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How to Be Nonlinear
Steve Keen
The basics
• Can‘t do nonlinear analysis without mathematics
– Nonlinearity banishes ―ceteris paribus‖
– Feedbacks too complicated to keep in mind verbally
• Though some aides to this—e.g., influence diagrams:
• Greenhouse gas rise
+
Absorption
of solar
radiation
+ • Causes temperature rise
• Causes fall in ice
Change in
solar Change in • Causes fall in reflection
temperature
reflection • Causes increase in
- Change in ice
area
- absorption of sun energy
– Positive feedback loop
• Mathematical methods provide means to quantify this
qualitative causal loop…
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
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
– Gaussian—presumes one ―atom‖ doesn‘t affect
others;
– Proper distribution ―fractal‖—one ―atom‖ does
affect others.
• Nature of ―differential‖ maths very different to
algebraic/differentiation you‘ve done to date
The basics
• Linear algebraic and differentiation problems normally
soluble
• Nonlinear differential equation problems normally
insoluble
• Summarising solvability of mathematical models (from
Costanza 1993: 33):
Linear Non-linear
One Several Many One Several Many
Equations equation equations equations equation equations equations
Most essentially
Algebraic economics
trivial easy impossible very difficult very difficult impossible
here
I work here
Ordinary essentially
Differential easy difficult impossible impossible
very difficult impossible
Partial essentially
Differential difficult impossible impossible impossible impossible impossible
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:
Lorenz‘s Butterfly
• An example: Lorenz‘s stylised model of 2D fluid flow
under a temperature gradient
• Lorenz‘s model derived by 2nd order Taylor expansion
of Navier-Stokes general equations of fluid flow. The
result:
x displacement
dx
a y x
dt
dy
b z x y y displacement
dt
temperature gradient
dz
xy cz
dt
• Looks pretty simple, just a semi-quadratic…
• First step, work out equilibrium: (try it now!)
Lorenz‘s Butterfly
dx y x,a 0
a y x 0
dt z b 1, b 1
dy
b z x y 0
dt x y b 1 c
dz x y z 0
xy cz 0
dt b 1 c
x
• Three equilibria result (for b>1):
y b 1 c
• Not so simple after all! But z
b 1
hopefully, one is stable and x 0
the other two unstable…
y 0
• Eigenvalue analysis gives the z 0
formal answer (sort of …) b 1 c
• But let‘s try a simulation x
first … y
b 1 c
z
b 1
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
• We‘ll try simulations with realistic parameter values,
starting a small distance from each equilibrium:
a 5 x 3 .7 4 2 3 .7 4 2 0
b 15 So that the y 3 .7 4 2 , 3 .7 4 2 , 0
c 1 equilibria are z 14 14 0
Lorenz‘s 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
Lorenz‘s 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
Lorenz‘s Butterfly
Lorenz's Strange Attractor
One small step for a butterfly, one enormous flap for mankind... Tiny error
8 in initial
6 readings
4
leads to
enormous
X displacement
difference
2
-0
in time path
-2
of system.
-4 And behind
-6 the chaos,
-8
strange
0 10 20 30 40 50 60 70 80 90 100 attractors...
Time
Lorenz‘s Butterfly
Lorenz's Strange Attractor
X,Y and Z displacement
Lorenz‘s 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.
Lorenz‘s Butterfly
• To understand systems like Lorenz‘s, 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 we‘ll
simulate)
• Initial examples non-economic (typical maths ones)
• Later we‘ll consider some economic/finance applications
before building full finance model
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 universe‘s 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
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
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 what‘s going
on in the insoluble set
– Hence, a crash course in ODEs, with some refreshers
on elementary calculus and algebra...
From Differentiation to Differential…
• You know to handle equations of the form
dy
Dependent variable f x Independent variable
dx
• Where f is some function. For
dy
sin x
dx
example dy
dx dx sin x dx
y cos x c
• On the other hand, differential equations are of the
form
dy • The rate of change of y is a function of its
f x, y value: y both independent & dependent
dx
• So how do we handle them? Make them look like the
stuff we know:
From Differentiation to Differential…
• The simplest differential equation is
dy
y (we tend to use t to signify time, rather than x
dt
for displacement as in simple differentiation)
• Try dy
y Divide both sides by y
solving dt
this for dy
yourself: dt 1
y
A trick :
d
ln y 1
dy
Rewrite the equation in this form :
dt y dt
d
ln y 1 Integrate both sides w.r.t. t :
dt
dt ln y dt
d
ln y 1dt t c Continued...
From Differentiation to Differential…
Because log of a negative ln y t c Take exponentia ls
number is not defined y e
tc
e e e C
t c t
Because an exponential ye
t
C
is always positive
y C e Exponentia l growth
t
• Another approach isn‘t quite so formal:
From Differentiation to Differential…
• Treat dt as a small quantity
• Move it around like a variable
• Integrate both sides w.r.t the relevant dy
y
―d(x)‖ term dt
– dy on LHS dy
dt
– dt on RHS y
• Some problems with generality of this dy
approach versus previous method, but OK dt
for economists & modelling issues y
ln y t c
y Ce
t
• So what‘s the relevance of this to economics
and finance? How about compound interest?
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:
dy s i p y dt p y dt Divide by y & Collect terms
Integrate
dy
s i p p dt
y
dy
y
s i p p dt
ln y s i p p t c Take exponentials y C e s i p p t
From Differential Equations to Finance
• Under what circumstances will our moneylender‘s assets
grow?
– C equals his/her initial assets: Known as ―eigenvalue‖;
tells how much the equation
y t C e
s i p p t
is ―stretching‖ space
y 0 C e
s i p p 0
C e C 1 C
0
• The moneylender will accumulate if the power of the
exponential is greater than zero:
t
If s i p p 0 then e
as t
• The moneylender will blow the lot if the power of the
exponential is less than zero:
t
If 0 then e 0 as t
Back to Differential Equations!
• The form of the preceding equation is the simplest
possible; how about a more general form:
Same basic idea applies: dy
f t y
dt
dy
f t dt
y
ln y f t dt
f t dt
y C e
• f(t) can take many forms, and all your integration
knowledge can be used…
• An example: compound interest
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
dA rA dt
An Example
• Work out the solution for A:
dA r Adt
dA
r dt
A
dA
A
ln A r dt r t c
r c r
A e Ce
t t
So what is the value of C? Work it out:
An Example
t
A Ce
rt
C
A e 1
0 C $
C1
r0
• Now let‘s 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
An Example
A
.
r 02 40
20002000
1e
2000 e1 1e
• Get out the calculators: what is this in decimal
format?
17
8370195
10
2.35385266
$ 6,837,019,
500
235,385,26
$235million
million
or million dollars
• How much gold is that at, say, $300 an
ounce? e .oz 2.224 1013 kg
40
300
• So how much space would that much gold
occupy? (Gold weighs 19,300 kg per cubic
metre)
An Example
A ( 2000 ) .
oz That’s 1.15 billion
300 9
1.152514691793 10 m
3 cubic metres
Gold density of gold
• So how large is that exactly... say, compared to the
volume of the earth? (The earth‘s radius is 6370 km)
4. . 3 21 3
V( R) p R V Earth radius 1.083 10 m
3
A ( 2000 ) .
oz So it’s not that big; 3
3 . .
300 R( V) Vm
just how big is it? 4 .p
Gold density 12
1.064 10 A ( 2000 ) .
oz
V Earth radius 300 2
R 650.408 m
Gold density
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 hasn‘t been one of continuous
accumulation!
– Current expected yields (4-6% p.a. min.) unsustainable…
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 dy
F t , y
dt
d
• And work on it till it is in the form t , y 0
dt
• Integration of this (with respect to t) yields t , y c
• The function 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 ?
– The answer is, not many!
Why most ODEs can‘t 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.
dy
• We start with a general ODE: N t , y M t , y or
dt
dy
M t , y N t , y 0
dt
Why most ODEs can‘t be solved
• Can this be put into the integrable form?
– Only if dy d
M t , y N t , y 0 t , y
dt dt
• The RHS of this can be expanded using the chain rule
for partial differentiation:
d dy
t , y
dt t y dt
• This lets us equate M and N to the partial derivatives
of :
• But this immediately imposes M t , y
t
conditions on the forms that M and N
can take:
N t , y
y
Why most ODEs can‘t be solved
• In (partial) differentiation, the order of
differentiation is irrelevant. Thus
2 2
t y y t
• 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:
2
• So, for a valid M and N to exist, it
M t , y must be true that
t y y
2 2
M t , y N t , y
2
N t , y t y y y t t
y t t
Why most ODEs can‘t be solved
• This condition will be true of the general relation
dy
N t , y M t , y or
dy
dy
M t , y N t , y 0
dy
• 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 it’s worse for higher order ODEs
Why that‘s not a problem anymore…
• The bad news…
– Incredibly hard work to massage minority of problems
into soluble forms
– Worse news
• Most real world problems can‘t 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
Why that‘s 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:
dy
F t , y
Rate at which stock
dt is a (often
y changes in volume complicated)
function of vessel
y‘s current volume
• y can be a vector of variables: a ―coupled‖ ODE
• No problem with modern computer mathematics software
– Difficulty lies in thinking dynamically…
An example…
• With (insincere…) apologies to those who‘ve 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‖ doesn‘t change
essentially barter nature of model
– From n to n+1 commodities; big deal!
• Instead, true money cannot be a commodity:
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)
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 can‘t
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, can‘t have C use its own tokens to
be a buyer
• Like paying for goods with ―IOU‖s
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 sided—buyer,
seller, banker. Banks are an essential aspect of
capitalism:
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:
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 B‘s 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
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
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…
Initial conditions
• Starting position is:
Bank Assets & Liabilities
Time Assets Liabilities
Firm Loan Firm Deposit Banker Worker Deposit
(FL) (FD) Deposit (WD)
(BD)
Initial 0 0 0 0
values
• Stage one: bank extends loan of L to capitalist:
Bank Assets & Liabilities
Time Assets Liabilities
Firm Loan Firm Deposit Banker Worker Deposit
(FL) (FD) Deposit (WD)
(BD)
Start of L L 0 0
loan
• Stage two: Loan involves obligations:
Stage two: obligations initiated by loan
• Loan obliges
– capitalist to pay interest on FL balance
– bank to pay interest on FD balance
Bank Assets & Liabilities
Time Assets Liabilities
Firm Firm Deposit Banker Worker Deposit
Loan (FL) (FD) Deposit (WD)
(BD)
Obligations +rL FL +rD FD 0 0
initiated by
loan
• Only sources of funds are Deposit Accounts
– FD for capitalist
– BD for bank…
– Now we‘re modelling flows of money into & out of the
stocks FD, BD, WD
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
Bank Assets & Liabilities
Flows Assets Liabilities SAM
Firm Firm Banker Worker Deposit Sum
Loan (FL) Deposit Deposit (WD)
(FD) (BD)
Interest flows +rL FL +rD FD +rL FL 0 0
initiated by - rL FL=0 - rL FL - rD FD
loan
• 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 stage three
• As a system of
Given Initial values Flow dynamics
d
FL ( t) = rL FL ( t) rL FL ( t )
equations, this is:
Firm loan account FL ( 0) = L
dt
d
Firm deposit account FD ( 0) = L FD ( t) = rD FD ( t) rL FL ( t)
dt
d
dt
FL 0 d
Bank deposit account BD ( 0) = 0 BD ( t) = rL FL ( t) rD FD ( t)
dt
d
dt
FD r D FD r L FL Worker deposit account W D ( 0) = 0
d
W D ( t) = 0
dt
d
dt
B D r L FL r D FD FL FL
FD FD
d
dt
WD 0 : Odesolve
BD
, t , Y
BD
W W
D D
• Using L=100, rD=3%, Circuit Model Step One: Interest payment only
rL=5%
100 100
Firm Loan
Firm Deposit
Account Balances
• Simulating in Mathcad:
Bank Deposit (RHS)
Worker Deposit (RHS)
• All money transferred
50 50
to BD after 30.5 years
• But model incomplete… 0
0 5 10 15 20 25 30
0
Time
FD ( Y) 0 BD ( Y) 100 W D ( Y) 0 FD ( Y) BD ( Y) W D ( Y) 100
Stage four: using the borrowed money
• Money borrowed to finance production:
– Workers hired & paid w FD:
Bank Assets & Liabilities
Flows Assets Liabilities SAM
Firm Loan (FL) Firm Deposit Banker Deposit Worker Deposit Sum
(FD) (BD) (WD)
Wage flow to initiate production -w. FD +w. FD 0
– Workers earn interest on balance in WD:
Interest income flows from - rD. WD +rD. WD 0
wages
• Stage five: workers & bankers buy goods from capitalists
Flows from sale +w.WD + b.BD -b.BD -w.WD 0
Complete model:
• Whole model is:
Bank Assets & Liabilities
Flows Assets Liabilities SAM
Firm Firm Banker Worker Sum
Loan Deposit Deposit Deposit
(FL) (FD) (BD) (WD)
Interest
flows +rD.FD +rL.FL
0 0 0
initiated - rL.FL - rD.FD
by loan
Wage flow -w. FD +w. FD
to initiate 0
production
Interest
income
- rD. WD +rD. WD 0
flows from
wages
Flows +w.WD + 0
-b.BD -w.WD
from sale b.BD
• Equations of motion read down the columns: e.g., FD:…
d
dt
FD rD FD rL FL w FD w W D b B D
Complete model:
• Complete set of equations: FL 0
d
dt
dFL r F Lr F w F w
FD W D b BD
• Simulation shows Circuit
F
d t D D
L L D
dFD FD
d t BD: r L FL r D FDY r D W D b BD
―works‖
Odesolve ,t,
BD BD
WW w F W r
d
W w W D
– Capitalists can d t D D D D D D
borrow money, pay
Basic Circuit Model
100 15
interest on it, &
Account Balances
operate indefinitely 95 10
– Activity continues Firm Loan
―forever‖ with single
90 5
Firm Deposit
Bank Deposit (RHS)
injection of money… 85
0 5 10 15
Worker Deposit (RHS)
20 25 30
0
• But these are just bank Time
account balances… FD ( Y) 85.83 BD ( Y) 4.255 W D ( Y) 9.915 FD ( Y) BD ( Y) W D ( Y) 100
– What about incomes?
Income dynamics
• Worker & bank income easy:
– Wages are the flow w FD w FD (Y ) 257.49
– Gross interest is the flow rL FL: rL FL (Y ) 5
• What about profits?
– Derive from w:
– w is part of net surplus from production accruing to
workers
– Surplus constituted by:
• Worker-capitalist split ([1-s:s]—sums to 1)
• Rate of turnover from M to M+
– Signified by P
• So we have w 1 s P conversely: p s P
• Profits are p FD (Y ) s P FD (Y ) 171.66
Income dynamics/debt repayment
• Confirming from simulation program:
Wages w F D ( Y ) 257.49
Interest rL F L ( Y ) 5
w
s : 40 % P :
(1 s)
Profit s P F D ( Y ) 171.66
Wages again ( 1 s ) P F D ( Y ) 257.49
• Yearly net income of 429.15 exceeds L by factor of four…
– Reflects turnover of capital—neglected 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‖
Model with repayment/growth
• Overall system still balanced:
Bank Assets & Liabilities
Time Assets Liabilities SAM
Firm Firm Banker Worker Income
Loan Deposit Deposit Deposit
(FL) (FD) (BD) (WD)
Repayment
-RL.FL -RL.FL -RL.FL
of debt
Relending
+LR.BR +LR.BR +LR.BR
of reserves
Bank Reserves
Time Reserve Account Capital
Repayment of debt RL.FL +RL.FL
Relending of reserves -LR.BR -LR.BR
• Final extension: growth
– Additional reserves/debt at rate FI
– Models Moore‘s ―Horizontalism‖
Model with Growth
• System is now ―dissipative‖
– Sum of SAM exceeds zero
Bank Assets & Liabilities
Flows Assets Liabilities SAM
Firm Loan Firm Banker Deposit Worker Deposit Income
(FL) Deposit (BD) (WD)
(FD)
Investment by
+FI.FD +FI.FD +FI.FD
firms
Bank Reserves
Time Reserves Capital
Investment by firms +FI.FD -FI.FD 0
SAM Sum
+FI.FD
• Accounts still balanced
– But ―Walras Law‖ violated in growing economy
• Sum of excess demands > 0
Model with Growth
d
F L L R B R R L F L FI F D
• Full dt
d
FD r D FD r L FL 1 s P F D w W D b B D L R B R R L F L FI F D
model dt
d
B D r L FL r D FD r D W D b B D
now is dt
d
W D 1 s P FD r D W D w W D
dt
d
dt
B R R L FL L R B R
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
But first a word from our saviours…
• Modelling wouldn‘t 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
Quick overview of Mathcad
• Program lets you type equations as you would write them:
1 1
1 ( n 1) 1 Q d P
1 ( n 1)
n P dQ
• Ugly, huh?
– Try reading it as a single line of unformatted text:
• =1/(1-(n-1)*)=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:
Quick overview of Mathcad
• Functions can be numerically simulated & graphed:
1 8 17
a : 800 b : C : 10 D : 10 E : 10 k : 1000000
10000000
2
2 b b n b D ( 2 b b n b D ) 4 [ E ( C a n a a ) ]
q max ( , n ) : n : 100
2 ( E n )
7
8 10
7
7 10
q ( , n) 6 10
7
max
7
5 10
7
4 10
0 0.2 0.4 0.6 0.8
Quick overview of Mathcad
• Many built-in functions
– & a simple (limited) programming language
• Key one for our purposes: Odesolve
– Arguments differential equations & initial conditions:
Given F L( 0 ) L F D ( 0) L B D ( 0) 0 W D ( 0) 0 B R ( 0) 0
d
F L( t ) F I F D ( t ) LR B R ( t ) R L F L( t )
dt
d
FD( t) b B D ( t ) rD F D ( t ) rL F L( t ) w W D ( t ) w F D ( t ) F I F D ( t ) LR B R ( t ) R L F L( t )
dt
d
BD ( t ) rL F L( t ) rD F D ( t ) b B D ( t ) rD W D ( t )
dt
d
W D( t) w F D ( t ) w W D ( t ) rD W D ( t )
dt
d
BR ( t ) R L F L( t ) LR B R ( t )
dt
Quick overview of Mathcad
• Function needs variable names, independent variable (t),
& number of time periods to simulate (Years)
– (Simulation in continuous time, not discrete)
FL F L • Result can be graphed, analysed…
FD F D Complete Circuit With Repayment
250 30
BD : Odesolve B D , t , Y
A ccount Balances
200
W WD
D 20
B B
R R 150
Firm Loan
10
Firm Deposit
100
Bank Deposit (RHS)
Worker Deposit (RHS)
50 0
0 10 20 30
Time
Quick overview of Mathcad
• Program includes some symbolic capabilities
– For example, system without repayment is
Given FL L
d
FL 0
dt
rD F D rL F L w F D w W D b BD 0
d
dt
FD r D FD r L FL w FD w W D b B D
rL F L rD F D rD W D b B D 0
d
dt
B D r L FL r D FD r D W D b B D
w F D rD W D w W D 0
d
dt
W D w FD r D W D w W D
• Equilibrium occurs when all FD BD W D FL
differentials equal zero L
L w r b r
D L
• FL remains at L with no FL
w rD w b rD
repayment
FD
Find L rD rL
factor
B
D b rD
• Sum of deposit accounts equal W
D
L w b rL
sum of FL
w rD w b rD
• Feed conditions into program &
ask for equilibrium solution:
And the competition
• Now many programs with these numerical symbolic
capabilities
– Mathematica
– Scientific Workplace
– Maple
– Scilab (free software—powerful 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
Coupled ODEs
• We‘ve just modelled a
– Fifth order
– Linear
– Set of coupled differential equations
• Goodwin‘s 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:
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
Predator-Prey Systems
• Rate of growth of fish is
– positive function of number of fish 1 dF
– negative function of the number of
a bS
F dt
sharks
• Rate of growth of sharks is
1 dS
– negative function of number of sharks
c d F
S dt
(starvation)
– positive function of the number of fish
• Together, a dF
aFbSF Can this
system: dt
dS be solved?
SdFS
c
dt
Predator-Prey Systems
• Well, yes; but it‘s the last 1100
nonlinear ODE we can
solve
Fish
1000
– any system with three
or more coupled ODEs 900
is insoluble
0 500 1000
Time
– first, a numerical
simulation: 102
101
Sharks
100
99
98
0 500 1000
Time
Predator-Prey Systems
dF
aFbSF
dt
dS
SdFS
c
dt
• 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
1 dF d
ln
F a b S
F dt dt
1 dS d
ln c d F
S
S dt dt
Predator-Prey Systems
d
ln F a b S
dt
• Notice how each
d ln F a b S dt variable is a function
d ln F a b S dt of the other:
ln F a b S t c
a b S t t
F C1 e Similarly,
c d F t t
S C2 e
Predator-Prey Systems
• What about the system‘s
equilibrium?
– How do you define it? dF
a F b S F 0
• When dF/dt=dS/dt=0 dt
b S a
– Is it stable or unstable?
a
S
• There are ways to work this out b
dS
(pertubation analysis: work out c S d F S 0
dt
the dynamics of behaviour a
d F c
short distance from equilibrium)
c
F
• It turns out that the equilibrium d
is neutral:
– neither attracts nor repels:
Predator-Prey Systems
• Generates a stable
―limit cycle‖:
– system orbits 1125
the equilibrium 1100
but never
converges to or 1050
diverges from it.
Fish
Z
n , 1 1000
• Such behaviour the
norm in complex 950
systems
900
875
98 99 100 101 102
98 Z 102
n,2
Sharks
Predator-Prey Systems
• Now an application of this to economics:
– Non-equilibrium predator-prey cycle can be derived
from Marx
• Check Ed‘s 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 wrong—dialectic
contradicts LTV…
• But ignoring that, dialectic when applied to
wages predicts cycles:
A predator-prey cycle in capitalism
• In capitalist, Exchange-Value of work brought to
foreground
– Exchange-Value of worker=subsistence wage
• Use-Value of worker in background: irrelevant to wage
– But Use-Value of worker to capitalist purchaser of
labour-time=ability 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?
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)
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)
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...
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 shareLow rate of
accumulationIncrease in unemploymentDrop in
wagesIncrease in accumulationIncrease in
employmentHigh wages share
• ―Phillips curve‖ part of Marx‘s logic (wage change a
function of the rate of unemployment)
• Goodwin built predator-prey model on this
foundation
– Try to work out a model:
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...
A predator-prey cycle in capitalism
• Level of output determines Y
L
employment a
• Differential equation of rate of 1 dw
L
w
w
change of wages determines wages w dt
N
,p 1w
Y W
• Output - Wages determines profits
• Profits determine investment
I I k
,
• Investment determines capital
dK
I
dt
• Capital determines output...
K
Y
v
• Can you see how to make a predator-prey system out
of this?
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)
1 dN
b
N dt
1 da
a dt
• Work out the differential equations for w and as
functions of themselves and each other…
A predator-prey cycle in capitalism
d d L
d t
t d N
1d d 1
LL
L w
Nt
d d
tN
1
L
1dY d a b
1
L
2 N N v
d
t d
N a Nt
1d Y a Ld
1 d1
Ya N N 1 w
1 This is w:
a
N
t
d
t
d N
d a b
t2
11
KYd
d 1 a
b
v
a vad
d
Nt a t
11
a
I L b
w wL W
w
a
Nv a
a aL Y
YW b
11
1w
b
d
Nav
dt v
Try same thing for w (it‘s easier!)
A predator-prey cycle in capitalism
dw
d W
Expand these
dt dt Y The end product is a
d w L
These cancel version of a predator-
dt L a
prey model:
d w
Apply chain rule 1d 1w
b
dt a dt v
1d w
1 d d 1
w
w w w dt
dt a
• Negative feedback from w to
a dt
1 1 d
w w w
a a
2 • a Positive feedback from to w
dt
w w
w 1 d
a – more complicated than basic predator-
a a dt
prey because of ―Phillips curve‖ relation
w w w
between rate of change of wages and
w w
level of employment
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
A predator-prey cycle in capitalism
• Simulation for given values of and b yields:
• Goodwin/Marx
0.977357 1
Basic Goodwin Growth Cycle
model thus
gives same
0.95 basic cycle as
biological
2
Z i
Z
1
i
0.9 predator-prey,
but for wages
0.85
share (income
0.831245
distribution) vs
0.8
0 2 4 6 8 10 12 14 16 18 20 employment
rather than
0 0 20
Z i
fish vs sharks
Time
Employment
Wages Share
A predator-prey cycle in capitalism
• As with biological model, trade cycle model traces out a
limit cycle:
• What causes Basic Goodwin Growth Cycle
this neither
0.977357 0.98
converging nor 0.975
diverging
behaviour? Employment
2
0.97
Z
– Nonlinearity
i
0.965
– Compare to a
linear model 0.96
2
with cycles: 0.95659
dy dy
a 2 c
b y0
0.955
0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9
dt dt 0.831245 1 0.899658
Z i
Wage Share
The importance of being nonlinear
• Characteristic equation is
a r
2
br c 0
• Roots are b ac
2
b 4
r
2a
• This bit causes
• General solution is of the form cycles
c
e cos 2 sin
t
c1
b t t
b
• If >0 then cycles get
infinitely large with time
• This bit • System must break down
– amplifies cycles if >0 (Tacoma bridge, Braun 1993:
– damps cycles if <0 173)
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
The importance of being nonlinear
• Number of fish
– positive function of number of
fish F (linear) dF
aFbSF
– negative function of F times S dt
(quadratic) dS
SdFS
c
• increasing fish+shark numbers dt
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 fish+shark numbers means this term
dominates linear death rate term
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
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…
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
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
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
Nonlinear investment function
Nonlinear investment
function means
desired (and
executed)
investment during
boom exceeds
profits
desired (and
executed)
investment during
slump less than
profits
Nonlinear Investment Function
Nonlinear investment function makes little change to nature of
basic model:
–Still closed cycle
But asymmetry much more obvious:
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
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!‖
• ―No it doesn‘t!
– It maximises profit by setting…
n 1
M R qi M C qi P Q M C q i
n
• Proving this to economists a bit like arguing with John
Cleese, so let‘s try a simulation
Multi-agent modelling
• Rather than predicting what profit-maximising firms will
do, let‘s ―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!
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
Multi-agent modelling
• P(Q)=100- 1/100000000 Q
• C(q) = 100000000 + 50 q
1 d
a : 100 b : P ( Q) : a b Q MR ( Q) : ( P ( Q) Q) MC : d
100000000 dQ
Demand Curve
100
Price • Cournot/Game theory
P ( Q) Marginal Revenue
prediction: firms
Market Price
Marginal Cost
equate MR & MC
MR ( Q)
50
MC
1 a d
0 qn
0 2 .10
9
4 .10
9
6 .10
9
8 .10
9
1 .10
10
n 1 b
Q
Industry Output
• ―Keen‖ profit maximisation
1 1 a d
prediction: firms produce where qn
MR–MC equals (n-1)/n times P-MC n 2 b
Multi-agent modelling
• Start with randomly allocated list of outputs by ten firms
• Random initial quantity
Q0 : round runif Firms , qK ( Firms) , qC ( Firms)
9
between Cournot & Keen
0
0 4.365·10 8 QC ( Firms) 4.545 10
predictions for each firm
1 3.988·10 8
2 3.866·10 8 9
QK ( Firms) 2.5 10
3 3.144·10 8
Q0 4 3.126·10 8
9
5 2.722·10 8 Q0 3.474 10
6 4.241·10 8
7 2.817·10 8
• Initial outputs 8
9
2.662·10 8
3.811·10 8
8
x : 0 , 10 .. 10 10
9
• Next step: work out Demand Curve
initial profits: 100
Demand curve
Profit0 : P
Q0 Q0 TC Q0
Marginal Revenue
Marginal Cost
0 Initial output level
0 6.559·10 9 50 Cournot prediction
1 5.985·10 9
2 5.799·10 9 Keen prediction
3 4.698·10 9
Profit0 4 4.669·10 9
P Q0 65.258
5 4.053·10 9
6 6.371·10 9
7 4.198·10 9 0
9 9 9 9 10
8 3.962·10 9 0 2 .10 4 .10 6 .10 8 .10 1 .10
9 5.715·10 9
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:
qC ( Firms)
dq : round rnorm Firms , 0 ,
Q : augment Q0 , Q0 dq
Firms
0 0
0 0 3.734·10 8 0 4.365·10 8
0 -6.31·10 7 1 3.012·10 8 1 3.988·10 8
1 -9.766·10 7 2 3.186·10 8 2 3.866·10 8
2 -6.803·10 7 1 3 1.777·10 8 0 3 3.144·10 8
dq 1.077 10
3 -1.368·10 8 8 Q Q
4 7.025·10 7 4 3.126·10 8
dq 4 -2.423·10 8 5 2.785·10 8 5 2.722·10 8
5 6.257·10 6 6 4.068·10 8 6 4.241·10 8
6 -1.734·10 7 7 3.617·10 8 7 2.817·10 8
7
8
9
7.998·10 7
3.15·10 8
1.162·10 8
dq Q 0 3.099 % 8
9
5.813·10 8
4.974·10 8
8
9
2.662·10 8
3.811·10 8
• Aggregate output drops a bit:
• Next what are new profit levels?
Multi-agent modelling
• Curiousity point: some firms lose profit by reducing
output; others increase!
1
P
Q 66.334
1 1 1
Profit1 : P
Q Q TC Q
0 0 0
0 5.999·10 9 0 -5.608·10 8 0 -6.31·10 7
1 4.82·10 9 1 -1.166·10 9 1 -9.766·10 7
2 5.104·10 9 2 -6.95·10 8 2 -6.803·10 7
3 2.802·10 9 3 -1.895·10 9 3 -1.368·10 8
Profit1 4 1.048·10 9 Profit1 Profit0 4 -3.621·10 9 dq 4 -2.423·10 8
5 4.449·10 9 5 3.953·10 8 5 6.257·10 6
6 6.544·10 9 6 1.733·10 8 6 -1.734·10 7
7 5.808·10 9 7 1.61·10 9 7 7.998·10 7
8 9.395·10 9 8 5.433·10 9 8 3.15·10 8
9 8.024·10 9 9 2.309·10 9 9 1.162·10 8
• 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
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:
0
0 -1
1 -1
2 -1
1 1 0 0 1 0
3 -1
sign P Q Q P Q Q TC Q TC Q 4 -1
5 1
6 1
7 1
8 1
9 1
• This is multiplied by the dq amounts and added to second
period output to work out third period…
Multi-agent modelling
• The entire program:
Cts ( f , r , s) : Seed ( s) Random number generator
Q0 round runif f , qK ( f) , qC ( f) Random initial outputs
qC ( f)
Random change amounts
dq round rnorm f , 0 ,
f
for i 1 .. r For r iterations
PThen P
Qi 1
Calculate market price
Qi Qi1 dq Change outputs
Calculate new market price
PNow P
Qi
for j 0 .. f 1 For each firm
dq j sign PNow Qi
j PThen Qi1 j tc Qi j tc Qi1 j dq j
Q Change direction if profit has fallen
f : 400 r : 250 Output : Cts ( f , r , 10)
Multi-agent modelling
• Result for 400 firms:
f : 400 r : 250 Output : Cts ( f , r , 10) i : 0 .. r 1
9
5 .10
Aggregate Output
9 Cournot
4.5 .10 Keen
9
4 .10
Quantity
9
3.5 .10
9
3 .10
9
2.5 .10
0 50 100 150 200 250
Iterations
• Converges towards Keen rather than Cournot…
• BUT doesn‘t quite reach it; apparent complex interaction
effects between firms…
Multi-agent modelling
Fi rm s : Se e d ( ran d )
• Can‘t avoid for i fir m sm in , fir m sm in fir m sst ep s .. fir m sm a x
programming in multi- Q
0
ro un d ru ni f i , q K ( i) , q C ( i) if i 1
agent work q C ( i) othe r wise
– Need to learn p
0
P
Q0 , a , b
if i 1
program
P q C ( i) , a , b othe r wise
structures q C ( i)
dq ro un d rn or m i , 0 , if i 1
• FOR loops 1 00
q C ( i)
• IF ELSE 1 00
ot he r wise
• Object for j 0 .. ru ns 1
Q dq
orientation
Q
j 1 j
P
Qj 1 , a , b if i 1
• Effort, but
p
j 1
interesting results… P Q j 1
,a,b ot her wise
– Slightly modified dq
sign p Q p Q tc Q
j 1
, i t c Q , i dq
j 1 j j j 1 j
program: F
j , i 1
Q
j
F
Multi-agent modelling
M ark et o u tco m e an d m o d el p red ictio n s
• Similar outcome:
9
4.5 1 0
S im ula tion
• But modified
K een
N eoclassical
program stores
9
41 0
A ggregat e output
results for each
firm at each time
9
3.5 1 0
step for each
industry 31 0
9
structure
(number of firms) 2.5 1 0
9
0 20 40 60 80 100
N umber of firm s
• 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 don‘t converge; the average does…
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
3 rando mly chosen firm s & average outcom e • Even more
curious:
7
4 .5 10
Fir m 1
―competitive‖
Fir m 2
Fir m 3
M ean
result appears to
7
4 10
Neoc lass ica l
Kee n
depend on
F irm 's output
3 .5 10
7
degree of
dispersal of
3 10
7
output…
• Not the number
2 .5 10
7
0 20 0 40 0 60 0 80 0 1 1 0
of firms
3
Iter ations
Testing divergence
• Program
F q .K( 1 , a , b , C, D , E)
0, 0
q .K( 1 , a , b , C, D , E)
iterates over
F
1, 0
for i 0 , dispersal .steps .. dispersal 1
standard for j 0 .. rand 1
deviation of Seed ( j 1)
Q round runif firms , q .K( firms , a , b , C, D , E) , q .C( firms , a , b , C, D , E)
dq from 1%
0
p P Q0 , a , b
to dispersal
0
dq round rnormfirms , 0 ,
1 i
% of Cournot
q .C( firms , a , b , C, D , E)
100
firm output for k 1 .. runs
Q Q dq
level
k k 1
p P
k
Qk , a , b
k
dq sign p Q p Q tc Q , firms , C, D , E, k tc Q , firms , C, D , E, k dq
k k k 1 k 1 k 1
Qk Qk1 Qk2 Qk3
Q.end
j 4
F
i, 0
mean Q.end
F stdev Q.end
i, 1
Q.end 0
F
―Nope, he‘s 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:
Market Output vs Dispersal (Rising Marginal Cost)
6 .10
9
Cournot
Keen
Average Final Market Output
Average of Simulations
5 .10
9
+/- 2 St. Dev.
4 .10
9
3 .10
9
2 .10
9
0 2 4 6 8 10 12 14 16 18 20
Per cent of qC
Goodbye to the ‗totem of the econ‘
• Neoclassical religion 1200
Market Functions, Predictions, Outcomes: 50 firms
teaches ―perfect Price Function
D em and, M arginal Revenue, M ar ginal Cost
competition good,
Marginal Revenue
1000
Marginal Cost
monopoly bad‖ 800
Cournot 50 Firms
Keen
– But maths wrong; 1% dispersal
10% dispersal
– & results
600
20% dispersal
contradicted by 400
multi-agent
modelling (MAM)…
200
0
2 . 10 4 . 10 6 . 10 8 . 10 1 . 10
9 9 9 9 10
0
Market Quantity
• So MAA powerful;
• But there are problems…
Multi-agent modelling
• Difficult to do
– Have to know how to write computer programs
• Sophisticated knowledge needed
– Object oriented concepts…
– We don‘t 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 macro—
even at level of single industry
• So MAM difficult to do in general
– Works best with well-defined problem
• Another example: Ormerod‘s Schumpeterian model of
competition…
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.
The motivation
• BUT many ―monopolists‖ complain
– Have reduced prices/increased quality
– Competition ―fierce‖
– Failure of new entrants natural part of competition
• Ormerod‘s 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
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
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
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 can‘t help it if we‘re better than the new guys…‖
• What is the truth?
– ―Anti-competitive behavior‖?; or
– ―That‘s just how the market works‖?
• Ormerod‘s 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)?
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.‖
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
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)
A ―Schumpeterian‖ model of competition
• Initial (monopolist) price highest (1) and quality lowest
(also 1 for convenience)
• New entrants offer different (randomly allocated)
price/quality combination between best (0,0) & (1,1)
• Consumers can switch if new entrant‘s deal more
appealing to them than current deal
– Switch probable only: each consumer has (randomly
allocated) propensity to switch
• Models real-world uncertainties
– Costs in switching (ignored in conventional
theory)
– Uncertainty re reliability of new supplier
– Heterogeneity of product means ―new deal‖
might not be relevant to one consumer
– Inertia: too many other things to do…
A ―Schumpeterian‖ model of competition
• ―product offers … are not perfect substitutes… the
lowest (p,q) supplier may specialise in an offer which
is not very important to a given consumer. Someone
who makes only local phone calls will not be interested in
a firm which provides only cheap international calls.
Second, … consumers may have doubts about the
reliability of a previously unknown supplier. … there
may be costs involved in switching. To take an obvious
example, if changing suppliers involved having to change
telephone number – staying with the telecomm example -
for most people the savings on price would have to be
considerable to offset the inconvenience involved …
consumers may simply exhibit inertia and stay with
their existing supplier, perhaps because the savings
involved are small.‖ (9)
A ―Schumpeterian‖ model of competition
• 40 iterations (like 40 quarters = 10 years)
– Possibility of new entrant(s) every quarter
– At each iteration, each firm can alter price/quality
offering to try to improve attractiveness to market
• Firms desire to move to most popular (on average)
price/quality combination
• Probability, not certainty of switch:
– ―The ability of the firm to do achieve the
desired (p,q) depends on the firm‘s flexibility
level ji.‖ (10)
– Models variations in internal flexibility, etc.
A ―Schumpeterian‖ model of competition
• After new price/quality offers made, consumers can
decide to switch again:
– ―Consumers then review their choice of suppliers given
the revised set of (p,q) from existing suppliers, and
given the (p,q) offered by new entrants (if any) in that
period.‖ (10)
• Process causes ―jiggling‖ of price/quality offers & market
shares over time
– Average price & quality tend to rise
– What happens to structure of industry?
• Is price lower, quality higher when market shares
small?
– Simulation run 1,000 times to see overall tendencies
A ―Schumpeterian‖ model of competition
• Price outcome:
– Not one single ―market‖ price
– Each firm offers different price
– Average price tends towards competitive (0) outcome:
• ―The single most
frequently observed
outcomes for the
market price is in the
range 0.05-0.10. In
other words, price
does fall to a level
close to the minimum
which is feasible.‖
• However occasionally
price is high…
A ―Schumpeterian‖ model of competition
• ―The mean level of market price after 40 periods is
0.145, with a minimum of 0.00007 and a maximum of
0.650. The inter-quartile range [from 25%-75% of
outcomes] is between 0.057 and 0.206.‖ (17)
• Quality behaves similarly: quality rises (tends towards 0)
• So outcomes ―competitive‖; what about structure?
• Not ―competitive‖, according to conventional theory
– Monopolist hangs on to substantial share of market
• ―Quite frequently, the incumbent monopolist retains
a very high market share… The average market
share of the monopolist after 40 periods is 53.5
per cent, with a minimum of 3.4 and a maximum of
100 per cent. The inter-quartile range is wide,
between 32.1 and 75.9 per cent.‖ (18)
A ―Schumpeterian‖ model of competition
• Important factor in eventual market share is
―flexibility‖ of monopolist—ability to match best
price/quality offer of new entrants
A ―Schumpeterian‖ model of competition
• ―A high level of flexibility is by no means a guarantee of a
high eventual market share, but the simple correlation
between the two variables is 0.712.‖ (19)
A ―Schumpeterian‖ model of competition
• Many new entrants ―fail‖ in that market share becomes
zero
• ―Grim‖ outcome in terms of
standard theory but very
similar to reality:
• ―The mean number of firms
is 8.2, so that on average
almost 12 out of the 20
firms fail completely i.e.
have no sales at all. This
seems compatible with the
outcomes which are
observed in practice (see,
for example, Carroll and
Hannan (2000)).‖ (20)
A ―Schumpeterian‖ model of competition
• Market share outcome ―uncompetitive‖ on standard
theory: • But results fit data:
• ―a good approximation to
the size distribution of
the largest 8 firms after
40 periods is provided by
a power law. [explained
later] Axtel (2001) shows
that this a general
characteristic of the
distribution of firm sizes
in the United States.‖ (21)
• ―A log-log least squares fit of average market share in
Figure 6 on the rank of the firm by market share … gives
an R2 of 0.983 and an estimated exponent of –2.09‖ (21)
A ―Schumpeterian‖ model of competition
• Would standard ―competition policy‖
– Reduce share of ex-monopoly/largest firm
• improve outcomes?
• ―it is often thought that reducing the market share of a
monopolist (for example by competition policy) will ensure
lower prices.‖ (23)
• Regression shows almost no relationship between
monopolist share of market and market price:
• ―We can examine whether there is any connection here
between the eventual market share of the monopolist and
the prevailing market price… The simple correlation
between the two is –0.014.‖ (23-24)
A ―Schumpeterian‖ model of competition
• Effectively no correlation between market share of
monopolist & market price:
• What about market price &
number of firms?
• ―standard economic theory
implies a relationship
between the equilibrium
market price and the
number of firms in the
market. The fewer the
number of firms, the more
the price will be above the
level which just covers
both costs and a normal
rate of profit.‖ (24)
A ―Schumpeterian‖ model of competition
• ―Figure 8 below plots the relationship between the
eventual market price and the number of firms in the
market. It is clear that there is little or no connection
between the two. The simple correlation is in fact 0.05.‖
• ―a very low price can
obtain with just one
or two firms in the
market. Equally, a
relative high price
may exist with 10 or
even 15 firms in the
market.‖ (26)
• Compare this to
Cournot oligopoly
theory:
A ―Schumpeterian‖ model of competition
• ―The key difference between our model and that of, say,
the Cournot model is that with the latter there is a
deterministic relationship between the number of firms
in the market and the market price which obtains. The
more the number of firms, the closer the price becomes
to the theoretical level of a perfectly competitive
market. In our model, in any particular solution of it
there is no necessary connection at all between price and
the number of firms… This difference between the
Cournot model and our own is much more important than
any similarities.‖ (26)
• However, despite this empirical difference, outcome of
model ―better than‖ conventional theory
A ―Schumpeterian‖ model of competition
• ―Purely by coincidence, given the average number of firms
which survive in 1,000 solutions of the model, the average
price across these solutions is very similar to that of the
in the standard Cournot model.
• On average, after 5 years there are 5.54 firms in total in
the market in the simulations of our model, rising to 8.21
after 10 years. Two widely used illustrations of the
Cournot model are with a linear and log-linear market
demand function, respectively. With a linear demand
function, the mark-up on cost is (1 +1/(N+1)), and with a
log-linear one it is (1 + 1/(N-1)). These imply, respectively,
a market price after 5 years which is 15 and 22 per cent
above cost. After 10 years the figures are 11 and 14 per
cent above cost.‖ (26-27)
A ―Schumpeterian‖ model of competition
• Conclusions:
– ―the market price generally falls from the level set by
the initial monopolist to close to the minimum which is
both technologically feasible and consistent with a
normal margin of profit…
– the market price is on average very similar to that
implied by the Cournot equilibrium given the average
number of firms with non-zero sales
– however, in any individual solution of the model, the
market price which eventually obtains is not really
influenced by the number of firms which remain in the
market
– the monopolist retains, in general, a substantial share
of the market
A ―Schumpeterian‖ model of competition
– judged on the conventional criterion of the
distribution of market shares, at any point in time the
market structure is, in general, anti-competitive. But
as the outcome on market price shows, the model is
highly competitive in any meaningful sense of the
word.‖
– ―the majority of new entrants fail, which seems to fit
empirical evidence
– the distribution of market share is approximated
closely by a power law, which again conforms with
empirical evidence.‖ (28-29)
A ―Schumpeterian‖ model of competition
• Implications for competition policy:
• The results of this approach to the issue should give
regulators and policy makers pause for thought when
considering contestable markets. For example, it is not
the case that a competitive market (in the sense of
having a competitive price), will necessarily have lots of
firms, or will have driven down the original incumbent‘s
market share. Further, although market share is often
used as an indicator – indeed as a primary indicator – of
the presence of monopoly power which may lead to anti-
competitive behaviour, these results show that this can
be seriously misleading. Finally, the existence of an
incumbent by itself does not necessarily tell us much
about whether the price is low or high and whether the
market is competitive or not.‖ (29)
A ―Schumpeterian‖ model of competition
• Ormerod/multi-agent model very different to
conventional economics
• Rather than single equations with simplifying assumptions,
computer program with many realistic assumptions
– some unrealistic ones can be altered later (e.g., no
change in firms‘ networks over time)
• Outcome comparable to actual statistics
– Majority of markets in USA dominated by top 8 firms
– Wide range of firm sizes in most industries
• Industries neither ―monopoly‖ nor ―oligopoly‖ nor
―competitive‖
Ormerod model
• Standard economic model a set of (unfortunately false!)
assumptions
– Rising marginal cost
– Homogeneous product, undifferentiated consumers
– Price competition only
• And mathematical maximisation equations
– (True formula) ―Maximise profit by setting gap
between marginal revenue & marginal cost equal to (n-
1)/n times gap between price and marginal cost‖
• Multi-agent model a computer simulation:
• Explaining the program:
Ormerod model
• (1) Creates arrays to store information about
– Consumers
• consumers(:,1) = rand(1000,1); % price/quality
weighting
• consumers(:,2) = rand(1000,1); % switch prob.
• consumers(:,3) = 1; % firm purchasing from
– Firms
• suppliers(1,1)=1; % Monopolist market penetration
• suppliers(1,2)=1; % Monopolist initial price
• suppliers(1,3)=1; % Monopolist initial quality
• suppliers(1,4)=mean(consumers(:,1))*suppliers(1,2) +
(1-mean(consumers(:,1))) *suppliers(1,3); %
Consumer rating of monopoly
• suppliers(1,5)=rand(1,1); % Monopolist flexibility
Ormerod model
• Network between them (which customers buy from which
firms)
– consfirm(:,1) = ones(1000,1); % All consumers
connected to monopolist
• Then for 40 iterations
– for k=2:40 % k=1 is when monopoly only firm
– IF there are less than 20 firms already
• if firms < 20
– THEN create new firm(s) (probabilistically)
• newfirms = round(1/5 + rand(1,1));
– Allocate initial properties of each new entrant:
Ormerod model
• for createnew = 1:newfirms
– suppliers(firms+createnew-1,1)=rand(1,1); % market
penetration; and also
• % Initial Price offered
• % Initial quality offered
• % Average consumer rating of initial offers
• % Firm Flexibility
– How many consumers on firm‘s network
• network_size = round(suppliers(firms+createnew-
1,1) * 1000);
• Each consumer then rates each firm according to own
preferences for price & quality:
Ormerod model
• for i=1:1000 % for each consumer
– Rate offerings of all firms
• offers = ( consumers(i,1) * suppliers(:,2) + (1-
consumers(i,1)) * suppliers(:,3) ) .*
full(consfirm(i,:)');
– Work out best offer
• [bestoffer,bestofferfirm] =
min(offers(find(offers)));
– ―Toss a coin‖ based on individual propensity to switch
• prob_change = sqrt(consumers(i,2)* rand(1,1));
– Switch if coin ―comes up heads‖
• if prob_change > 0.5
– consumers(i,3)=bestofferfirm;
• end
Ormerod model
• Next firms ―work out best practice‖:
– [C,I] = min(suppliers(actual_suppliers,4));
• ―Toss a coin‖
• for i=1:firms
– dice = rand(1,1);
– prob_change = suppliers(i,5) + dice;
• If coin ―comes up heads‖
– if prob_change > 1
• Copy offerings of ―best practice‖ firm:
– suppliers(i,2)=suppliers(I,2);
– suppliers(i,3)=suppliers(I,3);
• end
Ormerod model
• Process continues to next round and each stage graphed
• Ormerod‘s program repeats process 1,000 times and
records outcomes of each run.
• This program runs once (40 iterations) and shows market
outcome at each time step (1 iteration = 3 months ―real
time‖)
• Similar qualitative outcomes to Ormerod program
– Monopolist tends to hang on to lion‘s share of market
– But competitive outcome (falling price/rising quality)
independent of number of firms
– A few sample runs:
Ormerod model
• (1) A monopolist with ―the lot‖:
1 1
0.8 0.8
Average Quality
Average Price
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 10 20 30 40 0 10 20 30 40
Iterations Iterations
1 1
0.8 0.8
Market Share
Market Share
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 5 10 15 20 1
Firms Original Monopolist
Ormerod model
• ―Optus defeats Telstra‖…
1 1
0.8 0.8
Average Quality
Average Price
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 10 20 30 40 0 10 20 30 40
Iterations Iterations
1 1
0.8 0.8
Market Share
Market Share
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 5 10 15 20 1
Firms Original Monopolist
Ormerod model
• ―Yay‖ Competition at last…‖
1 1
0.8 0.8
Average Quality
Average Price
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 10 20 30 40 0 10 20 30 40
Iterations Iterations
1 1
0.8 0.8
Market Share
0.6 Market Share 0.6
0.4 0.4
0.2 0.2
0 0
0 5 10 15 20 1
Firms Original Monopolist
Ormerod model
• Wide range of structural outcomes
• Little difference in practical outcomes
– Price still falls
– Quality still rises
– The bottom line…
