2The Problem Economy is dynamic Exists in time Changes over time But economists analyse it “as if” static—ignore timeSome mathematicians (e.g., Blatt) see this as “immaturity”How to reconcile dynamic reality & static methods?Argue static determines long termShort-term cycles explained by external shocks to stable economic systemDon’t! Develop dynamic, nonequilibrium economics insteadBoth approaches compete in literature
3An Analogy Riding a bicycle do you need to know how to balance a stationary bicycle before you can ride it?Economist: “Yes!: you must learn statics before you can do dynamics”(1) Learn how to balance bike while stationary(2) Ride in straight line, using skills acquired in (1)(3) Turn bike…? How? Try handlebars(4) Fall flat on face!Real world: “No!”Dynamic art of riding bike exploits centripetal forces which don’t exist when bike is stationaryStatic art of balancing irrelevant to dynamic art of riding!So why do economists “do” Statics?
4The 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 : 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:
520th 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)
620th 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 DepressionFrisch and exogenous explanation for economic cyclesHarrod’s endogenous explanation
7The beginnings of dynamics Frisch 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 econometricsDominant method: fit “linear stochastic” model to economic data
8Harrod: Growth & cycle theory Criticised Frisch paradigmDivorces growth from cycles when “the trend of growth may itself generate forces making for oscillation” (OREF II 38)Has no explanation for growth or shocksDeveloped combined theory of growth/cycleBasic method: extension of Keynes’s GT into dynamicsHis dynamic equilibrium unstable: nonequilibrium modelDerivation starts from static Keynesian equality of S and I:
9Harrod’s “knife edge” Savings ratio Rate of growth Incremental stock tooutput ratio (“ICOR”)
10Harrod’s “knife edge” Types of growth Actual growth: g.cp=s cp actual ICOR: actual accumulation of stocks in given period“Warranted growth”: what fulfilled capitalist expectationsgw.c=sc desired ICOR: ratio of change in stocks to rate of growth that capitalists want“Natural”: maximum sustainable rate of growthgn
11Harrod’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=sCapitalists will increase orders to restore desired ICORGrowth acceleratesIf actual growth below warranted, then actual ICOR (accumulation of stocks) more than desired ICOR:If g < gw, then cp > cCapitalists decrease orders to restore desired ICORGrowth declinesDynamic equilibrium unstable
12Harrod’s “knife edge” Explains growth and cycles If g>gw economy boomseventually hits overfull employment constraintseconomy turns downIf g<gweconomy slumpshits rock bottomneed to replace equipment (depreciation) forces +ive investmentrestarts upward pattern
13Hicks “interprets” Harrod Hicks 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 asDesired I function of change in Y:Define actual consumption asTherefore actual saving isEquate the two (Keynesian S=I):2nd order difference equation:
14Hicks “interprets” Harrod Cycles alright, but whatever happened toGrowth?“”Knife-edge” instability?
15Hicks “interprets” Harrod c<1, cycles peter out:c>1, cycles explode:For realistic values of c, impossibly large cycles & eventually explosive negative collapse of Y
16Hicks “interprets” Harrod ProblemsEquation generates cycles, but not growth: Ye = zero!Cycles unstable for c > 1But c similar to v, the accelerator: ratio of capital stock to outputv between 2 & 3 for most countries“Solutions”Assume exogenous growth at “natural” rateAssume c < 1Assume exogenous shocks to explain persistence of cycle
17Hicks “interprets” Harrod Hicks interpretation dominates trade cycle theoryGrowth becomes separate topic, dominated by neoclassical modelsHicks approach extended/modified by Samuelson, DomarLed nowhere; interest in trade cycle declined over 60-70sRevival in 80s with neoclassical “real business cycle” modelsBut Hicks’s model based on an economic errorEquation results from equating desired investment to actual savingsKeynesian S=I applies ex-post: actual figures onlyCorrecting this:
18Hicks “interprets” Harrod Actual investment is defined as change in capital:orWe can relate this to output using the accelerator:Expanding, this is definitely not equal to Hicks’s relationorNor is his term for savings really actual savings:Savings a residual from this year’s income, not last year’s
19Hicks “interprets” Harrod Defining savings:Equate actual savings and actual investment:A first order growth equation: growth but no cycles!
20Hicks “interprets” Harrod Hicks’s 2nd order difference equation modelTherefore a complete waste of timeEffectively asks the question “what level of output will guarantee that desired investment equals actual savings?”The answer? Zero output! So-called trade cycle is just oscillations en route to what mathematicians call “the trivial solution”
21The rise & fall (& rise?) of dynamics Unfortunately, Hicks’s useless model dominated early attempts at economic dynamicsFailure to get useful results led to waning of initial 1950s enthusiasm for dynamicsEquilibrium analysis continued to dominate economicsPinnacle 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…
22Dynamics & Stability of General Equilibrium “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)
23Dynamics & 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)
24Dynamics & Stability of General Equilibrium Modern mathematics shows Walras’ belief incorrectIn model with production and growthAll input & net output quantities are non-negativeCan’t use negative quantities of an inputCan’t produce negative quantities of any output and have sustainable growthAll prices must be non-negativeProcess can be modelled with matrix of input-output quantities, all non-negativeMatrix plays two roles in model: it determines output dynamics; its inverse determines price dynamicsWill “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?
25Dynamics & Stability of General Equilibrium Answer 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—instabilityIf 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 equilibrium
26Dynamics & 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: ; (1961). 'Stability of a dynamic input-output system', Review of Economic Studies, 28: ; (1963) 'Stability of a dynamic input-output system: a reply', Review of Economic Studies, 30:McManus, M., (1963). 'Notes on Jorgenson’s model', Review of Economic Studies, 30:See Blatt, Dynamic Economic Systems, Ch. 6Response of GE modellers?Ignore issue of the time process by which a market economy does or does not approach equilibrium:
27Dynamics & Stability of General Equilibrium “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)Strong words?Consider Debreu on time...
28Dynamics & Stability of General Equilibrium “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)…”
29Dynamics & 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)For real dynamics...
30Time matters… Have to treat time seriously Processes occur in time Dynamic process over time need not converge to equilibriumBasic mathematical techniques for handling time-based processes are differential and difference equationsEssential aspect of mathematical models of real-world processes is nonlinearitySimplest relationship between two variables (apart from none at all) is linear: double independent, double the dependentReal world relationships between variables in real world dynamic systems are nonlinear (I.e., not that simple!)…
31The importance of being nonlinear Economist attitudes garnered from understanding of linear dynamic systemsStable linear systems do move from one equilibrium to anotherUnstable linear dynamic systems do break downStatics is the end point of dynamics in linear systemsSo economics correct to ignore dynamics if economic system islinear, ornonlinearities are minor;one equilibrium is an attractor; andsystem always within orbit of stable equilibriumOtherwiseNonlinearity necessary for proper dynamics
32The importance of being nonlinear Behavior 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 systemsNonlinear systems can have unstable equilibria and not break down
33The importance of being nonlinear Kaldor (1940) first economist to realise importance of nonlinearityBegan with linear modelRealised 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 determinateSystem therefore oscillates—never converges to equilibrium, but never reaches unattainable values either
34The importance of being nonlinear Linear models can be:Cycles in linear system requireFrisch/Hicks/Econometrics approachHarrod’s initial model
35The importance of being nonlinear Nonlinear systems can be:Cycles can occur because system is:Not so different from linear modelCompletely unlike linear modelAn example: Lorenz’s weather model
36The importance of being nonlinear Any complex model can be simulated using a polynomiala + bx + cx2 + dx3 + …Near equilibrium, linear components of model dominateIf equilibrium unstable, linear components push system away from equilibriumFor x<1, x>x2>x3>…Far from equilibrium, nonlinear components dominate & push system back towards equilibriumFor x>1, x3>x2>x> …Many non-mainstream nonlinear models developedOne example: Goodwin’s “predator-prey” model (1967)Based on Marx’s model, Capital I Ch. 25 (see lecture Week 10)
37Sample nonlinear model Marx’s modelHigh wages—>low investment—>low growth—>rising unemployment—>falling wage demands—>increased profit share—>rising investment—>high growth—>high employment—>High wages: cycle continuesGoodwin draws analogy with biology “predator-prey” modelsRate 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:
38Nonlinear bits stabilise far from equilibrium Predator-Prey cyclesFood supplyRate of growthof FishInteractions with SharksInteractions with FishNonlinear bits stabilise far from equilibriumRate of growthof SharksRate of deathin absence ofFish to eatLinear bitsunstablenear equilibriumGenerates a cycle:Lots of fish—>lots of interactions with Sharks—>rapid growthof Sharks—>Fall in Fish numbers—>less interactions with Sharks—>Fall in Shark numbers—>Lots of fish again...
39system will never reach it Predator-Prey cyclesEquilibrium here, butsystem will never reach itLinear forces push awayNonlinear forces push back inSystem cycles indefinitely
40Current state of dynamics in economics Undergoing revival since mid-80s3 streamsNeoclassical“real business cycle”Increasing returns to scale explanationStill using linear modelsNo longer major element of modern theoryNon-neoclassicalKaldor/Goodwin based nonlinear models“Complexity” analysisInspired by “chaos theory” in physics, evolution in biology
41Complexity TheoryNonlinear dynamic systems can develop complicated behaviour from interaction of simple rulesSystems live on border between “chaos” and “order”Tiny changes can push system from order into chaosUndermines “rational expectations” (Week 7):Impossible to predict course of complex systemExample: lemmingsRate of growth of lemmings+ive fn of current population-ive fn of overcrowdingCombining:
42overshoots equilibrium, Complexity TheoryFor low values of a,tapers tostable equilibrium:For a=2, a 2-valuedcycle: populationovershoots equilibrium,then undershoots, etc.For a > 2.7, apparentlyrandom pattern…But nothing random about it: deterministic nonlinear models can generate sustained, unpredictable aperiodic cycles—like those in real world systems
43The story today Dynamics now “hot” area of economics Much interaction with other sciencesBiologyComputingPhysicsNon-neoclassical models now match neoclassicals in mathematical sophisticationEconomics may finally “grow up”but most economists today still woefully ignorant of dynamics:
44Ignorance 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)
45Conclusion: Mathematicians & Physicists on… “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 physicsModern 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”…
46Conclusion: The debate goes on… Some physicists now developing “econophysics” to apply these new ideas to economicsSee for exampleInfluences on economics also coming from evolutionary theorySee for exampleCompeting schools still alive and wellTiny fragmented minority still developing what they call Marxian economicsSee for exampleSubstantial, more cohesive minority of Post KeynesiansQuasi-neoclassical Austrian school accepts rejects equilibrium analysis, focuses on uncertainty
47Conclusion: The end of history? These and other non-neoclassical approaches considered in Political Economy (next semester)Many economists believe History of Economic Thought is about the past of economicsBut current state of economics far from “cut & dried”Many competing paradigms (though one dominant)Many unresolved disputesHistory of Economics far from over…