Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Dynamical Systems and Financial Instability
Alternative
approaches
SFC models
Conclusions
M. R. Grasselli
Mathematics and Statistics - McMaster University
and Fields Institute for Research in Mathematical Sciences
ICTP-SAIFR Seminar, May 30, 2014
A brief history of Macroeconomics
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Classics (Smith, Ricardo, Marx): no distinction between
micro and macro, Say’s law, emphasis on long run.
Beginning of the 20th century (Wicksell, Fisher): natural
rate of interest, quantity theory of money.
Keynesian revolution (1936): shift to demand, fallacies of
composition, role of expectations, and much more!
Neoclassical synthesis - 1945 to 1970 (Hicks, Samuelson,
Solow): Keynesian consensus.
Rational Expectations Revolution - 1972 (Lucas, Prescott,
Sargent): internal consistency, microfoundations.
Start of Macro Wars: Real Business Cycles versus New
Keynesian.
1990’s: impression of consensus around DSGE models, but
with different flavours.
Dynamic Stochastic General Equilibrium
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Seeks to explain the aggregate economy using theories
based on strong microeconomic foundations.
Collective decisions of rational individuals over a range of
variables for both present and future.
All variables are assumed to be simultaneously in
equilibrium.
The only way the economy can be in disequilibrium at any
point in time is through decisions based on wrong
information.
Money is neutral in its effect on real variables.
Largely ignores uncertainty by simply subtracting risk
premia from all risky returns and treat them as risk-free.
Really bad economics: hardcore (freshwater) DSGE
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
The strand of DSGE economists affiliated with RBC
theory made the following predictions after 2008:
1
SFC models
Conclusions
2
3
Increases government borrowing would lead to higher
interest rates on government debt because of “crowding
out”.
Increases in the money supply would lead to inflation.
Fiscal stimulus has zero effect in an ideal world and
negative effect in practice (because of decreased
confidence).
Wrong prediction number 1
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Figure: Government borrowing and interest rates.
Wrong prediction number 2
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Figure: Monetary base and inflation.
Wrong prediction number 3
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Figure: Fiscal tightening and GDP.
Better (but still bad) economics: soft core
(saltwater) DSGE
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
The strand of DSGE economists affiliated with New
Keynesian theory got all these predictions right.
They did so by augmented DSGE with ‘imperfections’
(wage stickiness, asymmetric information, imperfect
competition, etc).
Still DSGE at core - analogous to adding epicycles to
Ptolemaic planetary system.
For example: “Ignoring the foreign component, or looking
at the world as a whole, the overall level of debt makes no
difference to aggregate net worth – one person’s liability is
another person’s asset.” (Paul Krugman and Gauti B.
Eggertsson, 2010, pp. 2-3)
Finance in DSGE models
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
The financial sector merely serve as intermediaries
channeling savings from households to business.
Banks provide indirect finance by borrowing short and
lending long (business loans), thereby solving the problem
of liquidity preferences (Diamond and Dybvig (1986)
model).
Financial market provide direct finance through shares,
thereby introducing market prices and discipline.
Financial Frictions (e.g borrowing constraints, market
liquidity) create persistence and amplification of real
shocks (Bernanke and Gertler (1989), Kiyotaki and Moore
(1997) models)
See Brunnermeier and Sannikov (2013) for a recent
contribution to this strand of literature in light of the
financial crisis, in particular in the context of
macro-prudential regulation.
Frictions literature still missing the point
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
Turner 2013 observes that:
“Quantitative impacts suggested by the models were far
smaller than those empirically observed in real world
episodes such as the Great Depression or the 2008 crisis”
“Most of the literature omits consideration of
behaviourally driven ‘irrational’ cycles in asset prices”.
SFC models
Conclusions
“the vast majority of the literature ignores the possibilities
of credit extension to finance the purchase of already
existing assets”.
“the dominant model remains one in which household
savers make deposits in banks, which lend money to
entrepreneurs/businesses to pursue ‘investment projects’.
The reality of a world in which only a small proportion
(e.g. 15%) of bank credit funds ‘new investment projects’
has therefore been left largely unexplored.”
Turner (2013) slide
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
A parallel history of Macroeconomics
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Classical 19th century monetarism (Bagehot, Allan
Young): role of banks in trade (Britain) and development
(U.S.), central banking.
Several prominent disciples of Keynes (Kaldor, Robinson,
Davidson) immediately rejected the Neoclassical synthesis
as “bastardized Keynesianism”.
Flow of Funds accounting - 1952 (Copeland): alternative
to both Y = C + I + G + X − M (finals sales) and
MV = PT (money transactions) by tracking exchanges of
both goods and financial assets.
Gurley, Shaw, Tobin, Minsky: financial intermediation at
centre stage.
Kindleberger (1978): detailed history of financier crises.
Stock-flow consistent models (Godley, Lavoie)
Revival of interest after the 2008 crisis.
Key insight 1: money is not neutral
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Money is hierarchical: currency is a promise to pay gold
(or taxes); deposits are promises to pay currency;
securities are promises to pay deposits.
Financial institutions are market-makers straddling two
levels in the hierarchy: central banks, banks, security
dealers.
The hierarchy is dynamic: discipline and elasticity change
in time.
Key insight 2: money is endogenous
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Banks create money and purchasing power.
Mainstream
Reserve requirements are never binding.
Alternative
approaches
SFC models
Conclusions
Key insight 3: private debt matters
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions
Figure: Change in debt and unemployment.
Key insight 4: finance is not just intermediation
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
Market never clear in all states: set of events is 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 real
economy.
SFC models
Conclusions
Figure: Cherny and Madan (2009)
Much better economics: SFC models
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Stock-flow consistent models emerged in the last decade
as a common language for many heterodox schools of
thought in economics.
Consider both real and monetary factors from the start
Specify the balance sheet and transactions between sectors
Accommodate a number of behavioural assumptions in a
way that is consistent with the underlying accounting
structure.
Reject silly (and mathematically unsound!) hypotheses
such as the RARE individual (representative agent with
rational expectations).
See Godley and Lavoie (2007) for the full framework.
Balance Sheets
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Balance Sheet
Firms
Households
current
Banks
Central Bank
Alternative
approaches
Cash
+Hh
Deposits
+Mh
+Hb
−H
+Mf
−L
Loans
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Bills
Equities
+Bh
−pf Ef
Sum (net worth)
0
+L
0
+Bc
−B
−pb Eb
−A
Advances
Capital
0
Vf
0
0
+A
0
+pK
Vh
Sum
0
−M
+Bb
+pf Ef + pb Eb
Government
capital
Mainstream
pK
Vb
0
−B
pK
Table: Balance sheet in an example of a general SFC model.
Transactions
Dynamical
Systems and
Financial
Instability
Transactions
current
Consumption
Mainstream
Firms
Households
M. R. Grasselli
−pCh
Banks
Central Bank
Government
Sum
capital
−pCb
+pC
0
−pI
Investment
+pI
Alternative
approaches
Gov spending
+pG
Acct memo [GDP]
[pY ]
SFC models
Wages
+W
−W
Taxes
−Th
−Tf
+rM .Mh
+rM .Mf
−rM .M
0
−rL .L
+rL .L
0
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Interest on deposits
Interest on loans
Interest on bills
Profits
Sum
0
−pG
0
0
+T
0
+rB .Bb
+rB .Bc
−rB .B
0
+Πd + Πb
−Π
+Πu
−Πb
−Πc
+Πc
0
Sh
0
Sf − pI
Sb
0
Sg
0
+rB .Bh
Table: Transactions in an example of a general SFC model.
Flow of Funds
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Flow of Funds
Firms
Households
current
Mainstream
Cash
+Ḣh
Alternative
approaches
Deposits
+Ṁh
SFC models
Bills
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Loans
Equities
+Ḣb
−Ḣ
−Ṁ
−L̇
+L̇
+Ḃh
+Ḃb
−pf Ėf
+pf Ėf + pb Ėb
Government
0
0
+Ḃc
−Ḃ
Sh
0
Sf
(Sf − ṗf Ef + ṗK − pδK )
0
0
+Ȧ
0
+pI
(Sh + ṗf Ef + ṗb Eb )
Sum
0
−pb Ėb
−Ȧ
Capital
Change in Net Worth
Central Bank
+Ṁf
Advances
Sum
Banks
capital
pI
Sb
(Sb − ṗb Eb )
0
Sg
pI
Sg
ṗK + p K̇
Table: Flow of funds in an example of a general SFC model.
General Notation
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Employed labor force: `
Mainstream
Alternative
approaches
Production function: Y = f (K , `)
Labour productivity: a =
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Y
`
Capital-to-output ratio: ν =
Employment rate: λ =
K
Y
`
N
Change in capital: K̇ = I − δK
Inflation rate: i =
ṗ
p
Goodwin Model (1967) - Assumptions
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Assume that
N = N0 e βt
a = a0 e
Mainstream
Alternative
approaches
(total labour force)
αt
Y = min
(productivity per worker)
K
, a`
ν
(Leontief production)
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Assume further that
K
Y =
= a`
ν
ẇ = Φ(λ, i, i e )w
pI = pY − w`
(full capital utilization)
(Phillips curve)
(Say’s Law)
Conclusions
NOTE: In the original paper, Goodwin assumed that w
above was the real wage rate, so all quantities were
normalized by p.
Goodwin Model - SFC matrix
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Balance Sheet
Firms
Households
current
Capital
Sum (net worth)
0
0
−pC
+pC
Sum
capital
+pK
pK
Vf
pK
Mainstream
Transactions
Alternative
approaches
Consumption
0
−pI
0
−Π
+Πu
0
0
0
0
0
+pI
pI
Sum
0
0
Change in Net Worth
0
Investment
+pI
Acct memo [GDP]
[pY ]
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Wages
+W
Profits
Sum
−W
0
Flow of Funds
Capital
Πu
pI + ṗK − pδK
pI
ṗK + p K̇
Table: SFC table for the Goodwin model.
Goodwin Model - Differential equations
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
Define
wL
w
=
pY
pa
L
Y
λ=
=
N
aN
ω=
(wage share)
(employment rate)
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
It then follows that
ω̇
w ṗ ȧ
= − − = Φ(λ, i, i e ) − i − α
ω
w p a
λ̇
1−ω
=
−α−β−δ
λ
ν
In the original model, all quantities were real (i.e divided
by p), which is equivalent to setting i = i e = 0.
Example 1: Goodwin model
Dynamical
Systems and
Financial
Instability
w0 = 0.8, λ0 = 0.9
1
M. R. Grasselli
Mainstream
0.98
Alternative
approaches
0.96
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
λ
SFC models
0.94
0.92
0.9
Conclusions
0.88
0.7
0.75
0.8
0.85
ω
0.9
0.95
1
Example 1 (continued): Goodwin model
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
ω
λ
Y
Mainstream
w0 = 0.8, λ0 = 0.9, Y0 = 100
1
6000
0.95
5000
0.9
4000
0.85
3000
0.8
2000
0.75
1000
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
ω, λ
SFC models
Conclusions
0.7
0
10
20
30
40
50
t
60
70
80
90
0
Y
Alternative
approaches
Goodwin Model - Extensions, structural instability,
and empirical tests
Dynamical
Systems and
Financial
Instability
Desai 1972: Inflation leads to a stable equilibrium.
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Ploeg 1985: CES production function leads to stable
equilibrium.
Goodwin 1991: Pro-cyclical productivity growth leads to
explosive oscillations.
Solow 1990: US post-war data shows three sub-cycles with
a “bare hint of a single large clockwise sweep” in the
(ω, λ) plot.
Harvie 2000: Data from other OECD confirms the same
qualitative features and shows unsatisfactory quantitative
estimations.
Testing Goodwin on OECD countries
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Figure: Harvie (2000)
Correcting Harvie
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Figure: Grasselli and Maheshwari (2012)
SFC table for Keen (1995) model
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Balance Sheet
Firms
Households
current
Deposits
Loans
−L
Capital
+pK
Sum (net worth)
Mainstream
Vh
0
−pC
+pC
Banks
Sum
−D
0
capital
+D
Vf
+L
0
pK
0
pK
Transactions
Alternative
approaches
Consumption
Investment
+pI
SFC models
Acct memo [GDP]
[pY ]
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Wages
+W
Interest on deposits
+rD
0
−pI
−W
0
Interest on loans
−rL
Profits
−Π
+Πu
0
Sf − pI
Sum
Sh
0
−rD
0
+rL
0
0
0
0
−Ḋ
0
Flow of Funds
Deposits
+Ḋ
Loans
−L̇
Capital
+pI
Sum
Sh
Change in Net Worth
Sh
0
Πu
(Sf + ṗK − pδK )
+L̇
0
pI
0
pI
ṗK + p K̇
Table: SFC table for the Keen model.
Keen model - Investment function
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Assume now that new investment is given by
K̇ = κ(1 − ω − rd)Y − δK
where κ(·) is a nonlinear increasing function of profits
π = 1 − ω − rd.
This leads to external financing through debt evolving
according to
Ḋ = κ(1 − ω − rd)Y − (1 − ω − rd)Y
Keen model - Differential Equations
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Denote the debt ratio in the economy by d = D/Y , the model
can now be described by the following system
ω̇ = ω [Φ(λ) − α]
κ(1 − ω − rd)
λ̇ = λ
−α−β−δ
(1)
ν
κ(1 − ω − rd)
ḋ = d r −
+ δ + κ(1 − ω − rd) − (1 − ω)
ν
Keen model - equilibria
Dynamical
Systems and
Financial
Instability
The system (1) has a good equilibrium at
ω =1−π−r
M. R. Grasselli
Mainstream
λ = Φ−1 (α)
ν(α + β + δ) − π
d=
α+β
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
ν(α + β + δ) − π
α+β
with
π = κ−1 (ν(α + β + δ)),
which is stable for a large range of parameters
It also has a bad equilibrium at (0, 0, +∞), which is stable
if
κ(−∞)
−δ <r
(2)
ν
Example 2: convergence to the good equilibrium in
a Keen model
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
ω
λ
Y
d
ω0 = 0.75, λ0 = 0.75, d0 = 0.1, Y0 = 100
7
8
x 10
1.3
1
2
1.8
7
1.2
Mainstream
0.95
1.6
Alternative
approaches
6
SFC models
5
Conclusions
1.4
0.9
1.2
1
1
ω
0.9
0.8
3
0.8
0.6
0.8
2
0.4
Y
1
0
0.75
0.7
0.2
0
50
100
150
time
200
250
300
0.7
0
d
0.85
λ
4
d
ω
Y
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
λ
1.1
Example 3: explosive debt in a Keen model
Dynamical
Systems and
Financial
Instability
ω0 = 0.75, λ0 = 0.7, d0 = 0.1, Y0 = 100
6000
6
35
M. R. Grasselli
Mainstream
30
5000
x 10
2.5
1
ω
λ
Y
d
0.9
0.8
Alternative
approaches
25
0.7
4000
SFC models
λ
0.6
1.5
d
0.5
λ
ω
3000
Y
d
20
Y
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
2
15
0.4
1
2000
0.3
10
0.2
1000
Conclusions
ω
0
0.5
5
0
0
50
100
0.1
150
time
200
250
300
0
0
Basin of convergence for Keen model
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
10
SFC models
8
6
d
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
4
2
0
0.4
0.5
0.5
0.6
Conclusions
0.7
1
0.8
0.9
1
1.1
λ
1.5
ω
Ponzi financing
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
To introduce the destabilizing effect of purely speculative
investment, we consider a modified version of the previous
model with
Ḋ = κ(1 − ω − rd)Y − (1 − ω − rd)Y + P
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Ṗ = Ψ(g (ω, d)P
where Ψ(·) is an increasing function of the growth rate of
economic output
g=
κ(1 − ω − rd)
− δ.
ν
Example 4: effect of Ponzi financing
Dynamical
Systems and
Financial
Instability
ω0 = 0.95, λ0 = 0.9, d0 = 0, p0 = 0.1, Y0 = 100
1
M. R. Grasselli
0.9
Mainstream
No Speculation
Ponzi Financing
0.8
Alternative
approaches
0.7
SFC models
0.6
ω
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
0.5
0.4
0.3
0.2
Conclusions
0.1
0
0
0.1
0.2
0.3
0.4
0.5
λ
0.6
0.7
0.8
0.9
1
Stock prices
Dynamical
Systems and
Financial
Instability
Consider a stock price process of the form
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
dSt
= rb dt + σdWt + γµt dt − γdN (µt )
St
where Nt is a Cox process with stochastic intensity
µt = M(p(t)).
The interest rate for private debt is modelled as
rt = rb + rp (t) where
rp (t) = ρ1 (St + ρ2 )ρ3
Example 5: stock prices, explosive debt, zero
speculation
Dynamical
Systems and
Financial
Instability
1
M. R. Grasselli
Mainstream
ω
λ
0.5
Alternative
approaches
0
SFC models
2
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
0
10
20
30
40
50
60
70
80
90
100
1000
p
d
1
0
500
0
10
20
30
40
50
60
70
80
90
0
100
90
100
200
St
150
100
50
0
0
10
20
30
40
50
60
70
80
Example 6: stock prices, explosive debt, explosive
speculation
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
3
ω
2
λ
1
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
0
0
10
20
10
8
6
4
2
0
30
40
50
60
70
80
90
1000
800
600
400
200
p
d
0
10
100
20
30
40
50
60
70
80
90
0
100
20
30
40
50
60
70
80
90
100
10000
St
5000
0
0
10
Example 7: stock prices, finite debt, finite
speculation
Dynamical
Systems and
Financial
Instability
1
M. R. Grasselli
ω
λ
0.9
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
0.8
0.7
0
10
20
30
40
50
60
70
80
90
0.011
0.5
p
d
0.01
0.009
100
0
10
20
0
30
40
50
60
70
80
90
−0.5
100
30
40
50
60
70
80
90
100
400
St
300
200
100
0
0
10
20
Stability map
Dynamical
Systems and
Financial
Instability
Stability map for ω0 = 0.8, p0 = 0.01, S0 = 100, T = 500, dt = 0.005, # of simulations = 100
1
0.9
0.55
0.85
λ
0.
0.665
0.5
0.55
45
0.
0
0.55 .5
0.5
0.6
0.9
0.95
0.5
0.5
0.55
5
55
0.6
0.6
0.6
0.5
5
0.55
0.55
0.8
0.6
5
5
0.
0.5
0.6
5
5
0.75
0.5
0.55
0.5
0.5
0.6
0.5
0.4
6
0.
6
0.
0
0.7
0.6
0.55
6
0.1
0.65
0.55
5
0.
0.
0.6
Conclusions
0.55
0.
65
55
0.
6
0.
0.
0.2
0.7
0.55
65
0.
0.6
0.7
0.3
0.
d
0.7
0.7
0.65
0.6
0.
55
0.
7
0.55
0.6
0.8
5
5
0.6 0.6
5
0.5
0.6
0.6 0.65
0.6
0.65
0.4
6
0.
5
0.65
0.85
0.75
0.55
0.5
0.5
0.6
0. 0.6
6 5
0.7
8
0.
.75
0. 0
7
0.6
65
6
0.7
SFC models
0.
0.6
7
0.
0.
85
00..8
6
0.6
0.8
0.8
Alternative
approaches
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
65
0.
0.7
Mainstream
0.
0.7
75
0.
0.6
5
0.9
0.9
8
0.
0.7
0.85
0.6
M. R. Grasselli
0.5
0.45
Introducing a government sector
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Following Keen (and echoing Minsky) we add discretionary
government subsidied and taxation into the original system
in the form
Mainstream
G = G1 + G2
Alternative
approaches
T = T1 + T2
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
where
G˙1 = η1 (λ)Y
T˙1 = Θ1 (π)Y
G˙2 = η2 (λ)G2
T˙2 = Θ2 (π)T2
Defining g = G /Y and τ = T /Y , the net profit share is
now
π = 1 − ω − rd + g − τ,
and government debt evolves according to
Ḃ = rB + G − T .
Differential equations - reduced system
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Notice that π does not depend on b, so that the last
equation can be solved separately.
Observe further that we can write
π̇ = −ω̇ − r ḋ + ġ − τ̇
(3)
leading to the five-dimensional system
ω̇ =ω [Φ(λ) − α] ,
λ̇ =λ [γ(π) − α − β]
ġ2 =g2 [η2 (λ) − γ(π)]
(4)
τ̇2 =τ2 [Θ2 (π) − γ(π)]
π̇ = − ω(Φ(λ) − α) − r (κ(π) − π) + (1 − ω − π)γ(π)
+ η1 (λ) + g2 η2 (λ) − Θ2 (π) − τ2 Θ2 (π)
Good equilibrium
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
The system (4) has a good equilibrium at
ω =1−π−r
ν(α + β + δ) − π η1 (λ) − Θ1 (π)
+
α+β
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
ν(α + β + δ) − π
,
α+β
Θ1 (π)
τ1 =
α+β
d=
g1 =
η1 (λ)
α+β
Bad equilibria - destabilizing a stable crisis
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Recall that π = 1 − ω − rd + g − τ .
Mainstream
The system (4) has bad equilibria of the form
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
(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 locally
unstable provided η2 (0) > r .
On the other hand, if g2 (0) < 0 (austerity), then these
equilibria are all locally stable.
Persistence results
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Proposition 1: Assume g2 (0) > 0, then the system (4) is
e π -UWP if either
1
λη1 (λ) is bounded below as λ → 0, or
2
η2 (0) > r .
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Proposition 2: Assume g2 (0) > 0 and τ2 (0) = 0, then the
system (4) is λ-UWP if either of the following three conditions
is satisfied:
1
λη1 (λ) is bounded below as λ → 0, or
2
η2 (0) > max{r , α + β}, or
3
r < η2 (0) ≤ α + β and
−r (κ(x) − x) + (1 − x)γ(x) + η1 (0) − Θ1 (x) > 0 for
γ(x) ∈ [η2 (0), α + β].
Hopft bifurcation with respect to government
spending.
Dynamical
Systems and
Financial
Instability
OMEGA
M. R. Grasselli
Mainstream
0.692
Alternative
approaches
0.69
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
0.688
0.686
0.684
0.682
Conclusions
0.68
0.28
0.285
0.29
0.295
0.3
0.305
eta_max
0.31
0.315
0.32
0.325
The Great Moderation in the U.S. - 1984 to 2007
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Figure: Grydaki and Bezemer (2013)
Possible explanations
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Real-sector causes: inventory management, labour market
changes, responses to oil shocks, external balances , etc.
Financial-sector causes: credit accelerator models, financial
innovation, deregulation, better monetary policy, etc.
Grydaki and Bezemer (2013): growth of debt in the real
sector.
Bank credit-to-GDP ratio in the U.S
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Figure: Grydaki and Bezemer (2013)
Cumulative percentage point growth of excess
credit growth, 1952-2008
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Figure: Grydaki and Bezemer (2013)
Excess credit growth moderated output volatility
during, but not before the Great Moderation
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
Figure: Grydaki and Bezemer (2013)
Example 8: strongly moderated oscillations
Dynamical
Systems and
Financial
Instability
ω = 0.9, λ = 0.91, d = 0.1, p = 0.01, Y = 100, κ’(π ) = 20
0
12
M. R. Grasselli
Mainstream
3500
3000
10
Alternative
approaches
0
0
0
eq
1
0.8
0.9
0.7
0.8
180
160
140
2500
8
120
ω
λ
Y
d
p
ω
Y
p
2000
6
0.5
0.7
100
0.6
d
0.6
λ
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
0
0.9
80
1500
0.4
0.5
0.3
0.4
0.2
0.3
4
60
1000
40
2
Conclusions
0
500
0
0.1
0
10
20
30
40
50
time
60
70
80
90
100
0.2
20
0
Example 9 (cont): Shilnikov bifurcation
Dynamical
Systems and
Financial
Instability
ω = 0.9, λ = 0.91, d = 0.1, p = 0.01, Y = 100, κ’(π ) = 20
0
0
0
0
0
eq
M. R. Grasselli
Mainstream
Alternative
approaches
12
SFC models
Conclusions
10
8
d
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
6
4
2
1
0.95
0
0.45
0.9
0.5
0.55
0.85
0.6
0.65
0.7
0.8
0.75
0.8
ω
0.75
0.85
0.9
0.7
λ
Shortcomings of Goodwin and Keen models
Dynamical
Systems and
Financial
Instability
No independent specification of consumption (and
therefore savings) for households:
M. R. Grasselli
C = W,
Sh = 0
(Goodwin)
Mainstream
Alternative
approaches
SFC models
Goodwin model
Keen model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Conclusions
C = (1 − κ(π))Y ,
Sh = Ḋ = Πu − I
(Keen)
Full capacity utilization.
Everything that is produced is sold.
No active market for equities.
Skott (1989) uses prices as an accommodating variable in
the short run.
Chiarella, Flaschel and Franke (2005) propose a dynamics
for inventory and expected sales.
Grasselli and Nguyen (2013) provide a synthesis, including
equities and Tobin’s portfolio choices.
Concluding remarks
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
Macroeconomics is too important to be left to
macroeconomists.
Since Keynes’s death it has developed in two radically
different approaches:
1
SFC models
Conclusions
2
The dominant one has the appearance of mathematical
rigour (the SMD theorems notwithstanding), but is based
on implausible assumptions, has poor fit to data in general,
and is disastrously wrong during crises. Finance plays a
negligible role
The heterodox approach is grounded in history and
institutional understanding, takes empirical work much
more seriously, but is generally averse to mathematics.
Finance plays a major role.
It’s clear which approach should be embraced by
mathematical finance “to boldly go where no man has
gone before” · · ·
Qatlho!
Dynamical
Systems and
Financial
Instability
M. R. Grasselli
Mainstream
Alternative
approaches
SFC models
Conclusions