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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

Simple Keynesian Model

National Income Determination

Three-Sector National Income Model

outline
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
outline3
Outline
  • Factors affecting Ye
  • Expenditure Multipliers k E
  • Tax Multipliers k T
  • Balanced-Budget Multipliers k B
  • Injection-Withdrawal Approach: Equilibrium National Income Ye
outline4
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
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 model6
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 model7
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 model8
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 model9
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
Tax Function
  • T = f(Y)
  • T = T’
  • T = tY
  • T = T’ + tY
tax function11
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 function12
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
Consumption Function
  • C = f(Yd)
  • C = C’

C = C’

  • C = cYd

C = c(Y - T)

  • C = C’ + cYd

C = C’ + c(Y - T)

consumption function 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 function c c c y t15
Consumption FunctionC = C’ + c (Y - T’)

Y-intercept = C’ - cT’

slope of tangent = c = MPC

slope of ray APC  when Y

consumption function c c c y ty
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
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 c t c c t 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
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 function20
Government Expenditure Function

Y-intercept = G’

slope of tangent = 0

slope of ray  when Y

aggregate expenditure function
Aggregate Expenditure Function
  • E = C + I + G

given C = 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 function22
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 function23
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 function24
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 function25
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 function26
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 function27
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 approach w t t ty w c c cyd
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

slide29

I, G, C, E, Y

Y=E

Y

Planned Y = Planned E

output expenditure approach i i exogenous function
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 approach i i iy endogenous function
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 approach t t exogenous function i i iy
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
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 ye34
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 
slide36

I, E, Y

I’

E’ = I’

 I’

Y

Ye = k E E’

slide37

G, E, Y

G’

Y

slide38

C, E, Y

C’

Y

slide39

C, E, Y

T’

C  by -cT’

Y

slide40

I, E, Y

 i

Y

digression
Digression
  • Differentiation
  • y = c + mx
  • differentiate y with respect to x
  • dy/dx = m
expenditure multiplier k e
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 e43
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
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 t45
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
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 b47
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
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 approach49
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 approach50
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 approach51
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]
fiscal policy
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
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 ye55
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 policy inflationary 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 policy inflationary gap ye yf57
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
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 stabilizers59
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
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
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 policy62
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 policy63
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 policy64
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 budget increasing taxes
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 budget printing more money
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 budget internal debt
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 budget external debt
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
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 e70
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 e71
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 e72
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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