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Simple Keynesian Model

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Simple Keynesian Model

National Income Determination

Three-Sector National Income Model

- 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

- Factors affecting Ye
- Expenditure Multipliers k E
- Tax Multipliers k T
- Balanced-Budget Multipliers k B
- Injection-Withdrawal Approach: Equilibrium National Income Ye

- Fiscal Policy (v.s. Monetary Policy)
- Recessionary Gap Yf - Ye
- Inflationary Gap Ye - Yf
- Financing the Government Budget
- Automatic Built-in Stabilizers

- 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

- 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

- National Income Personal Income Disposable Personal Income
- w/ direct income tax Ta and transfer payment Tr
- Yd Y
- Yd = Y - Ta + Tr

- 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

- 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

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

- 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

- C = f(Yd)
- C = C’
C = C’

- C = cYd
C = c(Y - T)

- C = C’ + cYd
C = 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

Y-intercept = C’ - cT’

slope of tangent = c = MPC

slope of ray APC when Y

Y-intercept = C’

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

slope of ray APC when Y

Y-intercept = C’ -cT’

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

slope of ray APC when 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

- 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’

Y-intercept = G’

slope of tangent = 0

slope of ray when Y

- 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

- 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

- 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

- 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)

- 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)

- 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 )

- 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)

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

- 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

- 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

- 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

- 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

- Change in E’
- If C’I’G’ E’ E Y
- If T’C’ - cT’ E’ by- cT’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 -cT’

Y

I, E, Y

i

Y

- Differentiation
- y = c + mx
- differentiate y with respect to x
- dy/dx = m

- 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

- 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

- 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

- 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.

- 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

- 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

- 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

- 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

- 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

- 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]

- 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

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

Y-line

T’ E’ by -c T’ E Y

E = E” + (c-ct) Y

E = E’ + (c -ct) Y

-cT’

Y= k E * E’ = k T *T’

Recessionary 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

Y = E

T’ E’ by -c T’ E Y

E = E’ + (c-ct) Y

E = E” + (c-ct) Y

-cT’

Y= k E * E’ = k T *T’

Nominal Y>Yf Inflationary Gap

Yf

Ye

- 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

- 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.

- 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’

- 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

- 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

- 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.

- 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.

- 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?

- 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

- 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.

- 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)

- 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

- 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

- 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

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