Honours Finance (Advanced Topics in Finance: Nonlinear Analysis)

1
Honours Finance (Advanced Topics in Finance
Nonlinear Analysis)

• Lecture 4 ODEs continuedCoupled ODEs,
  nonlinearity and the equilibria of nonlinear
  systems

2
Recap

• Last lecture we considered coupled differential
  equations, where the level of one variable (a
  species) depends on itself and another variable
• Before considering an economic application of
  these, Ill clear up one point concerning complex
  solutions to second order ODEs.
• Remember that
• a second order linear ODE can be solved by the
  characteristic equation method
• guess a solution of the form y(t)ert
• substitution converts equation into quadratic in
  r
• complex solution results when b2-4aclt0

3
Recap

• Last week we considered the Volterra
  predator-prey system

• This week we consider an application of this to
  economics and finance which provides a foundation
  for a model of the Dual Price Level Hypothesis.
• Non-equilibrium predator-prey cycle matches
  non-equilibrium foundation of Fishers
  debt-deflation theory
• Predator-prey model can be derived from Marx

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A predator-prey cycle in capitalism

• In capitalist, Exchange-Value of work brought to
  foreground
• Exchange-Value of workersubsistence wage
• Use-Value of worker in background irrelevant to
  wage
• But Use-Value of worker to capitalist purchaser
  of labour-timeability to produce commodities for
  sale
• Gap between (objective, quantitative) UV and EV
  of worker is source of surplus-value (SV)
• Analysis presumed labour bought and sold at its
  value
• cost of production of labour-power
• subsistence wage
• Is labour actually paid its value in practice?

5
A predator-prey cycle in capitalism

• Many Marxists (especially internationalists like
  Amin, etc.) argue labour paid less than its value
• But plenty of hints that Marx believed labour
  paid more than its value
• the value of the labour-power is equal to the
  minimum of wages (1861 I 46)
• the minimum wage, alias the value of
  labour-power (1861 II 233)
• For the time being, necessary labour supposed as
  such i.e. that the worker always obtains only
  the minimum of wages. (1857 817)

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A predator-prey cycle in capitalism

• No explanation given by Marx, but can be found in
  a dialectic of labour
• Worker both a commodity (labour-power) and
  non-commodity (person)
• Capitalism focuses on commodity aspect, pushes
  non-commodity aspects into background
• Pure commodity--paid subsistence wage only
• Non-commodity--demands share in surplus
• struggle over minimum wage, social wage, etc.
• Wage normally exceeds subsistence subsistence
  wage a minimum (when commodity aspect dominant
  and worker power minimal)

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A predator-prey cycle in capitalism

• Dialectic of labour puts into perspective a
  passage from Marx which is difficult to interpret
  for labour is paid less than its value analysts
• a rise in the price of labor resulting from
  accumulation of capital implies ... accumulation
  slackens in consequence of the rise in the price
  of labour, because the stimulus of gain is
  blunted. The rate of accumulation lessens but
  with its lessening, the primary cause of that
  lessening vanishes, i.e. the disproportion
  between capital and exploitable labour power. The
  mechanism of the process of capitalist production
  removes the very obstacles that it temporarily
  creates. The price of labor falls again to a
  level corresponding with the needs of the
  self-expansion of capital, whether the level be
  below, the same as, or above the one which was
  normal before the rise of wages took place...

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A predator-prey cycle in capitalism

• To put it mathematically, the rate of
  accumulation is the independent, not the
  dependent variable the rate of wages the
  dependent, not the independent variable. (Marx
  1867, 1954 580-581)
• Same passage used by Goodwin (1967) to devise a
  predator-prey model of cycles in employment and
  income distribution
• High wages shareLow rate of accumulationIncrease
  in unemploymentDrop in wagesIncrease in
  accumulationIncrease in employmentHigh wages
  share
• Phillips curve part of Marxs logic (wage
  change a function of the rate of unemployment)
• Goodwin built predator-prey model on this
  foundation
• Try to work out a model

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A predator-prey cycle in capitalism

• Capital stock determines output
• Level of output determines employment
• Level of employment determines rate of change of
  wages
• Differential equation of Rate of change of wages
  determines wages
• Output - Wages determines profits
• Profits determine investment
• Investment determines rate of change of capital
• Capital determines output...
• Try to work it out

10
A predator-prey cycle in capitalism

• Level of output determines employment

• Differential equation of rate of change of wages
  determines wages

• Output - Wages determines profits

• Profits determine investment

• Investment determines capital

• Capital determines output...

• Can you see how to make a predator-prey system
  out of this?

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A predator-prey cycle in capitalism

• System state variables are employment rate, and
  income distribution (use either w or p)
• Goodwin assumed exponential growth of population
  (N) and labour productivity (a)

• Work out the differential equations for w and l
  as functions of themselves and each other

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A predator-prey cycle in capitalism
This is w
Try same thing for w (its easier!)
13
A predator-prey cycle in capitalism
Expand these
The end product is aversion of a predator-prey
model
These cancel
Apply chain rule
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A predator-prey cycle in capitalism

• Negative feedback from wages share to employment
• Positive feedback from employment to wages share
• more complicated than basic predator-prey because
  of Phillips curve relation between rate of
  change of wages and level of employment
• Phillips recap 3 factors which might influence
  rate of change of money wages
• Level of unemployment (highly nonlinear
  relationship)
• Rate of change of unemployment
• Rate of change of retail prices when retail
  prices are forced up by a very rapid rise in
  import prices or agricultural products.
  Economica 1958 p. 283-4
• Latter two factors ignored in conventional
  treatment of Phillips

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A predator-prey cycle in capitalism

• Functional form for Phillips curve was log-log

• Well use raw linear-exponential form

(derived via a nonlinear regression onPhillipss
raw data)
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A predator-prey cycle in capitalism

• Put this Phillips curve in as the equation for
  w(l) in

• Simulation for given values of a and b yields

17
A predator-prey cycle in capitalism

• Goodwin/Marx model thus gives same basic cycle as
  biological predator-prey, but for wages share
  (income distribution) vs employment rather than
  fish vs sharks

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A predator-prey cycle in capitalism

• As with biological model, trade cycle model
  traces out a limit cycle

• What causes this neither converging nor diverging
  behaviour?
• Nonlinearity
• Compare to a linear model with cycles

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The importance of being nonlinear

• Characteristic equation is

• Roots are

• This bit causes cycles

• General solution is of the form

• If agt0 then cycles get infinitely large with time
• System must break down (Tacoma bridge, Braun
  1993 173)

• This bit
• amplifies cycles if agt0
• damps cycles if alt0

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The importance of being nonlinear

• In a linear system
• Forces determining oscillations (the trig
  functions) are distinct from forces determining
  magnitude of those oscillations (the exponential)
• In a nonlinear system
• Oscillation and magnitude are linked
• Magnitude is a function of deviation from
  equilibrium
• In predator prey system
• near equilibrium, linear term dominates
• far from equilibrium, power term dominates
• balance keeps cycles within check, but away from
  equilibrium

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The importance of being nonlinear

• Number of fish
• positive function of number of fish F (linear)
• negative function of F times S (quadratic)
• increasing fishshark numbers means this term
  dominates linear population growth term

• Number of sharks
• negative function of number of sharks S (linear)
• positive function of S times F (quadratic)
• increasing fishshark numbers means this term
  dominates linear death rate term

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The importance of being nonlinear

• Equilibria of nonlinear systems thus
  fundamentally different to those of linear
  systems
• If equilibrium of linear system is unstable,
  whole system is unstable
• If equilibrium of nonlinear system is unstable,
  whole system can still be stable
• If equilibrium of linear system is stable, whole
  system is stable and will converge to equilibrium
• If equilibrium of nonlinear system is stable,
  whole system may be stable or unstable and may or
  may not converge to equilibrium

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The importance of being nonlinear

• Three classes of equilibrium for linear systems

• Stable converges for all time (Re(r) lt 0)

• Unstable diverges for all time (Re(r)gt0)

• Marginal instability neither converges nor
  diverges (Re(r)0)

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The importance of being nonlinear

• Stable an attractor

• Unstable a repeller

• Others neither (limit cycles, etc.)

• Can have mix of stable unstable saddle node

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The importance of being nonlinear

• Nonlinear system can have complicated properties
• stable equilibrium, unstable about equilibrium,
  stable at greater displacement, etc...

• To understand system dynamics, have to
• Simulate
• Analyse stability properties of equilibria
  (stability analysis/ perturbation theory)

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Stability of nonlinear systems

• Simulation later today...
• How to analyse the stability of a nonlinear
  equilibrium?
• Linearise system about its equilibrium
• Properties of complete nonlinear system will
  match those of linearised version about
  equilibrium point (but not far from it)
• Can thus
• identify equilibria of nonlinear system
• characterise near equilibrium behaviour by
  behaviour of linear counterpart

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Stability of nonlinear systems

• Attractor

• Repeller

• Saddle node

• Neutral

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Stability of nonlinear systems

• With multiple equilibria, can have complicated
  landscape of dynamic system
• Knowledge of equilbria can give qualitative
  portrait of system dynamics

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Stability of nonlinear systems

• So how to linearise a nonlinear system?
• Take Taylor expansion about equilibrium point
• Convert complicated nonlinear system into
  polynomial
• Drop all but first two terms
• First is a constant
• Second is another constant times vector of
  variables
• Application of basic calculus property
• any nonlinear function can be converted into
  polynomial approximation (unless function not
  continuous)
• if function continuous on an interval a,b then
  its value at b can be estimated from its value at
  a and polynomials derived from its first order
  and higher differentials

30
Stability of nonlinear systems

• Consider cos(x)
• Set a0
• Cos(x) continuous for all x, so set bx (any
  value)
• Then what we get is...

• Lets look at this visually

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Stability of nonlinear systems

• To apply this to system of ODEs, have to
• convert into set of first order ODEs
• Apply matrix version of Taylor partial
  differentiation

32
Taylor Series

• Taylor Series are a delightful example of the
  application of the basic rules of mathematics
• what have we got that we dont want?
• A complex equation (sin, cos, exponential, log,
  etc.) that we cant easily numerically estimate
• what do we want that we can put in?
• something simple--like a polynomial
• keep it balanced
• So the basic question is can we estimate a
  difficult function using something simple like a
  polynomial?
• The answer is lets just assume the answer is
  yes and see what falls out
• and unlike economists, keep it balanced!

33
Taylor Series

• So we start with the assumption that we can
  represent a complex function, say f(x), as a
  polynomial

• Since all the powers of x are a known quantity,
  the only problem now is working out what the
  coefficients might be
• So how about mapping them to the differential of
  the function
• since we can easily (in most cases) derive the
  differential of a function (unlike integration,
  which is normally difficult and usually cant be
  done)
• starting with the first differential

34
Taylor Series

• Now for the second differential

• And in general for the nth derivative

• Finally we start to narrow down the coefficients
• Again we take the easy route lets substitute in
  a value we know
• If we use the value of f at zero, then its known
  as a Maclaurin series if any other value, its
  called a Taylor series (Brook Taylor rediscovered
  the method in 1715)

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Taylor Series

• Using the value of f at zero

• Notice a pattern here?

36
Taylor Series

• So the general rule is that the coefficient of
  the nth power of x in the polynomial expansion
  for a function is the nth derivative of the
  function, evaluated at zero, divided by n
  factorial

• So the Taylor series for a function is simply

• Try this for a few well-known functions

• Can you see a problem for some other functions?

37
Taylor Series

• Obviously we cant substitute in the value of
  these functions at zero, because they dont have
  any value (colloquially, 1/0 is infinity, the log
  of zero is minus infinity)
• Instead we have to choose a point at which there
  is a known value (this raises the issue known as
  the radius of convergence, but we wont worry
  about this here)
• Call this point a. Then we have the series

• Exactly the same logic applies to working out
  this more general Taylor series
• The unknowns are the coefficients, and we use
  differentiation to work out possible values.
• Cutting a long story short,...

38
Taylor Series

• The (fairly obvious) end result is that the
  constant a takes the place of 0 in the previous
  expansion

• The nth coefficient in the polynomial expansion
  for f about the point a is the nth derivative of
  f evaluated at a, divided by n factorial

• Try this for ln(x)

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Taylor Series

• The differentials of ln are

• We can substitute to give, in complete detail

40
Taylor Series

• Try this using a1, at which point ln(1)0

41
Taylor Series

• We get

• This works for xgt0 (since ln of a negative number
  is not defined) but it doesnt converge very
  well (for reasons we wont go into). Points other
  than 1 converge somewhat better.

42
Taylor Series

• So why are we bothering with all this stuff?
• The bad news to work out stability properties,
  we have to linearise a nonlinear system of ODEs
  in vicinity of (non-trivial, hence not zero)
  equilibrium point
• More bad news we have to apply the matrix
  equivalent of a Taylor series, which means powers
  of matrices, etc.
• The good news since were trying to linearise
  the system, all were interested in are the first
  two terms
• The first matrix of constants, which since its
  based on the equilibrium point of the system will
  be a matrix of zeros
• The second matrix of constants, which simply
  involves the first differentials of the system
  and multiplies the vector of variables
• More bad news the first differential is an array
  of differentials

43
Taylor Series

• Consider the predator-prey model

• Where x0 is the prey and x1 the predator

• This can be put in the form of a function

• Where both x and f are vectors
• It sounds messy, but all it means is that x has
  two values (x0 and x1) and f has two values

• So the first differential is actually an array
  of partial differentials f0 against x0, f0
  against x1, f1 against x0, f1 against x1. Its a
  matrix known as the Jacobian.

44
Taylor series

• The Jacobian of the predator-prey model is thus

• We use this to linearise the system in the
  vicinity of its equilibrium, to work out the
  stability properties of the equilibrium. As with
  the simple Taylor expansion about a point, the
  vector version is

45
Stability of equilibria of nonlinear systems

• What we do is
• choose a so that its an equilibrium point. The
  f(a) will be 0.
• Define a variable z(t) as the deviation of the
  current position x(t) from the equilibrium a.
• Then this vector version of a truncated Taylor
  expansion becomes

• Since a is the equilibrium point of the full
  nonlinear system, f(a) is 0. So the equation
  collapses to

• Where f(x)a is a matrix of constants, found by
  evaluating the Jacobian of f at the equilibrium
  point a. Now, just as for the linear systems we
  studied initially, the stability of this system
  depends on whether the real part of its
  eigenvalues exceed zero.

46
Stability of equilibria of nonlinear systems

• Trying this with the predator prey model, the
  Jacobian was

• The non-trivial equilibrium point was

• Feeding this into the Jacobian gives us the
  matrix

47
Stability of equilibria of nonlinear systems

• So is the system

stable?

• Same old story
• presume a solution of the form z(t)eltv.
• Rework equation, calling Jacobian A for the moment

• And as with the linear problems, the non-trivial
  solutions to this require that the determinant of
  the matrix lI-A equals zero.

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Stability of equilibria of nonlinear systems

• On we plug

For this to be zero we need the roots
And this isnt a bad place to give a reminder
about complex numbers
49
Complex numbers

• These are numbers which have a real component (as
  youre used to) and a component multiplied by the
  square root of minus one
• They were invented because without them, many
  quadratics did not have a solution--like the one
  were looking at now.
• The reason they matter to us in dynamic analysis
  is because they have the effect of generating
  cycles in mathematical models of real systems
• The reason they do this is because there is a
  very exact correspondence between exponentials,
  complex numbers, and the trigonometric functions
  sin and cos.
• This can be proven (dont ask me now!) but it can
  also be shown by using a Taylor series expansion

50
Complex numbers

• We can expand ex around x0 because it has a
  value there (e0 1)
• But we use xiq where i is the square root of
  minus one

• If you separate out the real (all terms without
  i) and imaginary (all terms with i), you find one
  is cos(q) and the other is i times sin(q)

51
Complex numbers

• And for Sin

• So from a Taylor expansion alone we know that

• And at a higher level this equation can be proven
• So a complex root to a polynomial can be
  factored

• The cos and sin bits mean that this solution
  generates cycles
• The magnitude of the real bit (a) tells us
  whether the cycles explode (agt0) or contract
  (alt0) with time

52
Stability of nonlinear systems

• Consider basic predator-prey model again

• Non-trivial equilibrium is

• Is this stable? Consider the point

• Treat (c/d, a/b) as the point about which the
  equation is converted into a polynomial
• like 0 in expansion of cos(x) about 0
• Then putting x0F,x1S, we have

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Stability of nonlinear systems

• In matrix terms

• Now define z(t) to be the difference between the
  current position x(t) and the equilibrium x(0)
  (which in this model occurs when x0c/d,x1a/b),
  where z(t) is a small distance. Then

• Now we can expand f(x) using the Taylor
  expansion. Set x(0) as the beginning of the
  interval (equivalent to 0 in the expansion for
  cos)

54
Stability of nonlinear systems

• Expanding, we get

• This is the linear bit of the Taylor expansion
  (so all coefficients are constants)

• This is zero, since dx/dt0 at x(0) where F and S
  are at equilibrium values of c/d and a/b

• This is the bit with powers of 2 and above of x,
  which can be ignored in the vicinity of x(0) when
  squared and above amounts are very small

55
Stability of nonlinear systems

• A visual explanation might help here

Taylor expansion forx(t) around equilibiumpoint
x(0)
If this goes to zero as t goesto infinity, then
the systemis stable.
x(0)
This distance is z(t)

• So question whether equilibrium is stable reduces
  to whether A.z grows with time or shrinks to
  zero.
• In a matrix Taylor expansion, A is the partial
  differentials of f evaluated at x(0) (as
  constants in cos expansion are evaluated at x0)

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Stability of nonlinear systems

• A is thus

Where f0 is the function for the first variable
(F), etc., and it is evaluated at the equilibrium
point (c/d)

• So we now define a new linear function

• And if this is stable then the equilibrium x(0)
  is stable

57
Stability of nonlinear systems

• What is A for our example of a predator-prey
  system?

58
Stability of nonlinear systems

• This reduces to

• So our system is

59
Stability of nonlinear systems

• Is this system stable? Analyse it using a guess
  (ansatz) of

For this to have a non-trivialsolution, the
determinant ofthe matrix must be zero
60
Stability of nonlinear systems

• The determinant is a quadratic in l

Oh oh complex numbers again!
So substituting this value for l backinto our
solution for z(t)
61
Stability of nonlinear systems

• This generates cycles, but no convergence
• for convergence, the real part of l must be less
  than zero, so that elt gt1, and thus elt0 as t
• for divergence, the real part of l gt0, so that
  elt is less than one, and thus elt as t.
• Here the real part of l is zero, so the system
  neither converges nor diverges

62
Stability of nonlinear systems

• This explains closed limit cycle behaviour of
  Goodwin model
• Many other behaviours possible for nonlinear
  model
• Technique works in vicinity of equilibrium, but
  doesnt tell you anything about behaviour away
  from equilibrium
• To characterise overall stability of nonlinear
  system, techniques used which explore parameter
  space
• How stable is system for varying values of
  constants, different initial conditions?
• Lyapunov exponents (large-scale eigenvalue
  analysis), etc.
• Can also simulate, as shown earlier
• Combination of two needed to fully characterise
  system
• Cant provide analytic solution

63
Simulation

• Numerical simulation of analytically insoluble
  ODEs/PDEs relies upon
• Approximation methods
• simplest Euler method
• more complex (and accurate) Runge-Kutta algorithm
• based on Taylor-style logic
• next value of function can be estimated from
  current value plus derivatives evaluated at
  current value
• b-shadowing hypothesis for chaotic systems
• Despite sensitive dependence on initial
  conditions, simulation can be guaranteed to
  shadow an actual trajectory of a system, though
  not the one intended to be simulated

64
Simulation

• Two broad classes of simulation tools
• Top down
• system specified as
• set of ODEs/PDEs
• system of flowcharts
• Bottom up
• characteristics of agents in such a system
  described in computer program
• environment of system constructed as experience
  space for agents
• system run as virtual world

65
Simulation

• Top Down programs
• ODE/PDE approach
• Mathematica, Mathcad, Maple, Matlab
• Flowchart approach
• Matlab/Simulink, Vissim, IThink, Vensim
• Bottom up programs
• Few commercial versions as yet (SimCity?)
• Many research programs, some public domain
  Swarm, Terra, Ecolab
• Illustration of both top down approaches for
  predator-prey model

66
Simulation

• Using Mathcad
• (1) Derive equations

• (2) Convert into vector form

• (3) Provide initial values run simulation
  function

• (4) Graph result

Number of pointsto store
Time range
67
Simulation

• (5) Run simulations with varying parameters many
  times for phase space analysis (Lyapunov, etc.),
  production of phase portraits...

68
Simulation

• Flowchart method
• (1) Express equations as integrals rather than
  differentials (integration more numerically
  robust)
• (No need to solve reduced form of equations, as
  with Goodwin model)

• (2) Build model using flowchart symbols for
  numerical operators

• (3) Run simulation dynamically

69
Simulation

• A completedflowchart model looks like this

• And a simulation run gives results like

70
Simulation
71
Next Week

• Derive a model of the Dual Price Level
  Hypothesis
• Simulate it and discuss results