Economic Dynamics

1
Economic Dynamics

• And the necessity of nonlinearity

2
Definitions

• Economic Dynamics
• the study of any economic process
• Huh?
• Its 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

3
The static-dynamics difference

• Consider standard micro supply and demand. We
  have a linear demand curve and a linear supply
  curve

• The static approach equate the two

• In more detail

4
The static-dynamics difference

• State (timeless) supply and demand formulae

• Work out equilibrium

• Draw graph

Notethisformula

• Break for lunch
• Problem agricultural markets not this stable

5
The static-dynamics difference

• Classic example is Minnesota Hog Cycle

• Not equilibrium but irregular cycles around
  long-term trend price

6
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 years returns
• Adaptive expectations
• Basic formulae

Producers expect next seasons price to be same
as last seasons or

• Yields difference equation for prices

Producers plant this seasons crop based on last
seasons price

• Gives same equilibrium result as static formula

7
The static-dynamics difference

• Set PtPt-1P

Notethisformula

• So eventual outcome same as statics?
• Statics is long-run dynamics?
• Depends on values of parameters

8
The static-dynamics difference

• If slope parameters bs/bdlt1, dynamicsstatics

• But for bs/bdgt1, dynamic instability

9
The static-dynamics difference

• No convergence to equilibrium price
• Crazy prices negative, tending to /- infinity

• Randomness no help
• System tends to impossible prices

10
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

11
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 Dtad-bdPt, feed in high Pt, youll get
  negative Dt
• But even neoclassical theory doesnt 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
  cyclesno need to hypothesise rational
  expectations to tame the cobweb

12
The static-dynamics difference

• Compare linear to nonlinear
• Simple nonlinear demand/supply curves
• Modified rectangular hyperbolas
• Basic hyperbola y1/x
• Area under hyperbola crucial to definition of
  log, exponential
• Used for illustration purposes only here

Generalised hyperbola formula is

• Used to derive S D curves

13
The static-dynamics difference

• Nonlinear demand

• Nonlinear supply

• Graphing them

• 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

14
The static-dynamics difference

• Solving for P as a function of time with these
  curves

• Generates sustained cycles

• More interesting deterministic dynamics
  possible with more complex functions
• Chaos can arise
• Impact of noise instructive

15
The static-dynamics difference

• Any pattern at all can result, without breakdown

16
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 modelproduct of being far
  from equilibrium

• So statics is not long run dynamics
• Dynamics can answer questions statics cant even
  pose

17
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

Statics
Dynamics
Evolution
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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
• But two generally consistent
• Results of dynamics more general than statics
• GENERALLY INCONSISTENT
• Dynamic results correct if actual system
  dynamic/evolutionary

Evolution
Dynamics
Statics

• In general, statics will give wrong answers to
  questions posed about economywhether questions
  posed in neoclassical or Post Keynesian terms

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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)

20
In summary

• Summarising validity of analytic techniques
  relations between them

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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)

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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

23
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

24
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
• Hickss trade cycle model
• Kaldors nonlinear explanation for cycle
• But first, the data

25
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
  post-1933
• Source NBER Macrohistorical database,
  http//www.nber.org/databases/macrohistory/data/01
  /a01007a.db (Index of manufacturing production)

26
Growth with cycles
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But what cycles!
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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

29
Propagation and impulse

• If cycles caused by exogenous shocks then
• propagation mechanism
• that which keeps disturbance at time t rippling
  through system till time tT, 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

30
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)

31
An example Hickss 2nd order model

• Investment a lagged function of change in income

• Consumption a lagged function of income

• Saving equals income minus consumption

• Equating I and S yields

• A 2nd order difference equation

32
2nd order difference equation

• Second order multiplier-accelerator model
  dominates theory of cycles in economics
  1950s-1960s
• But properties of model show all drawbacks of
  linear models
• Unrealistic cycles
• Too muchor too littleinstability
• No goldilocks here
• Zero equilibrium output level

33
2nd order difference equation

• 5 basic patterns, none realistic

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2nd order difference equation

• Adding noise doesnt help much

• The problem is linearity!
• But its also bad mathematics

35
2nd order difference equation

• Economists stuffed around with this model for
  decades
• A mathematician would have rejected it on day one
• Reason? Its only solution is the trivial
  solution
• YtYt-1Yt-20
• 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
  trivialzero.

36
The trivial solution

• In matrix form

• Special derived form of matrix can be inverted

• Means that only solution the trivial solution.
• Why is thiseconomically speaking?

37
Hickss error

• Because model equates desired I and actual S

??????????

• When does desired investment equal actual
  savings?
• When income equals zero!
• Actual investment is related to this periods
  output

or
38
A better (but still linear!) model

• Desired investment a function of change in output

• Capitalists carry out investment plans

• Investment adds to capital

• Capital determines output

• 3rd order difference equation

• A more interesting (but still linear!) model.
• Behaviour can be broken down into equilibrium
  trend cycle components

39
A better (but still linear!) model

• In matrix form, this is

• Special derived form of matrix cant be inverted

• As a result, non-trivial solutions possible
• Non-zero values for Y over time

40
Economic properties

• Cycles with growth
• Cv ratio determines nature of cycles growth
• Exponential with cgtv
• Linear with cv
• Damped with cltv
• Realistic period-independent values for c v
  feasible

41
Mathematical meaningful closed form
Equilibrium if c lt v
Growth term
Cycle term
42
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 notlong booms and short slumps
• Need nonlinearity to get asymmetry of real world
• First economist to realise the importance of
  being nonlinear was Kaldor

43
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 Keyness 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)

44
The endogenous critique
(1) Savings function steeper than
Investment (savings rises more than investment as
employment rises
SltI, system expands
SgtI, system contracts
Equilibrium stable
45
The endogenous critique
SltI, system expands
(2) Savings function flatter than
Investment (savings rises less than investment as
employment rises
SgtI, system contracts
Equilibrium unstable
46
The endogenous critique

• Kaldor
• In slope of Sgt 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 IgtS 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)

47
The endogenous critique

• Kaldors 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.

48
The importance of being nonlinear
Linear models can be
Cycles in linear system require
Frisch/Hicks/Econometrics approach
Harrods initial model
49
The importance of being nonlinear

• Nonlinear systems can be

• Cycles can occur because system is

Not so different from linear model
Completely unlike linear model
50
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 systems 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

51
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
• Systems 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

52
The importance of being nonlinear

• Advantages of nonlinear systems
• Realistic for most open systems
• Most open systemsones subject to evolutionary
  changeare far from equilibrium ones
• Nonlinear dynamics approximate this
• Evolution with fixed parameters
• Tractable compared to true evolutionary modelling

53
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

54
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 Steedmans critique of Kaleckian pricing
  theory

55
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

56
A brute fact

• the costs of any industry are constituted by the
  prices of industrial products and it would be ...
  one-sided 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

57
Steedmans 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

58
Equilibrium Prices

• Reworking this equation yields

• 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.)

59
What about dynamics?

• Steedman considers a once-only exogenous change
  (of du) in u.
• Then from

Note this equation
60
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

61
Their full effects

• Equation is

• As autonomous difference equation

• This is solved by breaking into two components
• First, homogeneous

• Presume solution of the form

• So that

Substituting
62
Solving difference equation
Dispense with
Collect terms in x
Factor

• Only possible for non-trivial x if

• So that

constant

• Second, particular

• Presume solution of the form

63
Solving difference equation

• Simple matrix manipulation

• Particular result same as Steedmans static
  solution

• General result sum of homogeneous plus particular
  solutions

• Static solution same as dynamic iff this?0 as t??

Skip eigenvalues
64
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 lt 1, matrix shrinks space and?0
  as t??
• If modulus of dominant eigenvalue of discrete
  dynamic system gt 1, matrix expands space and??
  as t??

How much does matrix stretch space?
in which direction?
Only possible for non-trivial v if
65
Eigenvalues eigenvectors

• is a polynomial in l.

• If the modulus of the dominant root of this
  polynomial lt 1, then this dynamic system will ?0
  as t?? and static price vector will be the final
  price vector

• If gt 1, then this dynamic system will ?? as t??
  and static price vector will be irrelevant
• If 1, then system marginally unstable

66
Steedmans stability

• Steedmans example system used

• With these values

• modulus of maximum eigenvalue of

67
Steedmans stability

• Convergence to equilibrium in Steedmans example
  system

68
Steedmans stability

• A different example system

• With these values

• Which static analysis would rule out for obvious
  reasons, but of which the modulus of maximum
  eigenvalue of

The consequence?
69
Steedmans stability

• With different input-output matrix, instability
• Permanent inflation away from the negative
  equilibrium price vector

70
Steedmans 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

71
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

72
Conclusion

• Static equilibrium not the end-product of dynamic
  processes
• Dynamicsnot staticsthe 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
  input-output 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

73
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
• Dont learn basics of dynamic modelling
• Dont appreciate importance of nonlinearity
• Next lecture some examples of how to be
  dynamically nonlinear

74
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 Jorgensons
  model', Review of Economic Studies, 30 141-147.