Download Lecture 12

The Multiplier
The Multiplier
Agenda
• What happens to Ye when G increases?
• Keynesian Cross (or Multiplier) Model
¾ Graphically, Ep line shifts up.
¾ The Multiplier
¾ Same for increases in C0, Ip, X or decreases in M.
• The (Simple) Spending Multiplier
• Multipliers with Income Tax Rates
• Multipliers with Endogenous Imports
• Or for any exogenous change in spending.
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The Multiplier
The Multiplier
Y=E
E, Ep
• By how much does Ye change?
Ep
¾ delta Ye > delta G
Ep
G1
2
¾This concept is the simple income multiplier
G0
• Multiplier for short
Y0
Y1
Y
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The Multiplier
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The Multiplier
• Algebraically,
• Ye, an endogenous variable, depends on:
¾Y=C+I+G+(X–M)
¾ And C = C0 + mpc( Y – T )
¾ Ap, the sum of several exogenous variables, and
¾ Y = C0 + mpcY – mpcT + I + G + ( X – M )
¾ Let Ap = C0 – mpcT + I + G + ( X – M )
¾ mpc, a parameter
¾ Y = Ap + mpcY
¾ Y = Ap / ( 1 – mpc )
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The Multiplier
The Multiplier
• If
• If
¾ Y = Ap / ( 1 – mpc )
¾ G = 400, Ap = 1,000 and mpc = 0.9,
• Then
• then
¾ delta Y = delta Ap / ( 1 – mpc )
¾ Y = Ap / ( 1 – mpc )
¾ = 1,000 / ( 1 – 0.9 )
¾ = 1,000 / 0.1
¾ = 10,000
• or
¾ delta Y / delta Ap = 1 / (1 – mpc )
• So
¾ k(Ap) = delta Y / delta Ap = 1 / ( 1 – mpc )
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The Multiplier
The Multiplier
• So
• Now if
¾ delta Y = delta Ap / ( 1 – mpc )
¾ delta Y = 100 / ( 1 – 0.9) = 1,000
¾ G = 500, Ap = 1,100 and mpc = 0.9,
• then
• Or
¾ Y = Ap / ( 1 – mpc )
¾ = 1,100 / ( 1 – 0.9 )
¾ = 1,100 / 0.1
¾ = 11,000
¾ delta Y / delta Ap = 1 / (1 – mpc )
¾ delta Y / delta Ap = 1 / ( 1 – 0.9 ) = 10
• And
• And delta Y = 1,000 > delta G = 100
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The Multiplier
¾ k(Ap) = delta Y / delta Ap = 1 / ( 1 – mpc )
¾ k(Ap) = delta Y / delta Ap = 1 / ( 1 – 0.9 ) = 10
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The Multiplier
• Continuing further
• Why Does This Work?
¾ 1 / ( 1 – mpc ) is called the simple multiplier
¾ When G increases, Y increase as well
• Because 0 < mpc < 1,
• ( 1 – mpc ) < 1, and
• 1 / ( 1 – mpc ) > 1
• G is a part of Y
¾ When Y increases, YD increase, so C increases
¾ When C increases, Y increases further
• C is a part of Y
¾ The larger is mpc, the greater is the multiplier
¾ This process continues but gets smaller and
smaller with each round of spending
• The steeper the spending line (the slope of Ep), the
greater is the multiplier and the change in Ye.
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The Multiplier
The Multiplier
mpc = 0.9
Spending
Round
Delta G
1
2
3
4
5
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100.0
-----------
Equilibrium
100.0
Delta C
---
Delta Y
• Why Does This Work?
Total
Delta Y
90.0
81.0
72.9
65.6
59.0
100.0
90.0
81.0
72.9
65.6
59.0
100.0
190.0
271.0
343.9
409.5
468.6
900.0
1,000.0
1,000.0
¾ The increase in Y is > the increase in G
because of the increase in C.
¾ The multiplier measures the amount of Y
stimulated by an increase in G or any other
categories of autonomous spending (that is not
itself sensitive to Y).
• This spending cycle takes place very quickly
• Less than a year
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The Multiplier
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Multipliers w/ Income Tax Rates
• Observations
• Multipliers with Income Tax Rates:
¾ Ye depends on the exogenous variables
¾ Ye is a multiple of these exogenous variables
¾ Relatively small changes in these exogenous
variables can lead to large changes in Ye
¾ The size of the multiplier depends on the
magnitude of the “leakages”.
¾ Assume T = tY
• t is the marginal tax rate
• 0<t<1
• tY are induced taxes
• The greater the leakages, the smaller the multiplier.
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Multipliers w/ Income Tax Rates
Multipliers w/ Income Tax Rates
• Multipliers with Income Tax Rates:
• Multipliers with Income Tax Rates:
¾ Y = Ap + mpc ( 1 – t ) Y
¾ Y = Ap / [ 1 – mpc ( 1 – t ) ]
¾ Reformulated Consumption Function
• C = C0 + mpc ( Y – T )
• C = C0 + mpc ( Y – tY )
• C = C0 + mpc ( 1 – t ) Y
¾ delta Y = delta Ap / [ 1 – mpc ( 1 – t ) ]
¾ delta Y / delta Ap = 1 / [ 1 – mpc ( 1 – t ) ]
¾ k(Ap) = 1 / [ 1 – mpc ( 1 – t ) ]
¾If mpc = 0.9 and t = 0.1, then
k(Ap) = 1 / [ 1 – 0.9 ( 1 – 0.1 ) ] = 1 / 0.19 = 5.26
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Multipliers w/ Income Tax Rates
Multipliers w/ Income Tax Rates
Y=E
Ep’ w/ t < t
E, Ep
• Implications of adding income tax rate, t
¾ The size of the multiplier is reduced.
• There are more leakages into taxes.
• A smaller multiplier => shallower business cycles.
• t serves as an “automatic” stabilizer.
Ep w/ t > t
¾ Ye changes when t changes.
• Higher Ye when t declines; lower Ye when t increases.
• Ep line rotates.
¾ Fixed on the vertical axis.
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Ye
Ye’
Y
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Multipliers w/ Income Tax Rates
Multipliers w/ Income Tax Rates
• If
• If
¾ Ap = 1,000, mpc = 0.9, and t = 0.2
¾ Ap = 1,000, mpc = 0.9, and t = 0.15
• then
• then
¾ Y = Ap / [ 1 – mpc ( 1 – t ) ]
¾ = 1,000 / [ 1 – 0.9 ( 1 – 0.2 ) ]
¾ = 1,000 / 0.28
¾ = 3,571
¾ Y = Ap / [ 1 – mpc ( 1 – t ) ]
¾ = 1,000 / [ 1 – 0.9 ( 1 – 0.15 ) ]
¾ = 1,000 / 0.235
¾ = 4,255
• And k(Ap) = 3.57
• And k(Ap) = 4.255
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Multipliers w/ Endogenous M
Multipliers w/ Endogenous M
• Algebraically,
• Multipliers with Endogenous Imports.
¾Y=C+I+G+(X–M)
where C = C0 + mpc ( 1 – t )Y and M = M0 + mY
¾ Assume M = M0 + mY
•
•
•
•
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M0 are autonomous imports.
m is the marginal propensity to import.
0<m<1
And mY are induced imports.
¾ Y = C0 + mpc ( 1 – t )Y + I + G + X – M0 – mY
¾ Let Ap = C0 + I + G + X – M0
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¾ Then Y = Ap + mpc ( 1 – t )Y – mY
¾ or Y = Ap + [ mpc ( 1 – t ) – m ]Y
¾ or Y = Ap / [ 1 – mpc ( 1 – t ) + m ]
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Multipliers w/ Endogenous M
Multipliers w/ Endogenous M
• Multipliers with Endogenous Imports:
• Implications of Endogenizing Imports:
¾ Y = Ap / [ 1 – mpc ( 1 – t ) + m ]
¾ delta Y = delta Ap / [ 1 – mpc ( 1 – t ) + m ]
¾ delta Y / delta Ap = 1 / [ 1 – mpc ( 1 – t ) + m ]
¾ k(Ap) = 1 / [ 1 – mpc ( 1 – t ) + m ]
¾ The size of the multiplier is reduced.
• There are more leakages into imports.
• A smaller multiplier => shallower business cycles.
¾ The is called the open economy multiplier.
¾If mpc = 0.9, t = 0.1, and m = 0.2, then
k(Ap) = 1 / [ 1 – 0.9 ( 1 – 0.1 ) + 0.2 ]
= 1 / [ 1 – 0.81 + 0.2 ] = 2.56
• Always smaller than the closed economy multiplier.
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Multipliers w/ Endogenous M
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Multipliers w/ Income Tax Rates
Y=E
E, Ep
• Implications of Endogenizing Imports:
Ep’ w/ m < m
¾ Ye changes when m changes.
• Higher Ye when m declines; lower Ye when m increases.
• Ep line rotates.
Ep w/ m > m
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Multipliers w/ Endogenous M
Ye
Ye’
Y
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Multipliers w/ Endogenous M
• If
• If
¾ Ap = 1,000, mpc = 0.9, t = 0.2, and m = 0.2
¾ Ap = 1,000, mpc = 0.9, t = 0.2, and m = 0.15
• then
• then
¾ Y = Ap / [ 1 – mpc ( 1 – t ) + m ]
¾ = 1,000 / [ 1 – 0.9 ( 1 – 0.2 ) + 0.2 ]
¾ = 1,000 / 0.48
¾ = 2083
¾ Y = Ap / [ 1 – mpc ( 1 – t ) + m ]
¾ = 1,000 / [ 1 – 0.9 ( 1 – 0.2 ) + 0.15 ]
¾ = 1,000 / 0.43
¾ = 2326
• And k(Ap) = 2.083
• And k(Ap) = 2.326
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The Multiplier
The Multiplier
• Summarizing the Multiplier:
• Implications for Business Cycles
¾ k(Ap) = 1 / marginal leakage rate
• Consumption Only: 1 / ( 1 – mpc )
• With Income Tax Rates: 1 / [ 1 – mpc ( 1 – t ) ]
• With Endogenous Imports: 1/[1 – mpc(1 – t) + m]
• Implications for Stabilization Policy
• If mpc = 0.9, t = 0.1, and m = 0.2, then
¾ Consumption: 1 / ( 1 – mpc ) = 10
¾ w/ Income Tax Rates: 1 / [ 1 – mpc ( 1 – t ) ] = 5.26
¾ w/ Endogenous Imports: 1/[1 – mpc(1 – t) + m] = 2.56
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The Multiplier
• Major Points:
¾ Ye is determined where E = Ep.
¾ The multiplier expresses the simple
relationship between changes in Ap and Ye.
¾ Introduction of income tax rates and
endogenous imports reduces the size of the
multiplier.
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