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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

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]

• 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]

### 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.