HKALE Macroeconomics
Chapter 2: Elementary
Keynesian Model (I)Two-sector
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
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.
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.
Business Cycle
GNP
Recovery
Boom
Recession
Depression
0
Time
Business Cycle
• It shows the recurrent fluctuations in
GNP around a secular trend
Trough
Recovery Peak
Recession
the
lowest
Rising
the
highest
Falling
Growth rate of Negative
real GNP
Rising
the
highest
Falling
Prices
Rising
the
highest
falling
Employment
level
the
lowest
HK’s Economic Performance
Assumptions behind
National Income Models
Assumptions behind
National Income Models
• Y = National income at constant price
• Potential/Full-employment national
income, Yf is constant
• Existence of idle resources, i.e.
unemployment
• 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
Equilibrium Income
Determination of Keynesian's
Two-sector Model (1)A Spendthrift Economy
John Maynard Keynes
Assumptions
• Two sectors: households and firms
• no saving, no tax and no imports
 no leakage/withdrawal
 Y=Yd while Yd = disposable income
• consumer goods only
 no investment or injection
Simple Circular Flow Model of a
Spendthrift Economy
C
Households
National income
National expenditure
Income
generated
Y
E
Firms
Payment for goods
and service
By
Income-expenditure Approach
• AD → (without S) E = C → Y (firms)
↑
↓
Y (households) ← AS ← D for factors
By
Income-expenditure Approach
• Equilibrium income, Ye is determined
when
– AS = AD
– Y = E Y = E = C
Equilibrium Income Determination
of Keynesian's Two-sector Model
(2)-A Frugal Economy
Assumptions
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
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
Simple Circular Flow Model of a
Frugal Economy
C
Households
S
National income
National expenditure
Financial markets
I
Income
generated
Y
E
Firms
Payment for goods
and service
Income Function:
Income line/45 line/Y-line
• an artificial linear function on which
each point showing Y = E
E
Y-line
E2
E1
45
0
Y1 Y2
Y
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
A Consumption Function, C
E
C = a + cYd
C2
C1
a
0
Y1
Y2
Y
Marginal Propensity to
Consume, MPC, c
• MPC = c =
C
Yd
E
M
C = a + cYd
△C
△Y
a
0
Y
Properties of MPC:
• the slope of the consumption function
• 1＞MPC＞0
• the value of 'c' is constant for all
income levels
Average Propensity to
Consume, APC
• APC =
C
Yd
E
M
C = a + cYd
C
a
Y
0
Y
Properties of APC:
• the slope of the ray from the origin
• APC falls when Y rises
• Since C = a + cYd
Then
i.e.
C
Yd
a

c
Yd
APC 
Yd
 (
Yd
a
C
Yd
) 
a
 (c )
Yd
 MPC
Yd
Thus, APC＞MPC for all income levels
Consumption Function
Without ‘a”
• If ‘a’ = 0, then C = cYd
E
a =0
C = cYd
＜45
Y
Consumption Function
Without ‘a”
• If ‘a’ = 0, then MPC = APC =
C
Yd
E
C = cYd
M
C = △C
Y
a =0
Y = △Y
Expenditure Function (2):
Investment Function, I
• showing the relationship between
planned investment expenditure
and disposable income level, Yd
Autonomous Investment
Function
• Autonomous investment function: I = I*
where I* = a constant representing
autonomous investment expenditure
E
I*
0
I = I*
Y
Induced Investment Function
• Induced investment function: I = I* + iYd
where i = Marginal Propensity to Invest
= MPI =
I
Yd
E
I = I* + iYd
I*
0
Y
Properties of MPI:
• the slope of the investment function
• 1＞MPI＞0
• the value of ‘i' is constant for all
income levels
Average Propensity to Invest,
API
• API =
I
Yd
E
I = I* + iYd
M
I
I*
Y
0
Y
Properties of API:
• the slope of the ray from the origin
• API falls when Y rises
• Since I = I* + iYd
Then
i.e.
I
Yd

I*
i
Yd
API 
I*
Yd
 (
Yd
I
Yd
)
I*
 (i )
Yd
 MPI
Yd
Thus, API＞MPI for all income levels
MPI under Autonomous
Investment Function
• If I = I*, then Y will not affect I
• Therefore, MPI =
E
I*
0
I
Yd

0
Yd
0
Slope = MPI = 0
I = I*
Y
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
Aggregate Expenditure
Function, E
•
Since
•
Then

C = a + cYd
I = I* (autonomous function)
E = C+I
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
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
Aggregate Expenditure
Function
E
E=C+I
C = a + cYd
E2
E1
(a+I*)
a
I = I*
I*
0
Y
Y1
Y2
Aggregate Expenditure
Function
E
E2
C = a + cYd
E1
I = I*+iYd
a
I*
0
Y
Y1
Y2
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
A Saving Function, S
E, S
S = -a + sYd
S2
S1
0
-a
Y
Y1
Y2
MPC (c) and MPS (s)
Marginal Propensity to
Saving, MPS, s
• MPS = s =
E, S
S
Yd
M
S = -a + sYd
△S
△Y
0
-a
Y
Properties of MPS:
• the slope of the saving function
• 1＞MPS＞0
• the value of ‘s' is constant for all
income levels
• Since Y ≡ C + S
Then
Yd
Yd

C
Yd

S
Yd
Hence 1 = c + s and s = 1 - c
Average Propensity to Save,
APS
• APS =
E, S
S
Yd
M
S = -a + sYd
S
0
-a
Y
Y
Properties of APS:
• the slope of the ray from the origin
• APS rises when Y rises
• Since S = -a + sYd
Then
i.e.
S
Yd

a
s
Yd
APS 
Yd
 (
Yd
a
S
Yd
)
a
 (s)
Yd
 MPS
Yd
Thus, APS＜MPS for all income levels
Saving Function Without ‘-a”
• If ‘-a’ = 0, then S = sYd
E, S
S = sYd
-a = 0
＜45
Y
Saving Function Without ‘-a”
• If ‘-a’ = 0, then MPS = APS =
S
Yd
E, S
M
S = sYd
S = △S
Y
-a = 0
Y = △Y
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
Given
C = a + cYd
Ye = Y
and I = I*
and Yd = Y
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*
Therefore
Ye 
aI*
1 c
or
aI*
s
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*
Therefore
Ye 
aI*
1 c  i
or
aI*
si
Graphical Representation of Ye
Y-line
E
E=C+I
C = a + cYd
Ee
(a+I*)
a
I = I*
I*
0
Y
Ye
If Investment Function is Induced….
E
Y-line
C = a + cYd
Ee
a
0
Y
Ye
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
Determination of Ye by
Injection-leakage Approach
• In equilibrium:
S=I
(-a + sYd) = (I *)
Then
sY = a + I*
Therefore
Ye 
aI*
s
or
aI*
1 c
If Investment Function is
Induced…
• In equilibrium:
Then
S=I
(-a + sYd) = (I *+iYd)
(s-i)Y = a + I*
Therefore
Ye 
aI*
si
or
aI*
1 c  i
Graphical Representation of Ye
E, S
I=S
S = -a + sYd
I*
0
-a

I = I*
Y
Ye
If Investment Function is
Induced…
E, S
S = -a + sYd
I=S

0
-a
Y
Ye
Graphical Representation of Ye
E($)
Y-line
E=C+I
C
S
45o
I
Ye
Y($)
If Investment Function is Induced…
E($)
Y-line
E=C+I
C
S
I
45o
Ye
Y($)
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
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
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
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.
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
Graphical Representation of Ye
E
Y-line
E = $80+0.8Yd
Ee
C = $30 + 0.8Yd
$(30+50)
I = $50
$50
$30
0
Y
Ye =$400
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
Graphical Representation of Ye
E, S
I=S
S = -$30 + 0.2Yd
$50
0
-$30

I = $50
Y
Ye=$400
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)
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 fullemployment level of income
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
Ex-post Saving Equals Expost Investment
• Actual income must be spent either on
consumption or saving
Y ≡ C + S
• Actual income must be spent buying
either consumer or investment goods
Y≡E≡C+I
Ex-post Saving Equals Expost 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
Ex-ante Saving Equals Exante 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
Ex-ante Saving Equals Exante Investment
• Therefore,
– Realized I = Planned I + Change in
unintended inventory
OR
– Realized I = Planned I – Unrealized
investment
Ex-ante Saving Equals Exante 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.
Ex-ante Saving Equals Exante Investment
• In equilibrium,
– By the Income-expenditure Approach,
• Actual Income = Planned Aggregate
Expenditure
 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)
Ex-ante Saving Equals Exante 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
Ex-ante Saving Equals Exante 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
Ex-ante Saving Equals Exante 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
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
An Illustration
Actual income or output level
(Y)
Planned aggregate
expenditure (E)
Ex-ante
Unintended change in stocks
Actual aggregate expenditure
Ex-post
200
300
400
240
300
360
E>Y
I>S
-40
E=Y
I=S
0
E<Y
I<S
40
240-40 300 360+40
=200
=400
YE YE
YE
Exercise 1
• Given: C = 60 + 0.8Y & I = 60
• Find the equilibrium level of national
income, Ye, by the incomeexpenditure and injection-leakage
approaches.
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 #
Answer 1
• Given: C = 60 + 0.8Y & I = 60
• By the Injection-leakage Approach:
I=S
60 = -60 + 0.2Y
Ye = 600 #
Exercise 2
• Given: C = 60 + 0.8Y & I = 60
•
Show the equilibrium level of national
income, Ye, in a diagram.
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
Exercise 4
• Given C = 10 + 0.8Y
• If Y = 1000, then
and I = 8
– What is the level of realized investment?
Exercise 4
• Given C = 10 + 0.8Y
• If Y = 1000, then
and I = 8
– 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
Exercise 4
• Given C = 10 + 0.8Y
• If Y = 1000, then
and I = 8
– What is the level of unplanned inventory
investment?
Exercise 4
• Given C = 10 + 0.8Y
• If Y = 1000, then
and I = 8
– What is the level of unplanned inventory
investment?
– Unplanned inventory investment =
actual I – planned I = 190 – 8 = 182
In Equilibrium…
• Actual Y = Planned aggregate E
• Ex-ante I = ex-ante S (=actual I =
actual S)
• Unplanned investment = 0
• Unrealized investment = 0
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.
• YC = a + cYd C
• YI = I* + iYd I
Movement Along a
Consumption Function
• YC = a + c Yd C
E
B
C2
A

C1

C = a + cYd
a
0
Y1
Y2
Y
Exercise 5
• Given C = 80 + 0.6Yd. How is
consumption expenditure changed
when Y rises from $100 to $150?
Show it in a diagram.
Answer 5
E
B
170
A

140

C = $80+0.6Yd
$80
0
100
150
Y
Exercise 6
• Given I = 40 + 0.2Yd. How is
investment expenditure changed
when Y rises from $100 to $150?
Show it in a diagram.
Answer 6
E
B
$70
$60

A
 I = $40+0.2Yd
$40
0
$100
Y
$150
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 + cYd
• I*  I = I* or I = I* + iYd
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
Shift of a Consumption
Function
• a  C = a + cYd
E, Y
C2=a2+cYd
C1=a1+cYd
a2
a1
0
Y
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.
Answer 7
E, Y
C2=100+0.6cYd
C1=80+0.6Yd
160
140
100
80
0
Y
100
Shift of an Investment
Function
• I*  I = I*
E, Y
I*2
I*1
0
I2=I*2
I1=I*1
Y
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
Rotation of a Consumption
Function
• c  C = a + cYd
E, Y
C2=a+c2Yd
C1=a+c1Yd
a
0
Y
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.
Answer 8
E, Y
C2=80+0.8Yd
160
C1=80+0.6Yd
140
80
0
Y
100
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.
The Multiplier
• a or I*  E
• E > Y
 planned I > planned S
 unintended inventory disinvestment
 AD > AS  excess demand occurs
 AD = AS (next round)
 E = Y (higher Ye)
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,
K 
change in Y
change in autonomous
expenditur
e
The Multiplier
The Multiplier
E or Y
In itia l
e x p e n d itu re
2 n d ro u n d
3 rd ro u n d
…
T o ta l
S
$1
$ 0 .6
$ 0 .4
$ 0 .3 6
$ 0 .2 4
…
…
$ 1 (1 /0 .4 )= $ 2 .5 $ 0 .4 (1 /0 .4 )= $ 1
The Multiplier
E, Y
Y-line

E2
E
E1 (with a1)

a2
E1
a1
0
E2 (with a2)
K=Y/E
Y
Y1
Y2
The Multiplier
Y 
a I*
or
a I*
1- c
 Y 
Then,
Δ(a  I*)
1 c
Y
 (a  I*)
s
or
Δ(a  I*)
s

1
1 c
or
Thus, by definition , k 
If s  1, then k  1
1
s
Y
E

ΔY
Δ(a  I*)

1
1 c
or
1
s
The Multiplier
• If I is an induced function, then...
Y 
a I*
or
a I*
1 - c -i
 Y 
Then,
Δ(a  I*)
1  c -i
Y
 (a  I*)
s -i
or
Δ(a  I*)
s -i

1
1  c -i
Thus, by definition , k 
or
1
s -i
Y
E

ΔY
Δ(a  I*)

1
1  c -i
or
1
s -i
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.
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
Exercise 9
• Given C = $80 + 0.6Yd
• Find the value of the multiplier if
– I = $40
– I = $40 + 0.1Yd
Exercise 10
• ‘By redistribute $1 from the rich to the
poor will help increase the level of
national income.’ Explain with the
following assumptions:
Exercise 11
• What is the size of the multiplier if the
economy has already achieved full
employment (i.e. Ye = Yf)?