Extensions of the Keen-Minsky Model for Financial...

Extensions ofthe

Keen-MinskyModel forFinancialFragility

M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Stabilizinggovernment

Ponzifinancing

Model withNoise

Extensions of the Keen-Minsky Model forFinancial Fragility

M. R. Grasselli

Sharcnet Chair in Financial MathematicsMathematics and Statistics - McMaster UniversityJoint work with B. Costa Lima, X.-S. Wang, J. Wu

University of Western Sydney, August 03, 2012

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Why another talk on financial crises?

Because they are a hardy perennial.

Because macroeconomics is too important to be left at thehands of macroeconomists.

Because Carthago delenda est

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Dynamic Stochastic General Equilibrium

Overwhelmingly dominant school in macroeconomics.

Seeks to explain the aggregate economy using theoriesbased on strong microeconomic foundations.

All variables are assumed to be simultaneously inequilibrium.

The only way the economy can be in disequilibrium at anypoint in time is through decisions based on wronginformation.

Money is neutral in its effect on real variables and onlyaffects price levels.

Largely ignores the role of irreducible uncertainty.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Hardcore (freshwater) DSGE

The strand of DSGE economists affiliated with RBCtheory made the following predictions after 2008:

1 Increases government borrowing would lead to higherinterest rates on government debt because of crowdingout.

2 Increases in the money supply would lead to inflation.3 Fiscal stimulus has zero effect in an ideal world and

negative effect in practice (because of decreasedconfidence).

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Wrong prediction number 1

Figure: Government borrowing and interest rates.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise


Figure: Monetary base and inflation.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise


Figure: Fiscal tightening and GDP.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Meanwhile in Britain...

Figure: Office for National Statistics (UK), April 2012

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Soft core (saltwater) DSGE

The strand of DSGE economists affiliated with NewKeynesian theory got all these predictions right.

They did so by augmented DSGE with imperfections(wage stickiness, asymmetric information, imperfectcompetition, etc).

Still DSGE at core - analogous to adding epicycles toPtolemaic planetary system.

For example: Ignoring the foreign component, or lookingat the world as a whole, the overall level of debt makes nodifference to aggregate net worth one persons liability isanother persons asset. (Paul Krugman and Gauti B.Eggertsson, 2010, pp. 2-3)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Then we can safely ignore this...

Figure: Private and public debt ratios.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Really?

Figure: Change in debt and unemployment.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Minskys alternative interpretation of Keynes

Neoclassical economics is based on barter paradigm:money is convenient to eliminate the double coincidence ofwants.

In a modern economy, firms make complex portfoliosdecisions: which assets to hold and how to fund them.

Financial institutions determine the way funds areavailable for ownership of capital and production.

Uncertainty in valuation of cash flows (assets) and creditrisk (liabilities) drive fluctuations in real demand andinvestment.

Economy is fundamentally cyclical, with each state (boom,crisis, deflation, stagnation, expansion and recovery)containing the elements leading to the next in anidentifiable manner.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Minskys Financial Instability Hypothesis

Start when the economy is doing well but firms and banksare conservative.

Most projects succeed - Existing debt is easily validated:it pays to lever.

Revised valuation of cash flows, exponential growth incredit, investment and asset prices.

Beginning of euphoric economy: increased debt toequity 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 pastand capitalist financial institutions is destabilizing.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Derivation

Example

Keen model


Ponzifinancing

Model withNoise

Goodwin Model (1967) - Assumptions

Assume that

N(t) = N0et (total labour force)

a(t) = a0et (productivity per worker)

Y (t) = K (t) = a(t)L(t) (total yearly output)

where K is the total stock of capital and L is the employedpopulation.

Assume further that

w = ()w (Phillips curve)

K = (Y wL) K (Says Law)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Derivation

Example

Keen model


Ponzifinancing

Model withNoise

Goodwin Model - Differential equations

Define

=wL

Y=

w

a(wage share)

=L

N=

Y

aN(employment rate)

It then follows that

= (() )

=

(1

)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Derivation

Example

Keen model


Ponzifinancing

Model withNoise

Example 1: Goodwin model

Basic_Goodwin_movie.aviMedia File (video/avi)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Derivation

Example

Keen model


Ponzifinancing

Model withNoise

Example 1 (continued): Goodwin model

0

1000

2000

3000

4000

5000

6000

Y

0 10 20 30 40 50 60 70 80 900.7

0.75

0.8

0.85

0.9

0.95

1

t

,

w0 = 0.8,

0 = 0.9, Y

0 = 100

Y

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Introducing a financial sector (Keen 1995)

Assume now that new investment is given by

K = (1 rd)Y K

where () is C 1(,) increasing function satisfyingcertain technical conditions.

Accordingly, total output evolves as

Y

Y=(1 rd)

:= g(, d)

This leads to external financing through debt evolvingaccording to

D = (1 rd)Y (1 rd)Y

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Keen model - Differential Equations

Denote the debt ratio in the economy by d = D/Y , the modelcan now be described by the following system

= [() ]

=

[(1 rd)

](1)

d = d

[r (1 rd)

+

]+ (1 rd) (1 )

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Good equilibrium

Define1 =

1(( + + ))

Then the following is an equilibrium for (1):

1 = 1 1 r( + + ) 1

+

1 = 1()

d1 =( + + ) 1

+

Moreover

g(1, d1) =(1 1 rd1)

= + .

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Bad equilibrium

If we rewrite the system with the change of variablesu = 1/d , we obtain

= [() ]

=

[(1 r/u)

](2)

u = u

[(1r/u)

r

]u2 [(1r/u)(1)] .

We now see that (0, 0, 0) is an equilibrium of (2)corresponding to the point

(2, 2, d2) = (0, 0,+)

for the original system.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Local stability

Analyzing the Jacobian of (1) and (2) we obtain thefollowing conclusions.

The good equilibrium (1, 1, d1) is stable if and only if

r

[(1)

(1 (1) + ( + )

) ( + )

]> 0.

The point (0, 0, 0) is a stable equilibrium for (2) if andonly if

0 < r .

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Example 2 : convergence to the good equilibriumin a Keen model

Goodwin_plus_banks_movie_convergent.aviMedia File (video/avi)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Example 2 (continued): convergence to the goodequilibrium in a Keen model

0.7

0.75

0.8

0.85

0.9

0.95

1

Yd

0

1

2

3

4

5

6

7

8x 10

7

Y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

d

0 50 100 150 200 250 300

0.7

0.8

0.9

1

1.1

1.2

1.3

time

0 = 0.75,

0 = 0.75, d

0 = 0.1, Y

0 = 100

d

Y

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Example 3: explosive debt in a Keen model

Goodwin_plus_banks_movie_divergent_70y.aviMedia File (video/avi)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Example 3 (continued): explosive debt in a Keenmodel

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1000

2000

3000

4000

5000

6000Y

0

0.5

1

1.5

2

2.5x 10

6

d

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

time

0 = 0.75,

0 = 0.7, d

0 = 0.1, Y

0 = 100

Yd

Y d

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise


Goodwin_plus_banks_movie_divergent_200y.aviMedia File (video/avi)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise


0

1

2

3

4

5

6

7

8

9

10

d

7

6

5

4

3

2

1

0

1dd

/dt

0 10 20 30 40 50 60 70 80 90

0.4

0.5

0.6

0.7

0.8

0.9

1

time

0 = 0.75,

0 = 0.7, d

0 = 0.1

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Data detour: debt

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Data detour: debt and employment

Figure: Source: Keen (2009)

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model

Derivation

Equilibria

Examples


Ponzifinancing

Model withNoise

Basin of convergence for Keen model

0.5

1

1.5

0.40.5

0.60.7

0.80.9

11.1

0

2

4

6

8

10

d

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Introducing a government sector

Following Keen (and echoing Minsky) we add discretionarygovernment spending and taxation into the original systemin the form

G = G1 + G2

T = T1 + T2

where

G1 = 1()Y G2 = 2()G2

T1 = 1()Y T2 = 2()T2

Defining g = G/Y and t = T/Y , the net profit share isnow

= 1 rd + g t,and government debt evolves according to

Dg = rDg + G T .

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 4: Start with initial conditions near thelocally stable equilibrium at infinite debt . . .

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

78

d

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

time

0 = 0.75,

0 = 0.8, d

0 = 5, g

0 = 0, t

0 = 0, r = 0.03,

max = 0.01

d

k

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 4 (continued): . . . then add governmentto drive it to the locally stable good equilibrium.

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4

5

6

d

0 50 100 1500.4

0.6

0.8

1

1.2

1.4

time

0 = 0.75,

0 = 0.8, d

0 = 5, g

0 = 0.1, t

0 = 0.1, r = 0.03,

max = 0.01

0.05

0

0.05

0.1

0.15

g T

0

0.5

1

1.5

2

d G

0 50 100 1500

0.05

0.1

0.15

0.2

time

g S

dk

dG

gT

gS

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 4 (continued): But the system stillcrashes for sufficiently bad initial conditions!

0

0.2

0.4

0.6

0.8

1

0

1

2

3x 10

201

d

0 50 100 1500

2

4

6

8

time

0 = 0.3,

0 = 0.3, d

0 = 5, g

0 = 0.1, t

0 = 0.1, r = 0.03,

max = 0.01

0.05

0

0.05

0.1

0.15

g T

0

1

2

3x 10

201

d G

0 50 100 1500

1

2

3x 10

198

time

g S

dk

gT

gS

dG

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 5: Make government spending highenough, however, and the system is persistent . . .

0

0.2

0.4

0.6

0.8

1

0

2

4

6

8

10

d

0 50 100 1500

0.5

1

1.5

2

2.5

3

time

0 = 0.3,

0 = 0.3, d

0 = 5, g

0 = 0.1, t

0 = 0.1, r = 0.03,

max = 0.5

0

0.02

0.04

0.06

0.08

0.1

g T

0

1

2

3

4

d G

0 50 100 1500

0.1

0.2

0.3

0.4

time

g S

d

k

dG

gS

gT

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 5 (continued): . . . no matter how bad itstarts.

0

0.2

0.4

0.6

0.8

1

0

1000

2000

3000

4000d

0 50 100 1500

2

4

6

8

10

time

0 = 0.1,

0 = 0.1, d

0 = 5, g

0 = 0.1, t

0 = 0.1, r = 0.03,

max = 0.5

2

0

2

4

6

8

g T

60

40

20

0

20

d G

0 50 100 1500

50

100

150

time

g S

d

k

dGg

Tg

S

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Hopft bifurcation with respect to governmentspending.

0.68

0.682

0.684

0.686

0.688

0.69

0.692

OMEGA

0.28 0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.32 0.325eta_max

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Ponzi financing

To introduce the destabilizing effect of purely speculativeinvestment, we consider a modified version of the previousmodel with

D = (1 rd)Y (1 rd)Y + PP = (g(, d)P

where () is an increasing function of the growth rate ofeconomic output

g(, d) =(1 rd)

.

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 4: effect of Ponzi financing

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 = 0.95,

0 = 0.9, d

0 = 0, p

0 = 0.1, Y

0 = 100

No SpeculationPonzi Financing

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 4 (continued): effect of Ponzi financing

0 20 40 60 80 100 120 140 160 180 2000.2

00.20.40.60.8

t

d

0 = 0.95,

0 = 0.9, d

0 = 0, p

0 = 0.1, Y

0=100

0 20 40 60 80 100 120 140 160 180 2000246810

x 104

d w

ith P

onziNo Speculation

Ponzi Financing

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8x 10

5

t

Y

0 20 40 60 80 100 120 140 160 180 2000

100

200

300

400

Y w

ith P

onziNo Speculation

Ponzi Financing

0 20 40 60 80 100 120 140 160 180 2000

5

10

p

t

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Stock prices

Consider a stock price process of the form

dStSt

= rbdt + dWt + tdt dN(t)

where Nt is a Cox process with stochastic intensityt = M(p(t)).

The interest rate for private debt is modelled asrt = rb + rp(t) where

rp(t) = 1(St + 2)3

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 6: stock prices, explosive debt, zerospeculation

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

0 10 20 30 40 50 60 70 80 90 1000

1

2

0 10 20 30 40 50 60 70 80 90 1000

500

1000

pd

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

St

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 6: stock prices, explosive debt, explosivespeculation

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

0 10 20 30 40 50 60 70 80 90 10002468

10

0 10 20 30 40 50 60 70 80 90 10002004006008001000

pd

0 10 20 30 40 50 60 70 80 90 1000

5000

10000

St

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Example 6: stock prices, finite debt, finitespeculation

0 10 20 30 40 50 60 70 80 90 1000.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 1000.009

0.01

0.011

0 10 20 30 40 50 60 70 80 90 1000.5

0

0.5

pd

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

St

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Stability map

0.5

0.5

0.55

0.55

0.55

0.55

0.55

0.55

0.55

0.550.550.

55

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.65

0.65

0.65

0.65 0.65

0.65

0.65

0.65

0.7

0.7

0.7

0.7

0.7

0.75

0.75

0.8

0.8

0.85

0.85

0.5

0.55

0.55

0.55

0.6

0.6

0.55

0.6

0.55

0.5

0.6

0.6

0.5

0.6

0.65

0.55

0.9

0.55

0.6

0.7

0.5

0.55

0.55

0.65

0.6

0.65 0.60.7

0.7

0.65

0.8

0.6

0.6

0.6

0.60.6

0.6

0.45 0.

5

0.45

0.6

0.55

0.7

0.5

0.8

0.65

0.5

0.6

0.7

0.5

0.5

0.6

0.6

d

Stability map for 0 = 0.8, p

0 = 0.01, S

0 = 100, T = 500, dt = 0.005, # of simulations = 100

0.7 0.75 0.8 0.85 0.9 0.950

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

Next steps

Investigate the effects of austerity versus deficit spendingfor depressed economies

Model prices for capital goods Pk and commodities Pcexplicitly (Kaleckian mark-up theory, inflation, etc)

Extend the stochastic model (stochastic interest rates,monetary policy, correlated market sectors, etc)

Extend to an open economy model (exchange rates,capital flows, etc)

Calibrate to macroeconomic time series

Extensions ofthe


M. R. Grasselli

Introduction

Goodwinmodel

Keen model


Ponzifinancing

Model withNoise

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.

IntroductionGoodwin modelDerivationExample

Keen modelDerivationEquilibriaExamples

Stabilizing governmentPonzi financingModel with Noise

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