How to Be Nonlinear

1
How to Be Nonlinear

• Steve Keen

2
The basics

• Cant do nonlinear analysis without mathematics
• Nonlinearity banishes ceteris paribus
• Feedbacks too complicated to keep in mind
  verbally
• Though some aides to thise.g., influence
  diagrams

• Greenhouse gas rise
• Causes temperature rise
• Causes fall in ice
• Causes fall in reflection
• Causes increase in absorption of sun energy
• Positive feedback loop

Absorption of solar radiation
Change in solar reflection
Change in temperature
-
-
Change in ice area

• Mathematical methods provide means to quantify
  this qualitative causal loop

3
The basics

• Basic mathematical tools for dynamic processes
  are
• Ordinary Differential Equations (ODEs)
• Partial Differential Equations (PDEs)
• Partials actually more complicated than
  Ordinaries
• ODEs one underlying variable (normally time)
• as if everything happens in one spot
• PDEs 2 or more (normally displacement)
• accounts for processes dispersed in space
• PDEs more realistic, but
• Maths much more complicated
• Range of problems that can be solved much more
  limited

4
The basics

• One special class of PDEs Stochastic
  Differential Equations (SDEs)
• Dispersal a function of stochastic distribution
• Literally rocket science
• Developed to model flight of rockets where
  exhaust from rocket spread over area of jet
  nozzle
• Applied to finance (Black-Scholes Options
  Pricing) but with wrong form of distribution
• Gaussianpresumes one atom doesnt affect
  others
• Proper distribution fractalone atom does
  affect others.
• Nature of differential maths very different to
  algebraic/differentiation youve done to date

5
The basics

• Linear algebraic and differentiation problems
  normally soluble
• Nonlinear differential equation problems normally
  insoluble
• Summarising solvability of mathematical models
  (from Costanza 1993 33)

Most economics here
I work here
6
The basics

• Simply ODEs used to model
• the decay of radioactive particles
• the growth of biological populations
• the spread of diseases
• the propagation of an electric signal through a
  circuit
• Equilibrium methods (simultaneous algebraic
  equations using matrices etc.) only tell us the
  resting point of a real-life process if the
  process converges to equilibrium (i.e., if the
  dynamic process is stable)
• ODEs tell us the dynamic path of a process
  whether stable or unstable
• Nonlinear ODEs can have unstable equilibria and
  not break down, contra standard economic belief

7
Lorenzs Butterfly

• An example Lorenzs stylised model of 2D fluid
  flow under a temperature gradient
• Lorenzs model derived by 2nd order Taylor
  expansion of Navier-Stokes general equations of
  fluid flow. The result

x displacement
y displacement
temperature gradient

• Looks pretty simple, just a semi-quadratic
• First step, work out equilibrium (try it now!)

8
Lorenzs Butterfly

• Three equilibria result (for bgt1)

• Not so simple after all! But hopefully, one is
  stable and the other two unstable
• Eigenvalue analysis gives the formal answer (sort
  of )
• But lets try a simulation first

9
Simulating a dynamic system

• Many modern tools exist to simulate a dynamic
  system
• All use variants (of varying accuracy) of
  approximation methods used to find roots in
  calculus
• Most sophisticated is 5th order Runge-Kutta
  simplest Euler
• The most sophisticated packages let you see
  simulation dynamically
• Well try simulations with realistic parameter
  values, starting a small distance from each
  equilibrium

So that the equilibria are
Over to Vissim...
10
Lorenzs Butterfly

• Now you know where the butterfly effect came
  from
• Aesthetic shape and, more crucially
• All 3 equilibria are unstable (shown later)
• Probability zero that a system will be in an
  equilibrium state (Calculus Lebesgue measure)
• Before analysing why, review economists
  definitions of dynamics in light of Lorenz
• Textbook the process of moving from one
  equilibrium to another. Wrong
• system starts in a non-equilibrium state, and
  moves to a non-equilibrium state
• not equilibrium dynamics but far-from equilibrium
  dynamics

11
Lorenzs Butterfly

• Founding father mathematical instability does
  not in itself elucidate fluctuation. A
  mathematically unstable system does not
  fluctuate it just breaks down. Wrong
• System with unstable equilibria does not break
  down but demonstrates complex behaviour even
  with apparently simple structure
• Not breakdown but complexity
• Researcher static analysis allows enough time
  for changes in prime costs, markups, etc., to
  have their full effects. Wrong
• Complex system will remain far from equilibrium
  even if run for infinite time
• Conditions of equilibrium never relevant to
  systemic behaviour

12
Lorenzs Butterfly
Tiny errorin initialreadingsleads
toenormousdifferencein time pathof
system.And behindthe chaos,strangeattractors..
.
13
Lorenzs Butterfly
14
Lorenzs Butterfly

• Lorenz showed that real world processes could
  have unstable equilibria but not break down in
  the long run because
• system necessarily diverges from equilibrium but
  does not continue divergence far from equilibrium
• cycles are complex but remain within realistic
  bounds because of impact of nonlinearities
• Dynamics (ODEs/PDEs) therefore valid for
  processes with endogenous factors as well as
  those subject to an external force
• electric circuit, bridge under wind and shear
  stress, population infected with a virus as
  before and also
• global weather, economics, population dynamics
  with interacting species, etc.

15
Lorenzs Butterfly

• To understand systems like Lorenzs, first have
  to understand the basics
• Differential equations
• Linear, first order (see Advanced Nonlinear
  Finance Lectures)
• Linear, second (and higher) order (ditto)
• Some nonlinear first order (ditto)
• Interacting systems of equations (ditto plus
  well simulate)
• Initial examples non-economic (typical maths
  ones)
• Later well consider some economic/finance
  applications before building full finance model

16
Maths and the real world

• Much of mathematics education makes it seem
  irrelevant to the real world
• In fact the purpose of much mathematics is to
  understand the real world at a deep level
• Prior to Poincare, mathematicians (such as
  Laplace) believed that mathematics could one day
  completely describe the universes future
• After Poincare (and Lorenz) it became apparent
  that to describe the future accurately required
  infinitely accurate knowledge of the present
• Godel had also proved that some things cannot be
  proven mathematically

17
Maths and the real world

• Today mathematics is much less ambitious
• Limitations of mathematics accepted by most
  mathematicians
• Mathematical models
• seen as first pass to real world
• regarded as less general than simulation models
• but maths helps calibrate and characterise
  behaviour of such models
• ODEs and PDEs have their own limitations
• most ODEs/PDEs cannot be solved
• however techniques used for those that can are
  used to analyse behaviour of those that cannot

18
Maths and the real world

• To model the vast majority of real world systems
  that fall into the bottom right-hand corner of
  that table, we
• numerically simulate systems of ODEs/PDEs
• develop computer simulations of the relevant
  process
• But an understanding of the basic maths of the
  solvable class of equations is still necessary to
  know whats going on in the insoluble set
• Hence, a crash course in ODEs, with some
  refreshers on elementary calculus and algebra...

19
From Differentiation to Differential

• You know to handle equations of the form

Independent variable
Dependent variable

• Where f is some function. For example

• On the other hand, differential equations are of
  the form

• The rate of change of y is a function of its
  value y both independent dependent

• So how do we handle them? Make them look like the
  stuff we know

20
From Differentiation to Differential

• The simplest differential equation is

(we tend to use t to signify time, rather than
xfor displacement as in simple differentiation)

• Try solving this for yourself

Continued...
21
From Differentiation to Differential
Because log of a negative number is not defined
Because an exponential is always positive

• Another approach isnt quite so formal

22
From Differentiation to Differential

• Treat dt as a small quantity
• Move it around like a variable
• Integrate both sides w.r.t the relevant d(x)
  term
• dy on LHS
• dt on RHS
• Some problems with generality of this approach
  versus previous method, but OK for economists
  modelling issues

• So whats the relevance of this to economics and
  finance? How about compound interest?

23
From Differential Equations to Finance

• Consider a moneylender charging interest rate i
  with outstanding loans of y.
• Who saves s of his income from borrowers
• Whose borrowers repay p of their outstanding
  principal each year
• Then the increment to bank balances each period
  dt will be dy

Divide by y Collect terms
Integrate
Take exponentials
24
From Differential Equations to Finance

• Under what circumstances will our moneylenders
  assets grow?
• C equals his/her initial assets

Known as eigenvaluetells how much the
equationis stretching space

• The moneylender will accumulate if the power of
  the exponential is greater than zero

• The moneylender will blow the lot if the power of
  the exponential is less than zero

25
Back to Differential Equations!

• The form of the preceding equation is the
  simplest possible how about a more general form

Same basic idea applies

• f(t) can take many forms, and all your
  integration knowledge can be used
• An example compound interest

26
Back to Differential Equations!

• Imagine that your ancestor deposited 1 in the
  year 0 in an account which was continuously
  compounded at a rate of 2 p.a.
• How much would be in the account in the year
  2000?
• Work out the formula

Rate of interest
Time period
Change in Asset
27
An Example

• Work out the solution for A

So what is the value of C? Work it out
28
An Example

• Now lets use the formula
• How much would that 1 invested at 2 p.a. be
  worth in the year 2000?
• Have a guess...
• Now work it out

29
An Example

• Get out the calculators what is this in decimal
  format?

• How much gold is that at, say, 300 an ounce?

• So how much space would that much gold occupy?
  (Gold weighs 19,300 kg per cubic metre)

30
An Example
Thats 1.15 billion cubic metresof gold

• So how large is that exactly... say, compared to
  the volume of the earth? (The earths radius is
  6370 km)

So its not that bigjust how big is it?
31
An Example

• So one dollar, invested at 2 p.a., turns into a
  ball of gold 1300 metres across in 2000 years
• And I bet you thought 2 was a lousy rate of
  return!
• What do you think 4 yields?
• 250,000 balls of gold the size of the earth, or a
  sphere of gold 400,000km across!
• With the knowledge imparted by this ODE, you
  should now be sceptical about the long term
  viability of growth rates which are currently
  taken as desirable in the modern world
• 10 p.a. for China, etc.
• World history hasnt been one of continuous
  accumulation!
• Current expected yields (4-6 p.a. min.)
  unsustainable

32
A little problem

• Most ODEs are insoluble impossible to find a
  closed form for y(t) from an expression for y(t)
• The general technique of solving an ODE is to
  take something in the form of

• And work on it till it is in the form

• Integration of this (with respect to t) yields

• The function f is then reworked to provide an
  expression for y in terms of t.
• The question now is, how many functions of the
  form F can we rework into a function of the form
  f?
• The answer is, not many!

33
Why most ODEs cant be solved

• It turns out that we can only process F into this
  form if we can break F down into two parts (M and
  N) which obey the condition that the differential
  of M with respect to y is the same as the
  differential of N with respect to t
• This is, as it sounds, a highly restrictive
  condition. The next couple of slides proves this,
  but are background only.
• We start with a general ODE

34
Why most ODEs cant be solved

• Can this be put into the integrable form?
• Only if

• The RHS of this can be expanded using the chain
  rule for partial differentiation

• This lets us equate M and N to the partial
  derivatives of f

• But this immediately imposes conditions on the
  forms that M and N can take

35
Why most ODEs cant be solved

• In (partial) differentiation, the order of
  differentiation is irrelevant. Thus

• But the LHS of the above is the differential of M
  with respect to y, and the RHS is the
  differential of N with respect to t

• So, for a valid M and N to exist, it must be true
  that

36
Why most ODEs cant be solved

• This condition will be true of the general
  relation

• Only in a very small minority of cases
• In some others, initially unsuitable equations
  can be processed to be in a more suitable form
• But in general most ODEs cannot be solved
• and its worse for higher order ODEs

37
Why thats not a problem anymore

• The bad news
• Incredibly hard work to massage minority of
  problems into soluble forms
• Worse news
• Most real world problems cant be so massaged
• Fundamentally insoluble
• Good news is
• Since most real world problems are fundamentally
  insoluble symbolically
• Engineers have worked out how to solve them
  numerically using computers
• Mathematicians have shown numerical simulations
  accurate even if system chaotic

38
Why thats not a problem anymore

• As a result, easier to do dynamics now than
  statics
• So long as you can think in terms of flows
• A differential equation fundamentally describes a
  flow into a stock

is a (often complicated) function of vessel ys
current volume
Rate at which stock y changes in volume

• y can be a vector of variables a coupled ODE
• No problem with modern computer mathematics
  software
• Difficulty lies in thinking dynamically

39
An example

• With (insincere) apologies to those whove done
  Financial Economics
• The Circuitist model of endogenous money
• With a different approach to thinking
  dynamically to Financial Economics
• First, a recap on the Circuitist School
• Attempt to model credit economy
• See neoclassical model as barter only
• Adding money commodity doesnt change
  essentially barter nature of model
• From n to n1 commodities big deal!
• Instead, true money cannot be a commodity

40
Conditions for money

• (1) Must be a token (otherwise still a barter
  model)
• The starting point of the theory of the circuit,
  is that a true monetary economy is inconsistent
  with the presence of a commodity money. A
  commodity money is by definition a kind of money
  that any producer can produce for himself. But an
  economy using as money a commodity coming out of
  a regular process of production, cannot be
  distinguished from a barter economy. A true
  monetary economy must therefore be using a token
  money, which is nowadays a paper currency (3)

41
Conditions for money

• (2) Must be money has to be accepted as a means
  of final settlement of the transaction (otherwise
  it would be credit and not money). (3)
• (3) Must not grant rights of seignorage (agents
  cant create it indefinitely at negligible cost
  as formally Governments can with fiat money)
• If seller A buyer B accept tokens issued by
  Bank C as final settlement, cant have C use its
  own tokens to be a buyer
• Like paying for goods with IOUs

42
Conditions for money

• The only way to satisfy those three conditions
  is to have payments made by means of promises of
  a third agent (3)
• Essential point in circuitist case (and
  endogenous money in general) transactions are
  all 3 sidedbuyer, seller, banker. Banks are an
  essential aspect of capitalism

43
Conditions for money

• When an agent makes a payment by means of a
  cheque, he satisfies his partner by the promise
  of the bank to pay the amount due.
• Once the payment is made, no debt and credit
  relationships are left between the two agents.
  But one of them is now a creditor of the bank,
  while the second is a debtor of the same bank.
• This insures that, in spite of making final
  payments by means of paper money, agents are not
  granted any kind of privilege.
• For this to be true, any monetary payment must
  therefore be a triangular transaction, involving
  at least three agents, the payer, the payee, and
  the bank. Real money is therefore credit money.
  (3)
• Second essential point of this school the
  minimum number of agents in a capitalist economy
  is three

44
Conditions for money

• (1) Seller A with commodity X to sell

• (2) Buyer B with money in a bank account AND

• (3) Bank C that records transfer from Bs account
  to A

• Essentially different to neoclassical barter
  vision of money as the money commodity
• Buyer/Seller A has commodity X, wants Y
• Buyer/Seller B has commodity Y, wants X
• They work out exchange ratio in terms of money
  commodity Y
• No bank involved
• Interesting model of primitive village
• But not a model of capitalism

45
One step forward, two steps back?

• So far, so good
• But Circuitists failed to model circuit
  dynamically
• Instead
• Tried static equilibrium methods (Graziani)
• Or fudged dynamics but shied away from actual
  processes in credit creation
• A dynamic innovation
• Possible to build coupled ODE model of monetary
  circuit using accounting double-entry
  book-keeping tables
• Transactions paradigm for dynamic modelling

46
Model Circuit Dynamically

• Starting point
• 3 classes
• Workers Work for wage in factories
• Capitalists Run factories profit from sale of
  output
• Bankers Lend money to capitalists
• No money anywhere at the start just the classes
• Banker maintains 3 deposit accounts (Firms FD,
  Workers WD, Bankers BD)
• Zero balance in all three
• One record of debt (Firms Debt FL)
• Not money vessel, but a record of obligation to
  repay
• Also zero

47
Initial conditions

• Starting position is

• Stage one bank extends loan of L to capitalist

• Stage two Loan involves obligations

48
Stage two obligations initiated by loan

• Loan obliges
• capitalist to pay interest on FL balance
• bank to pay interest on FD balance

• Only sources of funds are Deposit Accounts
• FD for capitalist
• BD for bank
• Now were modelling flows of money into out of
  the stocks FD, BD, WD

49
Stage three flow of interest payments

• Payment of interest keeps
• Loan balance at initial level L
• Transfers money from FD to BD
• Keeps balance in Deposit Accounts at L

• System of coupled ODEs can be read down columns
• Change in FL 0
• Change in FD rD FD - rL FL
• Change in BD rL FL - rD FD

Simulating...
50
Simulating stage three

• As a system of equations, this is

• Using L100, rD3, rL5
• Simulating in Mathcad

• All money transferred to BD after 30.5 years
• But model incomplete

51
Stage four using the borrowed money

• Money borrowed to finance production
• Workers hired paid w FD

• Workers earn interest on balance in WD

• Stage five workers bankers buy goods from
  capitalists

52
Complete model

• Whole model is

• Equations of motion read down the columns e.g.,
  FD

53
Complete model

• Complete set of equations

• Simulation shows Circuit works
• Capitalists can borrow money, pay interest on it,
  operate indefinitely
• Activity continues forever with single
  injection of money
• But these are just bank account balances
• What about incomes?

54
Income dynamics

• Worker bank income easy
• Wages are the flow w FD
• Gross interest is the flow rL FL

• What about profits?
• Derive from w
• w is part of net surplus from production accruing
  to workers
• Surplus constituted by
• Worker-capitalist split (1-sssums to 1)
• Rate of turnover from M to M
• Signified by P
• So we have

conversely

• Profits are

55
Income dynamics/debt repayment

• Confirming from simulation program

• Yearly net income of 429.15 exceeds L by factor
  of four
• Reflects turnover of capitalneglected by
  Circuitists
• What if loans repaid?
• Amount RL FL deducted from FD account
• No seignorage direct by bank into capital
  account
• Re-lent at rate LR
• The outcome repayment of loans creates reserves

56
Model with repayment/growth

• Overall system still balanced

• Final extension growth
• Additional reserves/debt at rate FI
• Models Moores Horizontalism

57
Model with Growth

• System is now dissipative
• Sum of SAM exceeds zero

• Accounts still balanced
• But Walras Law violated in growing economy
• Sum of excess demands gt 0

58
Model with Growth

• Full model now is

59
Economic modelling via transactions

• Transactions approach here may be sound way to
  model economy
• Actually captures economic exchanges
• All exchanges require transactions
• Either implicit in or ignored in models that
  start with income, etc.
• Flow accounting can therefore have errors
• Economic variables (profits, wages, employment)
  can be explicitly derived from transactions
  record
• May be best foundation for modelling actual
  economic dynamics

60
But first a word from our saviours

• Modelling wouldnt be possible without computer
  software
• 2 decades ago
• Programming unavoidable
• Really steep learning curve
• Computers extremely slow
• Output dodgy
• Now
• Really easy to use software
• No programming needed
• Very easy to learn
• Even laptops fast enough for single run
  simulations
• Brilliant graphics

61
Quick overview of Mathcad

• Program lets you type equations as you would
  write them

• Ugly, huh?
• Try reading it as a single line of unformatted
  text
• e1/(1-(n-1)q)1/(1-(n-1)(1/n)(-Q/P)(dP/dQ))
• No joke! This is how equations are formatted in
  programming languages
• Mathcad also uses keyboard shortcuts to make
  typing that simple

62
Quick overview of Mathcad

• Functions can be numerically simulated graphed

63
Quick overview of Mathcad

• Many built-in functions
• a simple (limited) programming language
• Key one for our purposes Odesolve
• Arguments differential equations initial
  conditions

64
Quick overview of Mathcad

• Function needs variable names, independent
  variable (t), number of time periods to
  simulate (Years)
• (Simulation in continuous time, not discrete)

• Result can be graphed, analysed

65
Quick overview of Mathcad

• Program includes some symbolic capabilities
• For example, system without repayment is

• Equilibrium occurs when all differentials equal
  zero
• FL remains at L with no repayment
• Sum of deposit accounts equal sum of FL
• Feed conditions into program ask for
  equilibrium solution

Beats doing it by hand!
66
And the competition

• Now many programs with these numerical symbolic
  capabilities
• Mathematica
• Scientific Workplace
• Maple
• Scilab (free softwarepowerful but poorly
  documented)
• Matlab
• Some much more powerful (but generally harder to
  use)
• Numerous ways to analyse complex dynamic systems
• Next, Goodwin trade cycle model as instance of
  importance of nonlinearity
• Then the bottoms-up approach to nonlinearity

67
Coupled ODEs

• Weve just modelled a
• Fifth order
• Linear
• Set of coupled differential equations
• Goodwins 1967 growth cycle model a second
• Second order nonlinear ODEs are common in
  mathematical modelling (but rare in economics)
• These model a system in which two variables
  affect each other a feedback system
• The most relevant example for us is the
  Lokta-Volterra predator-prey model

68
Predator-Prey Systems

• Fish and Sharks
• Fish eat seagrass (assumed unlimited supply)
• Sharks eat fish
• Together, a cycle
• Low numbers of fish, sharks die off
• Less sharks, more fish reproduce
• More fish available, shark numbers rise
• More sharks, fish population declines
• Low numbers of fish, sharks die off
• How to model it?
• Use F for Fish and S for Sharks

69
Predator-Prey Systems

• Rate of growth of fish is
• positive function of number of fish
• negative function of the number of sharks

• Rate of growth of sharks is
• negative function of number of sharks
  (starvation)
• positive function of the number of fish

• Together, a system

Can thisbe solved?
70
Predator-Prey Systems

• Well, yes but its the last nonlinear ODE we can
  solve
• any system with three or more coupled ODEs is
  insoluble
• first, a numerical simulation

71
Predator-Prey Systems

• How do we solve it?
• using the separable approach
• separate the equations into
• One side of sign that depends on F only
• Other side depends on S only

72
Predator-Prey Systems

• Notice how each variable is a function of the
  other

73
Predator-Prey Systems

• What about the systems equilibrium?
• How do you define it?
• When dF/dtdS/dt0
• Is it stable or unstable?

• There are ways to work this out (pertubation
  analysis work out the dynamics of behaviour a
  short distance from equilibrium)
• It turns out that the equilibrium is neutral
• neither attracts nor repels

74
Predator-Prey Systems

• Generates a stable limit cycle
• system orbits the equilibrium but never converges
  to or diverges from it.
• Such behaviour the norm in complex systems

75
Predator-Prey Systems

• Now an application of this to economics
• Non-equilibrium predator-prey cycle can be
  derived from Marx
• Check Eds notes for my interpretation of Marx
• Core analysis not Labour theory of value but
• Dialectic between use-value and exchange-value of
  commodity
• Labour theory of value (LTV) derived from this
  dialectic
• In fact, Marx got it wrongdialectic contradicts
  LTV
• But ignoring that, dialectic when applied to
  wages predicts cycles

76
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)
• LTV analysis presumes labour bought and sold at
  its value
• cost of production of labour-power
• subsistence wage
• Is labour actually paid its value in practice?

77
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)

78
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)

79
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...

80
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)
• Idea by Goodwin (1967) to devise a
  predator-prey model of cycles in employment and
  income distribution
• High wages share?Low rate of accumulation?Increase
  in unemployment?Drop in wages?Increase in
  accumulation?Increase in employment?High 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

81
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...

82
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?

83
A predator-prey cycle in capitalism

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

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

84
A predator-prey cycle in capitalism
This is ?
Try same thing for ? (its easier!)
85
A predator-prey cycle in capitalism
Expand these
The end product is aversion of a predator-prey
model
These cancel
Apply chain rule

• Negative feedback from w to l
• Positive feedback from l to w
• more complicated than basic predator-prey because
  of Phillips curve relation between rate of
  change of wages and level of employment

86
A predator-prey cycle in capitalism

• 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

87
A predator-prey cycle in capitalism

• Simulation for given values of ? and ? yields

• 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

88
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

89
The importance of being nonlinear

• Characteristic equation is

• Roots are

• This bit causes cycles

• General solution is of the form

• If ?gt0 then cycles get infinitely large with time
• System must break down (Tacoma bridge, Braun
  1993 173)

• This bit
• amplifies cycles if ?gt0
• damps cycles if ?lt0

90
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

91
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

92
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

93
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 dont
  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

94
Foundations

• Yet equilibrium of system is not average of
  system

• Divergence gets much more extreme with more
  complex models
• So time history matter cant 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

95
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

96
Nonlinear Investment Function

• Replacing linear with nonlinear investment
  function yields

Many possible forms, but basic property that
d(kp)/dt an increasing function of p. Well use
97
Nonlinear investment function

• Nonlinear investment function means
• desired (and executed) investment during boom
  exceeds profits
• desired (and executed) investment during slump
  less than profits

98
Nonlinear Investment Function

• Nonlinear investment function makes little change
  to nature of basic model
• Still closed cycle
• But asymmetry much more obvious

99
And now for something completely different

• ODEs are tops down dynamic models
• Many practitioners (Chiarella, Flaschel, Semmler,
  Skott, Keen)
• Computers (and games) make another form possible
• Bottoms up simulations
• Define some behaviour of individual agents in
  artificial economy
• Invent lots of them
• Let them interact and see what happens
• Complex behaviours should be emergent property
  of interactions between agents
• Relationships between agents in system often more
  important than individual definition of agents

100
And now for something completely different

• Example my critique of theory of the firm
• Standard theory A firm maximises profit by
  equating marginal cost and marginal revenue!

"Excuse me, is this the right room for an
argument?"

• No it doesnt!
• It maximises profit by setting

• Proving this to economists a bit like arguing
  with John Cleese, so lets try a simulation

101
Multi-agent modelling

• Rather than predicting what profit-maximising
  firms will do, lets find out with simulation
• Define profit maximisers in terms of behavior
  rather than calculus
• instrumental profit maximisers
• Try something (e.g., increase output)
• If profit increases, do same again
• If profit falls, reduce output
• Model single market, demand curve
• No assumptions about knowing/applying calculus,
  etc.
• Just computer programming
• Give computer precise instructions
• See what happens!

102
Multi-agent modelling

• Basic idea
• Define demand curve cost functions
• Create random list of initial outputs for n firms
• Work out initial price given sum of outputs
• Create random list of variations in output for n
  firms
• FOR a number of iterations
• Add variation to output of each firm
• Work out new price level
• FOR each firm
• Work out whether profit has risen or fallen
• IF rose, keep going the same way ELSE
• IF fell, reverse direction
• See whether output converges to Cournot or Keen
  prediction

103
Multi-agent modelling

• P(Q)100- 1/100000000 Q
• C(q) 100000000 50 q

• Cournot/Game theory prediction firms equate MR
  MC

• Keen profit maximisation prediction firms
  produce where MRMC equals (n-1)/n times P-MC

104
Multi-agent modelling

• Start with randomly allocated list of outputs by
  ten firms

• Random initial quantity between Cournot Keen
  predictions for each firm

• Initial outputs

• Next step work out initial profits

105
Multi-agent modelling

• Work out vector of changes in output (much
  smaller amounts than the initial output so that
  firms won't end up producing negative amounts)
• 6 firms reduce output and 4 increase

• Aggregate output drops a bit

• Next what are new profit levels?

106
Multi-agent modelling

• Curiousity point some firms lose profit by
  reducing output others increase!

• Firms 0-4 decreased output saw profit fall

• Firm 6 decreased output saw profit rise

• Cause elasticity interactions between size of
  aggregate price change size of individual
  output change

107
Multi-agent modelling

• Firms 0-4 saw profit fall, so they will alter the
  direction of their output changes
• Firms 4-9 increased profit, so they continue in
  the same direction
• If profit rose, this function returns 1 if it
  fell -1

• This is multiplied by the dq amounts and added to
  second period output to work out third period

108
Multi-agent modelling

• The entire program

Random number generator
Random initial outputs
Random change amounts
For r iterations
Calculate market price
Change outputs
Calculate new market price
For each firm
Change direction if profit has fallen
109
Multi-agent modelling

• Result for 400 firms

• Converges towards Keen rather than Cournot
• BUT doesnt quite reach it apparent complex
  interaction effects between firms

110
Multi-agent modelling

• Cant avoid programming in multi-agent work
• Need to learn program structures
• FOR loops
• IF ELSE
• Object orientation
• Effort, but interesting results
• Slightly modified program

111
Multi-agent modelling

• Similar outcome

• But modified program stores results for each firm
  at each time step for each industry structure
  (number of firms)

• Sample run of 3 firms plus average for all firms
  in 100 firm industry shows one feature of
  multi-agent modelling
• Competition (to my equilibrium) as emergent
  property
• Individual firms dont converge the average does

112
Multi-agent modelling

• Individual firms follow very different strategies
  despite identical costs simple behaviour
• Aggregate outcome matches my prediction, but as
  emergent property of the group rather than result
  of successful individual profit-maximising

• Even more curious competitive result appears
  to depend on degree of dispersal of output
• Not the number of firms

113
Testing divergence

• Program iterates over standard deviation of dq
  from 1 to dispersal of Cournot firm output
  level

114
Nope, hes still dead!

• Convergence to Cournot a function not of number
  of firms, but of dispersal of dq!
• Sample run with 50 firms and increasing dispersal

115
Goodbye to the totem of the econ

• Neoclassical religion teaches perfect
  competition good, monopoly bad
• But maths wrong
• results contradicted by multi-agent modelling
  (MAM)

• So MAA powerful
• But there are problems

116
Multi-agent modelling

• Difficult to do
• Have to know how to write computer programs
• Sophisticated knowledge needed
• Object oriented concepts
• We dont actually know what agents do!
• Tiny variations in micro behaviour (e.g., change
  in dispersal) can have major impacts on macro
  behaviour
• What we observe in economic statistics is
  macroeven at level of single industry
• So MAM difficult to do in general
• Works best with well-defined problem
• Another example Ormerods Schumpeterian model of
  competition

117
The motivation

• Many old monopolies/state enterprises being made
  competitive
• Entry deregulated
• Publicly owned assets privatised
• Success of policy judged according to
  conventional economics
• IF many firms enter AND original monopolist loses
  dominance THEN competitive
• The market works
• ELSE IF new entrants fail and monopolist remains
  dominant THEN uncompetitive
• The monopolist is exploiting its power
• Pro-competition regulations used to control
  monopolist, force lower market share, etc.

118
The motivation

• BUT many monopolists complain
• Have reduced prices/increased quality
• Competition fierce
• Failure of new entrants natural part of
  competition
• Ormerods approach produce computer model of
  industry with
• Differentiated firms (offering different
  price/quality combinations)
• Differentiated consumers (different price/quality
  tradeoffs)
• See what evolves
• IF instrumental outcome (price/quality) poor THEN
  industry uncompetitive
• IF outcome good then competitive
• Analyse correlation between standard taxonomic
  view of competition instrumental view

119
A Schumpeterian model of an industry

• Conventional micro models competition with
• Homogeneous product
• No quality differences between firms
• No technical change
• Quality costs constant
• Rising marginal costs and falling marginal
  revenue
• Schumpeter emphasises
• Differentiated products
• Quality differences between firms too
• Technical change
• Driving force of model/economy explanation for
  profits
• Shape of costs irrelevant when discontinuities
  apply
• innovator has lower costs, better quality than
  rivals

120
A Schumpeterian model of competition

• Archetypal industry telecommunications, post
  office
• Starts as monopolised industry
• Deregulation allows new firms to enter
• Conventional expectation competition will
• Drive price down quality up
• Result in original monopolist losing market share
• Result in many firms in industry
• Actual results
• Price often driven down
• Quality generally up (but sometimes reliability
  problems e.g., electricity in California,
  Queensland)
• BUT frequently also
• Original monopolist remains dominant (Telstra)
• Many entrants fail, industry remains concentrated

121
A Schumpeterian model of competition

• Regulators often claim negative outcomes mean
  ex-monopoly abusing market power
• Telstra v Optus, Qantas v Virgin
• Ex-monopolies often claim outcome evidence that
  industry competitive
• We cant help it if were better than the new
  guys
• What is the truth?
• Anti-competitive behavior? or
• Thats just how the market works?
• Ormerods approach model functionally
  competitive industry Rapid innovation in costs
  quality
• Is there a correlation between outcome (low price
  high quality) and structural picture of
  competition (lots of small firms)?

122
A Schumpeterian model of competition

• multi-agent modelling approach
• Define artificial agents
• Consumers who seek best price/quality
  combination
• Producers who seek most effective price/quality
  combination for gaining market share
• Run simulation and see what happens
• Model
• 1000 consumers
• Each has different linear preferences for price v
  quality
• Monopolist has 100 of market (1000 customers) at
  start
• By definition, a monopolist has a sales network
  which connects it to all consumers in the
  particular market.

123
A Schumpeterian model of competition

• New entrants come in offers are known by
  (randomly decided) fraction of consumers
• Consumers can only buy from those companies of
  whose product they are aware. The phrase 'sales
  network' in this paper means the set of
  connections from a firm to consumers.
• consumers on the network of firm fi are both
  aware of the offer from firm fi and are willing
  to consider buying from it.
• First new firm might have sales network of (e.g.)
  34 of market (340 customers)
• Sales network held constant during simulation

124
A Schumpeterian model of competition

• There are three obvious reasons why new firms in
  the market do not have potential access (in
  general) to all consumers, which can obtain
  either singly or in combination. First, the
  regulator could impose restrictions so that, for
  example and purely by way of illustration from
  the telephone market, a new entrant could be
  permitted to offer international calls but not
  domestic ones. Second, the marketing strategy of
  the firm may be such that not all consumers are
  aware that the firm is making an offer in the
  market. In reality, marketing strategies vary
  widely in effectiveness, and this is reflected in
  our model. Third, the firm itself may
  deliberately target only a small percentage of
  consumers. In the context of British land line
  phone calls, for example, several firms now
  specialise in offering cheap calls to India, say,
  or to the United States. (8)

125
A Schumpeterian mo