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

0

GNP

Time

Boom

Recession

Depression

Recovery

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

HK’s Economic Performance

Assumptions behind National Income Models

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

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

John Maynard Keynes

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

Simple Circular Flow Model of a Spendthrift Economy

Households

Firms

National income

National expenditure

Incomegenerated

Payment for goods and service

C

E Y

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

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

Assumptions

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

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

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

C1

Y1

C2

Y20

E

Y

C = a + cYd

a

Marginal Propensity to Consume, MPC, c

• MPC = c =dY

C

0

E

Y

C = a + cYd

a

△ C

△ Y

M

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 =

0

E

Y

C = a + cYd

a

C

Y

M

dY

C

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

Consumption Function Without ‘a”

• If ‘a’ = 0, then C = cYd

0

E

Y

C = cYd

a =＜ 45

Consumption Function Without ‘a”

0

E

Y

C = cYd

a =

C = △ C

Y = △ Y

M

• If ‘a’ = 0, then MPC = APC =dY

C

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

0 Y

I = I*I*

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

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 =

0

E

Y

I = I* + iYd

I*

I

Y

M

dY

I

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

*

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

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 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

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

E1

Y1

E2

Y20

E

YI* I = I*a

C = a + cYd

(a+I*)

E = C + I

Aggregate Expenditure Function

E1

Y1

E2

Y20

E

YI*

I = I*+iYda

C = a + cYd

(a+I*)

E = C + I

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

S1

Y1

S2

Y20

E, S

Y

S = -a + sYd

-a

MPC (c) and MPS (s)

Marginal Propensity to Saving, MPS, s• MPS = s =

△ S

△ Y

M

dY

S

S = -a + sYd

-a0

E, S

Y

• 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

Average Propensity to Save, APS• APS =

S

Y

MS = -a + sYd

-a0

E, S

Y

dY

S

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

Saving Function Without ‘-a”

• If ‘-a’ = 0, then S = sYd

0

E, S

Y

S = sYd

-a =＜ 45

Saving Function Without ‘-a”

0

E, S

Y

S = △ S

Y = △ Y

M

• If ‘-a’ = 0, then MPS = APS =

S = sYd

-a =

dY

S

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

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

*

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

*

Graphical Representation of Ye

0

E

YI* I = I*a

C = a + cYd

(a+I*)

E = C + IY-line

Ee

Ye

If Investment Function is Induced….

0

E

YI*

I = I*+iYd

a

C = a + cYd

(a+I*)

E = C + IY-line

Ee

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*

Thereforec

Iaor

s

IaYe

1

**

If Investment Function is Induced…

• In equilibrium:

S = I

(-a + sYd) = (I *+iYd)

Then (s-i)Y = a + I*

Thereforeic

Iaor

is

IaYe

1

**

Graphical Representation of Ye

0

E, S

Y

I* I = I*

-a

S = -a + sYd

Ye

I = S

If Investment Function is Induced…

0

E, S

YI*

I = I*+iYd

-a

S = -a + sYd

Ye

I = S

Graphical Representation of Ye

E($)

Y($)

I

C

E = C + I

S

Y-line

45o

Ye

If Investment Function is Induced…

E($)

Y($)

I

C

E = C + I

S

Y-line

45o

Ye

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

0

E

Y

$50 I = $50

$30

C = $30 + 0.8Yd

$(30+50)

E = $80+0.8Yd

Y-line

Ee

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

0

E, S

Y

$50 I = $50

-$30

S = -$30 + 0.2Yd

Ye=$400

I = S

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 full-employment 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 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

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

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

Ex-ante Saving Equals Ex-ante Investment

• Therefore,– Realized I = Planned I + Change in

unintended inventory

OR– Realized I = Planned I – Unrealized

investment

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.

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)

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

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

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

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)

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

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.

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 and I = 8

• If Y = 1000, then– What is the level of realized investment?

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

Exercise 4

• Given C = 10 + 0.8Y and I = 8

• If Y = 1000, then– What is the level of unplanned inventory

investment?

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

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.

YC = a + cYd CYI = I* + iYd I

Movement Along a Consumption FunctionYC = a + c Yd C

C = a + cYd

C1

Y10

E

Y

a

C2

Y2

A

B

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

C = $80+0.6Yd

170

150

140

1000

E

Y

$80

A

B

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

I = $40+0.2Yd

0

E

Y

$40

AB

$60

$100

$70

$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 + cYdI* 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 Functiona C = a + cYd

C2=a2+cYd

a2

C1=a1+cYd

a1

E, Y

Y0

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

C2=100+0.6cYd

C1=80+0.6Yd

80

E, Y

Y0

140

100

160

100

Shift of an Investment FunctionI* I = I*

I1=I*1I*1

E, Y

Y0

I2=I*2I*2

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 Functionc C = a + cYd

C2=a+c2Yd

C1=a+c1Yd

a

E, Y

Y0

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

C2=80+0.8Yd

C1=80+0.6Yd

80

E, Y

Y0100

160

140

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 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)

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

The Multiplier

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

The Multiplier

0

E1 (with a1)

a1E1

Y1

E, Y

Y

Y-line E2 (with a2)

a2

K=Y/E

Y

Y2

E

E2

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

The Multiplier

-ior

-i

-ior

-i

-ior

-i

-ior

-i

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

• If I is an induced function, then...

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)?