HKALE Macroeconomics

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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
GNP
Recovery
Boom
Recession
Depression
0
Time
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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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HKs Economic Performance
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Assumptions behind National Income Models
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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 PQ P 1, then Y (1)Q ? Y Q
• Price level tends to be rigid in downward
  direction

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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

• Two sectors households and firms

• no saving, no tax and no imports

• no leakage/withdrawal

• YYd while Yd disposable income

• consumer goods only

? no investment or injection
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Simple Circular Flow Model of a Spendthrift
Economy
Households
C

Income generated
Payment for goods and service
Y
E
Firms
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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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Assumptions

• 1. Households and firms
• 2. Saving, S, exists
• Income is either consumed or saved
• ? Y CS
• leakage, S, exists
• 3. Without tax, YYd

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Assumptions

• 4. Consumer and producer goods
• Injection (investment, I) exist
• 5. Investment is autonomous/exogenous
• 6. Saving and investment decisions
• made separately
• SI occurs only at equilibrium level of income

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Simple Circular Flow Model of a Frugal Economy
Households
C
S
Financial markets
I
Income generated
Payment for goods and service
Y
E
Firms
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Income Function Income line/45? line/Y-line

• an artificial linear function on which each point
  showing Y E

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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
E
C a cYd
a
Y
0
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Marginal Propensity to Consume, MPC, c

• MPC c

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Properties of MPC

• the slope of the consumption function
• 1gtMPCgt0
• the value of 'c' is constant for all income levels

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Average Propensity to Consume, APC

• APC

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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, APCgtMPC for all income levels
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Consumption Function Without a

• If a 0, then C cYd

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Consumption Function Without a

• If a 0, then MPC APC

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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

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Induced Investment Function

• Induced investment function I I iYd
• where i Marginal Propensity to Invest

MPI
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Properties of MPI

• the slope of the investment function
• 1gtMPIgt0
• the value of i' is constant for all income levels

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Average Propensity to Invest, API

• API

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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, APIgtMPI for all income levels
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MPI under Autonomous Investment Function

• If I I, then ?Y will not affect I

• Therefore, MPI

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 CI

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Aggregate Expenditure Function, E

• Since C a cYd
• I I (autonomous function)
• E CI
• 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 CI
• 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
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Aggregate Expenditure Function
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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
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MPC (c) and MPS (s)
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Marginal Propensity to Saving, MPS, s

• MPS s

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Properties of MPS

• the slope of the saving function
• 1gtMPSgt0
• the value of s' is constant for all income
  levels
• Since Y C S

Then
Hence 1 c s and s 1 - c
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Average Propensity to Save, APS

• APS

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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, APSltMPS for all income levels
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Saving Function Without -a

• If -a 0, then S sYd

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Saving Function Without -a

• If -a 0, then MPS APS

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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
• Given C 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

Therefore
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If Investment Function is Induced

• In equilibrium
• Y E C I
• (a cYd) (I iYd)
• ? Y- (ci)Y a I
• Then Y(1-c-i) a I

Therefore
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Graphical Representation of Ye
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If Investment Function is Induced.
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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

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Determination of Ye by Injection-leakage Approach

• In equilibrium
• S I
• (-a sYd) (I )
• Then sY a I

Therefore
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If Investment Function is Induced

• In equilibrium
• S I
• (-a sYd) (I iYd)
• Then (s-i)Y a I

Therefore
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Graphical Representation of Ye
?
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If Investment Function is Induced
?
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Graphical Representation of Ye
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If Investment Function is Induced
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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
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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
?
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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 lt 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 saving
• ?Y 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 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)

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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 gt Y, then
• ? AD gt AS
• ? planned I gt 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 lt Y, then
• ? AD lt AS
• ? planned I lt 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-ante EgtY EY EltY
Ex-ante IgtS IS IltS
Unintended change in stocks -40 0 40
Actual aggregate expenditure 240-40 200 300 36040 400
Ex-post Y?E Y?E Y?E
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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.
• ?Y?C a c?Yd? ?C
• ?Y?I I i?Yd? ?I

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Movement Along a Consumption Function

• ?Y?C a c ?Yd? ?C

C a cYd
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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 800.6Yd
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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
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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 cYd
• ?I ? ?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 Function

• ?a ? ?C ?a cYd

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Exercise 7

• Given C800.6Yd Y100. How is consumption
  function affected if autonomous consumption
  expenditure rises to 100? Show it in a diagram.

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Answer 7
160
100
108
Shift of an Investment Function

• ?I ? ?I ?I

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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 Function

• ?c ? ?C a ?cYd

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Exercise 8

• Given C800.6Yd Y100. How is consumption
  function affected if MPC rises to 0.8? Show it in
  a diagram.

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Answer 8
160
140
100
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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 Multiplier

• ?a or ?I ? ?E
• ?E gt Y
• ? planned I gt planned S
• ? unintended inventory disinvestment
• ? AD gt 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,

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The Multiplier
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The Multiplier
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The Multiplier
K?Y/?E
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The Multiplier
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The Multiplier

• 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.

122
Remarks on the Multiplier

• If MPS 1 or MPC 0 and MPI 0
• then, k1/1-c 1
• If MPS 0 or MPC 1 and MPI 0
• then, k1/1-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)?