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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 Income-expenditure 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
FirmsHouseholds
Composed of households and firms only
No government sector
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Features of households
Households provide factor services for income.
Factor Services Income
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Features of households
Consumption
Disposable Income
Disposable income is either consumed or saved.
Saving
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C = cYd + C*C = cYd + C*
Disposable IncomeDisposable Income
Consumption function
Autonomous ConsumptionAutonomous Consumption
Marginal Propensity to Consume (MPC)
Marginal Propensity to Consume (MPC)
Autonomous consumption (C*) is the consumption at zero disposable income (the minimum amount for subsistence). C* > 0.
Marginal propensity to consume (MPC or c) is the change in consumption resulting from a unit change in disposable income. c < 1.
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Graphical Illustration
Consumption function
C = cYd + C*
C
Yd
C*
c+1
0
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S = sYd + S*S = sYd + S*
Disposable IncomeDisposable Income
Saving function
Autonomous SavingAutonomous Saving
Marginal Propensity to Save (MPS)
Marginal Propensity to Save (MPS)
Autonomous saving is the saving at zero disposable income.
S* = -C*. Why?
Marginal propensity to save (MPS or s) is the change in saving resulting from a unit change in disposable income. s = 1 – c. Why?
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S = sYd + S*
S
Y
S* = -C*
s = (1-c)+1
0
Graphical Illustration
Saving function
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Features of firms
1. Firms employ factor services to produce goods.
2. Firms also consume final products (fixed investment & inventories) to help production.
Factor Services Firms
Products
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I = i Y + I*I = i Y + I*
National IncomeNational Income
Investment function
Autonomous InvestmentAutonomous Investment
Marginal Propensity to Invest (MPI)
Marginal Propensity to Invest (MPI)
Autonomous investment is the investment at zero income. I* > 0. Why?
Marginal propensity to invest (MPI or i) is the change in investment resulting from a unit change in income. i > 0. Why?
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I*
I = iY + I*
Y
r
0
Graphical illustration
Investment function
+1i
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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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AS = AD Y = E C + S = C + I S = I
Withdrawal
Equilibrium income (or equilibrium GNP) is reached when AS = AD
Injection =
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Withdrawal ( 撤出 , W)
In a 2 sector economy, saving is the withdrawal.
is the amount of income withdrawn from the circular flow, not being spent on final products.
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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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ABC Ltd.
Firms
Circular flow of a 2-sector economy
Investment Saving
Injection
Withdrawal
Consumption
Y
YdE
Households
When I = S, an equilibrium is achieved.
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Z (Y > E) U (Y = E)
V (Y < E)Y=E
Y
E
045o
Meaning of a 45o line
Meaning of a 45° line
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Aggregate expenditure function
Y=E
Y
E
045o
E= C+I
C
C*I*
Add consumption and investment functions vertically
I
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Adjustment mechanism
Income-expenditure approach
Injection-withdrawal approach
2 approaches:
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Y=E
Y
E
0
E=C+I
Ye Y1Y2
Income-expenditure approach
production
production
At Y2, Y < E
Unplanned decrease in inventories
At Y2, Y < E
Unplanned decrease in inventories
At Y1, Y > E
Unplanned increase in inv
entories
At Y1, Y > E
Unplanned increase in inv
entories
}
{
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Injection-withdrawal approach
At Y1, J < W Unintended inventory investment
At Y1, J < W Unintended inventory investment
At Y2, J > W Unintended inventory disinvestment
At Y2, J > W Unintended inventory disinvestment
Y
E
0
-C*
I*
S
I
Ye Y1Y2
production
production {}
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Multiplier effect
E
Y
Initial equilibrium
ΔEaE:
Y:
ΔEi ΔEi’
•••
Total change in Y = ΔE + c•ΔE + c•c• ΔE + c•c•c• ΔE + …
= (1 + c + c2 + c3 + …) • ΔE = •ΔE
: Change in Y brought by ΔEa
c11
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Multiplier (k)
Mathematical expression:
c11
EYk
Multiplier effect
is the ratio of the total change in equilibrium income to the initial change in autonomous expenditure (or autonomous withdrawal) that brought it about.
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Equilibrium Y = .(C* + I*)
Mathematical derivation of equil. income & multiplier
In a 2-sector economy
In equilibrium, Y = E = C + I
= cYd + C* + I*
= cY + C* + I*
E = C + I, where C = cYd + C* and I = I*
Y – cY = C* + I* (1 – c)Y= C* + I*
c11
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c11
EY
a
When C* or I* changes by ΔEa
Mathematical derivation of equil. income & multiplier
ΔY = . ΔEac11
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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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HouseholdsFirms
Government
Three-sector economy
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Government
Government’s expenditure is mainly financed by taxation
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Government expenditure function (G)
G = G*, where G > 0G = G*, where G > 0
is fixed by the budget at the beginning of a fiscal year.
G is a constant (G*) independent of any variables.
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Tax function
Only direct taxes are concerned. There exist three kinds of direct tax systems.
No indirect taxes is assumed.
Tax System Formula Example
Lump-sum tax system T = T’ Poll tax
Proportional tax system T = tY + T* Profits tax
Progressive tax system T = t*Y + T^ Salaries tax
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Aggregate supply (AS) of final products is GNP or Y.
Equilibrium condition (3-sector economy)
Yd = Y – T = C + S
Y = Yd + T = C + S + T
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Aggregate demand (AD) for final products is E.
Equilibrium condition (3-sector economy)
E = C + I + G
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AS = AD Y = E C + S + T = C + I + G S + T = I + G
Withdrawal
Equilibrium income (or equilibrium GNP) is reached when AS = AD
Injection =
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ABC Ltd.
HouseholdsFirms
Circular flow of a 3-sector economy
InvestmentSaving
Injection
Withdrawal
Consumption
Y
E
TaxGovernment expenditure
When I + G = S + T, equilibrium is achieved.
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Mathematical derivation of equil. income & multiplier
In a 3-sector economy with a lump-sum tax system
In equilibrium, Y = E = C + I + G = cYd + C* + I* + G*
E=C+I+G, where C = cYd + C*; Yd = Y – T’ and I = I* and G = G*
Y = c(Y-T’) + C* + I* + G* = cY- cT’+ C* + I* + G*
Equilibrium Y = .(C*+I*+G*- cT’)
(1–c)Y = C* + I*+ G*- cT’
c1
1
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Mathematical derivation of equil. income & multiplier
When C* or I* or G* changes by ΔEa
ΔY = . ΔEa
Multiplier =c1
1EY
a
c1
1
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Mathematical derivation of equil. income & multiplier
Under a lump-sum tax system
Equil. income: )cT'*G*I*(Cc1
1
)cT'*G*I*(C
c1
1
Multiplier:c1
1
c1
1
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Mathematical derivation of equil. income & multiplierIn a 3-sector economy with a proportional tax system
In equilibrium, Y = E = C + I + G = cYd + C* + I* + G*
E=C+I+G, where C = cYd + C*; Yd = Y - tY - T*; I = I* and G = G*
(1 - c + ct)Y = C* + I* + G* - cT*
Y = c(Y - tY - T*) + C* + I* + G* = cY - ctY - cT* + C* + I* + G*
Equilibrium Y = • (C*+I*+G*-cT*)ctc1
1
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Mathematical derivation of equil. income & multiplier
When C* or I* or G* changes by ΔEa
ΔY = • ΔEa
Multiplier =
ctc1
1
ctc1
1EY
a
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Under a proportional tax system
Equil. income: cT*)*G*I*(Cctc1
1
cT*)*G*I*(C
ctc1
1
Multiplier:ctc1
1
ctc1
1
The multiplier of proportional tax system is smaller than the multiplier of lump-sum tax system.
As t > 0, (1-c) < (1-c+ct)
c1
1
ctc1
1
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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*
Equilibrium Y = • (C*+I*+G*-cT*-cQ*)
Y = c(Y-tY-T*+qY+Q*) + C* + I* + G* (1–c+ct-cq) •Y = C* + I* + G* -cT* + cQ*
cq-ctc1
1
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ΔY = • ΔEa
Mathematical derivation of equil. income & multiplier
cq-ctc1
1
Multiplier =
When C* or I* or G* changes by ΔEa
cq-ctc1
1EY
a
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Under a proportional tax & transfer payment system
Equil. income: cQ*)*cT*G*I*(Ccq-ctc1
1
cQ*)*cT*G*I*(C
cq-ctc1
1
Multiplier:cq-ctc1
1
cq-ctc1
1
As q<0, (1-c+ct)<(1-c+ct-cq)
The multiplier with a proportional transfer payment is smaller than the multiplier without it.
ctc1
1
cq-ctc1
1
>>
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Four-sector Economy
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Four-sector economy
Foreign Sector
HouseholdsFirms
Government
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Export function:
X=X*, where X*>0X=X*, where X*>0
Export is determined by foreign economies, not the domestic economy
It is autonomous & is independent of Y
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Import function:
M= mY +M*M= mY +M*
Marginal Propensity to Import (MPM)
Marginal Propensity to Import (MPM) Autonomous ImportAutonomous 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.
All economic agents consume imports
Import is positively related to Y
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Aggregate supply (AS) of final products is GNP or Y.
Equilibrium condition (4-sector economy)
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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AS = AD
Y = E
C + S + T = C + I + G + X – M
S + T = I + G + X – M
S + T + M = I + G + X
Equilibrium income (or equilibrium GNP) is reached when
Injection= Withdrawal
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ABC Ltd.
HouseholdsFirms
Circular flow of a 4-sector economy
I - MISaving
InjectionWithdrawal
C - MC
Y
E
TaxG - MG
X - MXWhen I + G + X – M = S + T,
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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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 = •(C*+I*+G*+X*-M*-cT*-cQ*)
Mathematical derivation of equil. income & multiplier
In equilibrium, Y = E = C+I+G+X-M = cYd+C*+I*+G*+X*-mY-M*
mcq-ctc1
1
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Mathematical derivation of equil. income & multiplier
When C* or I* or G* or X* changes by ΔEa
mcq-ctc11
EY
a
ΔY = • ΔEamcq-ctc1
1
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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
ic1
1
cqctic1
1
mcqctic1
1
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Comparison of the size of expenditure multipliers (k)
As c>0, i>0, t>0, -q>0, and m>0,
mcqctic1cqctic1ic1
mcqctic1
1
cqctic1
1
ic1
1
k in a 2-sector economy > k in a 3-sector economy > k in a 4-sector economy
k in a 2-sector economy > k in a 3-sector economy > k in a 4-sector economy
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Explanation:
E
Y
Initial equilibrium
ΔEaE:
Y:
ΔEi
•••
ΔEi (2-sector economy) =c • ΔEa + i • ΔEa
: Change in Y brought by 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
mcqctic1
c
*T
Y
mcqctic1
c
*T
Y
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
mcqctic1
1
*M
Y
mcqctic1
1
*M
Y
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
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).
ΔEa < 0 Multiplier = Unchanged
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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-functionSlope of E-function E-interceptE-intercept
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Q4.7:
Derive the expenditure multiplier from the income-expenditure 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 induced in E
An autonomous in E
E
E’
E
Y
An autonomous change versus an induced change
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E
Y
Y=E
E’ (slope > 1)
ΔE’>ΔY
ΔYE (Slope < 1)
ΔE<ΔYΔY
45o
Can’t find the
equilibrium
Can’t find the
equilibrium
Slope of E-function cannot be greater than one
The equilibriumThe equilibrium
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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.
Deflationary Gap
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Deflationary income gap
45o
E1
}{Deflationary
AD gap
Ef
E
Y
Y=E
YfY1
Graphical illustration
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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.
Inflationary Gap
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45o
E2
}Inflationary income gap
{Inflationary AD gap
Ef
E
Y
Y=E
Yf Y2
Graphical illustration
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Deflationary income gap
Inflationary income gap
= Deflationary AD gap x multiplier
= 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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SP’
Ye’
}Unintended
inventory investment
Unintended inventory investment
0
SP
IP
S,I
YYe
Note: If investment is an autonomous expenditure, the results are Y, C & S unchanged (= I)
Saving is detrimental when Saving but the unspent income does not re-enter the circular flow
Saving but the unspent income does not re-enter the circular flow
Firms cut production
Firms cut production
Income (Y) Income (Y)
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SP’
0
SP
IP
S,I
YYe’ Ye
}Unintended inventory
investment
Unintended inventory
investment
Saving is detrimental
Note: If investment is an induced expenditure, the results are Y, C & S (= I)
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SP’
Ye=Ye’
IP’
Saving is beneficial when Saving and the unspent income can re-enter the circular flow as investment
Saving and the unspent income can re-enter the circular flow as investment
Then Y is unchanged. In addition, as I, productivity.
Then Y is unchanged. In addition, as I, productivity.
0
SP
IP
S,I
YNote: If investment is an autonomous expenditure, the results are Y unchanged, C, S & I
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SP’
0
SP
IP
S,I
YYe=Ye’
IP ’
Saving is beneficial
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
• 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.
Definition:
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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 savingIn 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 4.1Equality 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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With unintended inventory investment Y
With unintended inventory investment Y
0
SP
IP
S,I
YYe Y1
IP
Sp
(Sp>Ip)
Derivation
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Sp
0
SP
IP
S,I
YYeY2
IP
(Sp<Ip)
With unintended inventory disinvestment Y
With unintended inventory disinvestment Y
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0
SP
IP
S,I
YYe
(Sp=Ip)
No unintended change in inventories YeNo unintended change in inventories Ye
Equilibrium condition of a 2-sector economy
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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 ex-post saving is an identity.
As they must always be equal, the equality is a tautology without any economic meaning or implication.
Equality between investment and saving in a two-sector economy
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0
SP
S,I
YYe Y1
IP
[Sp (= Sa) >Ip] but
Ia = Ip + Iu (>0) = Sa
[Sp (= Sa) >Ip] but
Ia = Ip + Iu (>0) = Sa
Derivation
Y2
IP
Iu (>0)
IP
Iu (<0)
IP
[Sp (= Sa) < Ip] but
Ia = Ip + Iu (<0) = Sa
[Sp (= Sa) < Ip] but
Ia = Ip + Iu (<0) = Sa
[Sp (= Sa) = Ip]
Ia = Ip + Iu(=0) = Sa
[Sp (= Sa) = Ip]
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.
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Planned investment (Ip) is the planned change in fixed capital & inventories.
Different terms related to investment
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).
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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.
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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.
Correcting Misconceptions: