hkale macroeconomics chapter 2: elementary keynesian model (i)- two-sector
TRANSCRIPT
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HKALE Macroeconomics
Chapter 2: Elementary Keynesian Model (I)-
Two-sector
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References:
• CH 3, Advanced Level Macroeconomics, 5th Ed, Dr. LAM pun-lee, MacMillan Publishers (China) Limited
• CH 3, HKALE Macroeconomics, 2nd Ed., LEUNG man-por, Hung Fung Book Co. Ltd.
• CH 3, A-L Macroeconomics, 3rd Ed., Chan & Kwok, Golden Crown
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Introduction
• National income accounting can only provide ex-post data about national income.
• The three approaches are identities as they are true for any income level.
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Introduction
• In order to explain the level and determinants of national income during a period of time, we count on national income determination model, e.g. Keynesian Models.
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Business Cycle
0
GNP
Time
Boom
Recession
Depression
Recovery
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Business Cycle
• It shows the recurrent fluctuations in GNP around a secular trend
Trough Recovery Peak Recession
Employment level
the lowest
Rising the highest
Falling
Growth rate of real GNP
Negative Rising the highest
Falling
Prices the lowest
Rising the highest
falling
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HK’s Economic Performance
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Assumptions behind National Income Models
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Assumptions behind National Income Models
• The level of price is constant– as Y = P×Q & P = 1, then Y = (1)×Q Y = Q– Price level tends to be rigid in downward direction
• Existence of idle resources, i.e. unemployment
• Potential/Full-employment national income, Yf is constant
• Y = National income at constant price
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Equilibrium Income Determination of Keynesian's Two-sector Model (1)- A Spendthrift Economy
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John Maynard Keynes
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Assumptions
no investment or injection
• Two sectors: households and firms
• consumer goods only
• no saving, no tax and no imports
no leakage/withdrawal
Y=Yd while Yd = disposable income
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Simple Circular Flow Model of a Spendthrift Economy
Households
Firms
National income
National expenditure
Incomegenerated
Payment for goods and service
C
E Y
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By Income-expenditure Approach
• AD → (without S) E = C → Y (firms)
↑ ↓
Y (households) ← AS ← D for factors
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By Income-expenditure Approach
• Equilibrium income, Ye is determined when– AS = AD– Y = E Y = E = C
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Equilibrium Income Determination of Keynesian's Two-sector Model (2)-A Frugal Economy
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1. Households and firms
2. Saving, S, exists • Income is either consumed or saved
Y ≡ C+S• leakage, S, exists
3. Without tax, Y=Yd
Assumptions
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4. Consumer and producer goods • Injection (investment, I) exist
5. Investment is autonomous/exogenous
6. Saving and investment decisions
made separately• S=I occurs only at equilibrium level of
income
Assumptions
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Simple Circular Flow Model of a Frugal Economy
Households
Financial markets
Firms
National income
National expenditure
Incomegenerated
Payment for goods and service
C
S
I
E Y
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Income Function: Income line/45 line/Y-line
• an artificial linear function on which each point showing Y = E
E1
Y1
E2
Y245
0
E
Y
Y-line
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Expenditure Function (1): Consumption Function, C
• showing that planned consumption expenditure varies positively with but proportionately less than change in Yd
• A linear consumption function: C = a + cYd
where– a = a constant representing autonomous consumption expenditure– c = Marginal Propensity to Consume, MPC
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A Consumption Function, C
C1
Y1
C2
Y20
E
Y
C = a + cYd
a
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Marginal Propensity to Consume, MPC, c
• MPC = c =dY
C
0
E
Y
C = a + cYd
a
△ C
△ Y
M
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Properties of MPC:
• the slope of the consumption function
• 1 > MPC > 0
• the value of 'c' is constant for all income levels
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Average Propensity to Consume, APC• APC =
0
E
Y
C = a + cYd
a
C
Y
M
dY
C
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Properties of APC:
• the slope of the ray from the origin
• APC falls when Y rises
• Since C = a + cYd
Then
i.e.
Thus, APC > MPC for all income levels
)()( cY
a
Y
C
Y
Yc
Y
a
Y
C
ddd
d
dd
MPCY
aAPC
d
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Consumption Function Without ‘a”
• If ‘a’ = 0, then C = cYd
0
E
Y
C = cYd
a =< 45
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Consumption Function Without ‘a”
0
E
Y
C = cYd
a =
C = △ C
Y = △ Y
M
• If ‘a’ = 0, then MPC = APC =dY
C
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Expenditure Function (2): Investment Function, I
• showing the relationship between planned investment expenditure
and disposable income level, Yd
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Autonomous Investment Function
• Autonomous investment function: I = I*
where I* = a constant representing autonomous investment expenditure
E
0 Y
I = I*I*
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Induced Investment Function
• Induced investment function: I = I* + iYd
where i = Marginal Propensity to Invest
E
0 Y
I = I* + iYd
I*
= MPI = dY
I
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Properties of MPI:
• the slope of the investment function
• 1 > MPI > 0
• the value of ‘i' is constant for all income levels
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Average Propensity to Invest, API• API =
0
E
Y
I = I* + iYd
I*
I
Y
M
dY
I
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Properties of API:
• the slope of the ray from the origin
• API falls when Y rises
• Since I = I* + iYd
Then
i.e.
Thus, API > MPI for all income levels
)(*
)(*
iY
I
Y
I
Y
Yi
Y
I
Y
I
ddd
d
dd
MPIY
IAPI
d
*
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MPI under Autonomous Investment Function
• If I = I*, then Y will not affect I
E
0 Y
I = I*I*
• Therefore, MPI = 00
dd YY
I
Slope = MPI = 0
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Expenditure Function (3): Aggregate Expenditure Function, E
• Showing the relationship between planned aggregate expenditure and
disposable income level, Yd
• Aggregate expenditure function: E = C+I
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Aggregate Expenditure Function, E• Since C = a + cYd
I = I* (autonomous function)
E = C+I
• Then E = (a + cYd) + (I*)
E = (a + I*) + cYd
Where
• (a + I*) = a constant representing
the intercept on the vertical axis
• ‘c’ = slope of the E function
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Aggregate Expenditure Function, E
• Since C = a + cYd
I* + iYd (induced function)E = C+I
• Then E = (a + cYd) + (I* + iYd)
E = (a + I*) + (c + i)Yd
Where
• (a + I*) = a constant representing the intercept on the vertical axis
• ‘c + i’ = slope of the E function
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Aggregate Expenditure Function
E1
Y1
E2
Y20
E
YI* I = I*a
C = a + cYd
(a+I*)
E = C + I
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Aggregate Expenditure Function
E1
Y1
E2
Y20
E
YI*
I = I*+iYda
C = a + cYd
(a+I*)
E = C + I
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Leakage Function (1): Saving Function, S
• showing that planned saving varies positively with but proportionately less than change in Yd
• A linear saving function: S = -a + sYd
where– -a = a constant = autonomous saving– s = Marginal Propensity to save, MPS
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A Saving Function, S
S1
Y1
S2
Y20
E, S
Y
S = -a + sYd
-a
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MPC (c) and MPS (s)
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Marginal Propensity to Saving, MPS, s• MPS = s =
△ S
△ Y
M
dY
S
S = -a + sYd
-a0
E, S
Y
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• the slope of the saving function
• 1 > MPS > 0
• the value of ‘s' is constant for all income levels
• Since Y ≡ C + S
Properties of MPS:
ddd
d
Y
S
Y
C
Y
Y
Then
Hence 1 = c + s and s = 1 - c
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Average Propensity to Save, APS• APS =
S
Y
MS = -a + sYd
-a0
E, S
Y
dY
S
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Properties of APS:
• the slope of the ray from the origin
• APS rises when Y rises
• Since S = -a + sYd
Then
i.e.
Thus, APS < MPS for all income levels
)()( sY
a
Y
S
Y
Ys
Y
a
Y
S
ddd
d
dd
MPSY
aAPS
d
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Saving Function Without ‘-a”
• If ‘-a’ = 0, then S = sYd
0
E, S
Y
S = sYd
-a =< 45
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Saving Function Without ‘-a”
0
E, S
Y
S = △ S
Y = △ Y
M
• If ‘-a’ = 0, then MPS = APS =
S = sYd
-a =
dY
S
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Determination of Ye by Income-expenditure Approach
• Equilibrium income, Ye is determined when– AS = AD– Total Income = Total Expenditure
i.e. Y = E = C + I
GivenC = a + cYd and I = I*
Ye = Y and Yd = Y
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Determination of Ye by Income-expenditure Approach
• In equilibrium:
Y= E = C + I
= (a + cYd) + (I *)
Y- cY= a + I*
Then Y(1-c) = a + I*
Therefores
Iaor
c
IaYe
*
1
*
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If Investment Function is Induced …
• In equilibrium:
Y= E = C + I
= (a + cYd) + (I *+iYd)
Y- (c+i)Y= a + I*
Then Y(1-c-i) = a + I*
Thereforeis
Iaoric
IaYe
*
1
*
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Graphical Representation of Ye
0
E
YI* I = I*a
C = a + cYd
(a+I*)
E = C + IY-line
Ee
Ye
![Page 54: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/54.jpg)
If Investment Function is Induced….
0
E
YI*
I = I*+iYd
a
C = a + cYd
(a+I*)
E = C + IY-line
Ee
Ye
![Page 55: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/55.jpg)
Determination of Ye by Injection-leakage Approach
• Equilibrium income, Ye is determined when
– Total Leakage = Total Injection
• Given S = -a + sYd
I = I*
Ye = Y and Yd = Y
![Page 56: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/56.jpg)
Determination of Ye by Injection-leakage Approach
• In equilibrium:
S = I
(-a + sYd) = (I *)
Then sY = a + I*
Thereforec
Iaor
s
IaYe
1
**
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If Investment Function is Induced…
• In equilibrium:
S = I
(-a + sYd) = (I *+iYd)
Then (s-i)Y = a + I*
Thereforeic
Iaor
is
IaYe
1
**
![Page 58: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/58.jpg)
Graphical Representation of Ye
0
E, S
Y
I* I = I*
-a
S = -a + sYd
Ye
I = S
![Page 59: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/59.jpg)
If Investment Function is Induced…
0
E, S
YI*
I = I*+iYd
-a
S = -a + sYd
Ye
I = S
![Page 60: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/60.jpg)
Graphical Representation of Ye
E($)
Y($)
I
C
E = C + I
S
Y-line
45o
Ye
![Page 61: HKALE Macroeconomics Chapter 2: Elementary Keynesian Model (I)- Two-sector](https://reader034.vdocuments.us/reader034/viewer/2022051215/56649e4f5503460f94b4620e/html5/thumbnails/61.jpg)
If Investment Function is Induced…
E($)
Y($)
I
C
E = C + I
S
Y-line
45o
Ye
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A Two-sector Model: An Example
• Given:– C = $80 + 0.6Y– I = $40
• Since– E = C + I = ($80 + 0.6Y)+($40)
Then, E = $120 + 0.6Y
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A Two-sector Model: An Example
• By income-expenditure approach, in equilibrium:– Y = E = C + I
Then Y = ($120 + 0.6Y)
(1-0.6)Y = $120
Thus, Y = $120/0.4 = $300
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A Two-sector Model: An Example
• By injection-leakage approach, in equilibrium:– Total injection = Total leakage i.e. I = S
– Given I = $40 and S = -a + sYd
Then, $40 = (-$80 + 0.4Y) 0.4Y = $120 Thus, Y = $120/0.4 = $300
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A Two-sector Model: Exercise
• Given:– C = $30 + 0.8Y– I = $50
• Question: (1) Find the equilibrium national income level by the two approaches. (2) Show your answers in two separate diagrams.
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A Two-sector Model: Exercise
• By income-expenditure approach, in equilibrium:– Y = E = C + I
Then Y = ($30 + 50) + 0.8Y
(1-0.8)Y = $80
Thus, Y = $80/0.2 = $400
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Graphical Representation of Ye
0
E
Y
$50 I = $50
$30
C = $30 + 0.8Yd
$(30+50)
E = $80+0.8Yd
Y-line
Ee
Ye =$400
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A Two-sector Model: An Example
• By injection-leakage approach, in equilibrium:– Total injection = Total leakage i.e. I = S
– Given I = $50 and S = -a + sYd
Then, $50 = (-$30 + 0.2Y) 0.2Y = $80 Thus, Y = $80/0.2 = $400
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Graphical Representation of Ye
0
E, S
Y
$50 I = $50
-$30
S = -$30 + 0.2Yd
Ye=$400
I = S
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Aggregate Production Function
• It relates the amount of inputs, labor (L) and capital (K), used by the entire business sector to the amount of final output (Y) the economy can generate.– Y = f(L, K)
• Given the capital stock (i.e. K is constant), Y is a function of the employment of labor.– Thus, Y = 2L (the figure is assigned)
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An Application
• Given Ye = $300 and the labor force is 200. Find (1) the amount of labor (L) required to bring it happened; (2) the level of unemployment and (3) the full-employment level of income
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An Application
(1) Since Y = 2L
($300) = 2L
Then, L = 150
(2 Unemployment level = 200-150 = 50
(3) Since Yf = 2L = 2(200) = $400
Then, Ye < Yf by (400 – 300)$100
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Ex-post Saving Equals Ex-post Investment
• Actual income must be spent either on consumption or savingY ≡ C + S
• Actual income must be spent buying either consumer or investment goods Y ≡ E ≡ C + I
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Ex-post Saving Equals Ex-post Investment
• In realized sense, – Since Y ≡ C + S and Y ≡ C + I– Then, I ≡ S
• At any given income level, ex-post investment must be equal to ex-post saving, if adjustments in inventories are allowed
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Ex-ante Saving Equals Ex-ante Investment
• If planned investment is finally NOT realized (i.e. unrealized investment is positive), then past inventories must be used to meet the planned investment, thus leading to unintended inventory disinvestment.– Unrealized investment invites
unintended inventory disinvestment
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Ex-ante Saving Equals Ex-ante Investment
• Therefore,– Realized I = Planned I + Change in
unintended inventory
OR– Realized I = Planned I – Unrealized
investment
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Ex-ante Saving Equals Ex-ante Investment• As planned saving and investment
decisions are made separately, only when the level of national income is in equilibrium will ex-ante saving be equal to ex-ante investment.
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Ex-ante Saving Equals Ex-ante Investment• In equilibrium,
– By the Income-expenditure Approach, • Actual Income = Planned Aggregate Expen
diture Y = E = Planned C + Planned I Y = (a + cY) + (I*)
– By the Injection-leakage Approach.• Total Injection = Total Leakage Planned I = Planned S (= Actual I = Actual S)
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Ex-ante Saving Equals Ex-ante Investment• If planned aggregate expenditure is
larger than actual income or output level, i.e. E > Y, then AD > AS
planned I > planned S
unintended inventory disinvestment
AS (next round) = AD
Y = E
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Ex-ante Saving Equals Ex-ante Investment• If planned aggregate expenditure is
smaller than actual income or output level, i.e. E < Y, then AD < AS
planned I < planned S
unintended inventory investment
AS (next round) = AD
Y = E and unintended stock = 0
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Ex-ante Saving Equals Ex-ante Investment• If ex-ante saving and ex-ante
investment are not equal, income or output will adjust until they are equal.
• In equilibrium, therefore– Y = E or I = S– Unintended inventory = 0– Unrealized investment = 0
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An Illustration(1)
=(2)+(3)
(2)= (1)-(3)
(3)=(1)-(2)
(4)=I* (5)=(2)+(4)
(6)=(1)-(5)
(7)= -(6)
(8)=(4)+(6)
Y P. C. P. S. P. I. P. A. E. U.C.I. UR.I. A. I.Level of Income
Planned Consumption Expenditure
Planned Saving
Planned Investment Expenditure
Planned Aggregate
Expenditure
Unintended Change in Inventory
Unrealized Investment
Actual Investment
0 80 -80 40 120 -120 120 -80
100 140 -40 40 180 -80 80 -40
200 200 0 40 240 -40 40 0
300 260 40 40 300 0 0 40
400 320 80 40 360 40 -40 80
500 380 120 40 420 80 -80 120
•MPC, c = (140-80)/(100-0) = 0.6•C = a + cYd = 80 + 0.6Yd•I = 40 and E = C + I = 120 + 0.6Yd
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An Illustration
Actual income or output level (Y)
200 300 400
Planned aggregate expenditure (E)
240 300 360
Ex-anteE>Y E=Y E<Y
I>S I=S I<S
Unintended change in stocks -40 0 40
Actual aggregate expenditure 240-40
=200
300 360+40
=400
Ex-post YE YE YE
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Exercise 1
• Given: C = 60 + 0.8Y & I = 60
• Find the equilibrium level of national income, Ye, by the income-expenditure and injection-leakage approaches.
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Answer 1
• Given: C = 60 + 0.8Y & I = 60
• By the Income-expenditure Approach:Ye = E = C + I
Ye = (60 + 0.8Y) + (60)
Ye = 600 #
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Answer 1
• Given: C = 60 + 0.8Y & I = 60
• By the Injection-leakage Approach: I = S
60 = -60 + 0.2Y
Ye = 600 #
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Exercise 2
• Given: C = 60 + 0.8Y & I = 60• Show the equilibrium level of national
income, Ye, in a diagram.
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Exercise 3(1)
=(2)+(3)
(2)= (1)-(3)
(3)=(1)-(2)
(4)=I* (5)=(2)+(4)
(6)=(1)-(5)
(7)= -(6)
(8)=(4)-(7)
Y P. C. P. S. P. I. P. A. E. U.C.I. UR.I. A. I.Level of Income
Planned Consumption Expenditure
Planned Saving
Planned Investment Expenditure
Planned Aggregate
Expenditure
Unintended Change in Inventory
Unrealized Investment
Actual Investment
0 60 -60 60 120 -120 120 -60
200 220 -20 60 280 -80 80 -20
300 300 0 60 360 -60 60 0
400 380 20 60 440 -40 40 20
500 460 40 60 520 -20 20 40
600 540 60 60 600 0 0 60
700 620 80 60 680 20 -20 80
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Exercise 4
• Given C = 10 + 0.8Y and I = 8
• If Y = 1000, then– What is the level of realized investment?
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Exercise 4
• Given C = 10 + 0.8Y and I = 8
• If Y = 1000, then– What is the level of realized investment?
– As Y = 1000, C = 10 + 0.8(1000) = 810– As Y C + S
Actual S = I = 1000-810 = 190
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Exercise 4
• Given C = 10 + 0.8Y and I = 8
• If Y = 1000, then– What is the level of unplanned inventory
investment?
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Exercise 4
• Given C = 10 + 0.8Y and I = 8
• If Y = 1000, then– What is the level of unplanned inventory
investment?
– Unplanned inventory investment = actual I – planned I = 190 – 8 = 182
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In Equilibrium…
• Actual Y = Planned aggregate E
• Ex-ante I = ex-ante S (=actual I = actual S)
• Unplanned investment = 0
• Unrealized investment = 0
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Movement Along a Function
• A movement along a function represent a change in consumption or investment in response to a change in national income.
• While the Y-intercepting point and the function do NOT move.
YC = a + cYd CYI = I* + iYd I
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Movement Along a Consumption FunctionYC = a + c Yd C
C = a + cYd
C1
Y10
E
Y
a
C2
Y2
A
B
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Exercise 5
• Given C = 80 + 0.6Yd. How is consumption expenditure changed when Y rises from $100 to $150? Show it in a diagram.
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Answer 5
C = $80+0.6Yd
170
150
140
1000
E
Y
$80
A
B
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Exercise 6
• Given I = 40 + 0.2Yd. How is investment expenditure changed when Y rises from $100 to $150? Show it in a diagram.
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Answer 6
I = $40+0.2Yd
0
E
Y
$40
AB
$60
$100
$70
$150
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Shift of a Function
• A shift of a consumption or investment function is a change in the desire to consume(i.e. ‘a’) or invest(i.e. ‘I*) at each income level.
• As the change is independent of income, it is an autonomous change.
a C = a + cYdI* I = I* or I = I* + iYd
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Shift of a Function• A change in autonomous
consumption or investment expenditure (i.e. ‘a’ or ‘I*) will lead to a parallel shift of the entire function.
• The slope of the function remains unchanged.
• An upward parallel shift in C function implies a downward parallel shift of S function
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Shift of a Consumption Functiona C = a + cYd
C2=a2+cYd
a2
C1=a1+cYd
a1
E, Y
Y0
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Exercise 7
• Given C=80+0.6Yd & Y=$100. How is consumption function affected if autonomous consumption expenditure rises to $100? Show it in a diagram.
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Answer 7
C2=100+0.6cYd
C1=80+0.6Yd
80
E, Y
Y0
140
100
160
100
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Shift of an Investment FunctionI* I = I*
I1=I*1I*1
E, Y
Y0
I2=I*2I*2
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Rotation of a Function• A change in marginal propensities,
i.e. MPC and MPI, will lead to a rotation of the function on the Y-axis.
• The slope of the function rises with larger marginal propensities; vice versa.
• An upward rotation of C function implies a downward rotation of S function
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Rotation of a Consumption Functionc C = a + cYd
C2=a+c2Yd
C1=a+c1Yd
a
E, Y
Y0
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Exercise 8
• Given C=80+0.6Yd & Y=$100. How is consumption function affected if MPC rises to 0.8? Show it in a diagram.
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Answer 8
C2=80+0.8Yd
C1=80+0.6Yd
80
E, Y
Y0100
160
140
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The Multiplier• A n autonomous change in
consumption expenditure (‘a’) or investment expenditure (‘I*) will lead to a parallel shift of the aggregate expenditure function (E).
• The slope of E function rises with larger autonomous expenditure; vice versa.
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The Multipliera or I* EE > Y
planned I > planned S
unintended inventory disinvestment
AD > AS excess demand occurs
AD = AS (next round)
E = Y (higher Ye)
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The Multiplier• The (income) multiplier, K, measures
the magnitude of income change that results from the autonomous change in the aggregate expenditure function.
• If I is an autonomous function, then autonomous expenditure = (a + I*).
• Multiplier, eexpenditur autonomous in change
Yin changeK
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The Multiplier
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The Multiplier
E or Y S
Initialexpenditure
$1
2nd round $0.6 $0.4
3rd round $0.36 $0.24
… … …Total $1(1/0.4)=$2.5 $0.4(1/0.4)=$1
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The Multiplier
0
E1 (with a1)
a1E1
Y1
E, Y
Y
Y-line E2 (with a2)
a2
K=Y/E
Y
Y2
E
E2
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The Multiplier
1 k then 1, s If
s
1
c1
1
I*)Δ(a
ΔY
E
Yk ,definitionby Thus,
s
1
c1
1
I*)(a
Y Then,
s
I*)Δ(a
c1
I*)Δ(aY
s
*Ia
c-1
*IaY
or
or
or
or
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The Multiplier
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• If I is an induced function, then...
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Remarks on the Multiplier• If I is an induced function, then the
value of multiplier is smaller.
• The larger the value of MPC or MPI, the larger the value of the multiplier; vice versa.
• The smaller the value of MPS, the larger the value of the multiplier; vice versa.
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Remarks on the Multiplier• If MPS = 1 or MPC = 0 and MPI = 0
– then, k=1/1-c = 1
• If MPS = 0 or MPC = 1 and MPI = 0– then, k=1/1-c = 0, i.e. infinity– then there is an infinite increase in
income
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Exercise 9• Given C = $80 + 0.6Yd
• Find the value of the multiplier if – I = $40– I = $40 + 0.1Yd
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Exercise 10• ‘By redistribute $1 from the rich to the
poor will help increase the level of national income.’ Explain with the following assumptions:
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Exercise 11• What is the size of the multiplier if the
economy has already achieved full employment (i.e. Ye = Yf)?