1. Lecture 5: Modelling the Dual Price Hypothesis
Honours Finance (Advanced Topics in Finance: Nonlinear Analysis)
Lecture 5: Modelling the Dual Price Hypothesis

2. Recap
Last lecture we considered Goodwin’s predator-prey model of cyclical growth as a foundation for the “Dual Price Level” hypothesis
This lecture we’ll
Analyse the model in more detail
Add private debt, Prices and a Government sector to develop a series of models of the Dual Price Level hypothesis
Simulate the models and discuss the results
© Steve Keen 2005,

3. Foundations
The basic Goodwin model is
Properties of this simple model illustrate why nonlinear systems are so different to linear ones
Like predator-prey system, equilibrium is neutral: model neither converges to nor diverges from equilibrium;
Deviations above & below equilibrium don’t “cancel each other out”: equilibrium is NOT the average
Property not a result simply of “quirky” functions (like Phillips curve) but nature of nonlinear systems
E.g., simple predator-prey system has just 4 constants and 2 variables: no nonlinear functions…
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4. Foundations Yet equilibrium of system is not average of system:
Divergence gets much more extreme with more complex models
So time & history matter: can’t just treat ups & downs of trade cycle as on average equal to equilibrium!
Reason: asymmetries can apply because of nonlinear forces
System can go much further in one direction than other
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5. Foundations Asymmetry increases as more realism brought into model
Basic model
Only nonlinearity is Phillips curve
Capitalists assumed to invest all profits
But unrealistic:
implies capitalists destroy capital if profit falls below zero
Investment a function of (expectations of) profit
Keynes:
investors extrapolate existing conditions forward
Expectations low during times of low profit, high during times of high
Nonlinear investment function advisable
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6. Nonlinear Investment Function
Replacing linear with nonlinear investment function yields
Many possible forms, but basic property that d(k[p])/dt an increasing function of p. We’ll use
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7. Nonlinear investment function
Nonlinear investment function means
desired (and executed) investment during boom exceeds profits
desired (and executed) investment during slump less than profits
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8. Nonlinear Investment Function
Nonlinear investment function makes little change to nature of basic model:
Still closed cycle
But asymmetry much more obvious:
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9. (1) Fisher = Goodwin with debt
Basic Goodwin model has no debt finance
Debt introduced as stock buffer to firms investment plans
Aggregate position supported by, of all people, Fama & French:
“Debt seems to be the residual variable in financing decisions. Investment increases debt, and higher earnings tend to reduce debt.” (1997)
“The source of financing most correlated with investment is long-term debt… These correlations confirm the impression that debt plays a key role in accommodating year-by-year variation in investment.” (1998)
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10. A Banking Sector
Banking sector added by assuming all debt finances investment:
Work out new dynamics of debt/output ratio:
Product rule:
Substitute:
Debt ratio grows if investment exceeds profits…
Falls as growth rises
Complex interaction because growth encourages investment…
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11. A Banking Sector
This results in the following three-dimensional system:
Wages share of output
Employment ratio
Debt to output ratio
What are systemic dynamics now?
3 dimensional system…
Basic insight of Poincare; modern version from Robert May: “And Three Means Chaos”
Continuous time dynamic system with only 2 variables must either…
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12. A bit of chaos Converge to equilibrium; or
Explode to infinite cycles; or
Follow a fixed orbit; or
Converge to a limit cycle
Reason: basic property of ODE models
Solution gives a time path and initial conditions
Each solution & set of initial conditions unique
No two time paths can have the same initial conditions
No two time paths can intersect
2 dimensional system exists in a plane
Can’t have one time path cross another; so system either spirals in to equilibrium, blasts out indefinitely, or converges to closed path
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13. A bit of chaos
Good illustration of this is Van Der Pol equation for sustained signal in an electronic circuit
Also nice example of eigenvalue analysis of stability…
Currents with passive component (resistor) have damped oscillations: cycles die out when external current removed:
Currents with active component (transistor) replacing resistor have sustained oscillations: cycles continue even when external current removed:
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14. A bit of chaos Can also be written as 2nd order ODE by using I=dV/dt
Exercise: convert to pair of coupled 1st order equations…
OK, a hint: define Y0=V, Y1=dV/dt, Y2=d2Vdt2…
Coupled 1st order equations are
Work out equilibrium & stability…
Substitute terms in Y0 & Y1 only:
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15. A bit of chaos From 1st equation, Y1=0
Put into second; only valid for all t if Y0=0
So equilibrium is (0,0). Is it stable? Work it out!
Express model as matrix equation
Work out Jacobian
Calculate value at equilibrium
If dominant eigenvalue < 0, it is stable…
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16. A bit of chaos
Next, create matrix of partial derivatives if f1 & f2 with respect to Y0 and Y1:
Now evaluate this at the equilibrium (0,0):
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17. A bit of chaos We get a simple matrix:
Does it describe a stable or unstable system?
It’s now the matrix expression of a simple second order linear ODE.
Describes whether small divergence from equilibrium is damped or amplified
Work out its eigenvalues:
Express as matrix equation:
Work with these two bits…
Assume solution of form
Then
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18. A bit of chaos Rearrange using matrix rules:
Only possible for non-trivial v if determinant is zero…
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19. A bit of chaos Determinant is: Its roots are:
Equilibrium unstable for all values of m
Near equilibrium initial values deviate from equilibrium
Far from equilibrium ones converge to the limit cycle
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20. A bit of chaos
Phase plot shows instability and convergence to limit cycle
3rd order version of model (and experiments with vacuum tube circuits) gave initial instance of chaos in 1920s
Basic point here:
2 dimensional continuous systems cannot display chaos
Topography limits dynamic paths to convergence, divergence, or limit cycle
Can’t “cut through” space defined by limit cycle…
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21. “And three means chaos”
Third dimension means one time path can move anywhere in 3D volume without ever intersecting a different time path
Hence Lorenz’s complex dynamics
So basic Goodwin model can’t display chaos
But introduction of debt gives 3rd dimension…
Basics of chaos can (but not necessarily must) apply:
Sensitive dependence on initial conditions
Large scale divergence of time paths with very similar initial conditions
Complex (non limit cycle or aperiodic) cycles
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22. A Banking Sector
Without debt, the model gave us the following patterns:
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23. Analysing Goodwin’s model of cyclical growth
Eigenvalue analysis of Goodwin model:
Function:
Rate of growth equals labour productivity plus population growth
Non-trivial equilbrium:
Wage demands equal rate of growth of productivity
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24. Analysing Goodwin’s model of cyclical growth
Calculating the equilibrium (given parameter values):
Next, what are the stability properties of this non-trivial equilibrium?
Calculate Jacobian of model at equilibrium:
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25. These generate complex numbers
Analysing Goodwin’s model of cyclical growth
Jacobian at equilibrium:
These generate complex numbers
These two obviously zero
Eigenvalues have zero real part
Neutral equilibrium, closed limit cycle
Any initial value generates fixed orbit around equilibrium
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26. Properties of debt model
Equilibrium employment rate exists
Coefficients in Phillips curve function
Equilibrium profit share:
Coefficients in investment function
Investment/output ratio
Equilibrium debt level:
Profit share
Growth rate
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27. A Banking Sector Simulating the model near its equilibrium, we get:
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28. A Banking Sector For a more dynamic view…
All variables spiral towards equilibrium
But away from equilibrium…
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29. “And three means chaos”
Stability properties of
Symbolic answer would be roots of 3rd order polynomial…
Messy! For the record, roots of ax3+bx2+cx+d=0 are:
4th order (when government introduced) far worse; and impossible for 5th & above anyway: instead calculate numerical eigenvalues, Lyapunov exponents, etc.…
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30. Finance & Economic Breakdown
Numerical calculations give equilibrium of system as:
System stable around this equilibrium, but
Debt value implausible in real economy: equilibrium value equates to capitalist savings being 3.6 times GDP
Echoes of Fisher: away from this equilibrium, “instability ensues”…
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31. A Banking Sector With different initial conditions:
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32. A Banking Sector Debt cycles now rise over time:
Model replicates Fisher’s insight:
Stable near equilibrium
Unstable further away…
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33. Stable equilibrium...
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34. Unstable “far” from equilibrium
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35. Interesting properties of model
Equilibrium wage share a negative linear function of interest rate:
Slope depends on interest rate; narrow feasible range, but result
differs from neoclassical income distribution theory
consistent with Kaleckian
effect far more pronounced out of equilibrium when d can be very large
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36. Interesting properties of model
Equilibrium:
Now includes equilibrium debt to output ratio
Investment - profit ratio
Rate of growth
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37. A Stable Equilibrium… within limits
Jacobian is:
With equilibrium conditions:
Where we is a function of the rate of interest
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38. A Stable Equilibrium… within limits
As a function of r, the Jacobian is...
A mess! But the essential point is
The real part of the dominant eigenvalue is negative for almost all values of r
Therefore the equilibrium of the model is stable, and
The linearised version is stable
However outside the equilibrium, the nonlinear forces dominate…
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39. Of models and theories
To relate Dual Price Level Hypothesis model to the theory:
Fisher/Keynes/Minsky analysis argues
economy fundamentally cyclical
corporate debt crucial to understanding capitalism
deflation exacerbates impact of debt
government counter-cyclical spending can counteract tendency to corporate bankruptcy during slump after debt-financed boom
All these features emerge in these models
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40. Cyclical Economy Goodwin model has neutral equilibrium
No tendency towards or away from equilibrium, hence limit cycle behaviour
generalisations (endogenous technical change, variable capacity utilisation) make model unstable
equilibrium a repeller
Addition of debt changes nature of model:
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41. Debt Accumulation and Crisis
Combination of
Nonlinear investment function
Capitalists invest more than profit during booms, less than profit during slumps
Finance sector which lends at interest to finance investment
Generates system which
points out fundamental asymmetry in capitalism
firms commit to debt during booms
have to pay back debt during slumps
cyclical tendency towards debt accumulation and debt-induced Depression
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42. Debt Accumulation and Crisis
Model “bifurcates”
Close to equilibrium
rate of debt accumulation tapers till debt/output ratio a constant
income distribution and employment also converge to equilibrium values
Far from equilibrium
“exponential” component of debt relation overwhelms economy
a debt-induced depression
in practical terms, no escape without debt moratoria, etc.
Declining workers’ share of output (Minsky 1986) an emergent property of model
© Steve Keen 2005,

43. Inflation and Deflation
Two price levels (next lecture)
commodity prices, set by markup on prime cost
equipment prices, diverge from commodity prices
rise relatively during boom
fall relatively during slump
Complex price dynamics and distributional effects
General tendency towards deflation
Deflationary impact of technical progress on wages share
Tendency for employment to fall below level at which money wage rise exceeds growth in productivity
Surges of (cost-push) inflation during times of high employment
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44. Inflation and Deflation
At low interest rate
High profit share causes high investment
output and debt grow
High level of output causes high employment
High employment causes high wage demands
profit share eroded BUT
wage-push inflation reduces real debt burden
System cycles as inflation erodes accumulation of debt
A chaotic limit cycle in wages share, employment, debt and prices
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45. Inflation and Deflation
At high(er) interest rate
Faster growth of debt obligations reduces long run rate of growth
Lower long run rate of growth amplifies tendency to deflation
Deflation during slumps dramatically increases accumulation of debt
Model collapses in high debt/deflation/unemployment spiral as per Fisher 1933
Much longer cycles
similar to period of financial crises (10-20 years)
shorter cycles (2-4 years) may be inventory (Metzler), etc.
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46. Inflation, Distribution, Growth and Crisis
Capitalists beneficiaries of inflation
inflation reduces real debt burden
Bankers/rentiers losers/beneficiaries
inflation erodes real value of debt; but
avoidance of deflation means borrowers remain solvent
Workers losers/beneficiaries
erodes real value of wage; but
wage can be restored via wage bargains during booms
avoidance of depression maintains employment
Inflation improves growth by sidestepping debt-deflation
Matches 19th Century trade cycle (minimal government):
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47. Inflation, Distribution, Growth and Crisis
Procyclical prices
Frequent wage falls
Financial crises
roughly every
20 years
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48. Government and Reflation
Big Government crucial difference post-WWII
Fisher, Keynes & Minsky concur
Reflation only feasible means of escape from Depression
Keynes and Minsky
Government spending provides means to reflation during depression
financial basis for counter-cyclical stabilisation policy
Government counter-cyclical spending stabilises economy, not by eliminating cycles but by preventing a debt-deflation
Modelling this
firstly without prices...
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49. Stabilising an unstable economy
Government spending
rises if unemployment exceeds 5%
falls if less than 5%
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50. Government and Reflation
Government spending a function of employment:
Add to real model to get:
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51. Government and Reflation
Generates four-dimensional system:
Wages share of output
Employment ratio
Debt to output ratio
Subsidy to output ratio
Profit is now net of government:
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52. Government and Reflation
Generates system with cycles but not instability:
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53. Stabilising an unstable economy
Behaviour of model supports Minsky’s contention:
Government stabilises economy
But system remains cyclical
Stability means not the absence of cycles, but
The prevention of systemic collapse:
A “strange attractor”?
Technically, no: different initial conditions converge
But an unstable equilibrium
© Steve Keen 2005,

54. Government and Reflation
A “chaotic limit cycle” at the heart of sustained cyclical behaviour in the long run
The “long run” does not have to be an equilibrium position (c.f. Keynes “in the long run we are all dead...”)
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55. Leads to rising government spending (and vice versa)
Government and Reflation
Government spending counteracts tendency of private sector to accumulate excessive debt
A homeostatic stabiliser:
(with a phase difference)
Leads to rising government spending (and vice versa)
Falling employment
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56. Government and Reflation
“Real” model with government confirms Minsky on “Big Government”
Anti-cyclical spending and taxation of government enables debts to be repaid
Renewal of cycle once debt levels reduced
What about “the full Monty”?: Cyclical economy +
Debt
Prices
Government
A more complex picture still… but first, analysing the “simpler” one…
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57. Stabilising an unstable economy
Mathematically, adding government sector to the model converts it from one with locally stable equilibrium but globally unstable behaviour
to one with locally unstable equilibrium but globally stable behaviour
public debt counterbalances private
increased “taxation” during booms reduces size of booms
increased “subsidies” during slump finances repayment of private debt
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58. Summing Up How does FIH & simple dynamic model compare to US data?
1929 private debt/output ratio 60% (Fisher 1932)
Private non-financial debt/output ratio now 160+%
1929 employment high, economy booming, inflation low
2000 employment high, economy booming, inflation low…
Pattern of US debt to fiat money, debt to output ratio eerily resembles 3 dimensional model debt dynamics:
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59. Summing Up Cyclical “ratcheting up” of debt over time
Debt incurred during boom
Repaid during slump with less than expected cash flows
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60. Next Lecture Introducing prices and time lags
Overall systemic dynamics
Future development
Then introduction to econophysics…
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