Critical Mathematical Economics - Wikipage.mi.fu-berlin.de/jbuchner/cme.pdfCritical Mathematical...

CriticalMathematicalEconomics

JohannesBuchner

Introduction

Micro-/Meso-foundations

Goodwin andKeen models

The GreatModeration

Pseudo-Cycles

Conclusions

Critical Mathematical Economics

Johannes Buchner

Mathematics and Statistics, McMaster University

PhiMac Seminar, McMaster University, March 27th, 2015


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

The financial crisis and the crisis in economics

Steve Keen: Debunking Economics (2002)

The Financial Crisis and the Systemic Failure of AcademicEconomics, Dahlem Report 2008 (e.g. by Hans Foellmer)

Adair Turner: Economics after the Crisis (2012)

Institute for New Economic Thinking (INET, founded2009)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Mathematics for New Economic Thinking

MNET: Fields Institute, 2013 (organized by M. Grasselli)

Critical Mathematical Economics: Mathematics has beenpartly misused in mainstream economics to justify‘unregulated markets’ before the crisis

New approaches: e.g. Systemic Risk in Banking Systems,Revival of Keynes and Minsky in Macroeconomics


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Dynamic Stochastic General Equilibrium (DSGE)

Standard model in Macroeconomics

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

Collective decisions of rational individuals over a range ofvariables for both present and future.

All variables are assumed to be simultaneously inequilibrium.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

DSGE and ‘General Equilibrium Theory’ as aMicrofoundation

Theorem (from ‘The theory of general economic equilibrium: Adifferentiable approach’. A. Mas-Colell, 1985).

Let z : S → Rl be a C 3- vector field satisfying some boundaryconditions. Then for any ε > 0 we can find an economy E suchthat the excess demand function of E coincides with f on Sεand p is an equilibrium for E if and only if z(p) = 0

It is also known as ‘Sonnenschein-Mantel-Debreu Theorem’, or‘anything goes’ Theorem.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

SMD theorem: something is rotten in GE land


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Alternative Microfoundations: ABMs

‘Agent Based Models’ as a Microfoundation

simulate the behaviour of heterogenous agents on thecomputer numerically

Reference: e.g. ‘Handbook of Computational Economics’(Volumes 1-3, most recent 2014)

sometimes critized as ‘black box’, as until recently noanalytical solutions had been available

Idea: Use methods from statistical physics to provide thoseby a mean-field approximation (see e.g. S. Landini, 2014)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

A probabilistic Micro-/Mesofoundation ofMacroeconomics

‘The point is that precise behavior of each agent is irrelevant.Rather we need to recognize that microeconomic behavior isfundamentally stochastic.’ (Aoki and Yoshikawa 2006)

Idea:

identifying homogeneous sub-populations of agents

computing the mean-field variables (mean-fieldapproximation), on an intermediate ‘Meso-level’

master equation: study of the dynamics of the number ofagents in each cluster, i.e. the transition probabilities

Project with M.Grasselli and Patrick Li: Link this toStock-Flow-Consistent Maroeconomic Models


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Back to Macroeconomics: Dynamic StochasticGeneral Equilibrium (DSGE)

Equilibrium is only disrupted by exogenous shocks.

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.

Neoclassical economics is based on barter paradigm:money is convenient to eliminate the double coincidence ofwants. DSGE is ‘Macroeconomics without banks’ !


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Minsky’s alternative interpretation of Keynes

In a modern economy, financial institutions determine theway funds are available for ownership of capital andproduction.

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

“Stability - or tranquility -is destabilizing” (in a world witha cyclical past and capitalist financial institutions).


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Minsky’s Financial Instability Hypothesis

Start when the economy is doing well but firms and banksare conservative. Most projects succeed -it pays to lever.

Beginning of “euphoric economy”: increased debt toequity ratios, development of Ponzi/Speculative 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.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Stock-Flow Consistent models

Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.

They consider both real and monetary factorssimultaneously.

Specify the balance sheet and transactions betweensectors.

Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.

See Godley and Lavoie (2007) for the full framework.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Goodwin Model - SFC matrix

Balance Sheet HouseholdsFirms

Sum

current capital

Capital +pK pK

Sum (net worth) 0 0 Vf pK

Transactions

Consumption −pC +pC 0

Investment +pI −pI 0

Acct memo [GDP] [pY ]

Wages +W −W 0

Profits −Π +Πu 0

Sum 0 0 0 0

Flow of Funds

Capital +pI pI

Sum 0 0 Πu pI

Change in Net Worth 0 pI + pK − pδK pK + pK

Table: SFC table for the Goodwin model.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Goodwin Model (1967) - Assumptions

Assume that

N(t) = N0eβt (total labour force)

a(t) = a0eαt (productivity per worker)

Y (t) = νK (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 = Φ(λ)w (Phillips curve)

K = (Y − wL)− δK (Say’s Law)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Goodwin Model - Differential equations

Define

ω =wL

Y=

w

a(wage share)

λ =L

N=

Y

aN(employment rate)

It then follows that

ω = ω(Φ(λ)− α)

λ = λ

(1− ων− α− β − δ

)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: Goodwin model

0.7 0.75 0.8 0.85 0.9 0.95 10.88

0.9

0.92

0.94

0.96

0.98

1

ω

λw

0 = 0.8, λ

0 = 0.9


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

SFC table for Keen (1995) model

Balance Sheet HouseholdsFirms

Banks Sum

current capital

Deposits +D −D 0

Loans −L +L 0

Capital +pK pK

Sum (net worth) Vh 0 Vf 0 pK

Transactions

Consumption −pC +pC 0

Investment +pI −pI 0

Acct memo [GDP] [pY ]

Wages +W −W 0

Interest on deposits +rD −rD 0

Interest on loans −rL +rL 0

Profits −Π +Πu 0

Sum Sh 0 Sf − pI 0 0

Flow of Funds

Deposits +D −D 0

Loans −L +L 0

Capital +pI pI

Sum Sh 0 Πu 0 pI

Change in Net Worth Sh (Sf + pK − pδK ) pK + pK

Table: SFC table for the Keen model.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Keen model - allow Investment through debt

Assume now that new investment is given by

K = κ(π)Y − δK

where κ(·) is a nonlinear increasing function of profitsπ = 1− ω − rd .

This leads to external financing through debt evolvingaccording to

D = κ(π)Y − πY


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Keen model - Differential Equations

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

ω = ω [Φ(λ)− α]

λ = λ [g(π)− α− β]

d = κ(π)− π − dg(π)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: convergence to the good equilibrium in aKeen model

0.7

0.75

0.8

0.85

0.9

0.95

1

λ

ωλYd

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

ω

ω0 = 0.75, λ

0 = 0.75, d

0 = 0.1, Y

0 = 100

d

λ

ω

Y

Figure: Grasselli and Costa Lima (2012)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: explosive debt in a Keen model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

0

1000

2000

3000

4000

5000

6000

Y

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

ω

ω0 = 0.75, λ

0 = 0.7, d

0 = 0.1, Y

0 = 100

ωλYd

λ

Y d

ω



JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

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

ωλ

d


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Ponzi financing

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

D = κ(1− ω − rd)Y − (1− ω − rd)Y + P

P = Ψ(g(ω, d)P

where P stands for Ponzi (speculative) financing and Ψ(·) is anincreasing function of the growth rate of economic output.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: Destabilizing 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

λ

ω

ω0 = 0.95, λ

0 = 0.9, d

0 = 0, p

0 = 0.1, Y

0 = 100

No SpeculationPonzi Financing



JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

The Great Moderation in the U.S. - 1984 to 2007

Figure: Grydaki and Bezemer (2013)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Possible explanations

Real-sector causes: inventory management, labour marketchanges, etc.

Financial-sector causes: financial innovation, deregulation,better monetary policy, etc.

Grydaki and Bezemer (2013): growth of debt in the realsector.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Bank credit-to-GDP ratio in the U.S

Figure: Grydaki and Bezemer (2013)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Modelling the Great Moderation I: Keen modelwith Ponzi financing

Story: high debt → growth with low volatility → evenhigher debt → explosive volatility

Idea: choose initial conditions that are close to the basinof attraction of the equilibrium point with finite debt, butnot inside it

Result: long period of very low volatility, followed byexplosion and collapse


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: strongly moderated oscillations

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

0

500

1000

1500

2000

2500

3000

3500

Y

0

20

40

60

80

100

120

140

160

180

d

0

2

4

6

8

10

12p

0 10 20 30 40 50 60 70 80 90 1000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time

ω

ω0 = 0.9, λ

0 = 0.91, d

0 = 0.1, p

0 = 0.01, Y

0 = 100, κ’(π

eq) = 20

ωλYdp


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example (cont): Shilnikov bifurcation

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9 0.7

0.75

0.8

0.85

0.9

0.95

1

0

2

4

6

8

10

12

λ

ω0 = 0.9, λ

0 = 0.91, d

0 = 0.1, p

0 = 0.01, Y

0 = 100, κ’(π

eq) = 20

ω

d


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Modelling the Great Moderation II: Debt Dynamics

Joint with M.R. Grasselli (McMaster University and TheFields Institute) and Y. Choi (Montclair State University)

Consider the following Debt Dynamics of firms andhouseholds, where df = Df

Y and dh = DhY is the debt share:

Df = I − (1− ω − rf df )Y (investment - profits)

Dh = Ch + Qh − (ω − rhdh)Y (purchases - disp. income)

Ch is households’ consumption, and Qh their demand forreal estate


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Modelling the Great Moderation II: Consumption

Household consumption Ch is a multiple of wage lessinterest payment, plus a function of wealth:

Ch = γ1(W − rhDh) + γ2Xh

the wealth of households is given by

Xh = Mh + phEh − Lh,

where ph is the unit price of real estate and Eh is the totalstock of real estate (i.e the number of houses)

for the evolution of prices of houses, assume the followingsupply and demand relationship:

ph = η(Qh − phfhEh),

where η, fh are constants.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Modelling the Great Moderation II: Boom

high demand Qh (which can be financed by householddebt) can push prices up.

the overall wealth of households goes up, leading to higherconsumption

the increase in house prices ph also leads to high returnson real estate, which pushes demand Qh up as well.

So while house prices are going up, both consumption anddemand for houses go up.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Modelling the Great Moderation II: Crisis

eventually there is too much debt and consumption drops.

this brings overall demand Y down, together with profitsfor firms, leading to lower employment and lower wagesthrough the Philips curve mechanism.

this also negatively affects the demand for houses Qh,leading to lowers house prices, lower wealth, and evenlower consumption.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Modelling the Great Moderation II: Conclusion

the work on the model has just started :)

Note: This model is ‘demand-led’ in the sense thatinvestment is the accommodating variable (in the sameway that consumption was the accommodating variable inthe original Keen model)

In other words, we let consumption be determined by theequation above and then use the relations

K = I − δK , Y =K

ν, Y = C + I

to derive the necessary level of investment as

I =K

ν− C .


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Pseudo-Cycles and wage-led vs. profit-led

Goodwin / Keen models: ‘profit-led’ or ‘investment-driven’(investment is modelled, depends positively on profits, andconsumption adjusts)

Debt-Dynamics model: ‘wage-led’ or ‘consumption-driven’(consumption is modelled, depends positively on wages,and investment adjusts)

Research Question: Could there be a model that ‘lookslike’ profit-led (e.g. by showing Goodwin-Cycles), but it isindeed wage-led?

Joint with Charlie Brummit (Columbia), Torsten Heinrich(Bremen), Engelbert Stockhammer (Kingston)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

A simplified ‘toy’ Goodwin Model

We follow [Stockhammer, Mitchel 2014]. They consider thefollowing ‘toy’ Goodwin model

y = y(1− w)

w = w(−c + ry)

Note: This corresponds to the Goodwin model discussed abovewith parameters α = β = δ = 0 and ν = 1, where we havechosen a linear Phillips curve Φ(λ) = −c + rλ


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: ‘Goodwin Cycles’

outputy(t)

wagew(t)

2 4 6 8 10t

0.5

1.0

1.5

2.0

(A)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

outputy(t)

wagesharew(t)

(B)

Figure: Sample orbit of the toy Goodwin model for r = c = 1 andinitial condition y(0) = 0.6,w(0) = 0.5.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Recall: Minsky’s FIH - a Minsky Cycle

Start when the economy is doing well but firms and banksare conservative. Most projects succeed -it pays to lever.

Beginning of “euphoric economy”: increased debt toequity ratios, development of Ponzi/Speculative 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.


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

A toy 2d Minsky Model

Let’s consider the following ‘ toy Minsky Model’ model

y = y(1− f ) (1)

f = f (−1 + py) (2)

Note: f stands for ‘financial fragility’ (think debt), and playsthe role of the wage-share in the Goodwin model (the modelabove is equivalent to the toy Goodwin model for c=1, w=f,and renaming r to p)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: ‘Minsky Cycles’

outputy(t)

frag.f (t)

2 4 6 8 10t

0.5

1.0

1.5

2.0

(A)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

outputy(t)

financialfragilityf(t)

(B)

Figure: Sample orbit of the 2d Minsky model showing a ‘MinskyCycles’ (for p = 1 and initial condition y(0) = 0.6, f (0) = 0.5).


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

A 3d Minsky Model with wage share

Finally, consider the following 3d Minsky Model, by adding awage share variable (3) to equations (1) and (2):

y = y(1− f )

f = f (−1 + py)

w = w(−c + ry − w) (3)

A pseudo-Goodwin cycle is a counterclockwise closed orbit in(y(t),w(t)) such that y and w form a master-slave system(i.e. there is a feedback from y to w , but not vice-versa).


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Example: ‘Pseudo Goodwin Cycles’

0.5

1.0

1.5

2.0

2.5

(C)

outputy(t) wagew(t) frag. f (t)

01

2outputy(t)

0

1

2wagew(t)

0

1

2

frag. f (t)

(D)

Figure: Sample orbit in 3d Minsky model: ‘Pseudo Goodwin Cycle’(for p = 2, r = 5, c = 3/2 and y(0) = 0.6, f (0) = 0.5,w(0) = 0.4).


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

General Pseudo-Cycles

Goal 1: Try to find Pseudo Goodwin Cycles in a wage-ledMinsky model and make it rigorous

Goal 2: Generalize the ‘Pseudo-Cycles’ idea and relate todiscussion on observational equivalence

Note: Discussion if economy is profit-led’ or ‘wage-led’ isvery relevant for economic policy!


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Concluding remarks

We discussed stock-flow macro-models and theirmicro-/meso-foundations.

Applications included modelling the Great Moderation andthe identification of Pseudo-Cycles

The modelling framework is an alternative to the dominantmicrofounded DSGE paradigm in macroeconomics.

Our approach opens up new avenues for the application ofmodern mathematical techniques to economics


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Outlook

Complexity and the Art of Public Policy (Colander, 2014):

‘Complexity science - made possible by modern analyticaland computational advances - is changing the way wethink about social systems and social theory’

‘Unfortunately, economists policy models have not kept up’

Work has just begun . . . new collaborators welcome :)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

And now to something completely different...

... my research before I came to Canada :)


JohannesBuchner

Introduction



The GreatModeration

Pseudo-Cycles

Conclusions

Ancient Dynamics of the Einstein Equationsand the Tumbling Universe

The Einstein Field Equations

Ric(g)− 12Scal(g)g = T

geometry of space-time matter content

Bianchi Cosmological models

Spatially homogeneous but anisotropic

Spatial hypersurfaces: 3-dimensional Lie group,classification [Bianchi 1898]

the Einstein Equations reduce to ODEs, with chaoticoscillations towards the big bang: A Tumbling Universe

Critical Mathematical Economics - Wikipage.mi.fu-berlin.de/jbuchner/cme.pdfCritical Mathematical...

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