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Economic Dynamics

Economic Dynamics. And the necessity of nonlinearity. 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

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Economic Dynamics

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  1. Economic Dynamics And the necessity of nonlinearity

  2. 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…

  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 year’s returns: • Adaptive expectations • Basic formulae: Producers expect next season’s price to be same as last season’s; or… • Yields difference equation for prices: Producers plant this season’s crop based on last season’s price • Gives same equilibrium result as static formula

  7. The static-dynamics difference • Set Pt=Pt-1=P: 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/bd<1, “dynamics=statics” • But for bs/bd>1, “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 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…

  12. The static-dynamics difference • Compare linear to nonlinear • Simple nonlinear demand/supply curves • Modified rectangular hyperbolas • Basic hyperbola y=1/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 model—product of being “far from equilibrium” • So statics is not “long run dynamics” • Dynamics can answer questions statics can’t 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

  18. 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 economy—whether questions posed in neoclassical or Post Keynesian terms…

  19. 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:

  21. 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)

  22. 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: • Hicks’s trade cycle model • Kaldor’s 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

  27. But what cycles!

  28. 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 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

  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: Hicks’s 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 much—or too little—instability • No “goldilocks” here • Zero “equilibrium” output level

  33. 2nd order difference equation • 5 basic patterns, none realistic

  34. 2nd order difference equation • Adding noise doesn’t help much: • The problem is linearity! • But it’s 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? 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.

  36. The trivial solution • In matrix form: • Special derived form of matrix can be inverted: • Means that only solution the trivial solution. • Why is this—economically speaking?

  37. Hicks’s error • When does desired investment equal actual savings? • When income equals zero! • Actual investment is related to this period’s output: • Because model equates desired I and actual S: ?????????? 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 can’t be inverted: • As a result, non-trivial solutions possible: • Non-zero values for Y over time…

  40. Economic properties • Cycles with growth • C:v ratio determines nature of cycles & growth • Exponential with c>v • Linear with c=v • Damped with c<v • Realistic & period-independent values for c & v feasible

  41. Mathematical: meaningful closed form Equilibrium if c < 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 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:

  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 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)

  44. Y S I Employment The endogenous critique (1) Savings function steeper than Investment (savings rises more than investment as employment rises S<I, system expands S>I, system contracts Equilibrium stable

  45. Y I S Employment The endogenous critique S<I, system expands (2) Savings function flatter than Investment (savings rises less than investment as employment rises S>I, system contracts Equilibrium unstable

  46. 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)

  47. 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.

  48. The importance of being nonlinear Linear models can be: Cycles in linear system require Frisch/Hicks/Econometrics approach Harrod’s 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 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

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