stock flow consistent models an introduction to theory and ... 3_0.pdf · stock flow consistent...
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Stock Flow Consistent ModelsAn Introduction to Theory and Technique
Session 3
University of Pavia
Institute for New Economic Thinking -Young Scholars InitiativeTrento Festival of Economics
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Motivation
Allowing to simulate SFC models in an open source environment
◮ Very preliminary
◮ Only one numerical solver: Gauss-Seidel algorithm [Kinsella andO’Shea, 2010]
◮ Searching for Beta Tester ([email protected])
◮ http://www.antoinegodin.eu/sfc
Technical aspect
◮ R package
◮ EViews translator
◮ Read equation files
◮ Build model from console
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Model SIM I
Transaction Flow Matrix
Households Production Government∑
Consumption −C +C 0Govt. expenditures +G −G 0[Output] [Y ]Wages +WB −WB 0Taxes −T +T 0
Savings Sh 0 Sg 0Change in money stock −∆H +∆H 0∑
0 0 0 0
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Model SIM I
Transaction Flow Matrix
Households Production Government∑
Consumption −C +C 0Govt. expenditures +G −G 0[Output] [Y ]Wages +WB −WB 0Taxes −T +T 0
Savings Sh 0 Sg 0Change in money stock −∆H +∆H 0∑
0 0 0 0
Balance Sheet
Households Production Government Σ
Money +H −H 0
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Model SIM II
sim.sfcyd = w*n - t
t = theta*w*n
c = alpha1*yd + alpha2*h(-1)
h = h(-1) + yd - c
h c = h c(-1)+ g - t
y = c + g
n = y/w
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Model SIM II
sim.sfcyd = w*n - t
t = theta*w*n
c = alpha1*yd + alpha2*h(-1)
h = h(-1) + yd - c
h c = h c(-1)+ g - t
y = c + g
n = y/w
Loading the package and the model
>library(PKSFC)
>sim<-sfc.model("sim.sfc",modelName="sim")
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Checking the model I
>sim<-sfc.check(sim,fill=T)
Initial value for h [0]?
1:
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Checking the model I
>sim<-sfc.check(sim,fill=T)
Initial value for h [0]?
1:
Initial value for h c [0]?
1:
One or more exogenous variables are not defined in the
following equations w * n - t
theta * w * n
alpha1 * yd + alpha2 * h 1
h c 1 + g - t
c + g
y/w
do you want to insert these manually [Yes]/No?
1:
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Checking the model II
Insert name, value and description (return after each
value)
1:
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Checking the model II
Insert name, value and description (return after each
value)
1:
Insert name, value and description (return after each
value)
1: theta
2: 0.2
3: tax rate
Are there other variables to add [Yes]/No?
1:
Parameter values
w 1 Wage rate alpha1 0.6 Prop. to consume (income)g 20 Govt expenditure alpha2 0.4 Prop. to consume (wealth)
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Checking the model III
Years are not set, do you want to inser these manually
[Yes]/No?
1:
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Checking the model III
Years are not set, do you want to inser these manually
[Yes]/No?
1:
Insert initial period and final period (return after each
value)
1: 1945
2: 2010
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Checking the model III
Years are not set, do you want to inser these manually
[Yes]/No?
1:
Insert initial period and final period (return after each
value)
1: 1945
2: 2010
>datasim<-simulate(sim)
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Results
>years=c("1945","1946","1947","2010")
>variables=c("g","y","t","yd","c","h","h c")
>round(t(datasim$baseline[years, variables]), digits=1)
1945 1946 1947 2010
g 20 20.0 20.0 20
y NA 38.5 47.9 100
t NA 7.7 9.6 20
yd NA 30.8 38.3 80
c NA 18.5 27.9 80
h 0 12.3 22.7 80
h c 0 12.3 22.7 80
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Keynesian multiplier
Equations
Cd = α1 · YD + α2 · H−1 = α1 · YD (3.13)
Y = C + G = α1 · Y · (1− θ) + G
Y ⋆ =G
1− α1 · (1− θ)(3.14)
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Keynesian multiplier
Equations
Cd = α1 · YD + α2 · H−1 = α1 · YD (3.13)
Y = C + G = α1 · Y · (1− θ) + G
Y ⋆ =G
1− α1 · (1− θ)(3.14)
Short run vs. Long run
◮ (3.14) is the short run multiplier, depends on start-of-period stockvalues (i.e H
−1 = 0)
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Keynesian multiplier
Equations
Cd = α1 · YD + α2 · H−1 = α1 · YD (3.13)
Y = C + G = α1 · Y · (1− θ) + G
Y ⋆ =G
1− α1 · (1− θ)(3.14)
Short run vs. Long run
◮ (3.14) is the short run multiplier, depends on start-of-period stockvalues (i.e H
−1 = 0)
◮ Need to obtain the steady state to compute long-run multiplier
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Keynesian multiplier
Equations
Cd = α1 · YD + α2 · H−1 = α1 · YD (3.13)
Y = C + G = α1 · Y · (1− θ) + G
Y ⋆ =G
1− α1 · (1− θ)(3.14)
Short run vs. Long run
◮ (3.14) is the short run multiplier, depends on start-of-period stockvalues (i.e H
−1 = 0)
◮ Need to obtain the steady state to compute long-run multiplier
◮ Steady state: Y ⋆ = Gθ= 20
0.2 = 100
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Results
>years=c("1945","1946","1947","2010")
>variables=c("g","y","t","yd","c","h","h c")
>round(t(datasim$baseline[years, variables]), digits=1)
1945 1946 1947 2010
g 20 20.0 20.0 20
y NA 38.5 47.9 100
t NA 7.7 9.6 20
yd NA 30.8 38.3 80
c NA 18.5 27.9 80
h 0 12.3 22.7 80
h c 0 12.3 22.7 80
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Introducing expectation
Expected income
Cd = α1 · YDe + α2 · Hh,−1 (3.7E)
∆Hd = YDe− Cd (3.18)
YDe = YD−1 (3.20)
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Introducing expectation
Expected income
Cd = α1 · YDe + α2 · Hh,−1 (3.7E)
∆Hd = YDe− Cd (3.18)
YDe = YD−1 (3.20)
R code
>simex a<-sfc.addEqus(sim,list(
list(var="yde",eq="yd(-1)",desc="Expected income"),
list(var="h d",eq="h(-1)+yde-c",desc="Expected wealth")))
>simex a<-sfc.editEqu(simex a,var="c",eq="alpha1*yde+alpha2*h(
>simex a<-sfc.check(simex a,fill=T)
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Playing with expectations>timeline=seq(from=1945,to=2010)
>plot(timeline,datasimex$baseline[as.character(timeline),"yd"],type="l",
xlab="",ylab="",lty=2)
>lines(timeline,datasimex a$baseline[as.character(timeline),"yde"],lty=3)
>lines(timeline,vector(length=length(timeline))+datasimex a$baseline["2010","yd"])
>legend(x=1970,y=50,legend=c("Disposable Income","Expected Disposable
Income","Steady State"),lty=c(2,3,1),bty="n")
ResultsYDe = YD(−1) YDe = YD⋆
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Growth I
Sources of growth in SIM
◮ Government expenditure: G = G−1(1 + grg )
◮ Consumption: α1 or α2
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Growth I
Sources of growth in SIM
◮ Government expenditure: G = G−1(1 + grg )
◮ Consumption: α1 or α2
Growth steady state equations
∆H⋆
Y ⋆=
gr · (1− θ) · (1− α1)
gr + α2(A3.3.2)
H⋆
Y ⋆=
(1 + gr) · (1− θ) · (1− α1)
gr + α2(A3.3.3)
V T
Y ⋆= α3 =
1− α1
α2(A3.3.4)
H⋆
YD⋆=
(1 + gr) · (1− α1)
gr + α2(A3.3.5)
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Growth II
Source code
>simgr<-sfc.addEqu(sim,"g","g(-1)*(1+grg)","government
expenditures")
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Growth II
Source code
>simgr<-sfc.addEqu(sim,"g","g(-1)*(1+grg)","government
expenditures")
Steady state values (grg = 0%)
∆H⋆
Y ⋆
H⋆
Y ⋆
V T
Y ⋆
H⋆
YD⋆
Value 0 0.8 1 1Simulation 5.2e−7 0.7999992 1 0.999999
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Growth II
Source code
>simgr<-sfc.addEqu(sim,"g","g(-1)*(1+grg)","government
expenditures")
Steady state values (grg = 0%)
∆H⋆
Y ⋆
H⋆
Y ⋆
V T
Y ⋆
H⋆
YD⋆
Value 0 0.8 1 1Simulation 5.2e−7 0.7999992 1 0.999999
Steady state values (grg = 2%)
∆H⋆
Y ⋆
H⋆
Y ⋆
V T
Y ⋆
H⋆
YD⋆
Value 0.0152381 0.7771429 1 0.9714286Simulation 0.01523065 0.777154 1 0.9714425
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Results - No growth
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Results - Growing expenditures
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Model with Portfolio choice
Balance Sheet
Households Production Government Central Bank Σ
Money +H −H 0Bills +Bh −B +Bcb 0
Balance −V +V 0
Σ 0 0 0 0
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Model with Portfolio choice
Balance Sheet
Households Production Government Central Bank Σ
Money +H −H 0Bills +Bh −B +Bcb 0
Balance −V +V 0
Σ 0 0 0 0
Portfolio equation
Hh
V= (1− λ0)− λ1 · r + λ2 ·
(
YD
V
)
(4.6A)
Bh
V= λ0 + λ1 · r − λ2 ·
(
YD
V
)
(4.7)
Hh = V − Bh (4.6)
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From model SIM to model PC>pc<-sfc.addEqus(sim,list(
list(var="v",eq="v(-1)+yd-c",desc="households wealth"),
list(var="bs",eq="bs(-1)+g+r(-1)*bs(-1)-t-r(-1)*bcb(-1)",
desc="Supply of bonds"),
list(var="bcb",eq="bs-bh",desc="Bonds held by the central
bank"),
list(var="r",eq="r(-1)",desc="Interest rate"),
list(var="bh",eq="v*(lambda0+lambda1*r-lambda2*yd/v)",
desc="Bonds held by housholds")))
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From model SIM to model PC>pc<-sfc.addEqus(sim,list(
list(var="v",eq="v(-1)+yd-c",desc="households wealth"),
list(var="bs",eq="bs(-1)+g+r(-1)*bs(-1)-t-r(-1)*bcb(-1)",
desc="Supply of bonds"),
list(var="bcb",eq="bs-bh",desc="Bonds held by the central
bank"),
list(var="r",eq="r(-1)",desc="Interest rate"),
list(var="bh",eq="v*(lambda0+lambda1*r-lambda2*yd/v)",
desc="Bonds held by housholds")))
>pc<-sfc.editEqus(pc,list(
list(var="yd",eq="y-t+r(-1)*bh(-1)"),
list(var="t",eq="theta*(y+r(-1)*bh(-1))"),
list(var="c",eq="alpha1*yd+alpha2*v(-1)"),
list(var="h",eq="v-bh",desc="cash holding"),
list(var="h c",eq="h(-1)+bcb-bcb(-1)",desc="hidden
equation")))
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From model SIM to model PC>pc<-sfc.addEqus(sim,list(
list(var="v",eq="v(-1)+yd-c",desc="households wealth"),
list(var="bs",eq="bs(-1)+g+r(-1)*bs(-1)-t-r(-1)*bcb(-1)",
desc="Supply of bonds"),
list(var="bcb",eq="bs-bh",desc="Bonds held by the central
bank"),
list(var="r",eq="r(-1)",desc="Interest rate"),
list(var="bh",eq="v*(lambda0+lambda1*r-lambda2*yd/v)",
desc="Bonds held by housholds")))
>pc<-sfc.editEqus(pc,list(
list(var="yd",eq="y-t+r(-1)*bh(-1)"),
list(var="t",eq="theta*(y+r(-1)*bh(-1))"),
list(var="c",eq="alpha1*yd+alpha2*v(-1)"),
list(var="h",eq="v-bh",desc="cash holding"),
list(var="h c",eq="h(-1)+bcb-bcb(-1)",desc="hidden
equation")))
>pc<-sfc.rmEqu(pc,7)# Employment equation
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Adding expectations
Equation
C = α1 · YDe + α2 · V−1 (4.5E)
Hh
V e= (1− λ0)− λ1 · r + λ2 ·
(
YDe
V e
)
(4.6E)
Bh
V e= λ0 + λ1 · r − λ2 ·
(
YDe
V e
)
(4.7E)
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Adding expectations
Equation
C = α1 · YDe + α2 · V−1 (4.5E)
Hh
V e= (1− λ0)− λ1 · r + λ2 ·
(
YDe
V e
)
(4.6E)
Bh
V e= λ0 + λ1 · r − λ2 ·
(
YDe
V e
)
(4.7E)
Buffer Stocks
”money balance are the element of flexibility in a monetary system ofproduction”[Godley and Lavoie, 2007, p. 108]
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Adding expectations
Equation
C = α1 · YDe + α2 · V−1 (4.5E)
Hh
V e= (1− λ0)− λ1 · r + λ2 ·
(
YDe
V e
)
(4.6E)
Bh
V e= λ0 + λ1 · r − λ2 ·
(
YDe
V e
)
(4.7E)
Buffer Stocks
”money balance are the element of flexibility in a monetary system ofproduction”[Godley and Lavoie, 2007, p. 108]
Hd = V−1 + (YDe
− C )− Bh (4.13)
Hh = V−1 + (YD − C )− Bh
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Outline
1 PK-SFC Package
2 Model SIM3.4 A numerical example and the standard Keynesian multiplier3.7 ExpectationsAppendix 3.4 Government deficit and growth
3 Model PC4.4 Buffer stocks and expectations4.7 A government target for the debt to income ratio
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Debt to GDP
Maastricht treaty
The reference values referred to [...] are:
◮ 3% for the ratio of the planned or actual government deficit to grossdomestic product at market
◮ 60% for the ratio of government debt to gross domestic product atmarket prices.
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Debt to GDP
Maastricht treaty
The reference values referred to [...] are:
◮ 3% for the ratio of the planned or actual government deficit to grossdomestic product at market
◮ 60% for the ratio of government debt to gross domestic product atmarket prices.
Steady state
V ⋆
Y ⋆=
1−α1α2
1 +[
θ1−θ
]
− r ·[
(λ0 + λ1 · r) ·1−α1α2
− λ2
] (4.33)
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Implementing the Maastricht treaty I
Assuming rational government
θ =
1−α1α2
·
1M
+ r ·[
(λ0 + λ1 · r) ·1−α1α2
− λ2
]
− 1
1−α1α2
·
1M
+ r ·[
(λ0 + λ1 · r) ·1−α1α2
− λ2
]
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Implementing the Maastricht treaty I
Assuming rational government
θ =
1−α1α2
·
1M
+ r ·[
(λ0 + λ1 · r) ·1−α1α2
− λ2
]
− 1
1−α1α2
·
1M
+ r ·[
(λ0 + λ1 · r) ·1−α1α2
− λ2
]
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Implementing the Maastricht treaty II
Assuming (somewhat) rational government
G = G−1(1 + grg )
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Implementing the Maastricht treaty II
Assuming (somewhat) rational government
G = G−1(1 + grg )
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Implementing the Maastricht treaty III
Assuming (more) rational government
G = G−1(1 + grg ) grg = 3%
θ = θ−1 + dθ dθ = 4.5%
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Implementing the Maastricht treaty III
Assuming (more) rational government
G = G−1(1 + grg ) grg = 3%
θ = θ−1 + dθ dθ = 4.5%
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Take home message
◮ Open source package to simulate SFC models
◮ Two simple models: SIM and PC
◮ Role of buffer stocks and expectations
◮ Difference between short-run and long-run Keynesian multiplier
◮ Devastating impacts of the Maastricht treaty (Strong assumption ofGovernment expenditures)
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References I
W. Godley and M. Lavoie. Monetary Economics An Integrated Approach
to Credit, Money, Income, Production and Wealth. Palgrave MacMillan,New York, 2007.
Stephen Kinsella and Terence O’Shea. Solution and simulation of largestock flow consistent monetary production models via the gauss seidelalgorithm. Journal of Policy Modeling, 2010.
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