Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Dynamical systems and Financial Instability
Keen model
The Ultimate
Model
M. R. Grasselli
Mathematics and Statistics - McMaster University
Research in Options - Buzios, December 1, 2013
Outline
Dynamical
systems and
Financial
Instability
M. R. Grasselli
1
2
Introduction
Goodwin
model
Keen model
The Ultimate
Model
3
4
Introduction
DSGE models
Minskyian views
SFC models
Goodwin model
Original model
Inflation
Production function
Equity and savings
Monetary policy
Keen model
Original model
Ponzi financing
Noise and Stock Prices
Stabilizing government
Great Moderation
The Ultimate Model
Savings and investment
Inventories
Portfolios
Bringing it altogether
Dynamic Stochastic General Equilibrium
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
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
Introduction
DSGE models
Minskyian views
SFC models
The strand of DSGE economists affiliated with RBC
theory made the following predictions after 2008:
1
Goodwin
model
Keen model
2
The Ultimate
Model
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
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Figure: Government borrowing and interest rates.
Wrong prediction number 2
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Figure: Monetary base and inflation.
Wrong prediction number 3
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Figure: Fiscal tightening and GDP.
Better (but still bad) economics: soft core
(saltwater) DSGE
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
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)
Then we can safely ignore this...
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Figure: Private and public debt ratios.
Really?
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Figure: Change in debt and unemployment.
Minsky’s alternative interpretation of Keynes
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Neoclassical economics is based on barter paradigm:
money is convenient to eliminate the double coincidence of
wants.
In a modern economy, firms make complex portfolios
decisions: which assets to hold and how to fund them.
Financial institutions determine the way funds are
available for ownership of capital and production.
Uncertainty in valuation of cash flows (assets) and credit
risk (liabilities) drive fluctuations in real demand and
investment.
Economy is fundamentally cyclical, with each state (boom,
crisis, deflation, stagnation, expansion and recovery)
containing the elements leading to the next in an
identifiable manner.
Minsky’s Financial Instability Hypothesis
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Start when the economy is doing well but firms and banks
are conservative.
Most projects succeed - “Existing debt is easily validated:
it pays to lever”.
Revised valuation of cash flows, exponential growth in
credit, investment and asset prices.
Highly liquid, low-yielding financial instruments are
devalued, rise in corresponding interest rate.
Beginning of “euphoric economy”: increased debt to
equity ratios, development of Ponzi financier.
Viability of business activity is eventually compromised.
Ponzi financiers have to sell assets, liquidity dries out,
asset market is flooded.
Euphoria becomes a panic.
“Stability - or tranquility - in a world with a cyclical past
and capitalist financial institutions is destabilizing”.
Much better economics: SFC models
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
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
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
Cash
+Hh
Deposits
+Mh
+Hb
−H
+Mf
−L
Loans
Bills
Equities
+Bh
−pf Ef
Capital
Sum (net worth)
0
+L
0
+Bc
−B
−pb Eb
−A
Advances
The Ultimate
Model
0
Vf
0
0
+A
0
+pK
Vh
Sum
0
−M
+Bb
+pf Ef + pb Eb
Government
capital
Introduction
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
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Firms
Households
M. R. Grasselli
−pCh
Banks
+pC
Investment
+pI
Gov spending
+pG
Acct memo [GDP]
[pY ]
Central Bank
Government
Sum
capital
+pCb
0
−pI
0
−pG
0
Wages
+W
−W
Taxes
−Th
−Tf
+rM .Mh
+rM .Mf
−rM .M
0
−rL .L
+rL .L
0
Interest on deposits
Interest on loans
Interest on bills
Profits
Sum
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
+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
Introduction
DSGE models
Minskyian views
SFC models
Cash
+Ḣh
Deposits
+Ṁh
Loans
Goodwin
model
Bills
Keen model
Advances
The Ultimate
Model
Equities
Change in Net Worth
Central Bank
+Ḣb
−Ḣ
+Ṁf
−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
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: `
Introduction
DSGE models
Minskyian views
SFC models
Goodwin
model
Keen model
The Ultimate
Model
Production function: Y = f (K , `)
Labour productivity: a =
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
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
(total labour force)
αt
Y = min
(productivity per worker)
K
, a`
ν
Assume further that
K
Y =
= a`
ν
ẇ = Φ(λ, i, i e )w
pI = pY − w`
(Leontief production)
(full capital utilization)
(Phillips curve)
(Say’s Law)
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
current
Capital
Sum (net worth)
Introduction
Transactions
Goodwin
model
Consumption
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Firms
Households
0
0
−pC
+pC
Sum
capital
+pK
pK
Vf
pK
0
−pI
0
−Π
+Πu
0
0
0
0
Sum
0
0
Change in Net Worth
0
Investment
+pI
Acct memo [GDP]
[pY ]
Wages
+W
Profits
Sum
Keen model
Flow of Funds
The Ultimate
Model
Capital
−W
0
+pI
Πu
pI + ṗK − pδK
pI
pI
ṗK + p K̇
Table: Balance sheet, transactions, and flow of funds for the
Goodwin model.
Goodwin Model - Differential equations
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Define
wL
w
=
pY
pa
L
Y
λ=
=
N
aN
ω=
(wage share)
(employment rate)
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.
Original Goodwin Model - Properties
Dynamical
systems and
Financial
Instability
M. R. Grasselli
If we take Φ to be linear, this is a predator-prey model.
To ensure λ ∈ (0, 1) we assume instead that Φ is C 1 (0, 1)
and satisfies
Introduction
Φ0 (λ) > 0 on (0, 1)
Goodwin
model
Φ(0) < α
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
lim Φ(λ) = ∞.
λ→1−
The only non-trivial equilibrium is
(ω 0 , λ0 ) = 1 − ν(α + β + δ), Φ−1 (α)
is non-hyperbolic.
Moreover
g (ω 0 ) :=
Ẏ
1 − ω0
(ω 0 ) =
− δ = α + β,
Y
ν
Example 1: Goodwin model
Dynamical
systems and
Financial
Instability
w0 = 0.8, λ0 = 0.9
1
M. R. Grasselli
Introduction
0.98
Goodwin
model
Keen model
0.96
λ
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
0.94
0.92
The Ultimate
Model
0.9
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
Introduction
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
Keen model
The Ultimate
Model
0.7
0
10
20
30
40
50
t
60
70
80
90
0
Y
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
ω, λ
Goodwin
model
Inflation in the Goodwin model
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Assume first that i e = i, so that the Goodwin model with
inflation is
ω̇
w ṗ ȧ
ṗ
ṗ
= − − = Φ λ,
− −α
ω
w p a
p
p
1−ω
λ̇
(1)
=
−α−β−δ
λ
ν
ṗ
= i(p, ω, λ)
p
In general, we can define an instantaneous mark-up factor
m over unit labour costs by
w
p=m
a
Observe that it follows that
W
wL
w
1
ω=
=
=
=
pY
pY
ap
m
Desai (1978) model
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Desai postulates a price dynamics of the form
ap ṗ
ω
= −η log
= −η log
p
wm
ω
for an target mark-up factor m.
In addition, Desai considers a Philips curve of the form
ṗ
ṗ
ẇ
e
= Φ λ,
= Φ(λ) + η1
.
w
p
p
This leads to the system
ω̇
= Φ(λ) − α − (1 − η1 )η log(ωm)
ω
λ̇
1−ω
=
−α−β−δ
λ
ν
ṗ
= η log(ωm)
p
(2)
Equilibrium with inflation and money illusion
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
The fixed points for (2) are given by
ω 1 = 1 − ν(α + β + δ) = ω 0
λ1 (ω) = Φ(−1) (α + (1 − η1 )η log(ωm))
Define λ1 := λ1 (ω 1 ). Then
λ1 − λ0 = Φ(−1) (α + (1 − η1 )η log(ωm)) − Φ(−1) (α)
Therefore provided mω > 1 (positive inflation) and η1 < 1
(presence of money illusion), we see that λ1 − λ0 > 0.
Moreover, this equilibrium is locally stable for η1 < 1, a
centre for η1 = 1 (original Goodwin), and unstable for
η1 > 1.
Variable capital-to-output ratio
Dynamical
systems and
Financial
Instability
Desai (1978) also considers a cyclical ν of the form
ν = ν ∗ λ−µ
(3)
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
This leads to
λ̇
1
=
λ
1−µ
(1 − ω)λµ
−α−β−δ
ν∗
The new fixed point relations are
ω 2 (λ) = 1 − ν ∗ (α + β)λ−µ
λ2 = Φ(−1) (α + (1 − η1 )η log(ωm))
These two curves intersect at two distinct equilibria, which
are locally stable for typical parameters, that is, η1 < 1,
0 < µ < 1 and α + β sufficiently small. Higher values of µ
or η1 could make either or both equilibria unstable.
Inflation expectations
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Finally, Desai (1978) consider a Philips curve of the form
e
ẇ
ṗ
= Φ(λ) + η2
w
p
Moreover, inflation expectations evolve according to
e ṗ
d ṗ e
ṗ
= η3
−
,
dt p
p
p
This leads to
λ̇
α + β (1 − ω)λµ
=−
+ ∗
λ
1−µ
ν (1 − µ)
d ω̇
= η3 (Φ(λ) − α) + Φ0 (λ)λ̇ + η2 (1 − η3 )η log(mω)
dt ω
ω̇
− (1 + η3 )
ω
CES Production function, var der Ploeg (1985)
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
Consider a production function of the form
Y = f (K , `ef ) = A[µK −η + (1 − µ)`−η
ef ]
− η1
where `ef = A−1 a0 e αt ` is the effective labour.
This has an elasticity of substitution given by
s=
d log(k/`ef )
1
=
.
d log(f`ef /fK
1+η
The special cases for this function are
The Ultimate
Model
lim f (K , `ef ) = min(AK , A`ef ),
η→∞
lim f (K , `ef ) = AK µ `1−µ
ef ,
η→0
(Leontief)
(Cobb-Douglas)
(4)
Optimal capital-to-output and labour productivity
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Assume that firms optimize profits by setting
∂Y
= w.
∂L
Using (4), we find that the profit maximizing
capital-to-output ratio is
K
1
ν(ω) =
=
Y
A
1−ω
µ
− 1
η
.
(5)
a0 e αt
(6)
Similarly for the labour productivity we find
Y
a(ω) =
=
L
ω
1−µ
1
η
ODE system for the van der Ploeg 1985 model
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Using (5) and (6) we find
ω̇
Φ(λ) − α
=
ω
1 + 1/η
1
λ̇
1−ω η
Φ(λ) − α
= A(1 − ω)
−
− (α + β + δ)
λ
µ
(1 − ω)(1 + η)
The equilibrium is now
λ = Φ−(1) (α)
η
1
α + β 1+η 1+η
ω =1−
µ ,
A
(7)
and is locally stable for all 0 ≤ η < ∞.
We recover the original Goodwin model when η → ∞ and
A = 1/ν.
Example 2: van der Ploeg 1985 model
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Time with All variables
1
Wage Share
Employment
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0
50
100
150
Time
200
250
300
Figure: van der PLoeg 1985 model with η = 500.
Example 2 (continued): van der Ploeg 1985 model
Dynamical
systems and
Financial
Instability
Wage Share −− Employment
0.99
0.98
M. R. Grasselli
0.97
Introduction
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
0.96
Employment
Goodwin
model
0.95
0.94
0.93
Keen model
0.92
The Ultimate
Model
0.91
0.9
0.65
0.7
0.75
0.8
0.85
Wage Share
0.9
0.95
1
Figure: van der PLoeg 1985 model with η = 500.
A model with equities
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Balance Sheet
Firms
Households
current
Equities
+pe E
Capital
Sum (net worth)
Vh
0
−pC
+pC
Sum
capital
−pe E
0
+pK
pK
Vf
pK
Introduction
Transactions
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Consumption
Investment
+pI
Acct memo [GDP]
[pY ]
0
−pI
0
Wages
+W
−W
Profits
+Πd
−Π
+Πu
0
Sh
0
Sf − pI
0
Sum
0
Keen model
Flow of Funds
The Ultimate
Model
Equities
+pe Ė
Capital
Sum
Change in Net Worth
Sh
(Sh + ṗe E )
0
−pe Ė
0
+pI
pI
Sf
(Sf − ṗe E + ṗK − pδK )
pI
ṗK + p K̇
Table: SFC table for a model with equities.
Saving propensities, van der Ploeg (1983)
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Consider saving functions of the form
Sf = Πu = sp Π
Sh = sw (W + Πd ) = sw [W + (1 − sp )Π]
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
We would then have I = (sp + sw − sw sp )Π + sw W , so
sw + (1 − sw )sp (1 − ω)
K̇
=
−δ
K
ν
This leads to the modified system
ω̇
= Φ(λ) − α
ω
sw + (1 − sw )sp (1 − ω)
λ̇
=
−α−β−δ
λ
ν
Notice how investment is completely determined by
savings in this model.
The Di Matteo (1983) model
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Di Matteo considers a model identical to Goodwin’s,
expect for
ρ
ẇ
ṗ
=
−α
p
1+ρ w
!
w`
Ẏ
I =Y −
+ nK θ − µ
p
Y
S = S(Y )
M d = F (P, Y , r )
0 = P(I − S) + v (M d − M),
The crucial parameter here is an exogenous growth rate θ
for the money supply M.
Properties and extensions
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Original model
Inflation
Production
function
Equity and
savings
Monetary policy
Keen model
The Ultimate
Model
Using the new investment and price equations, one finds
ω̇
φ(λ) − α
=
ω
1+ρ
λ̇
1 − ω + (nθ − δ)ν
=
−α−β
λ
ν(1 + nµ)
For typical parameters, this behaves exactly like the OGM.
Assuming either variable growth rate in money supply of
the form
θ = θ0 + γ1 (λ − λ) − γ2 (ω − ω),
γ1 , γ2 ≥ 0
leads to a stable equilibrium provided γ2 <
1
nν .
A similar result holds for variable interest rates.
SFC table for Keen (1995) model
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Balance Sheet
current
Deposits
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
+D
−L
Capital
+pK
Vh
0
−pC
+pC
Banks
Sum
−D
0
capital
Loans
Sum (net worth)
Introduction
Firms
Households
Vf
+L
0
pK
0
pK
Transactions
Consumption
Investment
+pI
Acct memo [GDP]
[pY ]
Wages
+W
Interest on deposits
+rD
0
−pI
−W
0
Interest on loans
−rL
Profits
0
−Π
+Πu
0
Sf − pI
Sum
Sh
0
−rD
0
+rL
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: Balance sheet, transactions, and FOF for the Keen model.
Keen model - Investment function
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Assume now that new investment is given by
K̇ = κ(1 − ω − rd)Y − δK
where κ(·) is C 1 (−∞, ∞) satisfying
κ0 (π) > 0 on (−∞, ∞)
lim κ(π) = κ0 < ν(α + β + δ) < lim κ(π)
π→−∞
π→+∞
2 0
lim π κ (π) = 0.
π→−∞
Accordingly, total output evolves as
κ(1 − ω − rd)
Ẏ
=
− δ := g (ω, d)
Y
ν
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
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
(8)
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 (8) has a good equilibrium at
ω =1−π−r
M. R. Grasselli
Introduction
λ = Φ−1 (α)
ν(α + β + δ) − π
d=
α+β
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
(9)
ν
Example 1: 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
Introduction
0.95
1.6
Goodwin
model
6
Keen model
5
1.4
0.9
1.2
1
0.8
3
0.8
0.6
0.8
The Ultimate
Model
1
ω
0.9
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
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
λ
1.1
Example 2: 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
Introduction
30
5000
x 10
2.5
1
ω
λ
Y
d
0.9
0.8
Goodwin
model
25
0.7
4000
Keen model
λ
0.6
1.5
d
0.5
λ
ω
3000
Y
d
20
Y
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
2
15
0.4
1
2000
0.3
10
The Ultimate
Model
0.2
1000
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
Introduction
Goodwin
model
10
Keen model
The Ultimate
Model
8
6
d
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
4
2
0
0.4
0.5
0.5
0.6
0.7
1
0.8
0.9
1
1.1
λ
1.5
ω
Ponzi financing
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
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
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
Ṗ = Ψ(g (ω, d)P
where Ψ(·) is an increasing function of the growth rate of
economic output
The Ultimate
Model
g=
κ(1 − ω − rd)
− δ.
ν
Ponzi financing - Differential equations
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
With Ponzi financing the dynamical system becomes
ω̇ = ω [Φ(λ) − α]
κ(1 − ω − rd)
λ̇ = λ
−α−β−δ
(10)
ν
κ(1 − ω − rd)
+ δ + κ(1 − ω − rd) − (1 − ω) + p
ḋ = d r −
ν
κ(1 − ω − rd)
κ(1 − ω − rd)
ṗ = p Ψ
−δ −
+δ
ν
ν
Ponzi financing - Equilibria and stability
Dynamical
systems and
Financial
Instability
We find that (ω 1 , λ1 , d 1 , 0) is a stable equilibrium iff
Ψ(α + β) < α + β.
M. R. Grasselli
Introduction
Introducing u = 1/d we find that
Goodwin
model
(ω 2 , λ2 , d 2 , p) = (0, 0, +∞, 0)
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
is stable iff
Ψ(g0 ) < g0 .
Moreover, introducing , x = 1/p and v = p/d we find that
The Ultimate
Model
(ω 3 , λ3 , d 3 , p) = (0, 0, +∞, +∞)
is stable iff
g0 < Ψ (g0 ) < r .
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
Introduction
No Speculation
Ponzi Financing
0.8
Goodwin
model
0.7
Keen model
0.6
ω
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
0.5
0.4
0.3
The Ultimate
Model
0.2
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
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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 6: stock prices, explosive debt, zero
speculation
Dynamical
systems and
Financial
Instability
1
M. R. Grasselli
Introduction
ω
λ
0.5
Goodwin
model
0
Keen model
2
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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 6: stock prices, finite debt, finite
speculation
Dynamical
systems and
Financial
Instability
1
M. R. Grasselli
ω
λ
0.9
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
0.85
λ
0.
0.665
0.6
6
0.6
0.6
0.5
5
0.55
0.55
0.8
0.5
0.6
5
45
0.
0
0.55 .5
0.5
0.6
0.9
0.95
0.5
0.55
5
0.55
0.5
5
0.
0.5
0.6
5
5
0.55
55
0.5
0.55
0.5
0.5
0.4
6
0.
0.75
0.6
0.55
6
6
0.
0
0.7
0.6
0.5
0.65
0.55
5
0.
0.
0.6
0.1
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
0.
0.6
7
0.
0.
85
00..8
Keen model
The Ultimate
Model
0.6
0.8
0.8
Goodwin
model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
65
0.
0.7
Introduction
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
Introduction
G = G1 + G2
Goodwin
model
T = T1 + T2
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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 - full system
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Denoting γ(π) = κ(π)/ν − δ, a bit of algebra leads to the
following eight–dimensional system:
Introduction
ω̇ = ω[Φ(λ) − α]
Goodwin
model
λ̇ = λ[γ(π) − α − β]
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
ḋ = κ(π) − π − dγ(π)
ġ1 = η1 (λ) − g1 γ(π)
ġ2 = g2 [η2 (λ) − γ(π)]
τ̇1 = Θ1 (π) − gT1 γ(π)
τ̇2 = τ2 [Θ2 (π) − γ(π)]
ḃ = b[r − γ(π)] + g1 + g2 − τ1 − τ2
(11)
Differential equations - reduced system
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Notice that π does not depend on b, so that the last
equation in (11) can be solved separately.
Observe further that we can write
π̇ = −ω̇ − r ḋ + ġ − τ̇
(12)
leading to the five-dimensional system
ω̇ =ω [Φ(λ) − α] ,
λ̇ =λ [γ(π) − α − β]
ġ2 =g2 [η2 (λ) − γ(π)]
(13)
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
The system (13) has a good equilibrium at
ω =1−π−r
ν(α + β + δ) − π η1 (λ) − Θ1 (π)
+
α+β
α+β
Introduction
λ = Φ−1 (α)
Goodwin
model
π = κ−1 (ν(α + β + δ))
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
d=
α+β
η1 (λ)
g1 =
α+β
Θ1 (π)
τ1 =
α+β
Bad equilibria - destabilizing a stable crisis
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Recall that π = 1 − ω − rd + g − τ .
Introduction
The system (13) has bad equilibria of the form
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
(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 (13) is
e π -UWP if either
1
λη1 (λ) is bounded below as λ → 0, or
2
η2 (0) > r .
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Proposition 2: Assume g2 (0) > 0 and τ2 (0) = 0, then the
system (13) 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].
Example 3: Good initial conditions
Dynamical
systems and
Financial
Instability
ω(0) = 0.85, λ(0) = 0.85, d(0) = 0.5, gS (0) = 0.05, gT (0) = 0.05, gS (0) = 0.05, gT (0) = 0.05, dg(0) = 0, r = 0.03, η(2)
= 0.02
max
1
1
2
2
5
−5
1.5
0.6
0.8
λ
Goodwin
model
−10
1
0.4
−15
0.5
0.2
Keen model
−20
0
20
0.8
0.12
15
0.6
0.1
0.4
0.08
5
0.2
0.06
0
0
0
50
100
150
200
250
time
300
350
400
450
500
0
T
g +g
T
2
2
1
10
gS +gS
The Ultimate
Model
dg
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
1
d
Introduction
1
Keen Model
Model w/ Government
0
2
ω
M. R. Grasselli
0
50
100
150
200
250
time
300
350
400
450
500
0.04
Example 4: Bad initial conditions
Dynamical
systems and
Financial
Instability
ω(0) = 0.8, λ(0) = 0.8, d(0) = 0.5, gS (0) = 0.05, gT (0) = 0.05, gS (0) = 0.05, gT (0) = 0.05, dg(0) = 0, r = 0.03, η(2)
= 0.02
max
1
0
2
2
1
Keen Model
Model w/ Government
2
0.8
1.5
0.6
λ
ω
−5
d
Introduction
1
2.5
Goodwin
model
−10
1
0.4
−15
0.5
0.2
Keen model
−20
0
0.6
50
100
150
200
250
time
300
350
400
450
500
0.12
0.1
0.08
T
1
10
0
2
15
0
T
0.8
2
1
20
gS +gS
The Ultimate
Model
25
dg
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
0.4
5
0.2
0
0
1
M. R. Grasselli
g +g
5
0.06
0
50
100
150
200
250
time
300
350
400
450
500
0.04
Example 5: Really bad initial conditions with timid
government
Dynamical
systems and
Financial
Instability
6
M. R. Grasselli
4
20
λ
10
0
0.2
5
0
17
8
200
250
time
300
350
400
450
500
0
8
x 10
5
x 10
2
0
T
2
1
0
150
6
gS +gS
dg
1
0.5
100
10
x 10
2
1.5
50
4
T
2.5
0
−5
2
0
1
−2
g +g
Keen model
The Ultimate
Model
0.6
15
0.4
Goodwin
model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
2
ω
d
Introduction
ω(0) = 0.15, λ(0) = 0.15, d(0) = 3, gS (0) = 0.05, gT (0) = 0.05, gS (0) = 0.05, gT (0) = 0.05, dg(0) = 0, r = 0.03, η(2)
= 0.02
max
16
1
1
2
2
x 10
1
30
Keen Model
Model w/ Government
25
0.8
0
50
100
150
200
250
time
300
350
400
450
500
−10
Example 6: Really bad initial conditions with
responsive government
Dynamical
systems and
Financial
Instability
ω(0) = 0.15, λ(0) = 0.15, d(0) = 3, gS (0) = 0.05, gT (0) = 0.05, gS (0) = 0.05, gT (0) = 0.05, dg(0) = 0, r = 0.03, η(2)
= 0.2
max
1
1000
0.8
λ
0.6
15
10
0.2
5
−500
0
600
30
x 10
5
0.5
500
25
0
400
20
0
−0.5
0
50
100
150
200
250
time
300
350
400
450
500
0
15
200
10
100
5
0
0
2
2
1
1
1
300
g +g
+g
g
T
T
T
T
2
8
gS +gS
The Ultimate
Model
1
Keen Model
Model w/ Government
0.4
0
dg
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
2
20
ω
d
Goodwin
model
Keen model
2
25
500
Introduction
1
30
M. R. Grasselli
−1
−5
−1.5
0
50
100
150
200
250
time
300
350
400
450
500
−10
−2
Example 7: Austerity in good times: harmless
Dynamical
systems and
Financial
Instability
ω(0) = 0.8, λ(0) = 0.8, d(0) = 0.5, gS (0) = 0.05, gT (0) = 0.05, gS (0) = +−0.05, gT (0) = 0.05, dg(0) = 0, r = 0.03, η(2)
= 0.02
max
1
0
1
2
2
1
2.5
M. R. Grasselli
gS (0)>0
2
2
−5
gS (0)<0
0.8
The Ultimate
Model
0.2
λ
0.8
10
0.6
2
1
15
0
50
100
150
200
250
time
300
350
400
450
500
0.12
0.1
0.08
T
1
5
0
2
0
20
dg
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
0.5
T
−20
0.4
0.4
0
0.2
−5
0
1
Keen model
0.6
1
g +g
−15
1.5
ω
−10
Goodwin
model
gS +gS
d
2
Introduction
0.06
0
50
100
150
200
250
time
300
350
400
450
500
0.04
Example 8: Austerity in bad times: a really bad
idea
Dynamical
systems and
Financial
Instability
ω(0) = 0.15, λ(0) = 0.15, d(0) = 4, gS (0) = 0.05, gT (0) = 0.05, gS (0) = +−0.05, gT (0) = 0.05, dg(0) = 0, r = 0.03, η(2)
= 0.2
max
48
1
1
2
2
x 10
1
30
gS (0)>0
2
25
0.8
10
gS (0)<0
2
20
0.6
5
15
0.4
10
0
0.2
5
The Ultimate
Model
−5
λ
0
48
5
0
50
100
150
200
250
time
300
350
400
450
1
500
0
8
x 10
5
48
x 10
x 10
0
−5
−10
−2
−10
−15
−3
T
1
−5
T
2
2
0
−1
0
dg
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
1
Keen model
g +g
Goodwin
model
gS +gS
Introduction
d
M. R. Grasselli
ω
15
0
50
100
150
200
250
time
300
350
400
450
500
−15
Hopft bifurcation with respect to government
spending.
Dynamical
systems and
Financial
Instability
OMEGA
M. R. Grasselli
Introduction
Goodwin
model
0.692
0.69
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
0.688
0.686
0.684
0.682
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
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Figure: Grydaki and Bezemer (2013)
Possible explanations
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Figure: Grydaki and Bezemer (2013)
Cumulative percentage point growth of excess
credit growth, 1952-2008
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
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
Introduction
Goodwin
model
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
The Ultimate
Model
Figure: Grydaki and Bezemer (2013)
Example 3: weakly moderated oscillations
Dynamical
systems and
Financial
Instability
ω = 0.8, λ = 0.722, d = 0.1, Y = 100, κ’(π ) = 100
6
2.5
M. R. Grasselli
0
x 10
0
0
0
eq
4.5
1
800
0.9
4
700
Introduction
2
Goodwin
model
3.5
600
0.7
3
The Ultimate
Model
ω
λ
Y
d
ω
Y
500
0.6
2.5
2
1
0.5
0.4
λ
1.5
400
300
1.5
0.3
200
0.5
1
0.2
0.5
0
0
0.1
0
50
100
150
time
200
250
300
0
100
0
d
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
0.8
Example 3 (cont): weakly moderated oscillations in
3d
Dynamical
systems and
Financial
Instability
ω0 = 0.8, λ0 = 0.722, d0 = 0.1, Y0 = 100, κ’(πeq) = 100
M. R. Grasselli
Introduction
Goodwin
model
12
Keen model
10
The Ultimate
Model
8
d
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
6
4
1
2
0.9
0
0
0.8
0.7
0.5
1
1.5
0.6
2
2.5
0.5
3
3.5
ω
4
4.5
0.4
λ
Example 4: 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
Introduction
3500
3000
10
Goodwin
model
The Ultimate
Model
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
Keen model
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
0
0.9
80
1500
0.4
0.5
0.3
0.4
0.2
0.3
4
60
1000
40
2
0
500
0
0.1
0
10
20
30
40
50
time
60
70
80
90
100
0.2
20
0
Example 4 (cont): strongly moderated oscillations
in 3d
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
Introduction
Goodwin
model
12
Keen model
The Ultimate
Model
10
8
d
Original model
Ponzi financing
Noise and Stock
Prices
Stabilizing
government
Great
Moderation
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
M. R. Grasselli
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
No independent specification of consumption (and
therefore savings) for households:
C = W,
Sh = 0
C = (1 − κ(π))Y ,
Goodwin
Sh = Ḋ = Πu − I
Full capacity utilization.
Everything that is produced is sold.
No active market for equities.
Keen
Independent savings and investment, Skott (1989)
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
Skott assumes saving and investment of the form
S = g (π)Y ,
I = f (σ, π)Y ,
Ẏ = h(π, λ)Y ,
g 0 > 0,
fσ > 0,
g 0 > fπ ≥ 0,
hπ > 0, hλ < 0
where σ = Y /K = 1/ν is not constant.
Saving and investment decisions of firms are reconciled by
an implicit price level p satisfying
(W + rD + Πd − pC ) + Πu = pI ⇔ g (π) = f (σ, π)
It then follows that π = θ(σ) for some θ and

σ̇


 = h(θ(σ), λ) − σf (σ, θ(σ)) + ∆
σ
 λ̇

 = h(θ(σ), λ) − α − β
λ
(14)
Inventory changes, Chiarella, Flaschel and Franke
(2005)
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Given a realized demand Y d = C + I , consider an
inventory dynamics of the form
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
N d = αn d Y e
˙ = nN d + βn (N d − N)
In
˙
Y s = Y e + In
Ṅ = Y s − Y d
Ẏ e = nY e + βe (Y d − Y e )
Portfolio choices, Tobin (1969, 1980)
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Given a choice between holding equities E at a price pe
and money M, households allocate wealth according to
pe E = fe (ree )W
Introduction
M = fm (r )W
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
Here ree is the expected return on equity given by
rke pK
re
= k + πee
pe E
q
rke = (Y e − δK − W )/K
ṗe
π̇ee = βπe
− πee
pe
ree =
where the last equation can be further decomposed
between chartists and fundamentalists, etc.
A synthesis, Grasselli and Nguyen (2013)
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Consider
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
Y s = min{σK ; aL} = σK = aλ`
Ṅ = Y s − Y d
K̇ = I − δK , I = κY d
Y˙ e = αE Y e + βE (Y d − Y e )
ẇ = Φ(λ)w
Differential equations for firm
Dynamical
systems and
Financial
Instability
Defining τ := N/Y d , y := Y s /Y d , αN := γY e /N,
g := Y˙d /Y d , we obtain the following autonomous system
M. R. Grasselli
σ̇
= f (1 − y − (1 − αN )τ )
σ
τ̇
(y − 1)
=
−g
τ
τ
γ
α̇N
(y − 1)
= αE + β E
−1 −
αN
αn τ
τ
ẏ
κσ
= f (1 − y − αN τ ) +
−δ−g
y
y
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
(15)
Moreover, this system fully determines employment by
λ̇
σ̇ K̇
ȧ `˙
κσ
= + − − = f (1 − y − αT τ ) +
−δ−α−β
λ
σ K
a `
y
Prices and wages
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
Given a target mark-up factor m, firms set prices
according to
βP w
ω
ṗ
.
p−m
= βP 1 − m
i= =
p
p
a
y
It then follows that
ω̇
ẏ
= φ(λ) − α + − βP
ω
y
ω
1−m
,
y
which can also be fully determined by system (15).
Firms financing
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Introduction
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
Assume that firms retain a fraction sp of profits and seeks
to obtain a fraction ν of its external financing as new debt.
This leads to
ḋ = µ(κ − sp π) − d(i + g ).
Using π = 1 − ω − rd, we then find
ẏ
π̇ = −ω φ(λ) + − αL − i − r [ν(κ − sf π) − d(i + g )] .
y
We now have an autonomous system for (y , τ, σ, αN , λ, π).
Household savings
Dynamical
systems and
Financial
Instability
M. R. Grasselli
Assume that households allocate wealth and consumption
according to D = γD pC and eE = γE pC .
Introduction
This lead to Tobin’s valuation ratio of the form
Goodwin
model
Keen model
The Ultimate
Model
Savings and
investment
Inventories
Portfolios
Bringing it
altogether
q=
pe E
pe E σ
γE Dσ
γE σ(1 − κ)
=
=
=
,
p
d
pK
pY
y
γD pyY
(16)
which is therefore an auxiliary variable.
Finally, capital gains in this model are given by
κ − sp π (1 − ν)
p˙e
ν
µ=
=
−
.
pe
1−κ
γE
γD
and we can notice that µ is also an auxiliary variable.
(17)