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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 l.jpg

Simple Keynesian Model

National Income Determination

Three-Sector National Income Model


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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


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Outline

  • Factors affecting Ye

  • Expenditure Multipliers k E

  • Tax Multipliers k T

  • Balanced-Budget Multipliers k B

  • Injection-Withdrawal Approach: Equilibrium National Income Ye


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Outline

  • Fiscal Policy (v.s. Monetary Policy)

  • Recessionary Gap Yf - Ye

  • Inflationary Gap Ye - Yf

  • Financing the Government Budget

  • Automatic Built-in Stabilizers


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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


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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


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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


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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


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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]


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Tax Function

  • T = f(Y)

  • T = T’

  • T = tY

  • T = T’ + tY


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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


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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


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Consumption Function

  • C = f(Yd)

  • C = C’

    C = C’

  • C = cYd

    C = c(Y - T)

  • C = C’ + cYd

    C = C’ + c(Y - T)


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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


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

Y-intercept = C’ - cT’

slope of tangent = c = MPC

slope of ray APC  when Y


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

Y-intercept = C’

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

slope of ray APC  when Y


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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


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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


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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’


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Government Expenditure Function

Y-intercept = G’

slope of tangent = 0

slope of ray  when Y


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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


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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


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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


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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)


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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)


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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 )


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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)


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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


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I, G, C, E, Y

Y=E

Y

Planned Y = Planned E


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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


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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


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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


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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


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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 


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I, G, C, E, Y

Y=E

Y


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I, E, Y

I’

E’ = I’

 I’

Y

Ye = k E E’


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G, E, Y

G’

Y


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C, E, Y

C’

Y


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C, E, Y

T’

C  by -cT’

Y


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I, E, Y

 i

Y


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Digression

  • Differentiation

  • y = c + mx

  • differentiate y with respect to x

  • dy/dx = m


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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


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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


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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


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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.


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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


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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


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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


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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


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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


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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]



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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


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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


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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


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Contractionary Fiscal Policy Yf-YeInflationary 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


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Contractionary Fiscal Policy Yf-YeInflationary 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


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Automatic Built-in Stabilizers 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


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Automatic Built-in Stabilizers Yf-Ye

  • 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.


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Discretionary Fiscal Policy v.s. Yf-YeAutomatic 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’


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Discretionary Fiscal Policy Yf-Ye

  • 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


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Discretionary Fiscal Policy Yf-Ye

  • 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


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Discretionary Fiscal Policy Yf-Ye

  • 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.


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Discretionary Fiscal Policy Yf-Ye

  • 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.


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Financing the Government Budget Yf-YeIncreasing 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?


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Financing the Government Budget Yf-YePrinting 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


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Financing the Government Budget Yf-YeInternal 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.


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Financing the Government Budget Yf-YeExternal 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)


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The Problems of the Simple Keynesian Multiplier k Yf-YeE

  • 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


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The Problems of the Simple Keynesian Multiplier k Yf-YeE

  • 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


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The Problems of the Simple Keynesian Multiplier k Yf-YeE

  • 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


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The Problems of the Simple Keynesian Multiplier k Yf-YeE

  • Productivity of Government Projects

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


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