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Simple Keynesian Model. National Income Determination Three-Sector National Income Model. Outline. Three-Sector Model Tax Function T = f (Y) Consumption Function C = f (Yd) Government Expenditure Function G=f(Y) Aggregate Expenditure Function E = f(Y)

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Simple Keynesian Model

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Simple Keynesian Model

National Income Determination

Three-Sector National Income Model


Outline

  • Three-Sector Model

  • Tax Function T = f (Y)

  • Consumption Function C = f (Yd)

  • Government Expenditure Function G=f(Y)

  • Aggregate Expenditure Function E = f(Y)

  • Output-Expenditure Approach: Equilibrium National Income Ye


Outline

  • Factors affecting Ye

  • Expenditure Multipliers k E

  • Tax Multipliers k T

  • Balanced-Budget Multipliers k B

  • Injection-Withdrawal Approach: Equilibrium National Income Ye


Outline

  • Fiscal Policy (v.s. Monetary Policy)

  • Recessionary Gap Yf - Ye

  • Inflationary Gap Ye - Yf

  • Financing the Government Budget

  • Automatic Built-in Stabilizers


Three-Sector Model

  • With the introduction of the government sector (i.e. together with households C, firms I), aggregate expenditure E consists of one more component, government expenditure G.

    E = C + I+ G

  • Still, the equilibrium condition is

    Planned Y = Planned E


Three-Sector Model

  • Consumption function is positively related to disposable income Yd [slide 37 of 2-sector model],

    C = f(Yd)

    C= C’

    C= cYd

    C= C’ + cYd


Three-Sector Model

  • National Income  Personal Income  Disposable Personal Income

  • w/ direct income tax Ta and transfer payment Tr

  • Yd  Y

  • Yd = Y - Ta + Tr


Three-Sector Model

  • Transfer payment Tr can be treated as negative tax, T is defined as direct income tax Ta net of transfer payment Tr

  • T = Ta - Tr

  • Yd = Y - (Ta - Tr)

  • Yd = Y - T


Three-Sector Model

  • The assumptions for the 2-sector Keynesian model are still valid for this 3-sector model [slide 24-25 of 2-sector model]


Tax Function

  • T = f(Y)

  • T = T’

  • T = tY

  • T = T’ + tY


Tax Function

T = T’

Y-intercept=T’

slope of tangent=0

T = tY

Y-intercept=0

slope of tangent=t

T = T’ +tY

Y-intercept=T’

slope of tangent=t


Tax Function

  • Autonomous Tax T’

    • this is a lump-sum tax which is independent of income level Y

  • Proportional Income Tax tY

    • marginal tax rate t is a constant

  • Progressive Income Tax tY

    • marginal tax rate t increases

  • Regressive Income Tax tY

    • marginal tax rate t decreases


Consumption Function

  • C = f(Yd)

  • C = C’

    C = C’

  • C = cYd

    C = c(Y - T)

  • C = C’ + cYd

    C = C’ + c(Y - T)


Consumption FunctionC = C’ + c(Y - T)

  • T = T’

    C = C’ + c(Y - T’) C = C’- cT’ + cY

     slope of tangent = c

  • T = tY

    C = C’ + c(Y - tY) C = C’ + (c - ct)Y

    slope of tangent = c - ct

  • T = T’ + tY

    C = C’+c[Y-(T’+tY)]C = C’ - cT’ + (c - ct) Y

    slope of tangent = c - ct


Consumption FunctionC = C’ + c (Y - T’)

Y-intercept = C’ - cT’

slope of tangent = c = MPC

slope of ray APC  when Y


Consumption FunctionC = C’ + c (Y - tY)

Y-intercept = C’

slope of tangent = c - ct = MPC (1-t)

slope of ray APC  when Y


Consumption Function C = C’ + c [Y - (T’ + tY)]

Y-intercept = C’ -cT’

slope of tangent = c - ct = MPC (1-t)

slope of ray APC  when Y


Consumption Function C = C’ - cT’ + (c - ct)Y

  • C’ OR T’ 

     y-intercept C’ - cT’   C shift upward

  • t 

     c(1-t)   C flatter

  • c 

     c(1-t)  C steeper

     y-intercept C’ - cT’ C shift downward


Government Expenditure Function

  • G only includes the part of government expenditure spending on goods and services, i.e. transfer payments Tr are excluded.

  • Usually, G is assumed to be an exogenous / autonomous function

  • G = G’


Government Expenditure Function

Y-intercept = G’

slope of tangent = 0

slope of ray  when Y


Aggregate Expenditure Function

  • E= C + I + G

    givenC= C’ + cYd

    T= T’ + tY

    I= I’

    G= G’

  • E= C’ + c[Y -(T’+tY)] + I’ + G’

  • E= C’ - cT’ + I’+ G’ + (c-ct)Y

  • E= E’ + c(1-t) Y


Aggregate Expenditure Function

  • E= C’ - cT’ + I’ + G’ + (c - ct)Y

  • E= E’ + (c - ct)Y

    given E’ = C’ - cT’ + I’ + G’

  • E’ is the y-intercept of the aggregate expenditure function E

  • c - ct is the slope of the aggregate expenditure function E


Aggregate Expenditure Function

  • Derive the aggregate expenditure function E if T = T’

  • E = C’- cT’ + I’ + G’ + cY

  • y-intercept = C’- cT’ + I’ + G’

  • slope of tangent = c


Aggregate Expenditure Function

  • Derive the aggregate expenditure function E if T = tY

  • E = C’ + I’ + G’ + (c-ct)Y

  • y-intercept = C’ + I’ + G’

  • slope of tangent = (c-ct)


Aggregate Expenditure Function

  • Derive the aggregate expenditure function E if T = T’ and I = I’ + iY

  • E = C’- cT’ + I’ + G’ + (c + i)Y

  • y-intercept = C’- cT’ + I’ + G’

  • slope of tangent = (c + i)


Aggregate Expenditure Function

  • Derive the aggregate expenditure function E if T = tY and I = I’ +iY

  • E = C’ + I’ + G’ + (c - ct+i )Y

  • y-intercept = C’ + I’ + G’

  • slope of tangent = (c - ct+i )


Aggregate Expenditure Function

  • Derive the aggregate expenditure function E if T = T’ + tY and I = I’ +iY

  • E = C’- cT’ + I’ + G’ + (c - ct+i)Y

  • y-intercept = C’- cT’ + I’ + G’

  • slope of tangent = (c - ct+i)


Output-Expenditure Approachw/ T = T’ + tYw/ C = C’ + cYd

C

2-Sector

C = C’ + cYd = C’ + cY

Slope of tangent = c = MPC =C/Yd

Slope of tangent = c (1-t) = (1-t)*MPC  MPC

C = C’ - cT’ + c(1-t)Y

3-Sector

C’

C’ -cT’

Y


I, G, C, E, Y

Y=E

Y

Planned Y = Planned E


Output-Expenditure ApproachI = I’ exogenous function

  • E = E’ + (c - ct) Y[slide 21-22]

  • In equilibrium, planned Y = planned E

  • Y = E’+ (c - ct) Y

  • (1- c + ct) Y = E’

  • Y = E’

    E’ = C’ - cT’ + I’ + G’

    k E =

1

1 - c + ct

1

1 - c + ct


Output-Expenditure ApproachI= I’+iY endogenous function

  • E = E’ + (c - ct + i) Y[slide 27]

  • In equilibrium, planned Y = planned E

  • Y = E’ + (c - ct + i) Y

  • (1- c + ct - i) Y = E’

  • Y = E’

    E’ = C’ - cT’ + I’ + G’

    k E =

1

1 - c - i + ct

1

1 - c - i + ct


Output-Expenditure ApproachT = T’ exogenous functionI = I’ + iY

  • E = E’+ (c + i) Y[slide 25]

  • In equilibrium, planned Y = planned E

  • Y = E’+ (c + i) Y

  • (1 - c - i) Y = E’

  • Y = E’

    E’ = C’ - cT’ + I’ + G’

    k E =

1

1 - c - i

1

1 - c - i


Factors affecting Ye

  • Ye = k E * E’

  • In the Keynesian model, aggregate expenditure E is the determinant of Ye since AS is horizontal and price is rigid.

  • In equilibrium, planned Y = planned E

  • E = C’ - cT’ + I’ + G’ + (c - ct + i) Y

  • Any change to the exogenous variables will cause the aggregate expenditure function to change and hence Ye


Factors affecting Ye

  • Change in E’

  • If C’I’G’  E’  E Y 

  • If T’C’ - cT’ E’ by- cT’E Y

  • Change in k E / slope of tangent of E

  • If c i   E steeper  Y

  • If c   C’ - cT’  E’  E  Y 

  • If t   E steeper  Y 


I, G, C, E, Y

Y=E

Y


I, E, Y

I’

E’ = I’

 I’

Y

Ye = k E E’


G, E, Y

G’

Y


C, E, Y

C’

Y


C, E, Y

T’

C  by -cT’

Y


I, E, Y

 i

Y


Digression

  • Differentiation

  • y = c + mx

  • differentiate y with respect to x

  • dy/dx = m


Expenditure Multiplier k E

  • Y = k E * E’E’ = C’ - cT’ + I’ + G’

  • k E =if I=I’ & T=T’+tY

  • k E =if I=I’+iY & T=T’+tY

  • k E =if I=I’+iY & T=T’

1

1 - c + ct

1

1 - c + ct - i

1

1 - c - i


Expenditure Multiplier k E

  • Whenever there is a change in the autonomous spending C’I’ or G’ the national income Ye will change by a multiple of k E.

  • It actually measures the ratio of the change in national income Ye to the change in the autonomous expenditure E’

  • Ye/E’ = k E


Tax Multiplier k T

  • Y = k E * ( C’- cT’ + I’ + G’)

  • k T =if I=I’ & T=T’+tY

  • k T =if I=I’+iY & T=T’+tY

  • k T =if I=I’+iY & T=T’

-c

1 - c + ct

-c

1 - c + ct + i

-c

1 - c - i


Tax Multiplier k T

  • Any change in the lump-sum taxT’ will lead to a change in the national income Ye by a multiple of k T in the opposite direction since k T takes on a negative value

  • Besides, the absolute value of k T is less than the value of k E.


Balanced-Budget Multiplier k B

  • G’  E’   E   Ye  by k E times

  • T’  E’   E   Ye  by k T times

  • If G’  = T’  , the change in Ye can be measured by k B

  • Y/ G’ = k E

  • Y/ T’ = k T

  • k B = k E + k T

  • k B = += 1

1

1-c

-c

1-c


Balanced-Budget Multiplier k B

  • The balanced-budget multiplier k B = 1 when t=0 & i=0

  • What is the value of k B if t  0 ?

  • If k B = 1 an increase in government expenditure of $1 which is financed by a $1 increase in the lump-sum income tax, the national income Ye will also increase by $1


Injection-Withdrawal Approach

  • In a 3-sector model, national income is either consumed, saved or taxed by the government

  • Y = C + S + T

  • Given E = C + I + G

  • In equilibrium, Y = E

  • C + S + T = C + I + G

  •  S + T = I + G


Injection-Withdrawal Approach

  • Since S + T = I + G

  • S  I

  • T  G

  • I > S  T > G

  • I < S  T < G

  • (Compare with 2-sector model)

  • In equilibrium S = I


Injection-Withdrawal Approach

  • T = T’ + tY

  • S = -C’ + (1-c) Yd

  • S = -C’ + (1 - c)[Y -_(T’ + tY)]

  • S = -C’ + (1 - c)[Y - T’ - tY]

  • S = -C’ + Y - T’ - tY - cY + cT’ + ctY

  • S = -C’ + cT’ -T’ - tY + Y - cY + ctY

  • S = -C’ + cT’ - (T’ + tY) + Y - cY + ctY


Injection-Withdrawal Approach

  • S + T = -C’ + cT’ -(T’+ tY) + Y - cY + ctY +T

  • S + T = -C’ + cT’ + Y - cY + ctY

  • In equilibrium, S + T = I + G

  • -C’ + cT’ + Y - cY + ctY = I’ + G’

  • (1- c + ct)Y = C’ - cT’ + I’ + G’

  • Ye = k E * E’

  • E’ = C’ - cT’ + I’ + G’[slide 30]


Use the Injection-Withdrawal Approach to solve for Ye if T=T’


Fiscal Policy

  • The use of government expenditure and taxation to achieve certain goals, such as high employment, price stability.

  • Discretionary Fiscal Policy

    • Expansionary Fiscal Policy (when Yf > Ye)

    • Contractionary Fiscal Policy (when Yf < Ye)

  • Automatic Built-in Stabilizers

    • Proportional / Progressive Tax System

    • Welfare Schemes


Expansionary Fiscal Policy Recessionary/Deflationary Gap Yf-Ye

Y-line

G’  E’  E  Y

E = E” + (c-ct) Y

E = E’ + (c -ct) Y

G’

Y= k E * E’

Recessionary Gap

Ye

Yf


Expansionary Fiscal Policy Recessionary/Deflationary Gap Yf-Ye

Y-line

T’  E’ by -c T’  E  Y

E = E” + (c-ct) Y

E = E’ + (c -ct) Y

-cT’

Y= k E * E’ = k T *T’

Recessionary Gap

Ye

Yf


Contractionary Fiscal PolicyInflationary Gap Ye - Yf

Y = E

G’  E’  E  Y

E = E’ + (c-ct) Y

E = E” + (c-ct) Y

G’

Y= k E * E’

Nominal Y>Yf Inflationary Gap

Yf

Ye


Contractionary Fiscal PolicyInflationary Gap Ye - Yf

Y = E

T’  E’ by -c T’  E  Y

E = E’ + (c-ct) Y

E = E” + (c-ct) Y

-cT’

Y= k E * E’ = k T *T’

Nominal Y>Yf Inflationary Gap

Yf

Ye


Automatic Built-in Stabilizers

  • Proportional /Progressive Tax System

    • Recession: government’s tax revenue 

    • Boom: government’s tax revenue 

  • The more progressive the tax system, the greater is its stabilizing effect. But there will be greater dis-incentives to earn income

  • With t, k E  With proportional tax, the multiplying effect of a discretionary change in government expenditure G’ reduces


Automatic Built-in Stabilizers

  • Welfare Schemes

  • Unemployment benefits, public assistance allowances, agricultural support schemes

    • Recession: government’s expenditure

    • Boom: government’s expenditure 

  • Again, if the welfare schemes are generous, the incentives to work will be weakened.


Discretionary Fiscal Policy v.s.Automatic Built-in Stabilizers

  • If the economy is close to Yf, built-in stabilizers are useful as they can stabilize the economy around Yf or potential income level.

  • However, if the economy is far below Yf, discretionary fiscal policy is still necessary (Simple Keynesian model).

  • Another drawback of the built-in stabilizers is they may reduce the speed of recovery as

  • k E  Y = k E * E’


Discretionary Fiscal Policy

  • Government expenditure G’? Tax T’?

  • Location of effects

  • If a recession is localized in a particular industry  G’

  • Tax cut will have its impact on the entire economy


Discretionary Fiscal Policy

  • Government expenditure G’? Tax T’?

  • Duration of the time lag

    • Decision lag : time involved to assess a situation & decide what corrective actions should be taken

    • Executive lag : time involved to initiate corrective policies & for their full impact to be felt

       tax cut has a much shorter executive lag


Discretionary Fiscal Policy

  • Government expenditure G’? Tax T’?

  • Reversibility of the fiscal policy

    • Government expenditure can easily be increased but are not so easy to cut as the civil servants who have vested interests in the present allocation of government expenditure will resist

    • Tax is easier to be changed as the civil servants who administer income tax is independent of the rate being levied. Of course, voter resistance should also be considered.


Discretionary Fiscal Policy

  • Government expenditure G’? Tax T’?

  • Public reaction to short-term changes

  • A temporary tax cut raises Yd. Households, recognizing this situation, may not revise their current consumption. Instead, they save a large part of the tax cut.


Financing the Government BudgetIncreasing Taxes

  • By increasing taxes, the government transfers purchasing power from current taxpayers to itself

  • Current taxpayers bear the cost

  • If the revenue is spent on some investment project, (current / future) taxpayers may benefit when the project is completed.

  • How about the revenue is spent on transfer payment?


Financing the Government BudgetPrinting more Money

  • This will create inflationary pressure.

  • Households and firms will be able to buy less with each unit of money. Fewer resources are available for private consumption and investment.

  • Those whose incomes respond slowly to changes in price levels will bear most of the cost of the government activity


Financing the Government BudgetInternal Debt

  • The government can transfer purchasing power from any willing lenders to itself in return for the promise to repay equivalent purchasing power plus interest in future.

  • Since, repayment of the debt are made from tax revenue, future taxpayers will suffer.

  • However, if the debt raised today is spent on creating capital assets, the burden on future generation will be lighter.


Financing the Government BudgetExternal Debt

  • Borrowing from abroad transfers purchasing power from foreigners to the government.

  • The burden on future generations will once again depend on how the debt raised is used (investment project / transfer payment)


The Problems of the Simple Keynesian Multiplier k E

  • Y = k E * G’

  • There are several problems with this method of analysis, i.e., Y may be less

    • Sources of financing G’

    • Effects on private investment I’

    • Productivity of government projects


The Problems of the Simple Keynesian Multiplier k E

  • Sources of financing G’

  • Increasing Tax

    • will exert a contractionary effect on the economy

  • Increasing Money Supply

    • will generate an inflationary pressure

  • Increasing Debt

    • will increase the demand for loanable fund as well as interest rate  affect private investment


The Problems of the Simple Keynesian Multiplier k E

  • Effects on Private Investment I’

  • Private investment may be crowded out when government increases its expenditure

  • It is questionable that the government can really produce something which is desired by the consumers

  • Besides, government investment projects are usually less productive than private investment projects


The Problems of the Simple Keynesian Multiplier k E

  • Productivity of Government Projects

  • Government projects may not yield a rate of return (MEC / MEI) exceeding the market interest rate.


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