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So why do economists  do Statics?ZPZEZZZZ#ZPE 7*=#  (ZThe early days: Statics  because it was easy  . Historically, the  KISS principle:  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 1871 [1911]: 93)  ...dynamics includes statics... But the statical solution& is simpler...; 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) Founding fathers expected their successors to develop dynamic analysis, working from their static foundations:<$ZlZoZ$lo\ ' "  o )(20th Century as the Century of Dynamics? )  A point on which opinions differ is the capacity of the pure theory of Political Economy for progress. There seems to be a growing impression that, as a mere statement of principles, this science will sonn be fairly complete& It is with this view that I take issue. The great coming development of economic theory is to take place, as I venture to assert, through the statement and the solution of dynamic problems. (J.B. Clark [father of marginal productivity theory of distribution] 1898: 1) Why does dynamics matter (according to Clark)?:  A static state is imaginary. All actual societies are dynamic& Heroically theoretical is the study that creates, in the imagination, a static society. (Clark: 1898: 9)J  H5&   *(20th Century as the Century of Dynamics? )  It [dynamic analysis] will bring the society that figures in our theory into a condition that is like that of the real world. It will supply what a static theory openly and intentionally puts out of sight; namely, changes that alter the mode of production, and act on the very structure of society itself. (Clark 1898: 10-11) Great expectations& but little done until Great Depression Frisch and exogenous explanation for economic cycles Harrod s endogenous explanation6I;UI;U&   }The beginnings of dynamicsFrisch 1933: trade cycle explained  by the fact that certain exterior impulses hit the economic mechanism and thereby initiate more or less regular oscillations Underlying highly stable  propagation mechanism (like rocking horse) Random shocks from outside ( impulses ) Each shock sets up single regular harmonic pattern (like stone in pool of water) Overlay of many shocks gives irregular cycles (lots of stones) Gave rise to econometrics Dominant method: fit  linear stochastic model to economic dataL@@~Harrod: Growth & cycle theory4Harrod (1939) Criticised Frisch paradigm Divorces growth from cycles when  the trend of growth may itself generate forces making for oscillation (OREF II 38) Has no explanation for growth or shocks Developed combined theory of growth/cycle Basic method: extension of Keynes s GT into dynamics His dynamic equilibrium unstable: nonequilibrium model Derivation starts from static Keynesian equality of S and I:xN'T6)D *Harrod s  knife edge   *Harrod s  knife edge RTypes of growth Actual growth: g.cp=s cp actual ICOR: actual accumulation of stocks in given period  Warranted growth : what fulfilled capitalist expectations gw.c=s c desired ICOR: ratio of change in stocks to rate of growth that capitalists want  Natural : maximum sustainable rate of growth gnE;Y.  <; W.  $ *Harrod s  knife edge Reciprocal relation between g & gw: If actual growth exceeds warranted, then actual ICOR (accumulation of stocks) less than desired ICOR: If g > gw, then cp < c since both g.cp=gw.c=s Capitalists will increase orders to restore desired ICOR Growth accelerates If actual growth below warranted, then actual ICOR (accumulation of stocks) more than desired ICOR: If g < gw, then cp > c Capitalists decrease orders to restore desired ICOR Growth declines Dynamic equilibrium unstable4$ZfZzZdZ\ZZ!f    Qd  JPnp *Harrod s  knife edge Explains growth and cycles If g>gw economy booms eventually hits overfull employment constraints economy turns down If g WR$ 2Hicks  interprets HarrodfHicks could not accept that equilibrium unstable:  A mathematically unstable system does not fluctuate; it just breaks down (OREF II 56) Reworked Harrod s model: Define growth asL2X2X 2Hicks  interprets Harrod2Hicks  interprets Harrod2Hicks  interprets HarrodProblems Equation generates cycles, but not growth: Ye = zero! Cycles unstable for c > 1 But c similar to v, the accelerator: ratio of capital stock to outputP PF ,#F2Hicks  interprets HarrodHicks interpretation dominates trade cycle theory Growth becomes separate topic, dominated by neoclassical models Hicks approach extended/modified by Samuelson, Domar Led nowhere; interest in trade cycle declined over 60-70s Revival in 80s with neoclassical  real business cycle models But Hicks s model based on an economic error Equation results from equating desired investment to actual savings Keynesian S=I applies ex-post: actual figures only Correcting this:pLZZ- 3,|72Hicks  interprets Harrod 2Actual investment is defined as change in capital: 3 <2Hicks  interprets Harrod Defining savings:  =2Hicks  interprets Harrod XHicks s 2nd order difference equation model $- " - >%The rise & fall (& rise?) of dynamics & Unfortunately, Hicks s useless model dominated early attempts at economic dynamics Failure to get useful results led to waning of initial 1950s enthusiasm for dynamics Equilibrium analysis continued to dominate economics Pinnacle of this  general equilibrium & but even this raises dynamic issues: Is Walras general equilibrium dynamically stable? Walras himself abstracted from dynamic processes by assumptions of  tatonnement by an  auctioneer who did not allow any trades to take place until equilibrium was achieved& <*3*3`( >  ` /+Dynamics & Stability of General Equilibrium , F First, let us imagine a market in which only consumer goods and services are bought and sold... Once the prices or the ratios of exchange of all these goods and services have been cried at random& , each party to the exchange will offer at these prices those goods or services of which he thinks he has relatively too much, and he will demand those articles of which he thinks he has relatively too little for his consumption during a certain period of time. & the prices of those things for which the demand exceeds the offer will rise, and the prices of those things of which the offer exceeds the demand will fall. New prices now having been cried, each party to the exchange will offer and demand new quantities. And again prices will rise or fall until the demand and the offer of each good and each service are equal. Then the prices will be current equilibrium prices and exchange will effectively take place. (Walras 1874)JZac i[&   0+Dynamics & Stability of General Equilibrium , Presumed process would eventually converge because direct effects would outweigh indirect:  This will appear probable if we remember that the change from p b to p  b, which reduced the above inequality to an equality, exerted a direct influence that was invariably in the direction of equality at least so far as the demand for (B) was concerned; while the [consequent] changes from p c to p  c, p d to p  d, ..., which moved the foregoing inequality farther away from equality, exerted indirect influences, some in the direction of equality and some in the opposite direction, at least so far as the demand for (B) was concerned, so that up to a certain point they cancelled each other out. Hence, the new system of prices (p  b, p  c, p  d, ...) is closer to equilibrium than the old system of prices (p b, p c, p d, ...); and it is only necessary to continue this process along the same lines for the system to move closer and closer to equilibrium. (Walras 1874; my emphasis)zZTmCYzl            >      ?         1+Dynamics & Stability of General Equilibrium , ~Modern mathematics shows Walras belief incorrect In model with production and growth All input & net output quantities are non-negative Can t use negative quantities of an input Can t produce negative quantities of any output and have sustainable growth All prices must be non-negative Process can be modelled with matrix of input-output quantities, all non-negative Matrix plays two roles in model: it determines output dynamics; its inverse determines price dynamics Will  pull of direct effects exceed sum of  push & pull of indirect effects, as Walras surmised? Both price & quantity dynamics must be stable for stable outcome  will process converge to equilibrium price & quantities, or diverge?2Z$Z3ZvZ ZZcZZ2$3v  c@     2+Dynamics & Stability of General Equilibrium , 8Answer depends on key characteristic of matrix: its  dominant eigenvalue (German for  principal value ) tells how much the matrix  stretches space. If greater than certain value, matrix stretches space instability If less, matrix contracts space stability (key value 1 for discrete models, 0 for continuous ones) Perron-Frobenius theorem proves that the dominant eigenvalue of matrix with all non-negative entries is greater than zero. Matrix and its inverse have inverse eigenvalues: e.g., if dominant eigenvalue of production matrix is 1/10, then dominant eigenvalue of price matrix is 10. So either the quantity matrix or its inverse must have eigenvalue>1. So& ? Either price or quantity process will be unstable: either quantities or prices (possibly both) will not converge to equilibriumjjZZ9ZZj9a    >    "  c    ,  N   3+Dynamics & Stability of General Equilibrium , Mathematical results proved in early 1900s; considered in economic literature in 1960s: Jorgenson, D.W., (1960) 'A dual stability theorem', Econometrica 28: 892-899; (1961). 'Stability of a dynamic input-output system', Review of Economic Studies, 28: 105-116; (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. See Blatt, Dynamic Economic Systems, Ch. 6 Response of GE modellers? Ignore issue of the time process by which a market economy does or does not approach equilibrium:XZZZbZX4 DNB b&   4+Dynamics & Stability of General Equilibrium , t For Walras, general equilibrium theory was an abstract but nevertheless realistic description of the functioning of a capitalist economy. He was therefore more concerned to show that markets will clear automatically via price adjustments in response to positive or negative excess demand & than to prove that a unique set of prices and quantities is capable of clearing all markets simultaneously. By the time we got to Arrow and Debreu, however, general equilibrium theory had ceased to make any descriptive claim about actual economic systems and had become a purely formal apparatus about a quasi economy. It had become a perfect example of what Ronald Coase has called  blackboard economics , a model that can be written down on blackboards using economic terms like  prices ,  quantities ,  factors of production , and so on, but that nevertheless is clearly and even scandalously unrepresentative of any recognizable economic system. (Blaug 1998)Zt         5+Dynamics & Stability of General Equilibrium , D For any economic agent a complete action plan (made now for the whole future), or more briefly an action, is a specification for each commodity of the quantity that he will make available or that will be made available to him, i.e., a complete listing of the quantities of his inputs and of his outputs... For a producer, say the jth one, a production plan (made now for the whole future) is a specification of the quantities of all his inputs and all his outputs... The certainty assumption implies that he knows now what input-output combinations will be possible in the future (although he may not know the details of technical processes which will make them possible)&  n0     P p _ $JT6+Dynamics & Stability of General Equilibrium,+ +  As in the case of a producer, the role of a consumer is to choose a complete consumption plan... His role is to choose (and carry out) a consumption plan made now for the whole future, i.e., a specification of the quantities of all his inputs and all his outputs. The analysis is extended in this chapter to the case where uncertain events determine the consumption sets, the production sets, and the resources of the economy. A contract for the transfer of a commodity now specifies, in addition to its physical properties, its location and its date, an event on the occurrence of which the transfer is conditional. This new definition of a commodity allows one to obtain a theory of uncertainty free from any probability concept and formally identical with the theory of certainty developed in the preceding chapters. (Debreu 1953; my emphases)X ZHZ     0 .   8Time matters&   &Have to treat time seriously Processes occur in time Dynamic process over time need not converge to equilibrium Basic mathematical techniques for handling time-based processes are differential and difference equations Essential aspect of mathematical models of real-world processes is nonlinearity Simplest relationship between two variables (apart from none at all) is linear: double independent, double the dependent Real world relationships between variables in real world dynamic systems are nonlinear (I.e., not that simple!)& D P@     ,!The importance of being nonlinear! Economist attitudes garnered from understanding of linear dynamic systems Stable linear systems do move from one equilibrium to another Unstable linear dynamic systems do break down Statics is the end point of dynamics in linear systems So economics correct to ignore dynamics if economic system is linear, or nonlinearities are minor; one equilibrium is an attractor; and system always within orbit of stable equilibrium Otherwise Nonlinearity necessary for proper dynamicsJZZ?Z{Z Z+ZJF-?{ + <7  b 4 !The importance of being nonlinearHBehavior of modified Hicks-Harrod model a  quirk : Linear model (just constants and variables, no powers, etc.) But generates sustained cycles (for c>=v) Most linear models: Cycle to equilibrium (c<1 in Hicks) Rigid cycles (c=1 in Hicks) Unstable (c>1 in Hicks) Frisch/Hicks argument that  unstable system & just breaks down only true for linear systems Nonlinear systems can have unstable equilibria and not break downt3gX]B3gX]B ,!The importance of being nonlinearKaldor (1940) first economist to realise importance of nonlinearity Began with linear model Realised that this had only 2 states: either  dangerous instabilities or  more stability than the real world appears, in fact, to possess Deduced that therefore, economic relations  cannot be linear Nonlinearity makes it possible for equilibria to be unstable, and yet model to be determinate System therefore oscillates never converges to equilibrium, but never reaches unattainable values eitherDZ>ZfZ>Z^ZiZD>f>^iP3    !The importance of being nonlinear !The importance of being nonlinearNonlinear systems can be:;!The importance of being nonlinear! Any complex model can be simulated using a polynomial a + bx + cx2 + dx3 + & Near equilibrium, linear components of model dominate If equilibrium unstable, linear components push system away from equilibrium For x<1, x>x2>x3>& Far from equilibrium, nonlinear components dominate & push system back towards equilibrium For x>1, x3>x2>x> & Many non-mainstream nonlinear models developed One example: Goodwin s  predator-prey model (1967) Based on Marx s model, Capital I Ch. 25 (see lecture Week 10)x6ZZ6ZMZZ[ZZcZ>Z66M   [   c>H:    O Sample nonlinear modelMarx s model High wages >low investment >low growth >rising unemployment >falling wage demands >increased profit share >rising investment >high growth >high employment >High wages: cycle continues Goodwin draws analogy with biology  predator-prey models Rate of growth of prey (fish >capitalists!) depends +ively on food supply and -ively interactions with predator (shark >workers) Rate of growth of predator depends -ively on number of predators and +ively on interactions with prey: N : :P4QPredator-Prey cyclesPredator-Prey cycles$&Current state of dynamics in economicsUndergoing revival since mid-80s 3 streams Neoclassical  real business cycle Increasing returns to scale explanation Still using linear models No longer major element of modern theory Non-neoclassical Kaldor/Goodwin based nonlinear models  Complexity analysis Inspired by  chaos theory in physics, evolution in biology+ k&<+ k  &<r%Complexity Theory<Nonlinear dynamic systems can develop complicated behaviour from interaction of simple rules Systems live on border between  chaos and  order Tiny changes can push system from order into chaos Undermines  rational expectations (Week 7): Impossible to predict course of complex system&//1 &Complexity Theory'The story today"Dynamics now  hot area of economics Much interaction with other sciences Biology Computing Physics Non-neoclassical models now match neoclassicals in mathematical sophistication Economics may finally  grow up but most economists today still woefully ignorant of dynamics:LJo?Jo?+>Ignorance of dynamics the rule& Jevons/Marshall attitude still dominates most schools of economic thought, from textbook to journal: Taslim & Chowdhury, Macroeconomic Analysis for Australian Students:  the examination of the process of moving from one equilibrium to another is important and is known as dynamic analysis. Throughout this book we will assume that the economic system is stable and most of the analysis will be conducted in the comparative static mode. (1995: 28) Steedman, Questions for Kaleckians:  The general point which is illustrated by the above examples is, of course, that our previous '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. (Steedman 1992: 146)eZZe.ial_ I  -VConclusion: Mathematicians & Physicists on&  , N A baby is expected to first crawl, then walk, before running. But what if a grown-up man is still crawling? At present, the state of our dynamic economics is more akin to a crawl than a walk, to say nothing of a run. Indeed, some may think that capitalism as a social system may disappear before its dynamics are understood by economists. (Blatt 1983: 5) Mirowski, More Heat than Light claims neoclassical economics based on analogy to 19th century physics Modern physics based on completely different paradigm today (quantum mechanics, thermodynamics) Many physicists quite disparaging about modern economics  Unreal assumptions ,  absurd proposition/lemma style of argument & f+Z9ZDZo5q9D&e  ; :>Conclusion: The debate goes on&   Some physicists now developing  econophysics to apply these new ideas to economics See for example http://www.unifr.ch/econophysics/ Influences on economics also coming from evolutionary theory See for example http://www.orgs.bucknell.edu/afee/ Competing schools still alive and well Tiny fragmented minority still developing what they call Marxian economics See for example http://www.marx.org Substantial, more cohesive minority of Post Keynesians http://csf.colorado.edu/pkt Quasi-neoclassical Austrian school accepts rejects equilibrium analysis, focuses on uncertainty http://www.hayekcenter.org/TZ2Z=Z3Z'ZKZ$Z7ZZ`ZZT2=3'K $  7`   8        N          8      a     e!0d[!0]!0x^!0_!0?Z9Conclusion: The end of history?  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