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Week Thirteen Economic Dynamics
Week Thirteen
Economic Dynamics
The Problem• Economy is dynamic
– Exists in time– Changes over time
• But economists analyse it “as if” static—ignore time– Some mathematicians (e.g., Blatt) see this as
“immaturity”• How to reconcile dynamic reality & static methods?
– Argue static determines long term• Short-term cycles explained by external shocks to
stable economic system• Don’t! Develop dynamic, nonequilibrium economics
instead• Both approaches compete in literature
An 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 stationary
– Static art of balancing irrelevant to dynamic art of riding!
• So why do economists “do” Statics?
The 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:
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 The great coming development of economic theory is to take great coming development of economic theory is to take place, as I venture to assert, through the statement and place, as I venture to assert, through the statement and the solution of dynamic problemsthe solution of dynamic problems.” (J.B. Clark [father of marginal productivity theory of distribution] 18981898: 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)
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 explanation
The 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 econometrics
– Dominant method: fit “linear stochastic” model to economic data
Harrod: Growth & cycle theory• Harrod (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:
Harrod’s “knife edge” Divide both sides by YS I
Savings ratio
Incremental stock tooutput ratio (“ICOR”)
Rate of growth
S Multiply LHS by YI YY Y
S Restating:I YY Y Y
S Rearranging:YI YY Y
s c g
Harrod’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
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
• gn
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 unstable
Harrod’s “knife edge”• Explains growth and cycles
– If g>gw
• economy booms• eventually hits overfull employment
constraints• economy turns down
– If g<gw
• economy slumps• hits rock bottom• need to replace equipment (depreciation)
forces +ive investment• restarts upward pattern
Hicks “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 as 1
1
t t
t
Y YgY
– Desired I function of change in Y:
1 2t t tI c Y Y – Define actual consumption
as 11t tC s Y
– Therefore actual saving is
11t t t t tS Y C Y s Y
– Equate the two (Keynesian S=I):
1 2 11t t t tc Y Y Y s Y
– 2nd order difference equation:
1 21t t tY s c Y cY
Hicks “interprets” HarrodHicks's Difference Equation:
Cyclical Behaviour Through Lags
Periods0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
-200
-150
-100
-50
0
50
100
•Cycles alright, but whatever happened to•Growth?•“”Knife-edge” instability?
Hicks “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
Hicks “interprets” Harrod• Problems
– 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 output Kv
Y
– v between 2 & 3 for most countries
•“Solutions”–Assume exogenous growth at “natural” rate–Assume c < 1–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod• Hicks 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 errorBut 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:
Hicks “interprets” Harrod• Actual investment is defined as change in capital:
1t t tI K K
t tK v Y • We can relate this to output using the
accelerator:
1
1
1 2
t t t
t t
t t
I K Kv Y Yc Y Y
• Expanding, this is definitely not equal to Hicks’s relation
• Nor is his term for savings really actual savings:• Savings a residual from this year’s income, not last
year’s
1 1t t tK K I or
1
1
1 2
t t t
t t
t t
I K Kv Y Yc Y Y
or
Hicks “interprets” Harrod• Defining savings:
1
1t t t
t t
t
t
S Y CY s Ys Ys Y
• Equate actual savings and actual investment:
1t t t t tI v Y Y S s Y
1t tvY Y
v s
• A first order growth equation: growth but no cycles!
Hicks “interprets” Harrod• Hicks’s 2nd order difference equation model
1 21t t tY s c Y cY
• Therefore a complete waste of time• Effectively 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”
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…
Dynamics & 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)
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)
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?
Dynamics & 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—
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 equilibrium
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 Ignore issue of the time process by which a market economy does or does not approach market economy does or does not approach equilibrium:equilibrium:
Dynamics & 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)
Dynamics & 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)…”
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)
Time 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 Essential aspect of mathematical models of real-world processes is nonlinearityprocesses 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!)…
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 dynamics
The 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 systems– Nonlinear systems can have unstable equilibria
and not break down
The importance of being nonlinear• Kaldor (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 either
The importance of being nonlinear
Linear models can be:IInnddeetteerrmmiinnaanntt
Cycles in linear system require
SSttaabbllee eeqquuiilliibbrriiuummwwiitthh sshhoocckkss
Frisch/Hicks/Econometrics approach
UUnnssttaabbllee eeqquuiilliibbrriiuummwwiitthh bbaarrrriieerrss ((tthhuuss
nnoonnlliinneeaarr))
Harrod’s initial model
The importance of being nonlinear• Nonlinear systems can be:
GGlloobbaallllyy SSttaabbllee• Cycles can occur because system
is:LLooccaallllyy SSttaabbllee wwiitthh SShhoocckkss LLooccaallllyy uunnssttaabbllee:: ""FFaarrffrroomm EEqquuiilliibbrriiuumm""
Not so different from linear modelCompletely unlike linear modelAn example: Lorenz’s weather model Lorenz.vsm
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)
Sample nonlinear model• Marx’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:
Predator-Prey cycles
1
1
dF a b SF dt
dS c d FS dt
Rate of growthof Fish
Food supplyInteractions with Sharks
Rate of growthof Sharks
Rate of deathin absence of
Fish to eat
Interactions with Fish
• Generates 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...
dF a F b S FdtdS c S d F Sdt
Linear bitsLinear bitsunstableunstable
near equilibriumnear equilibrium
Nonlinear bitsNonlinear bitsstabilise far fromstabilise far from
equilibriumequilibrium
Predator-Prey cycles
0 200 400 600 8001000
900
1000
1100
Time
Fish
0 200 400 600 800100098
99
100
101
102
Time
Shar
ks
98 99 100 101 102
900
950
1000
1050
1100
Sharks
Fish
Equilibrium here, butsystem will never reach it
Linear forces push awayNonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics• Undergoing 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
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
•Example: lemmings–Rate of growth of lemmings
•+ive fn of current population 1 1t tL a L
•-ive fn of overcrowding 21t tL b L
•Combining: 21 1t t tL a L b L
Complexity Theory
For low values of a,tapers to
stable equilibrium:For a=2, a 2-valued
cycle: populationovershoots equilibrium,then undershoots, etc.
For a > 2.7, apparentlyrandom pattern…
But nothing random about it: deterministic nonlinear modelscan generate sustained, unpredictable aperiodic cycles—likethose in real world systems
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:
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 Throughout this book we will assume that the economic system is book we will assume that the economic system is stable and most of the analysis will be conducted in the stable and most of the analysis will be conducted in the comparative static modecomparative 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 'static' analysis does not 'ignore' time. To the contrary, that analysis allows 'ignore' time. To the contrary, that analysis allows enough time for changes in prime costs, markups, etc., enough time for changes in prime costs, markups, etc., to have their to have their fullfull effects effects.” (Steedman 1992: 146)
Conclusion: 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 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”…
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/
Conclusion: 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 economics• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)– Many unresolved disputes
