1. Chapter 2: Elementary Keynesian Model (I)- Two-sector
HKALE Macroeconomics
Chapter 2: Elementary Keynesian Model (I)-
Two-sector

2. 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

3. 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.

4. 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.

5. Business Cycle
GNP
Recovery
Boom
Recession
Depression
Time

6. 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
Prices
falling

7. HK’s Economic Performance

8. Assumptions behind National Income Models

9. 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

10. Equilibrium Income Determination of Keynesian's Two-sector Model (1)- A Spendthrift Economy

11. John Maynard Keynes

12. 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

13. Simple Circular Flow Model of a Spendthrift Economy
National income
National expenditure
Households C
Income
generated
Payment for goods
and service Y E
Firms

14. By Income-expenditure Approach
AD → (without S) E = C → Y (firms)
↑ ↓
Y (households) ← AS ← D for factors

15. By Income-expenditure Approach
Equilibrium income, Ye is determined when
AS = AD
Y = E Y = E = C

16. Equilibrium Income Determination of Keynesian's Two-sector Model (2)-A Frugal Economy

17. 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

18. 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

19. Simple Circular Flow Model of a Frugal Economy
National income
National expenditure
Households C S
Financial markets I
Income
generated
Payment for goods
and service Y E
Firms

20. Income Function: Income line/45 line/Y-line
an artificial linear function on which each point showing Y = E
45 E Y
Y-line E2 Y2 E1 Y1

21. 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

22. A Consumption Function, CE
C = a + cYd C2 Y2 C1 Y1 a Y

23. Marginal Propensity to Consume, MPC, cE Y
C = a + cYd a  M △C △Y

24. Properties of MPC: the slope of the consumption function 1＞MPC＞0
the value of 'c' is constant for all income levels

25. Average Propensity to Consume, APCE Y
C = a + cYd a  M C Y

26. 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

27. Consumption Function Without ‘a”
If ‘a’ = 0, then C = cYd E Y
C = cYd
a =
＜45

28. Consumption Function Without ‘a”
If ‘a’ = 0, then MPC = APC = E Y
C = cYd
a =  M
C = △C
Y = △Y

29. Expenditure Function (2): Investment Function, I
showing the relationship between
planned investment expenditure and disposable income level, Yd

30. Autonomous Investment Function
Autonomous investment function: I = I*
where I* = a constant representing
autonomous investment expenditure E Y
I = I* I*

31. Induced Investment Function
Induced investment function: I = I* + iYd
where i = Marginal Propensity to Invest
= MPI = E Y
I = I* + iYd I*

32. Properties of MPI: the slope of the investment function 1＞MPI＞0
the value of ‘i' is constant for all income levels

33. Average Propensity to Invest, APIE Y
I = I* + iYd I*  M I Y

34. 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

35. MPI under Autonomous Investment Function
If I = I*, then Y will not affect I
Therefore, MPI = E Y
Slope = MPI = 0
I = I* I*

36. 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

37. 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

38. 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

39. Aggregate Expenditure Function
(a+I*)
E = C + I E Y a
C = a + cYd E2 Y2 E1 Y1 I*
I = I*

40. Aggregate Expenditure FunctionE Y
(a+I*)
E = C + I E2 Y2 a
C = a + cYd E1 Y1 I*
I = I*+iYd

41. 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

42. A Saving Function, S
E, S Y
S = -a + sYd -a S2 Y2 S1 Y1

43. MPC (c) and MPS (s)

44. Marginal Propensity to Saving, MPS, s
E, S Y
S = -a + sYd -a  M △S △Y

45. 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
Hence 1 = c + s and s = 1 - c

46. Average Propensity to Save, APS
E, S Y
S = -a + sYd -a  M S Y

47. 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

48. Saving Function Without ‘-a”
If ‘-a’ = 0, then S = sYd
E, S Y
S = sYd
-a =
＜45

49. Saving Function Without ‘-a”
If ‘-a’ = 0, then MPS = APS =
E, S Y
S = sYd
-a =  M
S = △S
Y = △Y

50. 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 and I = I*
Ye = Y and Yd = Y

51. 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

52. 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

53. Graphical Representation of Ye
Y-line
(a+I*)
E = C + I E Y a
C = a + cYd Ee Ye I*
I = I*

54. If Investment Function is Induced….
Y-line
(a+I*)
E = C + I E Y a
C = a + cYd Ee Ye I*
I = I*+iYd

55. 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

56. Determination of Ye by Injection-leakage Approach
In equilibrium:
S = I
(-a + sYd) = (I *)
Then sY = a + I*
Therefore

57. If Investment Function is Induced…
In equilibrium:
S = I
(-a + sYd) = (I *+iYd)
Then (s-i)Y = a + I*
Therefore

58. Graphical Representation of Ye
E, S Y
I = S -a
S = -a + sYd I*
I = I*  Ye

59. If Investment Function is Induced…
E, S Y -a
S = -a + sYd
I = S I*
I = I*+iYd  Ye

60. Graphical Representation of Ye
E = C + I S
Y-line
45o Ye

61. If Investment Function is Induced…
Y($) I C
E = C + I S
Y-line
45o Ye

62. A Two-sector Model: An Example
Given:
C = $ Y
I = $40
Since
E = C + I = ($ Y)+($40)
Then, E = $ Y

63. A Two-sector Model: An Example
By income-expenditure approach, in equilibrium:
Y = E = C + I
Then Y = ($ Y)
(1-0.6)Y = $120
Thus, Y = $120/0.4 = $300

64. 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, $40 = (-$ Y)
0.4Y = $120
Thus, Y = $120/0.4 = $300

65. A Two-sector Model: Exercise
Given:
C = $ Y
I = $50
Question: (1) Find the equilibrium national income level by the two approaches. (2) Show your answers in two separate diagrams.

66. A Two-sector Model: Exercise
By income-expenditure approach, in equilibrium:
Y = E = C + I
Then Y = ($ ) + 0.8Y
(1-0.8)Y = $80
Thus, Y = $80/0.2 = $400

67. Graphical Representation of YeE Y
$50
I = $50
$30
C = $ Yd
$(30+50)
E = $80+0.8Yd
Y-line Ee
Ye =$400

68. 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, $50 = (-$ Y)
0.2Y = $80
Thus, Y = $80/0.2 = $400

69. Graphical Representation of Ye
E, S Y
I = S
-$30
S = -$ Yd
$50
I = $50 
Ye=$400

71. 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)

72. 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

73. An Application (1) Since Y = 2L ($300) = 2L Then, L = 150
(2 Unemployment level = = 50
(3) Since Yf = 2L = 2(200) = $400
Then, Ye < Yf by (400 – 300)$100

74. Ex-post Saving Equals Ex-post 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

75. 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

76. 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

77. Ex-ante Saving Equals Ex-ante Investment
Therefore,
Realized I = Planned I + Change in unintended inventory OR
Realized I = Planned I – Unrealized investment

78. 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.

79. Ex-ante Saving Equals Ex-ante 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)

80. 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

81. 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

82. 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

83. An Illustration MPC, c = (140-80)/(100-0) = 0.6
(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 80
-80 40
120
-120
100
140
-40
180
200
240
300
260
400
320
360
500
380
420
MPC, c = (140-80)/(100-0) = 0.6
C = a + cYd = Yd
I = 40 and E = C + I = Yd

84. An Illustration Actual income or output level (Y) 200 300 400
Planned aggregate expenditure (E)
240
360
Ex-ante
E>Y
E=Y
E<Y
I>S
I=S
I<S
Unintended change in stocks
-40 40
Actual aggregate expenditure
240-40
=200
360+40
=400
Ex-post
YE

85. 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.

86. Answer 1
Given: C = Y & I = 60
By the Income-expenditure Approach:
Ye = E = C + I
Ye = ( Y) + (60)
Ye = 600 #

87. Answer 1
Given: C = Y & I = 60
By the Injection-leakage Approach:
I = S
60 = Y
Ye = 600 #

88. Exercise 2 Given: C = 60 + 0.8Y & I = 60
Show the equilibrium level of national income, Ye, in a diagram.

90. Exercise 3 Y P. C. P. S. P. I. P. A. E. U.C.I. UR.I. A. I. 60 -60 120
(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 60
-60
120
-120
200
220
-20
280
-80 80
300
360
400
380 20
440
-40 40
500
460
520
600
540
700
620
680

92. Exercise 4 Given C = 10 + 0.8Y and I = 8 If Y = 1000, then
What is the level of realized investment?

93. 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 = (1000) = 810
As Y  C + S
 Actual S = I = = 190

94. Exercise 4 Given C = 10 + 0.8Y and I = 8 If Y = 1000, then
What is the level of unplanned inventory investment?

95. 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

96. In Equilibrium… Actual Y = Planned aggregate E
Ex-ante I = ex-ante S (=actual I = actual S)
Unplanned investment = 0
Unrealized investment = 0

97. 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

98. Movement Along a Consumption Function
YC = a + c Yd C C1 Y1 E Y
C = a + cYd  A B C2 Y2

99. Exercise 5
Given C = Yd. How is consumption expenditure changed when Y rises from $100 to $150? Show it in a diagram.

100. Answer 5
140
100 E Y
C = $80+0.6Yd  A B
170
150

101. Exercise 6
Given I = Yd. How is investment expenditure changed when Y rises from $100 to $150? Show it in a diagram.

102. Answer 6
I = $40+0.2Yd E Y  A B
$70
$150
$60
$100

103. 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

104. 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

105. Shift of a Consumption Function
a  C = a + cYd
C1=a1+cYd a1
E, Y Y
C2=a2+cYd a2

106. 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.

107. Answer 7
C1=80+0.6Yd 80
E, Y Y
140
100
C2= cYd
160
100

108. Shift of an Investment Function
I*  I = I*
I1=I*1
I*1
E, Y Y
I2=I*2
I*2

109. 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

110. Rotation of a Consumption Function
c  C = a + cYd
C1=a+c1Yd a
E, Y Y
C2=a+c2Yd

111. 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.

112. Answer 8
C1=80+0.6Yd 80
E, Y Y
C2=80+0.8Yd
160
140
100

113. 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.

114. 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)

115. 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,

116. The Multiplier

117. The Multiplier

118. The Multiplier E, Y Y-line E2 (with a2) E2  E E1 (with a1) a2 E1
E1 (with a1) a1 E1 Y1
E, Y Y
Y-line
E2 (with a2) a2  E E2
K=Y/E  Y Y2

119. The Multiplier

120. The Multiplier
If I is an induced function, then...

121. 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.

122. 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

123. Exercise 9 Given C = $80 + 0.6Yd Find the value of the multiplier if
I = $ Yd

124. Exercise 10
‘By redistribute $1 from the rich to the poor will help increase the level of national income.’ Explain with the following assumptions:

125. Exercise 11
What is the size of the multiplier if the economy has already achieved full employment (i.e. Ye = Yf)?