Economic Dynamics
And the necessity of nonlinearity
Steve Keen
Definitions
• Economic Dynamics:
– the study of any economic process
– Huh?
– It’s easier to define by considering what you have
already studied:
• Economic Statics
– the study of the determination of points of
economic equilibrium
– no consideration of the time path taken to get
there
• Nonlinearity:
– Realism in functional representation of a system
– First step towards evolutionary modelling…
©Steve Keen 2007
University of Western Sydney
2
The static-dynamics difference
• Consider standard micro supply and demand. We have
a linear demand curve and a linear supply curve:
S  P  a  b  P
D P  c  d  P
• The static approach: equate the two:
a b P  c  d  P
ca
P
bd
©Steve Keen 2007
• In more detail…
University of Western Sydney
3
The static-dynamics difference
• State (timeless) supply and demand formulae;
• Work out equilibrium
1
as  200
bs 
ad  1000 bd  1
• Draw graph:
3
S ( P)  as  bs P
D ( P)  ad  bd  P
Se  as  b s Pe
as  b s Pe
ad  b d  Pe
Se  as  b s Pe
Pe  900
Se  100
1500
Supply
Note
as  ad


this
formula
Pe 
Demand
ad  b d  Pe
bs  bd
De  ad  b d  Pe
1000
Price
De
Static Supply & Demand Analysis
500
De  100
0
0
50
100
150
200
Quantity
• Break for lunch…
• Problem: agricultural markets not this stable…
©Steve Keen 2007
University of Western Sydney
4
The static-dynamics difference
• Classic example is “Minnesota Hog Cycle”
• Not
“equilibrium”
but irregular
cycles
around longterm trend
price…
©Steve Keen 2007
University of Western Sydney
5
The static-dynamics difference
• Attempted dynamic explanation: cobweb model…
– Recast supply and demand as time-lagged (actually
time-delayed) functions:
• Demand now reflects prices now
• Supply now reflects prices last season
– Farmers plant based on last year’s returns:
Producers expect
• Adaptive expectations
next season’s price
Dt ad  bd Pt
• Basic formulae:
to be same as last
St  as  bs Pt1
Dt
St
season’s; or…
• Yields difference equation for prices:
Producers plant
• Gives same
ad  as bs
this season’s crop
Pt

 Pt1
equilibrium result
bd
bd
based on last
as static formula
season’s price
©Steve Keen 2007
University of Western Sydney
6
The static-dynamics difference
• Set Pt=Pt-1=P:
P
P
P
ad  as
bd
bs
bd
P
ad  as
bd

bs
bd

1 


P
bs 
 P
bd 



 1


bs 

bd 

ad  as
bd
ad  as
bd
 bd



b
 d
bs 

bd 

ad  as
bd
 bd  bs  aNote
 as
d



 b
 bthis
d  formula
d  bs

• So eventual outcome same as statics?
– “Statics is long-run dynamics?”
• Depends on values of parameters…
©Steve Keen 2007
University of Western Sydney
7
The static-dynamics difference
• If slope parameters bs/bd<1, “dynamics=statics”
b s 
bs
bd
b d 
40
 0.615
Pe 


 as  ad
bs  bd
b s  0.4
65
b d  0.65
 ad  as

Pt 

 b
d

Pe  1143


 Pt1

bd

bs
Price as function of time
1800
Dynamic Price
Equilibrium Price
1600
Price
1400
1200
1000
800
600
0
©Steve Keen 2007
5
10
Time
15
20
• But for bs/bd>1,
“dynamic instability”
University of Western Sydney
8
The static-dynamics difference
• No convergence to equilibrium price;
• “Crazy” prices: negative, tending to +/- infinity…
b s 
bs
bd
b d 
40
 1.081
Pe 
1  10


 as  ad
bs  bd
37
Pe  1558
b s  0.4
• Randomness no help…
 ad  as bs



– System tends to
Pt 

 Pt1
 b

bd
d


impossible prices
Price as function of time
4
b d  0.37
T  1  120 Pr  1558
0
Dynamic Price
Equilibrium Price
Dynamic Price
Price
4
1  10
0
Equilibrium Price
Price
5000
5000
1  10

  rnd ( 1)
 Pr
b d T 1 

bs
Price as function of time
4
1.5  10
5000
 ad  as

Pr 

 b
T
d

0
5000
4
0
©Steve Keen 2007
5
10
Time
15
20
1  10
4
1.5  10
4
0
University of Western Sydney
20
40
60
Time
80
100
120
9
The static-dynamics difference
• In cobweb model, dynamic answer diverges from
static answer if suppliers are more responsive to price
than consumers. (Which group do you think is more
responsive?)
• “Mad” result of negative prices is result of “mad”
assumption of linear functions (which allow negative
supply, and negative demand!).
• Effect disappears with “sensible” nonlinear functions
– Why “sensible”?
– Because linearity an abstraction
• Nothing in the real world is really linear
– Not even neoclassical economics…
©Steve Keen 2007
University of Western Sydney
10
The static-dynamics difference
• Markets (and models of markets) cannot be linear
– Crazy results (negative prices & quantities) product
of linear form for demand & supply curves
• Given Dt=ad-bdPt, feed in high Pt, you’ll get
negative Dt
• But even neoclassical theory doesn’t justify linear
demand & supply curves
– “Non-satiation” implies D∞ as P0
– Ditto supply: stops at 0, reaches finite
maximum as marginal cost∞
• Nonlinear, time-delayed models give realistic cycles—
no need to hypothesise “rational” expectations to tame
the cobweb…
©Steve Keen 2007
University of Western Sydney
11
The static-dynamics difference
• Compare linear to nonlinear
– Simple nonlinear
demand/supply curves
• Modified rectangular
hyperbolas
1
x
– Basic hyperbola
y=1/x
– Area under
hyperbola crucial to
definition of log,
exponential;
– Used for illustration
purposes only here…
©Steve Keen 2007
Simple Rectangular Hyperbola
0.8
0.6
0.4
0.2
0
20
40
60
80
100
x
Generalised
hyperbola formula
isy)
( x  h) ( k 
c
• Used to derive S & D
curves:
12
University of Western Sydney
The static-dynamics difference
• Nonlinear demand:
• Nonlinear supply:
• Graphing them:
 PmaxD



qhd P  PmaxD  PminD  Dmin 
 Dmin
PP

minD


 PmaxS



qhs P  PmaxS  Smax  PminS 
 Smax
P

P
minS






The Importance of Being Nonlinear
Price
• More realistic even in
terms of neoclassical
theory than standard
“linear” curves used
– Linear obsession mainly
due to lazy pedagogy
– But had real impact on
development of theory
©Steve Keen 2007
2000
Demand
Supply
1500
1000
500
0
100
University of Western Sydney
200
Quantity
300
400
13
The static-dynamics difference
• Solving for P as a function of time with these curves:


PmaxD

Ph 
 PminD


P
t
maxS


 Smax  Dmin
 PminS  Ph

t

1



Price Dynamics with Nonlinear Functions
540

520
Price
• Generates sustained cycles:
• More “interesting”
deterministic dynamics
possible with more complex
functions
– Chaos can arise
• Impact of noise instructive:
500
480
460
0


PmaxD
Ph  
 PminD  rnd ( 1)


PmaxS
t


 Smax  Dmin
 PminS  Ph

t

1
 2007
 of Western Sydney
©Steve Keen
University

5
10
15
20
Time

14
The static-dynamics difference
• Any pattern at all can result, without breakdown:
Price Dynamics with Nonlinear Functions & Noise
Price Dynamics with Nonlinear Functions & Noise
504
506
503
504
502
501
Price
Price
502
500
498
499
496
498
504
497
500
Price Dynamics with Nonlinear Functions & Noise
0
20
40
503
60
80
Price Dynamics with Nonlinear Functions & Noise
503
494
100
Time
0
20
40
60
80
100
60
80
100
Time
502
501
Price
Price
502
500
501
500
499
499
498
497
0
20
©Steve Keen 2007
40
60
Time
80
100
498
0
20
University of Western Sydney
40
Time
15
The static-dynamics difference
• “Looks like” empirical data too, even though model
incredibly simple:
• Apparent “volatility
clustering”…
– Very difficult to get
from linear models
– Simple with nonlinear
model—product of
being “far from
equilibrium”
• So statics is not “long run dynamics”
• Dynamics can answer questions statics can’t even pose
©Steve Keen 2007
University of Western Sydney
16
Why the economic obsession with statics?
• Neoclassical economists tend to think:
– “Evolution leads to optimising
behaviour”
– “Dynamics explains movement from
one equilibrium point to another”
– So “statics is long run dynamics”
• also believed by Sraffian
economists
• implicit in Post Keynesian or
Marxian analysis using
comparative static or
simultaneous equation methods
©Steve Keen 2007
University of Western Sydney
Statics
Dynamics
Evolution
17
Why the economic obsession with statics?
• Modern mathematics reverses this
– Field of evolution larger than
dynamics
– Dynamics larger than statics
• Results of evolutionary analysis more
general than dynamics
Dynamics
– But two generally consistent
• Results of dynamics more general than
Statics
statics
– & GENERALLY INCONSISTENT
• Dynamic results correct if actual
system dynamic/evolutionary
• In general, statics will give wrong answers to questions
posed about economy—whether questions posed in
neoclassical or Post Keynesian terms…
Evolution
©Steve Keen 2007
University of Western Sydney
18
General Disequilibrium
• ‘There exist known systems, therefore, in which the
important and interesting features of the system are
“essentially dynamic”, in the sense that they are not
just small perturbations around some equilibrium
state, perturbations which can be understood by
starting from a study of the equilibrium state and
tacking on the dynamics as an afterthought.’
• ‘If it should be true that a competitive market
system is of this kind, then… No progress can then be
made by continuing along the road that economists
have been following for 200 years. The study of
economic equilibrium is then little more than a waste
of time and effort…’ Blatt (1983: 5-6)
©Steve Keen 2007
University of Western Sydney
19
In summary…
• Summarising validity of analytic techniques & relations
between them:
Validity of Techniques
Approach
Statics
Technique
Linear Algebra, Calculus,
Optimisation…
Differential/Difference
Dynamics
Evolution
Valid When
Invalid When
Equilibrium stable, System nonlinear or Very low to
system linear
equilibrium unstable inconsistent
Parameters stable, Parameters evolve
equations, multi-agent
dimensionality
or dimensionality
simulations
stable
changes
Open-dimensional analysis,
multi-agent simulations
©Steve Keen 2007
Consistency with
next level
Moderate to
high
System evolves
University of Western Sydney
20
Why did economics start with statics?
• Because it was easier!
• Marshall (of Micro “fame”)
– The modern mathematician is familiar with the
notion that dynamics includes statics. If he can
solve a problem dynamically, he seldom cares to
solve it statically also... But the statical solution
has claims of its own. It is simpler than the
dynamical; it may afford useful preparation and
training for the more difficult dynamical solution;
and it may be the first step towards a provisional
and partial solution in problems so complex that a
complete dynamical solution is beyond our
attainment. (Marshall, 1907 in Groenewegen 1996:
432)
©Steve Keen 2007
University of Western Sydney
21
Why did economics start with statics?
• Jevons (one of the founders of General Equilibrium
analysis)
– “If we wished to have a complete solution … we
should have to treat it as a problem of dynamics.
But it would surely be absurd to attempt the
more difficult question when the more easy one
is yet so imperfectly within our power.” [Jevons,
Theory of Political Economy, Ch. 4, 4th edition,
p. 93]
• So statics regarded as easier way to reach the same
answers as the more general dynamics would give.
• Now known to be incorrect outside economics, but still
not common knowledge within economics
©Steve Keen 2007
University of Western Sydney
22
Why study dynamics?
• Many real world processes
– do not have an equilibrium;
– or do not have a single equilibrium;
– or do not have stable equilibria
• globally,
• or locally
• Examples:
– weather patterns; animal population growth/decline;
…
©Steve Keen 2007
University of Western Sydney
23
Why study dynamics?
• In these systems, equilibrium values will never apply.
• Equilibrium (and therefore static analysis) irrelevant
to system in both short and long term
– system will not be at equilibrium now
– it is not moving towards equilibrium over time
• Economics?
– When did you last see an economy at rest?…
– Question is whether the economy is stable subject
to shocks, or unstable…
• Two examples of linear vs nonlinear thinking:
– Hicks’s trade cycle model
– Kaldor’s nonlinear explanation for cycle
– But first, the data…
©Steve Keen 2007
University of Western Sydney
24
The pre-1933 Trade cycle
• Pre-1933 trade cycle predates “Big Government”
• Cycles and growth performance therefore closer to
“pure market economy” results than data for post1933
• Source: NBER Macrohistorical database,
http://www.nber.org/databases/macrohistory/data/01
/a01007a.db (Index of manufacturing production)
©Steve Keen 2007
University of Western Sydney
25
Growth with cycles
Index of USA Output
200
180
160
140
Index
120
100
80
60
40
20
1927
1923
1919
1915
1911
1907
1903
1899
1895
1891
1887
1883
1879
1875
1871
1867
1863
0
Year
©Steve Keen 2007
University of Western Sydney
26
But what cycles!
The USA Trade Cycle
40%
30%
10%
1930
1926
1922
1918
1914
1910
1906
1902
1898
1894
1890
1886
1882
1878
1874
1870
-10%
1866
0%
Year
Rate of Growth
20%
-20%
-30%
Years
©Steve Keen 2007
University of Western Sydney
27
What causes these cycles?
• 2 classes of possible explanations
– Exogenous shocks to stable system
• economy stable, but disturbed by weather
patterns, wars, etc.
– Endogenous fluctuations generated by dynamics of
the economy itself
• can also have exogenous shocks imposed on this
class of systems, of course
• First interpretation dominated early work in “economic
dynamics”
©Steve Keen 2007
University of Western Sydney
28
Propagation and impulse
• If cycles caused by exogenous shocks then
– “propagation mechanism”
• that which keeps disturbance at time t rippling
through system till time t+T, at which time
impact of disturbance completely dissipated
– differs from “impulse mechanism”
• source of random shocks from outside the
economy
• This interpretation dominated early work because
economists believed (wrongly) that endogenous cycles
were not possible
©Steve Keen 2007
University of Western Sydney
29
The exogenous shocks interpretation
• Frisch in 1933 (depth of Great Depression)
– “The majority of the economic oscillations which we
encounter seem to be explained most plausibly as
free oscillations. In many cases they seem to be
explained by the fact that certain exterior
impulses hit the economic system and thereby
initiate more or less regular oscillations” (Economic
essays in honour of Gustav Cassel: 171)
– “If you hit a rocking horse with a club, the
movement of the horse [stable propagation
mechanism] will be very different to that of the
club [exogenous shocks]” (198)
©Steve Keen 2007
University of Western Sydney
30
An example: Hicks’s 2nd order model
Investment a lagged function of change in income:
It  c  (Yt 1  Yt 2 );
• Consumption a lagged … function of income:
Ct  (1  s ) Yt 1 ;
• Saving equals income minus consumption:
St  Yt  (1  s ) Yn t
• Equating I and S yields
c  (Yt 1 Yt 2 )  Yt  (1  s ) Yt 1
• A 2nd order difference equation:
Yt  (1  s  c ) Yt 1  c Yt 2
©Steve Keen 2007
University of Western Sydney
31
2nd order difference equation
• Second order multiplier-accelerator model
dominates theory of cycles in economics 1950s1960s
– But properties of model show all drawbacks of
linear models…
• Unrealistic cycles
• Too much—or too little—instability
– No “goldilocks” here
• Zero “equilibrium” output level
©Steve Keen 2007
University of Western Sydney
32
2nd order difference equation
• 5 basic patterns, none realistic
Hicks's 2nd order difference equation model; c<1, s large
150
100
100
50
50
Output
Output
Hicks's 2nd order difference equation model; c<1, s small
150
0
 50
 100
0
 50
0
20
40
60
80
 100
100
0
20
40
Time
Hicks's 2nd order difference equation model; c<1
4
80
100
Time
Hicks's 2nd order difference equation model; c=1
110
60
200
Hicks's 2nd order difference equation model; c>2.1
19
110
100
0
Output
Output
Output
0
0
19
 110
4
 110
19
 100
4
 210
0
20
40
60
Time
©Steve Keen 2007
80
100
 200
 210
19
0
20
40
60
80
100
Time
University of Western Sydney
 310
0
20
40
60
Time
80
100
33
2nd order difference equation
• Adding noise doesn’t help much:
Y t 2  ( 1  s  c)   Y t 1  rnorm 1 0 
Y t 1
 0  c   Yt  rnorm 1 0 
 


Hicks's 2nd order difference equation model; c<1, s small
 0
 
Hicks's 2nd order difference equation model; c>2.1
18
400
Yt
510
200
0
Output
Output
0
 200
18
 510
19
 110
 400
 600
0
20
40
60
80
100
19
 1.510
0
Time
• The problem is linearity!
– But it’s also bad mathematics…
©Steve Keen 2007
University of Western Sydney
20
40
60
80
100
Time
34
2nd order difference equation
• Economists stuffed around with this model for
decades
– A mathematician would have rejected it on day one
• Reason? It’s only solution is “the trivial solution”
– Yt=Yt-1=Yt-2=0
• Takes “elementary” mathematical analysis to show this
– Convert model into matrix form
• If matrix non-invertible, model has meaningful
solutions
• If non-invertible, only solution is “trivial”—zero.
©Steve Keen 2007
University of Western Sydney
35
The trivial solution
• In matrix form:
1   x1 t  
 x1 t  1   0





x2 t  1 c 1  s  c  x2 t 
• Special derived form of matrix can be inverted:
1 c

1
s
 1 0   0

geninv 

 simplify  
0
1

c
1

s

c

 

 c
 s

s 

1

s 
1
• Means that only solution the trivial solution.
• Why is this—economically speaking?
©Steve Keen 2007
University of Western Sydney
36
Hicks’s error
• Because model equates desired I and actual S:
I  c  (Yn 1 Yn 2 )  S  Yn  (1  s ) Yn 1
??????????
• When does desired investment equal actual savings?
– When income equals zero!
• Actual investment is related to this period’s output:
It  Kt  Kt 1
Yt  Yt 1
It 
v
©Steve Keen 2007
or
Kt  Kt 1  It 1
Yt 1  Yt
It 
v
University of Western Sydney
37
A better (but still linear!) model


Desired investment a function Idt  c  Yt 1  Yt 2 
of change in output
Capitalists carry out investment plans: I  I
t
dt
• Investment adds to capital:
• Capital determines output



3rd order difference equation
Kt  Kt 1  It 1
1
Yt   Kt
v
c
Yt  Yt 1   Yt 2  Yt 3 
v
A more interesting (but still linear!) model.
Behaviour can be broken down into equilibrium + trend
+ cycle components:
©Steve Keen 2007
University of Western Sydney
38
A better (but still linear!) model
• In matrix form, this is:
 x1 
 t 1 
 x2 
 t 1 
 x3 
 t 1 
 0
 0
 c
 v

 x1 
  t
0 1 x 
 2
  t
c
1 

v
  x3t 
1 0
• Special derived form of matrix can’t be inverted:
 1 0 0
 0


0


geninv 0 1 0   

  c
0
0
1

  v


1 0 

0 1 

1 
v

simplify  undefined
c
• As a result, non-trivial solutions possible:
• Non-zero values for Y over time…
©Steve Keen 2007
University of Western Sydney
39
Economic properties
3rd order linear multiplier-accelerator model: growth with cycles•
Output
500
400
300
200
100
0
10
©Steve Keen 2007
20
30
Years
40
50
Cycles with growth
• C:v ratio
determines nature
of cycles & growth
– Exponential with
c>v
– Linear with c=v
– Damped with
c<v
– Realistic &
periodindependent
values for c & v
feasible
University of Western Sydney
40
Mathematical: meaningful closed form
Equilibrium if c < v
Yt

1
 
2
Growth term
1
 
2
Cycle term
v Y2 c Y0


v c

 
v c  Y1 Y0 v  Y2 Y1
c  v c

 
v c  Y1 Y0 v  Y2 Y1
c  v c
Output
Growth


 

c

v
 
t
c
v
Cycles
350
7.5
300
5
2.5
250
0
200
-2.5
150
-5
100
-7.5
0
©Steve Keen 2007
10
20
30
40
50
0
10
Time
University
of Western Sydney
20
30
40
50
41
t
But the limitations of being linear
• Model itself a “quirk”
– Cycle size perfectly synchronised with growth of
output
– Mathematically, eigenvalue for growth exactly same
magnitude as eigenvalue for cycles
• Normally, these differ in linear models
• Cycles also symmetrical
– Trade cycle is not—long booms and short slumps
– Need nonlinearity to get asymmetry of real world
• First economist to realise “the importance of being
nonlinear” was Kaldor:
©Steve Keen 2007
University of Western Sydney
42
The endogenous critique
• Kaldor 1940, “A model of the trade cycle”
– Considered static model based on interaction of
ex-ante savings and ex-ante investment:
– “the basic principle underlying all these theories
may be sought in the proposition … derived from
Mr Keynes’s General Theory … that economic
activity always tends towards a level where Savings
and Investment are equal… in the ex-ante …
sense.” (78)
– Savings and Investment both assumed to be
positively sloped functions of activity level
(employment as proxy).
– “If we assume the S and I functions as linear, we
have two possibilities:” (79)
©Steve Keen 2007
University of Western Sydney
43
The endogenous critique
Y
(1) Savings
function
steeper than
Investment
(savings rises
more
than investment
as employment
rises
©Steve Keen 2007
S
S<I, system
expands
I
Equilibrium
stable
S>I,
system
contracts
Employment
University of Western Sydney
44
The endogenous critique
(2) Savings
function
flatter than
Investment
(savings rises less
than investment
as employment
rises
©Steve Keen 2007
S<I,
system
expands
I
Y
S>I, system
contracts
S
Equilibrium
unstable
University of Western Sydney
Employment
45
The endogenous critique
• Kaldor
– In “slope of S”> “slope of I” situation
• “any disturbances … would be followed by the
re-establishment of a new equilibrium, with a
stable level of activity… this … assumes more
stability than the real world, in fact, appears to
possess.” (80)
– In I>S situation
• “the economic system would always be rushing
either towards a state of hyper-inflation … or
towards total collapse… Since recorded
experience does not bear out such dangerous
instabilities, this possibility can be dismissed”
(80)
©Steve Keen 2007
University of Western Sydney
46
The endogenous critique
• Kaldor’s solution
– “Since thus neither of these two assumptions can
be justified, we are left with the conclusion that
the I and S functions cannot both be linear.” (81)
– Insight: nonlinear functions make endogenous
fluctuations possible, and limit size to meaningful
levels
– Endogenous fluctuations and nonlinearity are
inseparable elements of dynamic analysis.
©Steve Keen 2007
University of Western Sydney
47
The importance of being nonlinear
Linear models can be:
Ind e tem
r inan t
U
G
G
lob
ly
al
ns
tab
le
lob
al
ly
le
b
a
St
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
©Steve Keen 2007
University of Western Sydney
48
The importance of being nonlinear
• Nonlinear systems can be:
L
• Cycles can occur
because system is:
L o ca lly S tab lew ith S h o ck s
Not so different
from linear model
©Steve Keen 2007
ly
l
a
oc
U
ns
tab
le
G loba lly S tab le
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Completely unlike
linear model
University of Western Sydney
49
The importance of being nonlinear
• Advantages of linear systems:
– Easily analysed (closed form solutions exist)
– Powerful analytic maths (linear algebra)
– Proof by theorem
– Stable linear dynamic system’s behaviour a function
of parameter values of system only
– Behaviour can be broken down into
• Equilibrium value
• Growth component
• Cyclical component
• Disadvantages of linear systems:
– Unrealistic for most open systems
©Steve Keen 2007
University of Western Sydney
50
The importance of being nonlinear
• Disadvantages of nonlinear systems:
– Difficult to analyse (no closed form solutions)
– No analytic maths
• Many high level forms of maths needed to
characterise, but no analytic results possible
– Proof by simulation rather than theorem
– System’s behaviour a function of both parameter
values and initial conditions
• Path dependent behavior
– Behaviour cannot be broken down into growth and
cyclical components
• Instead, magnitude of cycles a function of
deviation from equilibrium; equilibria often
“repellers” rather than “attractors”
©Steve Keen 2007
University of Western Sydney
51
The importance of being nonlinear
• Advantages of nonlinear systems
– Realistic for most open systems
– Most “open systems”—ones subject to evolutionary
change—are “far from equilibrium” ones
• Nonlinear dynamics approximate this;
– “Evolution” with fixed parameters
– Tractable compared to true evolutionary
modelling
©Steve Keen 2007
University of Western Sydney
52
Statics vs. Dynamics
• Economics unique amongst mathematically-oriented
disciplines in reliance upon static methodology
(simultaneous equations rather than differential
equations)
• Reliance on statics not limited to Neoclassicals
– Many Keynesian/Kaleckian theorists (including the
masters) use simultaneous equations
– “Sraffian” economists criticise all other schools
using advanced equilibrium-oriented methodology
– Why?
• Belief that economic system will settle down to
equilibrium “in the long run”
• “Dynamics simply describes transients”
©Steve Keen 2007
University of Western Sydney
53
Statics vs. Dynamics
• Long ago shown to be untrue even for “general
equilibrium” neoclassical models (Jorgenson 1960,61,
63; McManus 1963; Blatt 1983)
– Linear component of input-output system with
growth must be unstable in either price or output
vector
– Reliance on static methods a hangover from past
practice and faith
• Dynamic answers to economic questions
fundamentally different to static ones
– EVEN IF model “Keynesian”
• Example: Steedman’s critique of Kaleckian
pricing theory
©Steve Keen 2007
University of Western Sydney
54
Steedman on Kalecki
• A (mathematical/methodological) critique of Kaleckian
microfoundations:
• A “Kalecki after Sraffa”?
• No consideration of macro (“capitalists get what they
spend...”)
• Input-output analytic critique of markup pricing
theory and related theory of distribution
©Steve Keen 2007
University of Western Sydney
55
A “brute fact”
• “the costs of any industry are constituted by the
prices of industrial products and it would be ... onesided to say that ‘prices are largely cost determined’
without saying also that ‘costs are to a significant
degree price determined’”
• Justified attack on lack of analytic consideration of
input-output relations in Kaleckian tradition...
• Unjustified attack on Kaleckian analysis of the
process of price setting
©Steve Keen 2007
University of Western Sydney
56
Steedman’s Crucible
• A model of price setting which takes account of
input-output relations
• Circulating capital only; no overhead labour
• Equilibrium analysis, quantities taken as given, which
leaves prices only:
p  (u  pA )(I  mˆ) where u  wE  fM :
p a vector of prices, w wage rates, f import prices
A an n.n input-output matrix, E labor, M imports
m a vector of markups (non-uniform)
u "exogenous costs" (wages and imported
inputs)
©Steve Keen 2007
University of Western Sydney
57
Equilibrium Prices
• Reworking this equation yields:
1
ˆ
ˆ
p  u (I  m)(I  A  Am)
Price can be expressed as a function of markup, but
• Given input-output relations, price in industry j will at
least depend on all 1...n industries which are basic
• QED I: prices in industry j cannot be set without
regard to conditions in other industries
– (Followed by critiques of averages, vertical
integration, wages share, etc.)
©Steve Keen 2007
University of Western Sydney
58
What about dynamics?
• Steedman considers a once-only exogenous change (of
du) in u.
• Then from
p  (u  pA )(I  mˆ) we get
p1  (u  du  pA )(I  mˆ) or
Note this
equation
p1  p  du (I  mˆ) and
p2  (u  du  p1A )(I  mˆ) or
p2  p  du (I  mˆ)(I  A(I  mˆ))
leading to
pt 1  p  du (I  mˆ)(I  ...  A(I  mˆ  dmˆ)t
©Steve Keen 2007
University of Western Sydney
59
“Their full effects”
• QED II: Price converges to a new equilibrium vector
where initial interdependence of (each) price on many
(at least basic industries) markups is restored.
Steedman concludes that
– QED III: “‘static’ analysis does not ignore time. To
the contrary, that analysis allows enough time for
changes in prime costs, markups, etc., to have
their full effects.”
– Really?
• Like most economists, Steedman is apparently
unaware of basic methods of mathematical
dynamical analysis
• Reworking his equation into a standard
difference equation:
©Steve Keen 2007
University of Western Sydney
60
“Their full effects”
• Equation is
p2  (u  du  p1A )(I  mˆ) or
pt 1  (u  pt A )(I  mˆt )
• As autonomous difference pt 1  u I  m   pt A I  m 
equation
• This is solved by breaking into two components:
pt 1  pt A I  m 
• First, “homogeneous”
pt 1  pt A I  m   0
t
p

x
• Presume solution of the form t
t 1
p

x
• So that t 1
Substituting:
©Steve Keen 2007
University of Western Sydney
61
Solving difference equation
t 1
t
with
ptDispense

p
A
I

m

x

x
A I  m   0


1
t
x t x  x t A I  m   0 Collect terms in x
x t  x  A  I  m    0 Factor
• Only possible for non-trivial x if x  A I  m   0
x  c A I  m   • So that pt  x  c A I  m  
t
t
t
t
constant
pt 1  u I  m 
• Second, “particular”:
pt  K
• Presume solution of the form
pt 1  u I  m   pt A I  m   K  u I  m   KA I  m   0
©Steve Keen 2007
University of Western Sydney
62
Solving difference equation
• Simple matrix manipulation:
K  u I  m   KA I  m   0
K I  A  Am   u I  m 
• Particular result same as K  u  I  m  I  A  Am 
Steedman’s static solution: p  u (I  m
ˆ)(I  A  Amˆ)1
1
• General result sum of homogeneous plus particular
solutions:
pt  u I  m I  A  Am   c A I  m  
1
t
• Static solution same as dynamic iff this0 as t
Skip eigenvalues
©Steve Keen 2007
University of Western Sydney
63
Eigenvalues & eigenvectors
• “Eigen” (German for “characteristic”) values tell you
how much a matrix is stretching space
– If modulus of dominant eigenvalue of discrete
dynamic system < 1, matrix “shrinks” space and0
as t
– If modulus of dominant eigenvalue of discrete
dynamic system > 1, matrix “expands” space
and as t
How much does
matrix ‘stretch A  I  m  v   v & in which direction?
space’? A  I  m  v  v  0
A  I
 m   I v  0
Only possible for nontrivial v if
A  I  m   I  0
©Steve Keen 2007
University of Western Sydney
64
Eigenvalues & eigenvectors
A  I  m   I  0 is a polynomial in .
• If the modulus of the dominant root of this
polynomial < 1, then this dynamic system will
0 as t and static price vector will be the
final price vector
• If > 1, then this dynamic system will  as
t and static price vector will be irrelevant
• If =1, then system “marginally unstable”
0
 1
t

if   1
A I  m   v  t v 
marginally unstable if  =1
©Steve Keen 2007
University of Western Sydney
if
65
Steedman’s stability
• Steedman’s example system used

0

A  0

1

 6
1
2

0

1
1 1 1 
1 1 1 
0
,m  
,u  

3
2 2 2 
 2 6 3 

0 0

1
u
I

m
I

A

Am
• With these values 

  1 1 1
• & modulus of maximum eigenvalue of A I  m   1
©Steve Keen 2007
University of Western Sydney
66
Steedman’s stability
Good 1 Price
Good 2 Price
Good 3 Price
Prices
3
2
• Convergence
to equilibrium
in Steedman’s
example
system…
1
0
1
2
3
4
5
6
7
Periods
©Steve Keen 2007
University of Western Sydney
67
Steedman’s stability
• A different example system

0

A0

7

 10
3
5
0
0

0

4
1
,m  
5
2

0

• With these values
1
2
1
1
,u  

2
2
1
6
1
3 
u  I  m  I  A  Am    11.866 10.429 12.015
1
• Which static analysis would rule out for obvious
reasons, but of which the modulus of maximum
eigenvalue of A I  m   1.38 The consequence?
©Steve Keen 2007
University of Western Sydney
68
Steedman’s stability
Prices
30
Good 1 Price
Good 2 Price
Good 3 Price
20
10
0
2
©Steve Keen 2007
7
12
17
Periods
22
27
University of Western Sydney
• With
different
input-output
matrix,
instability:
– Permanent
inflation
away from
the
negative
equilibrium
price
vector
69
Steedman’s stability
• Continuous price inflation
– Negative equilibrium price vector irrelevant since
equilibrium unstable and prices will always diverge
from it.
• Static analysis does not describe the “full effects” of
a dynamic system unless the dynamic system is stable
– In real-world systems, instability/marginal
instability rather than stability seems to be the
rule
• Complex systems/evolutionary intepretation:
“evolution to the edge of chaos”
©Steve Keen 2007
University of Western Sydney
70
With more reality?
• Increased realistic complexity would introduce add
quantity, banks, effective demand, nonlinear wage &
investment functions, etc., to prices & markups
• Each additional element of reality brings increased
nonlinearity (even with no explicit nonlinear functions)
• Full system almost certainly has unstable (multiple)
equilibria, hence exhibits far-from-equilibrium
dynamic behaviour
©Steve Keen 2007
University of Western Sydney
71
Conclusion
• Static equilibrium not the end-product of dynamic
processes
– Dynamics—not statics—the true crucible of
economics:
• Not so much “Kalecki after Sraffa” as “Sraffa
after Lorenz”
• Kaleckian price-setting process fully consistent with
dynamic input-output analysis; but
– Kaleckian results require nonlinear dynamic inputoutput analysis for full expression
• Kaleckian analysis insufficiently developed on this
front to date; but on the other hand,
– Sraffians unjustifiably reliant upon statics
• Time for some cross-pollination…
©Steve Keen 2007
University of Western Sydney
72
Conclusion
• Non-neoclassical economists almost have as much to
learn about dynamics as do neoclassicals
– Most Post Keynesian/Marxian/Sraffian economists
still only learn maths from other economists
– Don’t learn basics of dynamic modelling
– Don’t appreciate importance of nonlinearity
• Next lecture: some examples of “how to be
dynamically nonlinear”
©Steve Keen 2007
University of Western Sydney
73
References
• Blatt, J.M., (1983). Dynamic Economic Systems, ME
Sharpe, Armonk.
• Jorgenson, D.W., (1960). 'A dual stability theorem',
Econometrica 28: 892-899.
• Jorgenson, D.W., (1961). 'Stability of a dynamic
input-output system', Review of Economic
Studies, 28: 105-116.
• Jorgenson, D.W., (1963). 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
• McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
©Steve Keen 2007
University of Western Sydney
74