*- cT’
 Equilibrium Y =
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1
1 c
.(C*+I*+G*- cT’)
Mathematical derivation of equil. income & multiplier
When C* or I* or G* changes by ΔEa

ΔY
=
1 . ΔE
a
1 c
 Multiplier =  Y  1
Ea
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1 c
Mathematical derivation of equil. income & multiplier
Under a lump-sum tax system
Equil. income:
Multiplier:
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

1
1 c
 (C *  I *  G *  cT' )
1
1 c
Mathematical derivation of equil. income & multiplier
In a 3-sector economy with a proportional tax system
 E=C+I+G, where C = cYd + C*; Yd = Y - tY - T*;
I = I* and G = G*
In equilibrium, Y = E = C + I + G
= cYd + C* + I* + G*
Y = c(Y - tY - T*) + C* + I* + G*
= cY - ctY - cT* + C* + I* + G*

(1 - c + ct)Y = C* + I* + G* - cT*
 Equilibrium Y =
1
1  c  ct
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• (C*+I*+G*-cT*)
Mathematical derivation of equil. income & multiplier
When C* or I* or G* changes by ΔEa

ΔY
1
=
1  c  ct
 Multiplier =  Y 
Ea
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• ΔEa
1
1  c  ct
Under a proportional tax system
Equil. income: 
Multiplier:

1
1  c  ct
 (C *  I *  G *  cT*)
1
1  c  ct
As t > 0, (1-c) < (1-c+ct) 
1
1c

1
1  c  ct
 The multiplier of proportional tax system is smaller
than the multiplier of lump-sum tax system.
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Mathematical derivation of equil. income & multiplier
With the addition of a proportional transfer payment,
where Yd = Y - tY - T* + qY + Q*
In equilibrium, Y = E = C + I + G
= cYd + C* + I* + G*

Y = c(Y-tY-T*+qY+Q*) + C* + I* + G*
 (1–c+ct-cq) •Y = C* + I* + G* -cT* + cQ*
1
Equilibrium Y =
1  c  ct - cq
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• (C*+I*+G*-cT*-cQ*)
Mathematical derivation of equil. income & multiplier
When C* or I* or G* changes by ΔEa

ΔY
=
• ΔEa
1
1  c  ct - cq
 Multiplier =  Y
Ea
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
1
1  c  ct - cq
Under a proportional tax & transfer payment system
Equil. income:
Multiplier:
1
1  c  ct - cq
 (C *  I *  G *  cT *  cQ*)
1
1  c  ct - cq
As q<0, (1-c+ct)<(1-c+ct-cq) 
1
1  c  ct
>
1
1  c  ct - cq
 The multiplier with a proportional transfer payment is
smaller than the multiplier without it.
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Four-sector Economy
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Four-sector economy
Firms
Households
Foreign Sector
Government
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Export function:
 Export is determined by foreign economies, not
the domestic economy
 It is autonomous & is independent of Y
X=X*, where X*>0
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Import function:
 All economic agents consume imports
 Import is positively related to Y
M= mY +M*
Marginal Propensity to
Import (MPM)
Autonomous Import
MPM is the change in the value of imports resulting
from a unit change in national income. MPM > 0.
M* is the value of imports at zero income. M* > 0.
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Equilibrium condition (4-sector economy)
 Aggregate supply (AS) of final products is GNP or Y.
 Y = Yd + T
=C+S+T
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Equilibrium condition (4-sector economy)
 Aggregate demand (AD) for final products is
aggregate expenditure (E).
E = (C-MC) + (I-MI) + (G-MG) + (X-MX)
= C + I + G + X - (MC+MI+MG+MX)
=C+I+G+X-M
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Equilibrium income (or equilibrium GNP) is reached
when
AS = AD
Y=E
C+S+T=C+I+G+X–M
S+T=I+G+X–M
S+T+M = I+G+X
Withdrawal
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=
Injection
Injection
Withdrawal
Circular flow of a 4-sector economy
Y
ABC Ltd.
Firms
Households
E
C - MC
Saving
I - MI
Tax
G - MG
When I + G + X – M = S + T,
X - MX
equilibrium is achieved.
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Mathematical derivation of equil. income & multiplier
In a four-sector economy, E = C + I + G + X – M
where
C = cYd + C*;
Yd = Y - tY - T* + qY + Q*;
I = I*;
G = G*;
X = X*;
M = mY + M*
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Mathematical derivation of equil. income & multiplier
In equilibrium, Y = E = C+I+G+X-M
= cYd+C*+I*+G*+X*-mY-M*
Y = c(Y-tY-T*+qY+Q*)+C*+I*+G*+X*-mY-M*
(1-c+ct-cq+m)•Y = C*+I*+G*+X*-M*-cT*+cQ*
Equil.Y =
1
1  c  ct - cq  m
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•(C*+I*+G*+X*-M*-cT*-cQ*)
Mathematical derivation of equil. income & multiplier
When C* or I* or G* or X* changes by ΔEa
ΔY
=
1
1  c  ct - cq  m
• ΔEa
1
Y

Ea
1  c  ct - cq  m
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Q4.5
Derive the equilibrium income and the multiplier in a
four-sector economy if investment is an induced
expenditure (I = iY + I*).
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Different Kinds of Multipliers
in Different Economies
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Expenditure multiplier (k)
Two-sector
economy
Three-sector
economy
Four-sector
economy
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1
1ci
1
1  c  i  ct  cq
1
1  c  i  ct  cq  m
Comparison of the size of expenditure multipliers (k)
As c>0, i>0, t>0, -q>0, and m>0,
1  c  i  1  c  i  ct  cq  1  c  i  ct  cq  m
1
1c  i

1
1  c  i  ct  cq

1
1  c  i  ct  cq  m
k in a 2-sector economy > k in a 3-sector economy
> k in a 4-sector economy
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Explanation:
E:
E
ΔEa
ΔEi
•••
Y:
Y
Initial equilibrium
: Change in Y brought by ΔEa
ΔEi (2-sector economy) =c • ΔEa + i • ΔEa
ΔEi (3-sector economy) =c•(1-t+q) • ΔEa + i • ΔEa
ΔEi (4-sector economy) =c•(1-t+q) • ΔEa+ i •ΔEa– m •ΔEa
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Tax multiplier
Y
T *

c
1  c  i  ct  cq  m
 Tax is a withdrawal  Its multiplier is negative.
 When T*:  by $1  Yd:  by $1  C:  by -$c
 ∆Y = -c x expenditure multiplier.
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Q4.6:
Derive the transfer payment multiplier and explain
why it is smaller than the expenditure multiplier.
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Import multiplier
Y
M *

1
1  c  i  ct  cq  m
 Import is a withdrawal  Its multiplier is negative.
When M*:  by $1  AE:  by $1
 ΔY = -1 x expenditure multiplier.
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Multipliers at full employment
 ΔEa > 0
 Multiplier = 0
 ΔEa < 0
 Multiplier = Unchanged
 At full employment, as all resources have been used
efficiently, real income can no longer be raised
(when ΔEa>0, ΔY=0) but it can be lowered
(when ΔEa<0, ΔY<0).
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Other Points to be Noticed
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More about aggregate expenditure function
E=C+I+G+X-M
= c(Y-tY-T*+qY+Q*)+C* +iY+I* +G* +X* - mY- M*
E = (c + i - ct + cq-m) Y + (C*+I*+G*+X*-M*-cT*+ cQ*)
Slope of E-function
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E-intercept
Q4.7:
Derive the expenditure multiplier from the incomeexpenditure diagram.
Q4.8:
Expenditure multiplier is ΔY/ΔE and slope of E-function
is ΔE/ΔY. Is the expenditure multiplier equal to the
inverse of the slope of E-function?
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An autonomous change versus an induced change
E
An autonomous  in E E’
E
An induced  in E
Y
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Slope of E-function cannot be greater than one
E
E’ (slope > 1)
Y=E
Can’t find the
equilibrium
ΔE’>ΔY
ΔY
E (Slope < 1)
ΔE<ΔY
ΔY
45o
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The equilibrium
Y
Deflationary Gap
Deflationary AD gap is the amount of
expenditure by which the present expenditure
falls short of the expenditure achieving
full employment.
Deflationary income gap is the amount of
income by which the equilibrium income
falls short of the full employment income.
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Graphical illustration
E
Y=E
Ef
E1
{
Deflationary
AD gap
45o
Y1
Yf
Deflationary income gap
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Y
Inflationary Gap
Inflationary AD gap is the amount of
expenditure by which the present expenditure
exceeds the expenditure achieving
full employment.
Inflationary income gap is the amount of
income by which the equilibrium income
exceeds the full employment income.
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Graphical illustration
E
Y=E
E2
Ef
Inflationary
AD gap
{
45o
Yf
Y2
Inflationary income gap
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Y
Deflationary income gap
= Deflationary AD gap x multiplier
Inflationary income gap
= Inflationary AD gap x multiplier
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Paradox of thrift
 The puzzle why national income falls (the
society gets poorer) when people as a whole
save more.
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Saving is detrimental when
S,I
 Saving  but the unspent
income does not re-enter
the circular flow
SP’
}
0
Ye’
Ye
SP
Unintended
inventory investment
 Firms cut
production
IP
Y
Income (Y) 
Note: If investment is an autonomous expenditure,
the results are  Y, C & S unchanged (= I)
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Saving is detrimental
S,I
}
SP’
SP
IP
Unintended
inventory
investment
0
Ye’
Ye
Note: If investment is an induced expenditure,
the results are  Y, C & S (= I)
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Y
 Saving  and the
unspent income can
re-enter the circular
flow as investment
Saving is beneficial when
S,I
SP’
SP
IP’
IP
0
Ye=Ye’
Then Y is unchanged.
In addition, as I,
productivity.
Y
Note: If investment is an autonomous expenditure,
the results are  Y unchanged, C, S & I
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Saving is beneficial
S,I
SP’
SP
IP ’
IP
Ye=Ye’
0
Y
Note: If investment is an induced expenditure,
the results are  Y unchanged, C, S & I
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Implications of private saving, public saving & national saving
Definition:
• Private saving (SP or S, 私人儲蓄) is the saving of
households, i.e., SP = Yd – C = Y – T – C.
• Public saving (SG, 公共儲蓄) is the saving of the
government, also called fiscal surplus. SG = T – G.
• National saving (SN, 國民儲蓄) is the saving of the
economy as a whole. SN = SP + SG.
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Implications of private saving
In equilibrium, total withdrawal = total injection.
SP + T + M = I + G + X
 SP = I + (G – T) + (X – M) ………………. (1)
In equilibrium, AS = AD. Resources not consumed by
households (private saving) must be consumed by other
economic agents – by firms as investment (I), and/or by
the government creating fiscal deficit (G - T), and/or by
the foreign sector as net exports (X - M).
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Implications of public saving
In equilibrium, total withdrawal = total injection.
SP + T + M = I + G + X
 Fiscal surplus = SG = T – G = (I – SP) + (X – M) …….. (2)
 Fiscal deficit = -SG = G – T = (SP – I) + (M – X) …….. (3)
Equation (2): In equilibrium, AS = AD. Resources not consumed
by the government (public saving) must be consumed by other
economic agents – by private sector (I – SP), and/or by the foreign
sector (X - M).
Equation (3): If there exists fiscal deficit, the resources have to be
supplied by the private sector and/or the foreign sector, through the
issuance of internal debt (SP - I), and/or external debt (which
enables the economy to have net imports, M – X).
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Implications of natioinal saving
By definition, SN = SP + SG = [I + (G-T) + (X-M)] + (T-G)
 SN = I + (X – M) …….. (4)
In equilibrium, CA + KA = 0  CA = X – M = -KA.
From equation (4), SN – I = X – M = CA = -KA …….. (5)
Equation (4): In equilibrium, AS = AD. Resources not consumed
by households and the government (national saving) must be
consumed by other economic agents – by firms as investment (I),
and/or by the foreign sector as net exports (X - M).
Equation (5): In equilibrium, AS = AD. Resources not consumed
by our economy (SN - I) must be consumed by foreign economies
as net exports (X - M) and illustrated by our current account
surplus. To have external balance, the capital account must have
deficit (CA = -KA).
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Advanced Material 4.1
Equality between investment and saving in a
two-sector economy
Meaning of equality between
ex-ante investment & ex-ante saving
 In a two-sector economy, equality between ex-ante
(or planned or desired) investment and ex-ante saving
is the equilibrium condition.
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Derivation
S,I
SP
With unintended inventory
investment  Y
Sp
IP
0
Ye
Y1
(Sp>Ip)
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IP
Y
S,I
SP
IP
With unintended inventory
disinvestment  Y
IP
Sp
0
Y
Y2
(Sp<Ip)
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Ye
S,I
SP
No unintended change in inventories  Ye
IP
0
Ye
Y
(Sp=Ip)
Equilibrium condition of a 2-sector economy
© Pilot Publishing Company Ltd. 2005
Equality between investment and saving in a
two-sector economy
 Meaning of equality between
ex-post investment and ex-post saving
 In a two-sector economy, equality between ex-post
(or actual or realized or observed) investment and expost saving is an identity.
 As they must always be equal, the equality is a
tautology without any economic meaning or implication.
© Pilot Publishing Company Ltd. 2005
Derivation
S,I
[Sp (= Sa) = Ip]
Ia = Ip + Iu(=0) = Sa
Iu (<0)
SP
Iu (>0)
IP
IP
IP
0
Y2
Ye
IP
Y1
Y
[Sp (= Sa) < Ip] but
[Sp (= Sa) >Ip] but
Ia = Ip + Iu (<0) = Sa
Ia = Ip + Iu (>0) = Sa
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Implications
1. In a 4-sector economy, equality between planned
total withdrawal and planned total injection is the
equilibrium condition.
2. In a 4-sector economy, equality between actual
total withdrawal and actual total injection is an
identity and is meaningless in economics.
© Pilot Publishing Company Ltd. 2005
Different terms related to investment
 Planned investment (Ip) is the planned change in fixed
capital & inventories.
 Unplanned investment (Iu) is the unplanned change in
inventories, which is the amount not purchased by any
economic agents.
 Realized investment is the amount of actual investment
(= Ip + Iu).
 Unrealized investment is the amount of actual investment
falling short of the amount of planned investment (= Iu < 0).
© Pilot Publishing Company Ltd. 2005
Correcting Misconceptions:
1. The multiplier of an autonomous decrease
in expenditure is negative.
2. The import function is represented by the
linear equation: M = mYd + M*.
3. Multipliers must be positive.
4. An increase in aggregate expenditure would
shift the E-curve upward in a parallel manner.
© Pilot Publishing Company Ltd. 2005
Correcting Misconceptions:
5. Equilibrium income is the same as full
employment income.
6. Equality between total injection & total
withdrawal is the equilibrium condition of
goods market.
7. An increase in saving (a withdrawal) is
detrimental to an economy.
© Pilot Publishing Company Ltd. 2005