© Pilot Publishing Company Ltd. 2005 Chapter 4 Income Determination I --- Income-expenditure Model.

© Pilot Publishing Company Ltd. 2005

Chapter 4 Income Determination I ---Income-expenditure Model


Contents:• Assumptions of income-expenditure Model• Two-sector economy • Three-sector economy• Four-sector economy• Different kinds of multipliers in different economies • Other points to be noticed• Paradox of thrift• Implications of private saving, public saving & national saving• Advanced Material 4.1 : Equality between investment and saving

in a two-sector economy


Assumptions of Income-expenditure Model


Assumptions of income-expenditure model (or elementary Keynesian model)

The amount of resources and the state of technology remain unchanged, i.e., Yf is a constant.

There exists an unemployment of resources. The model is to find out the determinants of equilibrium GNP and the ways to eliminate unemployment.

No indirect taxes, subsidies, depreciation or net factor income from abroad, i.e., Y = GDP = GNP.

The interest rate and the price level are fixed. So nominal variables = real variables and nom. r = real r.


Two-sector Economy


Two-sector economy

FirmsHouseholds

Composed of households and firms only

No government sector


Features of households

Households provide factor services for income.

Factor Services Income


Features of households

Consumption

Disposable Income

Disposable income is either consumed or saved.

Saving


C = cYd + C*C = cYd + C*

Disposable IncomeDisposable Income

Consumption function

Autonomous ConsumptionAutonomous Consumption

Marginal Propensity to Consume (MPC)

Marginal Propensity to Consume (MPC)

Autonomous consumption (C*) is the consumption at zero disposable income (the minimum amount for subsistence). C* > 0.

Marginal propensity to consume (MPC or c) is the change in consumption resulting from a unit change in disposable income. c < 1.


Graphical Illustration

Consumption function

C = cYd + C*

C

Yd

C*

c+1

0


S = sYd + S*S = sYd + S*

Disposable IncomeDisposable Income

Saving function

Autonomous SavingAutonomous Saving

Marginal Propensity to Save (MPS)

Marginal Propensity to Save (MPS)

Autonomous saving is the saving at zero disposable income.

S* = -C*. Why?

Marginal propensity to save (MPS or s) is the change in saving resulting from a unit change in disposable income. s = 1 – c. Why?


S = sYd + S*

S

Y

S* = -C*

s = (1-c)+1

0

Graphical Illustration

Saving function


Features of firms

1. Firms employ factor services to produce goods.

2. Firms also consume final products (fixed investment & inventories) to help production.

Factor Services Firms

Products


I = i Y + I*I = i Y + I*

National IncomeNational Income

Investment function

Autonomous InvestmentAutonomous Investment

Marginal Propensity to Invest (MPI)

Marginal Propensity to Invest (MPI)

Autonomous investment is the investment at zero income. I* > 0. Why?

Marginal propensity to invest (MPI or i) is the change in investment resulting from a unit change in income. i > 0. Why?


I*

I = iY + I*

Y

r

0

Graphical illustration

Investment function

+1i


Equilibrium condition (2-sector economy)

Aggregate supply (AS) of final products is GNP or Y.

Without government or taxation

AS = Y = Yd = C + S



Aggregate demand (AD) for final products is aggregate expenditure (E).

AD = E = C + I


AS = AD Y = E C + S = C + I S = I

Withdrawal

Equilibrium income (or equilibrium GNP) is reached when AS = AD

Injection =


Withdrawal ( 撤出 , W)

In a 2 sector economy, saving is the withdrawal.

is the amount of income withdrawn from the circular flow, not being spent on final products.


Injection (注入 , J)

is the amount of expenditure on final products injected into the circular flow

not financed by income earned from production.

In a 2 sector economy, investment is the injection.


ABC Ltd.

Firms

Circular flow of a 2-sector economy

Investment Saving

Injection

Withdrawal

Consumption

Y

YdE

Households

When I = S, an equilibrium is achieved.


Z (Y > E) U (Y = E)

V (Y < E)Y=E

Y

E

045o

Meaning of a 45o line

Meaning of a 45° line


Aggregate expenditure function

Y=E

Y

E

045o

E= C+I

C

C*I*

Add consumption and investment functions vertically

I


Adjustment mechanism

Income-expenditure approach

Injection-withdrawal approach

2 approaches:


Y=E

Y

E

0

E=C+I

Ye Y1Y2

Income-expenditure approach

production

production

At Y2, Y < E

Unplanned decrease in inventories

At Y2, Y < E

Unplanned decrease in inventories

At Y1, Y > E

Unplanned increase in inv

entories

At Y1, Y > E

Unplanned increase in inv

entories

}

{


Injection-withdrawal approach

At Y1, J < W Unintended inventory investment

At Y1, J < W Unintended inventory investment

At Y2, J > W Unintended inventory disinvestment

At Y2, J > W Unintended inventory disinvestment

Y

E

0

-C*

I*

S

I

Ye Y1Y2

production

production {}


Multiplier effect

E

Y

Initial equilibrium

ΔEaE:

Y:

ΔEi ΔEi’

•••

Total change in Y = ΔE + c•ΔE + c•c• ΔE + c•c•c• ΔE + …

= (1 + c + c2 + c3 + …) • ΔE = •ΔE

: Change in Y brought by ΔEa

c11


Multiplier (k)

Mathematical expression:

c11

EYk

Multiplier effect

is the ratio of the total change in equilibrium income to the initial change in autonomous expenditure (or autonomous withdrawal) that brought it about.


Equilibrium Y = .(C* + I*)

Mathematical derivation of equil. income & multiplier

In a 2-sector economy

In equilibrium, Y = E = C + I

= cYd + C* + I*

= cY + C* + I*

E = C + I, where C = cYd + C* and I = I*

Y – cY = C* + I* (1 – c)Y= C* + I*

c11


c11

EY

a

When C* or I* changes by ΔEa


ΔY = . ΔEac11


Q4.1:

Calculate the value of multiplier and explain its meaning in each of the following cases.

(a) MPC = 1

(b) MPS = 1


Q4.2:

If the autonomous expenditure decreases by ΔE, what will be the change in equilibrium income?


Three-sector Economy


HouseholdsFirms

Government

Three-sector economy


Government

Government’s expenditure is mainly financed by taxation


Government expenditure function (G)

G = G*, where G > 0G = G*, where G > 0

is fixed by the budget at the beginning of a fiscal year.

G is a constant (G*) independent of any variables.


Tax function

Only direct taxes are concerned. There exist three kinds of direct tax systems.

No indirect taxes is assumed.

Tax System Formula Example

Lump-sum tax system T = T’ Poll tax

Proportional tax system T = tY + T* Profits tax

Progressive tax system T = t*Y + T^ Salaries tax




Yd = Y – T = C + S

Y = Yd + T = C + S + T


Aggregate demand (AD) for final products is E.


E = C + I + G


AS = AD Y = E C + S + T = C + I + G S + T = I + G

Withdrawal

Equilibrium income (or equilibrium GNP) is reached when AS = AD

Injection =


ABC Ltd.

HouseholdsFirms


InvestmentSaving

Injection

Withdrawal

Consumption

Y

E

TaxGovernment expenditure

When I + G = S + T, equilibrium is achieved.



In a 3-sector economy with a lump-sum tax system

In equilibrium, Y = E = C + I + G = cYd + C* + I* + G*

E=C+I+G, where C = cYd + C*; Yd = Y – T’ and I = I* and G = G*

Y = c(Y-T’) + C* + I* + G* = cY- cT’+ C* + I* + G*

Equilibrium Y = .(C*+I*+G*- cT’)

(1–c)Y = C* + I*+ G*- cT’

c1

1



When C* or I* or G* changes by ΔEa

ΔY = . ΔEa

Multiplier =c1

1EY

a

c1

1



Under a lump-sum tax system

Equil. income: )cT'*G*I*(Cc1

1

)cT'*G*I*(C

c1

1

Multiplier:c1

1

c1

1


Mathematical derivation of equil. income & multiplierIn a 3-sector economy with a proportional tax system


E=C+I+G, where C = cYd + C*; Yd = Y - tY - T*; I = I* and G = G*

(1 - c + ct)Y = C* + I* + G* - cT*

Y = c(Y - tY - T*) + C* + I* + G* = cY - ctY - cT* + C* + I* + G*

Equilibrium Y = • (C*+I*+G*-cT*)ctc1

1




ΔY = • ΔEa

Multiplier =

ctc1

1

ctc1

1EY

a


Under a proportional tax system

Equil. income: cT*)*G*I*(Cctc1

1

cT*)*G*I*(C

ctc1

1

Multiplier:ctc1

1

ctc1

1

The multiplier of proportional tax system is smaller than the multiplier of lump-sum tax system.

As t > 0, (1-c) < (1-c+ct)

c1

1

ctc1

1



With the addition of a proportional transfer payment, where Yd = Y - tY - T* + qY + Q*


Equilibrium Y = • (C*+I*+G*-cT*-cQ*)

Y = c(Y-tY-T*+qY+Q*) + C* + I* + G* (1–c+ct-cq) •Y = C* + I* + G* -cT* + cQ*

cq-ctc1

1


ΔY = • ΔEa


cq-ctc1

1

Multiplier =


cq-ctc1

1EY

a


Under a proportional tax & transfer payment system

Equil. income: cQ*)*cT*G*I*(Ccq-ctc1

1

cQ*)*cT*G*I*(C

cq-ctc1

1

Multiplier:cq-ctc1

1

cq-ctc1

1

As q<0, (1-c+ct)<(1-c+ct-cq)

The multiplier with a proportional transfer payment is smaller than the multiplier without it.

ctc1

1

cq-ctc1

1

>>


Four-sector Economy


Four-sector economy

Foreign Sector

HouseholdsFirms

Government


Export function:

X=X*, where X*>0X=X*, where X*>0

Export is determined by foreign economies, not the domestic economy

It is autonomous & is independent of Y


Import function:

M= mY +M*M= mY +M*

Marginal Propensity to Import (MPM)

Marginal Propensity to Import (MPM) Autonomous ImportAutonomous Import

MPM is the change in the value of imports resulting from a unit change in national income. MPM > 0.

M* is the value of imports at zero income. M* > 0.

All economic agents consume imports

Import is positively related to Y




Y = Yd + T

= C + S + T



Aggregate demand (AD) for final products is aggregate expenditure (E).

E = (C-MC) + (I-MI) + (G-MG) + (X-MX)

= C + I + G + X - (MC+MI+MG+MX)

= C + I + G + X - M


AS = AD

Y = E

C + S + T = C + I + G + X – M

S + T = I + G + X – M

S + T + M = I + G + X

Equilibrium income (or equilibrium GNP) is reached when

Injection= Withdrawal


ABC Ltd.

HouseholdsFirms


I - MISaving

InjectionWithdrawal

C - MC

Y

E

TaxG - MG

X - MXWhen I + G + X – M = S + T,

equilibrium is achieved.



In a four-sector economy, E = C + I + G + X – M

where C = cYd + C*;

Yd = Y - tY - T* + qY + Q*;

I = I*;

G = G*;

X = X*;

M = mY + M*


Y = c(Y-tY-T*+qY+Q*)+C*+I*+G*+X*-mY-M*

(1-c+ct-cq+m)•Y = C*+I*+G*+X*-M*-cT*+cQ*

Equil.Y = •(C*+I*+G*+X*-M*-cT*-cQ*)


In equilibrium, Y = E = C+I+G+X-M = cYd+C*+I*+G*+X*-mY-M*

mcq-ctc1

1



When C* or I* or G* or X* changes by ΔEa

mcq-ctc11

EY

a

ΔY = • ΔEamcq-ctc1

1


Q4.5

Derive the equilibrium income and the multiplier in a four-sector economy if investment is an induced expenditure (I = iY + I*).


Different Kinds of Multipliers in Different Economies


Expenditure multiplier (k)

Two-sector economy

Three-sector economy

Four-sector economy

ic1

1

cqctic1

1

mcqctic1

1


Comparison of the size of expenditure multipliers (k)

As c>0, i>0, t>0, -q>0, and m>0,

mcqctic1cqctic1ic1

mcqctic1

1

cqctic1

1

ic1

1

k in a 2-sector economy > k in a 3-sector economy > k in a 4-sector economy

k in a 2-sector economy > k in a 3-sector economy > k in a 4-sector economy


Explanation:

E

Y

Initial equilibrium

ΔEaE:

Y:

ΔEi

•••

ΔEi (2-sector economy) =c • ΔEa + i • ΔEa

: Change in Y brought by ΔEa

ΔEi (3-sector economy) =c•(1-t+q) • ΔEa + i • ΔEa

ΔEi (4-sector economy) =c•(1-t+q) • ΔEa+ i •ΔEa– m •ΔEa


Tax multiplier

mcqctic1

c

*T

Y

mcqctic1

c

*T

Y

Tax is a withdrawal Its multiplier is negative.

When T*: by $1 Yd: by $1 C: by -$c

∆Y = -c x expenditure multiplier.


Q4.6:

Derive the transfer payment multiplier and explain why it is smaller than the expenditure multiplier.


Import multiplier

mcqctic1

1

*M

Y

mcqctic1

1

*M

Y

Import is a withdrawal Its multiplier is negative.

When M*: by $1 AE: by $1 ΔY = -1 x expenditure multiplier.


Multipliers at full employment

ΔEa > 0 Multiplier = 0

At full employment, as all resources have been used efficiently, real income can no longer be raised (when ΔEa>0, ΔY=0) but it can be lowered (when ΔEa<0, ΔY<0).

ΔEa < 0 Multiplier = Unchanged


Other Points to be Noticed


More about aggregate expenditure function

E = C + I + G + X - M

= c(Y-tY-T*+qY+Q*)+C* +iY+I* +G* +X* - mY- M*

E = (c + i - ct + cq-m) Y + (C*+I*+G*+X*-M*-cT*+ cQ*)

Slope of E-functionSlope of E-function E-interceptE-intercept


Q4.7:

Derive the expenditure multiplier from the income-expenditure diagram.

Q4.8:

Expenditure multiplier is ΔY/ΔE and slope of E-function is ΔE/ΔY. Is the expenditure multiplier equal to the inverse of the slope of E-function?


An induced in E

An autonomous in E

E

E’

E

Y

An autonomous change versus an induced change


E

Y

Y=E

E’ (slope > 1)

ΔE’>ΔY

ΔYE (Slope < 1)

ΔE<ΔYΔY

45o

Can’t find the

equilibrium

Can’t find the

equilibrium

Slope of E-function cannot be greater than one

The equilibriumThe equilibrium


Deflationary AD gap is the amount of expenditure by which the present expenditure falls short of the expenditure achieving full employment.

Deflationary income gap is the amount of income by which the equilibrium income falls short of the full employment income.

Deflationary Gap


Deflationary income gap

45o

E1

}{Deflationary

AD gap

Ef

E

Y

Y=E

YfY1



Inflationary AD gap is the amount of expenditure by which the present expenditure exceeds the expenditure achieving full employment.

Inflationary income gap is the amount of income by which the equilibrium income exceeds the full employment income.

Inflationary Gap


45o

E2

}Inflationary income gap

{Inflationary AD gap

Ef

E

Y

Y=E

Yf Y2



Deflationary income gap

Inflationary income gap

= Deflationary AD gap x multiplier

= Inflationary AD gap x multiplier


Paradox of thrift The puzzle why national income falls (the society gets poorer) when people as a whole save more.


SP’

Ye’

}Unintended

inventory investment

Unintended inventory investment

0

SP

IP

S,I

YYe

Note: If investment is an autonomous expenditure, the results are Y, C & S unchanged (= I)

Saving is detrimental when Saving but the unspent income does not re-enter the circular flow

Saving but the unspent income does not re-enter the circular flow

Firms cut production

Firms cut production

Income (Y) Income (Y)


SP’

0

SP

IP

S,I

YYe’ Ye

}Unintended inventory

investment

Unintended inventory

investment

Saving is detrimental

Note: If investment is an induced expenditure, the results are Y, C & S (= I)


SP’

Ye=Ye’

IP’

Saving is beneficial when Saving and the unspent income can re-enter the circular flow as investment

Saving and the unspent income can re-enter the circular flow as investment

Then Y is unchanged. In addition, as I, productivity.

Then Y is unchanged. In addition, as I, productivity.

0

SP

IP

S,I

YNote: If investment is an autonomous expenditure, the results are Y unchanged, C, S & I


SP’

0

SP

IP

S,I

YYe=Ye’

IP ’

Saving is beneficial

Note: If investment is an induced expenditure, the results are Y unchanged, C, S & I


Implications of private saving, public saving & national saving

• Private saving (SP or S, 私人儲蓄 ) is the saving of households, i.e., SP = Yd – C = Y – T – C.

• Public saving (SG, 公共儲蓄 ) is the saving of the government, also called fiscal surplus. SG = T – G.

• National saving (SN, 國民儲蓄 ) is the saving of the economy as a whole. SN = SP + SG.

Definition:


Implications of private saving

In equilibrium, total withdrawal = total injection.

SP + T + M = I + G + X SP = I + (G – T) + (X – M) ………………. (1)

In equilibrium, AS = AD. Resources not consumed by households (private saving) must be consumed by other economic agents – by firms as investment (I), and/or by the government creating fiscal deficit (G - T), and/or by the foreign sector as net exports (X - M).


Implications of public savingIn equilibrium, total withdrawal = total injection.

SP + T + M = I + G + X Fiscal surplus = SG = T – G = (I – SP) + (X – M) …….. (2) Fiscal deficit = -SG = G – T = (SP – I) + (M – X) …….. (3)

Equation (2): In equilibrium, AS = AD. Resources not consumed by the government (public saving) must be consumed by other economic agents – by private sector (I – SP), and/or by the foreign sector (X - M).

Equation (3): If there exists fiscal deficit, the resources have to be supplied by the private sector and/or the foreign sector, through the issuance of internal debt (SP - I), and/or external debt (which enables the economy to have net imports, M – X).


Implications of natioinal saving

By definition, SN = SP + SG = [I + (G-T) + (X-M)] + (T-G) SN = I + (X – M) …….. (4)

In equilibrium, CA + KA = 0 CA = X – M = -KA.

From equation (4), SN – I = X – M = CA = -KA …….. (5)

Equation (4): In equilibrium, AS = AD. Resources not consumed by households and the government (national saving) must be consumed by other economic agents – by firms as investment (I), and/or by the foreign sector as net exports (X - M).Equation (5): In equilibrium, AS = AD. Resources not consumed by our economy (SN - I) must be consumed by foreign economies as net exports (X - M) and illustrated by our current account surplus. To have external balance, the capital account must have deficit (CA = -KA).


Advanced Material 4.1Equality between investment and saving in a

two-sector economy

Meaning of equality between ex-ante investment & ex-ante saving

In a two-sector economy, equality between ex-ante (or planned or desired) investment and ex-ante saving is the equilibrium condition.


With unintended inventory investment Y

With unintended inventory investment Y

0

SP

IP

S,I

YYe Y1

IP

Sp

(Sp>Ip)

Derivation


Sp

0

SP

IP

S,I

YYeY2

IP

(Sp<Ip)

With unintended inventory disinvestment Y

With unintended inventory disinvestment Y


0

SP

IP

S,I

YYe

(Sp=Ip)

No unintended change in inventories YeNo unintended change in inventories Ye

Equilibrium condition of a 2-sector economy


Meaning of equality between ex-post investment and ex-post saving

In a two-sector economy, equality between ex-post (or actual or realized or observed) investment and ex-post saving is an identity.

As they must always be equal, the equality is a tautology without any economic meaning or implication.

Equality between investment and saving in a two-sector economy


0

SP

S,I

YYe Y1

IP

[Sp (= Sa) >Ip] but

Ia = Ip + Iu (>0) = Sa

[Sp (= Sa) >Ip] but

Ia = Ip + Iu (>0) = Sa

Derivation

Y2

IP

Iu (>0)

IP

Iu (<0)

IP

[Sp (= Sa) < Ip] but

Ia = Ip + Iu (<0) = Sa

[Sp (= Sa) < Ip] but

Ia = Ip + Iu (<0) = Sa

[Sp (= Sa) = Ip]

Ia = Ip + Iu(=0) = Sa

[Sp (= Sa) = Ip]

Ia = Ip + Iu(=0) = Sa


Implications

1. In a 4-sector economy, equality between planned total withdrawal and planned total injection is the equilibrium condition.

2. In a 4-sector economy, equality between actual total withdrawal and actual total injection is an identity and is meaningless in economics.


Planned investment (Ip) is the planned change in fixed capital & inventories.

Different terms related to investment

Unplanned investment (Iu) is the unplanned change in inventories, which is the amount not purchased by any economic agents.

Realized investment is the amount of actual investment (= Ip + Iu).

Unrealized investment is the amount of actual investment falling short of the amount of planned investment (= Iu < 0).


Correcting Misconceptions:

1. The multiplier of an autonomous decrease in expenditure is negative.

2. The import function is represented by the linear equation: M = mYd + M*.

3. Multipliers must be positive.

4. An increase in aggregate expenditure would shift the E-curve upward in a parallel manner.


5. Equilibrium income is the same as full employment income.

6. Equality between total injection & total withdrawal is the equilibrium condition of goods market.

7. An increase in saving (a withdrawal) is detrimental to an economy.

Correcting Misconceptions:

© Pilot Publishing Company Ltd. 2005 Chapter 4 Income Determination I --- Income-expenditure Model.

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Transcript of © Pilot Publishing Company Ltd. 2005 Chapter 4 Income Determination I --- Income-expenditure Model.