An introduction to stock-flow consistent models in macroeco- nomics M. R. Grasselli Introduction Discrete-time SFC models Continuous- time SFC models Extensions Conclusions An introduction to stock-flow consistent models in macroeconomics M. R. Grasselli Mathematics and Statistics - McMaster University Masterclasses on New Approaches to Economic Challenges OECD-NAEC, April 17, 2019
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Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
An introduction to stock-flow consistent modelsin macroeconomics
M. R. Grasselli
Mathematics and Statistics - McMaster University
Masterclasses on New Approachesto Economic Challenges
OECD-NAEC, April 17, 2019
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
1 Introduction
2 Discrete-time SFC modelsBenchmark model
3 Continuous-time SFC modelsGoodwin modelKeen model
4 ExtensionsStabilizing governmentSpeculationStock PricesGreat ModerationEffective Demand and Inventories
5 Conclusions
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Stock-Flow Consistent models
Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.
They consider both real and monetary factorssimultaneously.
Specify the balance sheet and transactions betweensectors.
Accommodate a number of behavioural assumptions in away that is consistent with the underlying accountingstructure.
Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Stock-Flow Consistent models
Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.
They consider both real and monetary factorssimultaneously.
Specify the balance sheet and transactions betweensectors.
Accommodate a number of behavioural assumptions in away that is consistent with the underlying accountingstructure.
Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Stock-Flow Consistent models
Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.
They consider both real and monetary factorssimultaneously.
Specify the balance sheet and transactions betweensectors.
Accommodate a number of behavioural assumptions in away that is consistent with the underlying accountingstructure.
Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Stock-Flow Consistent models
Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.
They consider both real and monetary factorssimultaneously.
Specify the balance sheet and transactions betweensectors.
Accommodate a number of behavioural assumptions in away that is consistent with the underlying accountingstructure.
Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Stock-Flow Consistent models
Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.
They consider both real and monetary factorssimultaneously.
Specify the balance sheet and transactions betweensectors.
Accommodate a number of behavioural assumptions in away that is consistent with the underlying accountingstructure.
Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 1: money is not neutral
Money is hierarchical: currency is a promise to pay gold(or settle taxes); deposits are promises to pay currency;securities are promises to pay deposits.
Financial institutions are market-makers straddling twolevels in the hierarchy: central banks, banks, securitydealers.
The hierarchy is dynamic: discipline and elasticity changein time.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 1: money is not neutral
Money is hierarchical: currency is a promise to pay gold(or settle taxes); deposits are promises to pay currency;securities are promises to pay deposits.
Financial institutions are market-makers straddling twolevels in the hierarchy: central banks, banks, securitydealers.
The hierarchy is dynamic: discipline and elasticity changein time.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 1: money is not neutral
Money is hierarchical: currency is a promise to pay gold(or settle taxes); deposits are promises to pay currency;securities are promises to pay deposits.
Financial institutions are market-makers straddling twolevels in the hierarchy: central banks, banks, securitydealers.
The hierarchy is dynamic: discipline and elasticity changein time.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 2: money is endogenous
Banks create money and purchasing power.
Reserve requirements are never binding.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 2: money is endogenous
Banks create money and purchasing power.
Reserve requirements are never binding.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 3: private debt matters
Figure: Change in debt and unemployment.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 4: finance is not justintermediation
Market never clear in all states of the world: set of eventsis larger than what can be contracted.
The financial sector absorbs the risk of unfulfilled promises.
The cone of acceptable losses defines the size of the realeconomy.
Figure: Cherny and Madan (2009)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 4: finance is not justintermediation
Market never clear in all states of the world: set of eventsis larger than what can be contracted.
The financial sector absorbs the risk of unfulfilled promises.
The cone of acceptable losses defines the size of the realeconomy.
Figure: Cherny and Madan (2009)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Heterodox insight 4: finance is not justintermediation
Market never clear in all states of the world: set of eventsis larger than what can be contracted.
The financial sector absorbs the risk of unfulfilled promises.
The cone of acceptable losses defines the size of the realeconomy.
Figure: Cherny and Madan (2009)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Balance Sheets
Households Firms Banks Government Sum
Deposits +D −D 0
Loans −L +L 0
Bills +B −B 0
Capital goods +pK pK
Equities +peE −peE 0
Sum (net worth) Vh Vf 0 −B pK
Table: Aggregate balance sheets in the ‘benchmark’ SFC model ofDos Santos and Zezza (2008)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Transactions
TransactionsHouseholds
FirmsBanks Government Sum
current capital
Consumption −pC +pC 0
Gov spending +pG −pG 0
Investment +pI −pI 0
Acct memo [GDP] [pY ]
Depreciation −pδK +pδK 0
Wages +W −W 0
Taxes −Th −Tf +T 0
Interest on loans −rLt−1Lt−1 +rLt−1Lt−1 0
Interest on bills +rBt−1Bt−1 −rBt−1Bt−1 0
Interest on deposits +rDt−1Dt−1 −rDt−1Dt−1 0
Dividends +Πd + Πb −Πd −Πb 0
Sum Sh Sf −p(I − δK ) 0 Sg 0
Table: Transactions in the ‘benchmark’ SFC model of Dos Santosand Zezza (2008)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Flow of Funds
Households Firms Banks Government Sum
Change in Deposits +∆D −∆D 0
Change in Loans −∆L +∆L 0
Change in Bills +∆B −∆B 0
Change in Capital +p(I − δK ) p(I − δK )
Equities +pe∆E −pe∆E 0
Sum Sh Sf 0 Sg p(I − δK )
Change in Net Worth (Sh + ∆peE ) (Sf −∆peE + ∆pK ) 0 Sg ∆pK + p∆K
Table: Flow of funds in the ‘benchmark’ SFC model of Dos Santosand Zezza (2008)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules - Dos Santos and Zezza (2008)
Assume that the price level is given by
pt = (1 + τ)uct = (1 + τ)Wt
Yt= (1 + τ)
wtLtatLt
= (1 + τ)wt
at
It follows that the wage share of nominal output is
ω =Wt
ptYt=
wtLtat(1 + τ)wtYt
=1
1 + τ
and the corresponding profit share is πt = 1− ωt = τ1+τ .
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules - Dos Santos and Zezza (2008)
Assume that the price level is given by
pt = (1 + τ)uct = (1 + τ)Wt
Yt= (1 + τ)
wtLtatLt
= (1 + τ)wt
at
It follows that the wage share of nominal output is
ω =Wt
ptYt=
wtLtat(1 + τ)wtYt
=1
1 + τ
and the corresponding profit share is πt = 1− ωt = τ1+τ .
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Consumption and investment are assumed to be given by
Ct = Wt − (Th)t + (1− s)(Vh)t−1
It = (g0 + (απ + β)ut − θrLt )Kt−1
where ut = Yt/Kt−1 is a proxy for capacity utilization.
Households decide to invest in equity and depositsaccording to
(pe)tEt = ϕ · (Vh)t−1
Dt = (1− ϕ) · (Vh)t−1
Firms try to keep Et/Kt = χ constant and borrow theremainder of the funds needed to finance investment.Consequently, the equilibrium price for equities is
pet =ϕ · (Vh)tχKt
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Consumption and investment are assumed to be given by
Ct = Wt − (Th)t + (1− s)(Vh)t−1
It = (g0 + (απ + β)ut − θrLt )Kt−1
where ut = Yt/Kt−1 is a proxy for capacity utilization.Households decide to invest in equity and depositsaccording to
(pe)tEt = ϕ · (Vh)t−1
Dt = (1− ϕ) · (Vh)t−1
Firms try to keep Et/Kt = χ constant and borrow theremainder of the funds needed to finance investment.Consequently, the equilibrium price for equities is
pet =ϕ · (Vh)tχKt
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Consumption and investment are assumed to be given by
Ct = Wt − (Th)t + (1− s)(Vh)t−1
It = (g0 + (απ + β)ut − θrLt )Kt−1
where ut = Yt/Kt−1 is a proxy for capacity utilization.Households decide to invest in equity and depositsaccording to
(pe)tEt = ϕ · (Vh)t−1
Dt = (1− ϕ) · (Vh)t−1
Firms try to keep Et/Kt = χ constant and borrow theremainder of the funds needed to finance investment.
Consequently, the equilibrium price for equities is
pet =ϕ · (Vh)tχKt
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Consumption and investment are assumed to be given by
Ct = Wt − (Th)t + (1− s)(Vh)t−1
It = (g0 + (απ + β)ut − θrLt )Kt−1
where ut = Yt/Kt−1 is a proxy for capacity utilization.Households decide to invest in equity and depositsaccording to
(pe)tEt = ϕ · (Vh)t−1
Dt = (1− ϕ) · (Vh)t−1
Firms try to keep Et/Kt = χ constant and borrow theremainder of the funds needed to finance investment.Consequently, the equilibrium price for equities is
pet =ϕ · (Vh)tχKt
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Banks are assumed to meet the demand for loans by firmsand deposits by households.
In addition, banks set the interest rate on deposits as equalto the interest rate on government bills and the interestrate on loans as a fixed on markup on the rate on deposits.
The government chooses the level of spending Gt , theinterest rate rbt and the level of taxes Tt , with the amountof debt determined as a residual.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Banks are assumed to meet the demand for loans by firmsand deposits by households.
In addition, banks set the interest rate on deposits as equalto the interest rate on government bills and the interestrate on loans as a fixed on markup on the rate on deposits.
The government chooses the level of spending Gt , theinterest rate rbt and the level of taxes Tt , with the amountof debt determined as a residual.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Behavioural rules (continued)
Banks are assumed to meet the demand for loans by firmsand deposits by households.
In addition, banks set the interest rate on deposits as equalto the interest rate on government bills and the interestrate on loans as a fixed on markup on the rate on deposits.
The government chooses the level of spending Gt , theinterest rate rbt and the level of taxes Tt , with the amountof debt determined as a residual.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Dynamics
Using the SFC tables and the behavioural rules, one canreduce the dynamics of the model to system of differenceequations for the variables
gt =It
Kt−1bt =
Bt
ptKt
vht =(Vh)tptKt
ut =Yt
Kt−1
It is very difficult to establish the properties of thedynamics analytically, although the model is relatively easyto simulate.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Benchmarkmodel
Continuous-time SFCmodels
Extensions
Conclusions
Dynamics
Using the SFC tables and the behavioural rules, one canreduce the dynamics of the model to system of differenceequations for the variables
gt =It
Kt−1bt =
Bt
ptKt
vht =(Vh)tptKt
ut =Yt
Kt−1
It is very difficult to establish the properties of thedynamics analytically, although the model is relatively easyto simulate.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Goodwin Model - SFC matrix
Balance Sheet HouseholdsFirms
Sum
current capital
Capital +pK pK
Sum (net worth) 0 0 Vf pK
Transactions
Consumption −pC +pC 0
Investment +pI −pI 0
Acct memo [GDP] [pY ]
Depreciation −pδK +pδK 0
Wages +W −W 0
Sum 0 Sf p(I − δK ) 0
Flow of Funds
Change in Capital +p(I − δK ) p(I − δK )
Sum 0 Sf p(I − δK )
Change in Net Worth 0 Sf + pK pK + pK
Table: SFC table for the Goodwin model.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Goodwin Model - Differential equations
Define
ω =w`
pY=
w
pa(wage share)
λ =`
N=
Y
aN(employment rate)
It then follows that
ω
ω=
w
w− p
p− a
a= Φ(λ, i , ie)− i − α
λ
λ=
1− ων− α− β − δ
In the original model, all quantities were real (i.e dividedby p), which is equivalent to setting i = ie = 0.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Goodwin Model - Differential equations
Define
ω =w`
pY=
w
pa(wage share)
λ =`
N=
Y
aN(employment rate)
It then follows that
ω
ω=
w
w− p
p− a
a= Φ(λ, i , ie)− i − α
λ
λ=
1− ων− α− β − δ
In the original model, all quantities were real (i.e dividedby p), which is equivalent to setting i = ie = 0.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Goodwin Model - Differential equations
Define
ω =w`
pY=
w
pa(wage share)
λ =`
N=
Y
aN(employment rate)
It then follows that
ω
ω=
w
w− p
p− a
a= Φ(λ, i , ie)− i − α
λ
λ=
1− ων− α− β − δ
In the original model, all quantities were real (i.e dividedby p), which is equivalent to setting i = ie = 0.
Apart from the interior equilibrium (ω1, λ1, d1) and theexplosive equilibria of the form (ω2, λ2, d2) = (0, 0,±∞),the system has a new undesirable equilibrium of the form(ω3, 0, b3) where
ω3 =1
ξ+
Φ(0)− αξηp(1− γ)
and b3 solves the nonlinear equation
b [i(ω3) + g(1− ω3 − rb)− r ] = κ(1−ω3− rb)− 1 +ω3 .
Notice that
i(ω1) =Φ(λ1)− α
1− γ>
Φ(0)− α1− γ
= i(ω3) ,
so that this type of equilibrium is necessarily deflationary.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Keen model with inflation- equilibria
Apart from the interior equilibrium (ω1, λ1, d1) and theexplosive equilibria of the form (ω2, λ2, d2) = (0, 0,±∞),the system has a new undesirable equilibrium of the form(ω3, 0, b3) where
ω3 =1
ξ+
Φ(0)− αξηp(1− γ)
and b3 solves the nonlinear equation
b [i(ω3) + g(1− ω3 − rb)− r ] = κ(1−ω3− rb)− 1 +ω3 .
Notice that
i(ω1) =Φ(λ1)− α
1− γ>
Φ(0)− α1− γ
= i(ω3) ,
so that this type of equilibrium is necessarily deflationary.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Example 3: convergence to the good equilibrium
Figure: Trajectories for λ for different values of price adjustment ηpand money illusion (1− γ), Grasselli and Nguyen Huu (2015)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Example 4: convergence to (new) bad equilibrium
Figure: Trajectories for ω for different values of mark-up ξ, Grasselliand Nguyen Huu (2015)
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Goodwin model
Keen model
Extensions
Conclusions
Example 5: explosive debt and ‘great moderation’
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Introducing a government sector
Following Keen (and echoing Minsky) we add discretionarygovernment subsidied and taxation into the original systemin the form
G = G1 + G2
T = T1 + T2
where
G1 = η1(λ)Y G2 = η2(λ)G2
T1 = Θ1(π)Y T2 = Θ2(π)T2
Defining g = G/Y and τ = T/Y , the net profit share isnow
π = 1− ω − rd + g − τ,and government debt evolves according to
B = rB + G − T .
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Introducing a government sector
Following Keen (and echoing Minsky) we add discretionarygovernment subsidied and taxation into the original systemin the form
G = G1 + G2
T = T1 + T2
where
G1 = η1(λ)Y G2 = η2(λ)G2
T1 = Θ1(π)Y T2 = Θ2(π)T2
Defining g = G/Y and τ = T/Y , the net profit share isnow
π = 1− ω − rd + g − τ,and government debt evolves according to
B = rB + G − T .
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Good equilibrium
The system (??) has a good equilibrium at
ω = 1− π − rν(α + β + δ)− π
α + β+η1(λ)−Θ1(π)
α + β
λ = Φ−1(α)
π = κ−1(ν(α + β + δ))
g2 = τ2 = 0
and this is locally stable for a large range of parameters.
The other variables then converge exponentially fast to
d =ν(α + β + δ)− π
α + β
g1 =η1(λ)
α + β
τ1 =Θ1(π)
α + β
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Good equilibrium
The system (??) has a good equilibrium at
ω = 1− π − rν(α + β + δ)− π
α + β+η1(λ)−Θ1(π)
α + β
λ = Φ−1(α)
π = κ−1(ν(α + β + δ))
g2 = τ2 = 0
and this is locally stable for a large range of parameters.The other variables then converge exponentially fast to
d =ν(α + β + δ)− π
α + β
g1 =η1(λ)
α + β
τ1 =Θ1(π)
α + β
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Bad equilibria - destabilizing a stable crisis
Recall that π = 1− ω − rd + g − τ .
The system has bad equilibria of the form
(ω, λ, g2, τ2, π) = (0, 0, 0, 0,−∞)
(ω, λ, g2, τ2, π) = (0, 0,±∞, 0,−∞)
If g2(0) > 0, then any equilibria with π → −∞ is locallyunstable provided η2(0) > r .
On the other hand, if g2(0) < 0 (austerity), then theseequilibria are all locally stable.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Bad equilibria - destabilizing a stable crisis
Recall that π = 1− ω − rd + g − τ .
The system has bad equilibria of the form
(ω, λ, g2, τ2, π) = (0, 0, 0, 0,−∞)
(ω, λ, g2, τ2, π) = (0, 0,±∞, 0,−∞)
If g2(0) > 0, then any equilibria with π → −∞ is locallyunstable provided η2(0) > r .
On the other hand, if g2(0) < 0 (austerity), then theseequilibria are all locally stable.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Bad equilibria - destabilizing a stable crisis
Recall that π = 1− ω − rd + g − τ .
The system has bad equilibria of the form
(ω, λ, g2, τ2, π) = (0, 0, 0, 0,−∞)
(ω, λ, g2, τ2, π) = (0, 0,±∞, 0,−∞)
If g2(0) > 0, then any equilibria with π → −∞ is locallyunstable provided η2(0) > r .
On the other hand, if g2(0) < 0 (austerity), then theseequilibria are all locally stable.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Bad equilibria - destabilizing a stable crisis
Recall that π = 1− ω − rd + g − τ .
The system has bad equilibria of the form
(ω, λ, g2, τ2, π) = (0, 0, 0, 0,−∞)
(ω, λ, g2, τ2, π) = (0, 0,±∞, 0,−∞)
If g2(0) > 0, then any equilibria with π → −∞ is locallyunstable provided η2(0) > r .
On the other hand, if g2(0) < 0 (austerity), then theseequilibria are all locally stable.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Example 3: Good initial conditions
0
0.2
0.4
0.6
0.8
1
λ
−20
−15
−10
−5
0
5d
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
1.5
2
time
ω
ω(0) = 0.85, λ(0) = 0.85, d(0) = 0.5, gS
1
(0) = 0.05, gT
1
(0) = 0.05, gS
2
(0) = 0.05, gT
2
(0) = 0.05, dg(0) = 0, r = 0.03, η
max(2) = 0.02
Keen ModelModel w/ Government
0.04
0.06
0.08
0.1
0.12
g T1+
g T2
0
5
10
15
20
d g
0 50 100 150 200 250 300 350 400 450 5000
0.2
0.4
0.6
0.8
time
g S1+
g S2
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Example 4: Bad initial conditions
0
0.2
0.4
0.6
0.8
1
λ
−20
−15
−10
−5
0
5d
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
1.5
2
2.5
time
ω
ω(0) = 0.8, λ(0) = 0.8, d(0) = 0.5, gS
1
(0) = 0.05, gT
1
(0) = 0.05, gS
2
(0) = 0.05, gT
2
(0) = 0.05, dg(0) = 0, r = 0.03, η
max(2) = 0.02
Keen ModelModel w/ Government
0.04
0.06
0.08
0.1
0.12
g T1+
g T2
0
5
10
15
20
25
d g
0 50 100 150 200 250 300 350 400 450 5000
0.2
0.4
0.6
0.8
1
time
g S1+
g S2
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Example 5: Really bad initial conditions with timidgovernment
0
0.2
0.4
0.6
0.8
1
λ
−2
0
2
4
6x 10
16
d
0 50 100 150 200 250 300 350 400 450 5000
5
10
15
20
25
30
time
ω
ω(0) = 0.15, λ(0) = 0.15, d(0) = 3, gS
1
(0) = 0.05, gT
1
(0) = 0.05, gS
2
(0) = 0.05, gT
2
(0) = 0.05, dg(0) = 0, r = 0.03, η
max(2) = 0.02
Keen ModelModel w/ Government
−10
−5
0
5x 10
8
g T1+
g T2
0
0.5
1
1.5
2
2.5x 10
17
d g
0 50 100 150 200 250 300 350 400 450 5000
2
4
6
8x 10
10
time
g S1+
g S2
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Example 6: Really bad initial conditions withresponsive government
0
0.2
0.4
0.6
0.8
1
λ
−10
−5
0
5x 10
8
g T1+
g T2
0
0.2
0.4
0.6
0.8
1
λ
−500
0
500
1000d
0 50 100 150 200 250 300 350 400 450 5000
5
10
15
20
25
30
time
ω
ω(0) = 0.15, λ(0) = 0.15, d(0) = 3, gS
1
(0) = 0.05, gT
1
(0) = 0.05, gS
2
(0) = 0.05, gT
2
(0) = 0.05, dg(0) = 0, r = 0.03, η
max(2) = 0.2
Keen ModelModel w/ Government
−2
−1.5
−1
−0.5
0
0.5
g T1+
g T2
0
100
200
300
400
500
600
d g
0 50 100 150 200 250 300 350 400 450 5000
5
10
15
20
25
30
time
g S1+
g S2
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Example 7: Austerity in good times: harmless
0
0.2
0.4
0.6
0.8
1
λ
−20
−15
−10
−5
0d
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
1.5
2
2.5
time
ω
ω(0) = 0.8, λ(0) = 0.8, d(0) = 0.5, gS
1
(0) = 0.05, gT
1
(0) = 0.05, gS
2
(0) = +−0.05, gT
2
(0) = 0.05, dg(0) = 0, r = 0.03, η
max(2) = 0.02
g
S2
(0)>0
gS
2
(0)<0
0.04
0.06
0.08
0.1
0.12
g T1+
g T2
−5
0
5
10
15
20
d g
0 50 100 150 200 250 300 350 400 450 5000
0.2
0.4
0.6
0.8
1
time
g S1+
g S2
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Example 8: Austerity in bad times: a really badidea
0
0.2
0.4
0.6
0.8
1
λ
−5
0
5
10
15x 10
48
d
0 50 100 150 200 250 300 350 400 450 5000
5
10
15
20
25
30
time
ω
ω(0) = 0.15, λ(0) = 0.15, d(0) = 4, gS
1
(0) = 0.05, gT
1
(0) = 0.05, gS
2
(0) = +−0.05, gT
2
(0) = 0.05, dg(0) = 0, r = 0.03, η
max(2) = 0.2
g
S2
(0)>0
gS
2
(0)<0
−15
−10
−5
0
5x 10
8
g T1+
g T2
−15
−10
−5
0
5x 10
48
d g
0 50 100 150 200 250 300 350 400 450 500−3
−2
−1
0
1x 10
48
time
g S1+
g S2
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Stabilizinggovernment
Speculation
Stock Prices
GreatModeration
EffectiveDemand andInventories
Conclusions
Hopft bifurcation with respect to governmentspending.
Solow (1990): The true test of a simple model is whetherit helps us to make sense of the world. Marx was, ofcourse, dead wrong about this. We have changed theworld in all sorts of ways, with mixed results; the point isto interpret it.
Schumpeter (1939): Cycles are not, like tonsils, separablethings that might be treated by themselves, but are, likethe beat of the heart, of the essence of the organism thatdisplays them.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Concluding thoughts
Solow (1990): The true test of a simple model is whetherit helps us to make sense of the world. Marx was, ofcourse, dead wrong about this. We have changed theworld in all sorts of ways, with mixed results; the point isto interpret it.
Schumpeter (1939): Cycles are not, like tonsils, separablethings that might be treated by themselves, but are, likethe beat of the heart, of the essence of the organism thatdisplays them.
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Concluding thoughts (continued)
Since Keynes’s death it has developed in two radicallydifferent approaches:
1 The dominant one has the appearance of mathematicalrigour (the SMD theorems notwithstanding), but is basedon implausible assumptions, has poor fit to data in general,and is disastrously wrong during crises. Finance plays anegligible role
2 The heterodox approach is grounded in history andinstitutional understanding, takes empirical work muchmore seriously, but is generally averse to mathematics.Finance plays a major role.
Stock-flow consistent agent-based models, complementedby mean-field approximations and other techniques(including mean-field games), have the potential toredefine the role of mathematics in macroeconomics.
Thank you!
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Concluding thoughts (continued)
Since Keynes’s death it has developed in two radicallydifferent approaches:
1 The dominant one has the appearance of mathematicalrigour (the SMD theorems notwithstanding), but is basedon implausible assumptions, has poor fit to data in general,and is disastrously wrong during crises. Finance plays anegligible role
2 The heterodox approach is grounded in history andinstitutional understanding, takes empirical work muchmore seriously, but is generally averse to mathematics.Finance plays a major role.
Stock-flow consistent agent-based models, complementedby mean-field approximations and other techniques(including mean-field games), have the potential toredefine the role of mathematics in macroeconomics.
Thank you!
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Concluding thoughts (continued)
Since Keynes’s death it has developed in two radicallydifferent approaches:
1 The dominant one has the appearance of mathematicalrigour (the SMD theorems notwithstanding), but is basedon implausible assumptions, has poor fit to data in general,and is disastrously wrong during crises. Finance plays anegligible role
2 The heterodox approach is grounded in history andinstitutional understanding, takes empirical work muchmore seriously, but is generally averse to mathematics.Finance plays a major role.
Stock-flow consistent agent-based models, complementedby mean-field approximations and other techniques(including mean-field games), have the potential toredefine the role of mathematics in macroeconomics.
Thank you!
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Concluding thoughts (continued)
Since Keynes’s death it has developed in two radicallydifferent approaches:
1 The dominant one has the appearance of mathematicalrigour (the SMD theorems notwithstanding), but is basedon implausible assumptions, has poor fit to data in general,and is disastrously wrong during crises. Finance plays anegligible role
2 The heterodox approach is grounded in history andinstitutional understanding, takes empirical work muchmore seriously, but is generally averse to mathematics.Finance plays a major role.
Stock-flow consistent agent-based models, complementedby mean-field approximations and other techniques(including mean-field games), have the potential toredefine the role of mathematics in macroeconomics.
Thank you!
Anintroductionto stock-flow
consistentmodels inmacroeco-
nomics
M. R. Grasselli
Introduction
Discrete-timeSFC models
Continuous-time SFCmodels
Extensions
Conclusions
Concluding thoughts (continued)
Since Keynes’s death it has developed in two radicallydifferent approaches:
1 The dominant one has the appearance of mathematicalrigour (the SMD theorems notwithstanding), but is basedon implausible assumptions, has poor fit to data in general,and is disastrously wrong during crises. Finance plays anegligible role
2 The heterodox approach is grounded in history andinstitutional understanding, takes empirical work muchmore seriously, but is generally averse to mathematics.Finance plays a major role.
Stock-flow consistent agent-based models, complementedby mean-field approximations and other techniques(including mean-field games), have the potential toredefine the role of mathematics in macroeconomics.