Business cycles and long waves : a behavioral disequilibrium perspective

HD28

.M414

no.

c|-5

WORKING

PAPER

ALFRED

P.

SLOAN SCHOOL

OF

MANAGEMENT

Business Cycles

and Long

Waves:

A

Behavioral

Disequilibriun^

Perspective

John

D.

Sterman

and

Erik

Mosekilde

WP#

3528-93-MSA

January,

1993

MASSACHUSETTS

INSTITUTE

OF

TECHNOLOGY

50

MEMORIAL

DRIVE

CAMBRIDGE,

MASSACHUSETTS

02139

Business

Cycles

and Long

Waves:

A

Behavioral

Disequilibrium Perspective

John

D.

Sterman

and

Erik

Mosekilde

WP#

3528-93-MSA

January,

1993

^

D-4308

Business

Cycles

and

Long

Waves:

A

Behavioral

Disequilibrium

Perspective*

John

D.

Sterman

SloanSchoolof

Management

MassachusettsInstituteof

Technology

Cambridge,

MA

02139

USA

and ErikMosekilde PhysicsLaboratory

m

Technical University ofDenmaric

2800

Lyngby

Denmark

December

1992

*

Forthcoming

in

Semmler,

W.

(ed.)BusinessCycles:

Theory

and

EmpiricalMethods.

EX)rdrecht:

Kluwer Academic

Pubhshers.

PleaseaddressccMrespondencetoJohn Stermanattheaddress

above

orjstennan@mit.edu.

This

work

was

supportedin part

by

the Sponsors ofthe

MTT

System Dynamics

National

Model

0^308

1.

Introduction

The

evolution ofthe

macrocconomy

reflects theinteractionof multiple

modes

ofbehavior.

By

a

mode

ofbehavior

we mean

a particularpatternof

dynamic

behavior, such asgrowthor fluctuation,

caused bya particularsetoffeedback processes.

The

mostimportant

mode

is thelong-term

expo-nential growthofthe world

economy.

This exponential growth, both cause andconsequenceof

industrialization, population growth,capitalaccumulation, technologicaladvance, and historical

accident, has accelerated dramatically since thebeginning ofthe industrial revolution,transforming

virtuallyevery aspectof ourworld, including economic, pxjlitical,cultural,and even

biogeochemi-caJ systems.'

Yet

economic

development aroundthegrowthtrend isfarfromsteady. Indeed,cyclical

fluctua-tions are a persistent featureof

economic

life.

Economic

historianshaveidentifiedseveral distina

cycles, including the shon-term businesscycle(3-7 years), theconstructionorKuznetscycle

(15-25 years),and the long

wave

or Kondratieff cycle(40-60years).

The

existence ofthesecycles is

notwithout controversy, however. Debatecontinuestoday about thecauses ofthe short-term businesscycle, the

most

extensively studied

mode.

The

causesand even theexistenceofthe

longer cyclesare still

more

controversial. In part,theuncertainty isempirical:

we

necessarilyhave dataforfewerlong cycles thanshortones. Yetinlarge

measure

thecontroversy is

due

toa lackof appropriate theory toaccountfordisequilibrium

dynamics

thatcanpersist foryearsoreven

decades.

Of

course, theoryanddataare entwinedin a feedbackloop: without theoryto guide empiricaltests, littleevidence fordisequilibrium

dynamics

suchaslong cycles

was

collected; with-outcompelling evidenceof longcycles, there

was

litdemotivationtodevelop

new

theory. Recent yearshave

wimessed

adramaticchange in both the theories available to

model

nonlinear,

disequi-librium

dynamics

such aslong

waves

and thedata supportingtheirexistence.

The

existence and

' _{Ultimaiely,ofcourse,growthof populationandmaterial production} willcease astheworldmakesa transiuonto a post-industriaJeconomyconsistentwith varioussocial,environmental andecologicallimits. Debatecontinuesas

tothe proximityofthe limits togrowth,the likelydynamicsofthe transition from growth,andthe susiainabilityof

IM308

causesofthelong

wave

are

now

reasonably wellestablished,

and

theoryis emergingtounderstand

how

the different cyclical

modes

inthe

economy

interactwithoneanother.

One

ofthe principalmysteriestheoristshavefacedis

why

there

seem

tobeonly a

few

distinct

periodicitiesratherthan cyclesatallfrequencies.

And how

mightthe different cyclical

modes

interact? Could, as

Schumpeter

(1939)argued, the coincident

downturn

ofthe businesscycle,

constructicMicycle,andlong

wave

accountforthe severityoftheGreatDepression?

More

fiindamentaUy,

why

should the frequenciesofthese cycleshave(roughly)

commensurate

periods,

so thattheirdownturns mightcoincide? Indeed,even if

one

admitsthe possibility that individual

firmsmightgeneratecycUcal

movements,

the differentparameterscharacterizing the structureand decision

making

processesofdifferentfirms

would

cause

them

to oscillate withdifferent

frequenciesandphases.

Why

then shouldtherebe aggregatecychcal

movements

at all?

UnfOTtunately,

macroeconomic

theoryhasbeenlargelysilent

on

theissuesof multiplemodes,

syn-chrcHiization,

and

entrainment.

The

problem

residesboth intheprevailingassumptionsof

rational-ityandequilibrium, neitherof

which

are

good

approximationsto actual

economic

systems,and in

thetoolsusedtoanalyze

economic

dynamics.

Over

thepast

few

decadesan impressive

body

of

evidencehasaccumulated

documenting

the

bounds on

human

rationality

(Simon

1982). Experimentalandfieldstudies inpsychology,

economics

andothersocial scienceshave

docu-mented

a

wide

rangeofheuristicspeopleuseto

make

decisionsin

complex

environments, andthe

many

systematicerrors

and

biasesthat result

(Kahneman,

Slovic,

and Tversky

1982,Hogarth

1987). Appropriatetheoriesof

economic

dynamics, and

economic

behavioringeneral,should

embody

models

ofdecision

making

consistentwith empirical

knowledge

(includingqualitativedata

andfieldstudy as well aseconometric analysis)oftheprocessesof

judgment

andchoice

managers

actually use

(Simon

1979,

Sterman

1987,Morecroft 1985).

The

analytical tools traditionally usedtostudy

economic

dy-namicshavealsoslowed

D^308

muluplccyclical modes. Inpan, this is becausedifferenceequationshave dominated

dynamic

analysis (see

Samuelson

1947, p. 380), and

many

difference equation models

do

not explicidy

identify the unit of time

between

periods' (e.g.

Samuelson

1939,

GoodvMn

1951) sothatthe

structures, parameters,and behaviorofsuch

models

cannotbe validated. It issimply

presumed

that the cyclesofthese

models

are theshort-term business cycle(sec

Low

1980foracritique).

More

important, despitenotable earlyexceptions (e.g.

Goodwin

1951, Kaldor 1940), until

recently

most

nxxlelsof

econonuc

cycles

were

linearornearlylinear(seee.g.

Day

1982, Lorenz

1989,

Semmler

1989for

modem

nonlinear approaches). But lineartheoryis notan appropriate foundation forthe studyof

economic dynamics

(Forrester 1987). First,

economic

systems distin-gtiishthemselves

from

most

systemsconsideredin the naturalsciencesby the prevalenceof

fKjsitivefeedback loops.

Well

kiwwn

examples

include theacceleratorand multiplierloops of Keynesiantheory. Otherpositive loops operate throughextrapolativeexpectations, agglomeration

effects, increasingreturns, the effectofinflationexpectationson realinterestratesandthus

aggregate

demand,

speculationand financialcrises,andsynergies and standards formation

among

and withintechnologiesforproduction,communication, andorganization (Sterman 1986a,

Graham

and

Senge

1980,Arthur 1988,

Semmler

1989).

Such

positivefeedbackscreate the

possibilityof strongly nonlinear behavior, the positiveloops

may

destabilizeotherwise convergent

processesofadjustment

which

then

grow

in amplitudeuntilconstrainedby variousnonlinearities.

Such

phenomena

cannot be understoodby

means

oflinearornearly-linearmodels.

Furthermore,ifthe

economic

systemwere linear,thecyclesproduced bydifferentfirms,

indus-tries,and nations

would

evolve independentiyofoneanotherandthe totalbehavior

would

bethe

linearsuperpositionoftheindependent modes.

While

individualfirms mightexhibit fluctuations,

the aggregateof

many

independentlyoscillatingfirms might bequiteconstant

-

there

would

beno business cycle as a

macroeconomic phenomenon. While

diffusionof business cycles has received

considerable empiricalattention, theoretical understanding of synchronization has lagged.

Thus

D^308

government

monetary andfiscalpoUcies,changesinaggregate

demand,

or highlycorrelated

shocks

and

expectations

(Bums

1969, Mitchell 1927;

Zamowitz

1985provides asurvey).

Modem

dynamicaltheoryoffersanother explanation: nonlinear

mode

locking. InnonUnear

sys-tems, superpositiondoes nothold. Instead,theperiodicitiesofcoupledoscillators

may

adjust to

one

anothertoachieve arationalratio,orwinding number.

Mode

lockinghasrecendyattracted

considerableinterestinthe natural sciences,especiallysinceithasbeenestablishedthat

mode

locking possesses a

number

ofuniversalfeaturesindependentof theparticularsystem understudy (Jensen,Bak,and

Bohr

1983, 1984).

The same

processesof entrainmenthavebeenobserved, for

instance, inpacednervecells (Colding-Jorgensen 1983),externallystimulatedheartcells(Glass,

Shrier,

and

Belair 1986),fluid

dynamics

(Glazieretal. 1986),coupledthermostaticallycontrolled

radiators (Togeby,etal. 1988),

and

forced

microwave

diodes (Mosekildeetal. 1990).

Mode

lockingprovidesanexplanationfortheentrainmentof

economic

fluctuations thatis

more

robust

thanpriorexplanations, andcreates the possibilityof nonlinear

phenomena

such as

period-doublingbifurcations, simultaneousmultiple periodicsolutions,anddeterministicchaos.

Mode

locking also givesrisetothe'devil's staircase',an unusualfractal structure

we

describe below.

We

beginby reviewingthe stylized factsof thedifferentcycles,then discussthe behavioral

foun-dationsforeach

mode

atthemicrolevel.

We

focus

on

thelonger cycles as theseare the

most

controversial andleastunderstood,particularlythe

economic

long wave.

We

surveythe principal

theoriesof the

economic

long

wave

thathave

emerged

inthe pastdecade, specificallytheintegrated

thcOTy devel(^)ed

by

the

MIT

System Dynamics

Group

andthe neo-Schumpeterianinnovation

theories.

To

illustratethetypeofbehavioral, nonlinear disequilibrium theory

we

advocate,

we

presentasimplenxxlelofthe long

wave

and

show

how

the

wave

arisesthroughinteractions

anoonglocally rationaldecision rules

embedded

ina nonlinearfeedbacksystem.

Next,

we

usethe

model

toconsiderinteractions

among

themodes.

The

theory ofnonlinear en-trainment

and

mode

lockingshedslight

on

why

there area small

number

of

modes

ratherthan

cy-0^308

clcsofallfrequencies, and

why

there areaggregate

movements

at all ratherthan firm orindustry

level cyclesthat

wash

out at the

macroeconomic

level.

We

conclude with implicationsfor the

development of empiricallygrounded,behavioral, disequilibriumtheoriesof

economic

dynamics.

2.

Economic

Dynamics:

Multiple

Modes

of

Behavior

The most

thoroughly analyzedcyclical

mode

inthe

economy

isthe short-term(3-7 year) business cycle (Mitchell 1927.

Gordon

1951,

Moore

1961.

Zamowitz

1985).illustrated in figure 1 by

US

industrialproduction andcivilian

unemployment

for theperiod 1947 to 1992.

With

characteristic

phaseshifts andampliuides,the short-termbusiness cycle manifestsclearlyin ahostof aggregates andindustry leveldataincluding capacityutilization, inventorycoverage, helpwanted advertising,

interest rates, etc.

Among

the

well-known

characteristicsoftheshort-termcycleisthe

phenomenon

ofamplification, in

which

tbe amplitudeofthecycle increasesasone

moves

fromthe

productionof

consumer

goodstointermediatesto

raw

materials (figure2). Theories ofthe shon-termcycleshould explain these details aswellasgenerate a fluctuationinoutput withthe

appro-priate period,amplitude, phaserelations, andvariability.

Many

time series alsoprovide evidencefor theexistence ofa 15-25 year construction (orKuznets)

cycle

(Riggleman

1933,

Hoyt

1933.

Long

1940.Kuznets 1973).

An

example

is givenin figure 3

showing

thevacancyrateof

commercial

officespace in Bostonfrom 1952 to 1990. Similar cycles can alsobe found,forinstance, inproduction capacity ofthepaperindustry (Randers 1984) orin

capacity utilizationofthe worldoil tankerfleet

(Bakken

1992).

At

theindustryor regional level,

the amplitudeoftheconstructioncycleisoften so highthatnonlinearitiesare clearly involved,For example,during bust periodsthe rateof

new

construction fallsnearlyto zeroforextended periods,

andexcess capacity declinesatarateconstrainedby the lifetimeofcapital stocks.

fib.

3

t^^^

_{Let usconsiderthe processesthatproduce these}

_two

_{distinctmodes.}

_We

focus, initially,on the

dynamics

ofindividual firms, andlater consider

how

such firms

may

become

entrained withone anotherandwith thegovernment, consumer, and financialsectors toproduceacoherent aggregate

D^308

cycle. Consideramanufacturingfirminequilibrium,assumingforsimplicity that thefirmissmall

relative tothe labor, capital andother input markets, sothatfactorpricescanbe considered

con-stant.

Now

consider the firm'sresponsetoanunanticipatedstepincreaseinincomingorders.

The

company

willeventually

expand

outputto

meet

orders. In thelongrun,production, material con-sumption,

woik

force

and

capital stockallrise inprqK>Ttion toincomingorders.

The

questionis

how

the transient willunfold.

Ifallinputscould beadjustedimmediately,the transient

would

befastand nonoscillatory.

However,

factorinputscannotchangeinstantly. Backlogs andinventories buffertheproduction

line

from

short termvariationsin

demand

toprovide timefor efficientadjustment ofinputs. In

fact,immediatelyfollowingthe

demand

shock thefirm

may

notchangeproductionatall, untilit

becomes

clear thatincomingorderswilljjemain atthe new,higherlevel. Inventoriesnecessarily

fall. Optimalinventory

may

also increaseastheexpectedthroughputrises.

To

restore inventoryto

desiredlevels,thefirm

must

increaseproductionabovethe rateofincomingordersforat least

some

periodoftime.

As

aconsequence,ordersformaterialsandintermediate goods

must

also

in-crease

above

therateof

incoming

orders,passing a largerdisturbance

on

tothe supplying

indus-tries. Thisprocess, thefamiliarinventoryaccelerator,providesanexplanationfor theamplification ofthebusinesscycle

from

the

consumer goods

sectorthrough theintermediate

goods

andfinally to

the

raw

materials sector(T.Mitchell 1923,Metzler 1941, Forrester

1%1, Mass

1975).

The

amplificationof

demand

shocksateachstageofproductionis aninevitableconsequence of

threefundamentalfeaturesof production: (1) theexistenceof decision-making

and

physical delays

inadjustingproductionto

demand

shocks(e.g.forecastingandadministrativelags, lags in factor

acquisition); (2) theexistenceofstockssuchas inventories,

work

inprocess,and backlogs

which

buffer thedifference

between

ordersandoutput; and (3)theneedto adjustthese stocks towards

targetvalues

when

shocksoccur(torestoreinventorytoinitiallevels afteranunanticipated

demand

D-1308

The

inevitabilityofamplification,however, doesnot

mean

thatoscillation issimilarlyinevitable

(byoscillation is

meant

a systemthat isless thancriticallydamped).

The

existence, stability, and

frequency ofoscillatory responseto

demand

shocksdepends on thenatureofthefeedback processesby

which

afirmadjustsoutput to

demand,

aswellas the myriadcouplings

among

the

firm,itssuppliers, customers,and otheractorsin the

economy.

Providedthat neededmaterials are available,smallchangesinoutputtrjaybe accomplishedquickly through

more

intensiveuseofexisting

employees

(overtime).

From

acontrol-theoreticpx)intof view, theuse of

workweek

toregulate inventorycreatesan effectively first-ordernegative feedback loop

which

is non-oscillatoryand adds

damping

tothe system (Sterman 1988).

However,

the

workweek

response is nonlinear: itis limitedbythecostof overtime, by decreasingworker

pro-ductivityafterlong

work

weeks, and ultimatelyby thelength oftheday. Thus, while small

ampli-tudechangesin

demand

can be

ascommodated

through overtime, larger and

more

persistent

changes saturatethe

workweek

feedback, requiringexpansionofthe

work

force.

Expanding

the

work

force, however, involvessignificantdelays. Vacancies

must

be authorized,

new

employees

hired andtrained, andrime

must

pass before productivityrisesto thatof

experi-enced workers (comparabledelaysexistin thecaseof an unexpecteddecreaseindemand).

The

useof

employment

tocontrol inventorylevelsand respond to

demand

shockscreates anegative feedbackloop, but unlikethe

work

week

loop,the

employment

adjustmentloop involves delays on

theorder ofseveral

months

ormore. Negative feedbackloops with such phase lagelements are

oscillatory.

The

characteristic behaviorof

models

thatportray

workweek

and

work

force

adjust-mentswithrealistic decisionparameters is

damped

oscillationswithaperiodof 3to7 years

(Forrester 1961,

Mass

1975). These

models

also generatethe phase (leadandlag) and amplitude

relationshipsobservedin thedata foroutput,

employment,

inventories,deliverydelay, vacancies,

labor accessionand separation flows,andothervariables.

The

business cycle thesemodels

D^308

monetary

and

fiscalpolicies,andotherelementsofthe traditionalaggregate supply/aggregate

demand

model (Mass

1975, N. Forrester 1982).

Regulationofoutputby workforceadjustmentisalsolimited

due

todiminishingreturns aslabor expandsrelative to existing plantand

equipment

In thelongrun, capital stocks

must

alsobe

increased.

However,

capitalinvestmentinvolvesevenlonger delaysarising

from

theprocessof

planningfor,ordering

and

constructing

new

plantand

equipment

Adjustment ofcapitalstocks thusinvolves a negativefeedback loopwithsubstantiallylongerdelays.

Models

thatintegrate

capi-talinvestmentwith inventoryand

work

fence

management

tendtoproduceoscillations with periods of 15-25yearsinadditionto the short-termcycle

(Mass

1975, N. Foirester 1982,

Low

1980).

The

theorydescribed sofar

assumes

agentshave

bounded

rationalityinthe senseof

Simon

(1979, 1982).

Agents

seektotakeappropriatedecisions,but

do

not possess the cognitiveandother

re-sourcesnecessarytoapproachoptimality, eveninthe

weak

rationalexpectationssense,due tothe

complexity ofthehighorder,nonlinear,randomly-excited

dynamic

systemin

which

theyoperate.

The

theoryof

bounded

raticmality,asappliedhere,recognizesthatfirmspartition thetotalproblem ofoptimizingthe enterprise into subproblems. Production istypicallyinfluencedbydecisionsat

theplantlevel,while pricing

may

betheresponsibilityof seniordivisional

management

andcapital

investment

may

be decidedatcorporate headquarters.

Due

tolimitationsoftime, information

availability,

and

attentional resources,

management

ofthe subsystems

may

be imperfectly

coordi-nated.

The

theoryof

bounded

rationalitydoesnot

assume

thattheindividual

managers

are

irrationalbutrather locallyor intendedlyrational

-

thatis,they useheuristics that

would

work

well

ifthecouplingsanoong subsystems

were

weak

andthe separability assumptionimplicitintask

factoring

and

decision

making

withinthefirm

were

valid(Sterman 1985, 1987;Morecroft 1985).

Extensive experimentalevidence

shows

thatthe

bounds

on

rationaldecision

making

in

dynamic

systemsare severe. In simple experimentaleconomies suchas the classical multiplier-accelerator

0^308

(Sierman 1989b), subjects performwell

below

optimalandgenerate systematic, persistentand

costlyoscillations. Thesesystematic decision errors

become

nx)re severe as the feedback

com-plexity oftheenvironment increases, particularly asdelays lengthen (Diehl 1992, Paich and

Sterman 1992,

Brehmer

1990). Experience,incentives, and marketinstitutionsmoderate butdo

not eliminate theseerrors (Paichand Sterman 1992,

Kampmann

and Sierman 1992, Smith,

Suchanek

and

WUliams

1988).

3.

The

economic

long

wave

The

third

main

cyclical

mode

of

economic

behavioristhe

economic

long

wave

orKondratieff

cycle.

The

long

wave

is the

most

controversialand leastunderstoodofthethree cyclical modes. It

isalso the

most

important TTie long

wave

is farlargerin amplitude thanthe businesscycle,andof

such greatduration thatthe stressesitgeneratescannotbe contained within the marketsystem, but

ratherinfluence theevolutionof,and sometimes therevolutions in, theinstitutional strucmre ofthe

world

economic

andpolitical system (Sterman 1992, 1986a).

The

Russian economist N.D. Kondratieff(1928/1984, 1935)

was one

ofthefirst to

draw

attentiontothe wave-like character of

industrialdevelopment,withalternating periodsofrelativeaffluence and

economic

hardship.

Usingdata

on

commodity

prices, interestrates, industrialproduction,

raw

materialsconsumption, and foreigntrade, Kondratieffarguedfortheexistenceofaroughly

60

yearcyclic motion,and speculatedthat it

was

relatedtoinvestmentinlong-livedcapital.

The

economic

stagnation andcrisesofthelast

two

decades and theinabilityof conventional

eco-nomic

pobcies to restoreformerbalances haveprompted

renewed

interest in the long

wave

and

many

new

theoriesofits origin

(Freeman

1982,van Duijn 1983,

Vasko

1987).

However,

the

long

wave

remainscontroversial

among

economists.

Most

havetakena ratheragnostic stance concerningtheexistenceoflongwaves, maintainingthathistoricalevidence forlongfluctuations

ofsufficient regularity tobe consideredcyclicisunconvincing (Garvy 1943, Mansfield 1983,

Fifc

4-D-4308 10

experiencessignificantlongtermvariations,

many

economistssee these

more

as the

outcome

of

particular historicaleventssuchaswarsorgold discoveries thanasaresultof

endogenous

processes. In contrast,recentstudies

by

Bieshaarand Kleinknecht(1984) and

by Rasmussen

et

al. (1989)designedto testtheKondratieff hypothesis in realseriesarriveatgenerallypositive

results,and

Sterman

(1986a)repcmsa

wide

range ofdata consistentwiththe long

wave

hypothe-sis. Today,

most

studentsoflong cycles agreethat the historicdepression periods

were

the 1830s

and

1840s, the 1870s through late 1890s, the 1920sand 1930s,and the periodfiromabout 1974 through (at least)the eariy 1990s(vanDuijn 1983,

Vasko

1987, Goldstein 1988).

To

illustrate.Figure4

shows

detrendedreal

GNP

inthe UnitedStates

from

1947 to 1992. After removal ofthe long-term exponentialgrowthtrendwhat remainsarethe cyclicalmodes,

particu-larlytheshort-term business cycleandthe longwave.

The

post-warlong

wave

isclearly visible,

with

GNP

growing

fasterthan trend

from

theend of

World

War

IIthrough about 1970, and slowerthan trendsince.

The

business cycle,with

much

smalleramplitudethanthelong wave, appearsas

smaU

ripples

on

the great swellofthelongwave.

Note

also

how

thephaseofthe long

wave

conditionsthe apparentseverity ofthebusinesscycle. During theexpansion ofthelong

wave,periodsofbusiness cycleexpansion

seem

tobelong andvigorous,whilerecessionsare

thoughttobeshortandmild,astherising tideofthe long

wave

lifts all boats. Duringthe

down-turnphase ofthelongwave,recessions

seem

tobelongeranddeeper, andthe growth phaseof the business cycle appearstobe weaker.

An

analyst

unaware

ofthelong

wave

would

concludethat thecharacterofthebusiness cyclehad

changed

as thelong

wave

peaked and beganto decline.

Kondratieff

viewed

thelong

wave

asa manifestationofessentialforcesinthe capitalist

economy,

and arguedthatabroadspectrumofsocial and

economic

phenomena

were

shaped

by

the wave. In

particular,eachburstofcapitalexpansion

would

allow a

new

setoftechnologiestobeexploited.

While

acceptingthe generalideaof endogenouslygenerated longwaves,

Schumpeter

(1939)

articulatedtheoppositecausality between

economic

growthandtechnological innovation. For

D-1308 1

1

Both linesof thought continue today.

One

oftheearliestand nx)Stthoroughly testedformal

mod-elsofthe long

wave

has beendeveloped at

MITs

System

Dynamics Group

(Forrester 1976, 1977, 1979, 1981,

Graham

and

Sengc

1980,

Sicrman

1985. 1986a, 1986b, 1987, 1988, 1989a, 1990,

1992). TTic theory integrates a varietyof

economic

processes,both real andnominal, including

capital investment,

employment,

work

force participation, wages, inflation,interest rates,

mone-tary policy, debt, and

consumer demand,

among

others.

The

MIT

model

endogenouslygenerates

thelong

wave

aswell as the short-term businesscycle,constructioncycle, andother

modes

includ-ing

economic

growth andthe expansionofthe governmentsector relativeto theprivate

economy.

A

simple version ofthis

model

isanalyzed below.

In parallel withthis lineof

economic

modeling, neo-Schumpeterian theories stressing the roleof technological innovation ascausesofthelong

wave

have beendeveloped.

Mensch

(1979) argues fundamentalscientificdiscoveriesand

new

inventionsoccur

more

orlessrandomly. Butforan

in-ventiontoacquire

economic

significance,innovation, orthecommercialization oftheinvention,

must

occur.

The

rateof basic innovations, those

which

plant theseedsof

new

industries,is

con-ditionedby the stateofthe

economy.

During long

wave

upturns,

economic

growthisrapidandthe

existing infrastrucmre is highly productive: incentivestoinvestin

new

technologies aresmall.

At

the

same

time,fx>sitive networkexternalitiesand

commitment

to existing infrastructure

make

it

dif-ficult tointroducealternative transport,

communication

orenergysystems. Lx)ng

wave

downturns

arise

when

the potentialofexistingtechnologiessaturates. Switchingcoststhen decline,

produc-inga burstof basic innovationas

many

oftheinventionsaccumulatedduring the

upswing

now

find practical application.

The

resulting

swarm

of innovationslaunches

new

industriesand

pro-vides theimpetus forthenextupswing.

Formal

mathematical

models

oftheseneo-Schumpeterian

theoriesinclude Montarioand Ebeling (1980), Mosekilde and

Rasmussen

(1986), Silverberg

(1988) and Dosi(1988); Kleinknecht (1984) provides

some

empiricaltests.

One

difficultyin

in-novation theoriesofthe long

wave

isexplaining

why

disparatetechnologies indisparatecontexts and markets should reach saturationin synchronyafter40-60years,cycle aftercycle. Addressing

D^308

12

thisproblem, Graharn and

Senge

(1980)integratedinnovation theorieswiththe

MIT

model

and

ar-gueinnovaticmratesareentrained

by

the

endogenous

economic

processesthatgeneratethelong wave. Otherauthorshaverelatedthelong

wave

tochangesin

employment and wages

(Freemanet

al. 1982),resourcescarcity

(Rostow

1978),class struggle

(Mandel

1980),

and war

(Goldstein 1988).

4.

A

simple behavioral

model

of the

long

wave

A

control-theoreticexplanationforthelong

wave

emergingfiromthe

MIT

theorycanbe divided

into

two

parts: first,asdescribed above, acquisitionofcapacityinindividualfirms involves

inher-endy

oscillatoryprocesses. In isolation, theseprocessesarestable, producing

damped

oscillations

when

excited

by exogenous

changesin

demand. However,

a

wide

range ofself-reinforcing pro-cessesexistinthelinkages

between

firmsand

among

theproduction,financial,householdand

government

sectorsofthe

economy,

destabilizing thecycleandlengtheningitsperiod.

Demand

for capitalincreasesthecapacityneedsofthe capitalproducingindustries,furtherboosting orders

forcapital. For example, expansion

by

capitalproducersraiseslabor

demand

and wages,leading

tosubstitutionofcapital forlaborandstill greater

demand

forcapital. Rising aggregate

demand

boostsprices,reducingreal interest ratesand furtherstimulatinginvestment. Rising output boosts

income and

aggregate

demand,

furtherboostingoutput. Expansionleads toexpectations offuture

growth,leadingtofurtherinvestmentandoutput growth. Risingcredit

demand

tofinancethe

boom

causesmonetary

accommodation,

additionalinflation,andstill lowerrealinterest rates.

And

so on.

These

positiveloops include

many

familiarprocesses includingthe Keynesian

income

multiplier,the

Mundell

effect,

and

Fisher's(1933)debt/deflationspiral.

The

full

MTT

national

model

integratesthese

and

otherfeedbackprocesses(Sterman 1986a

and

1988 providedetails).

Model

analyses(Rasmussen, Mosekilde and

Sterman

1985,

Br0ns and

Sturis 1991)

show

that

thesepositivefeedbacks causeaHopf-bifurcation through

which

theequilibriumofthe

economy

becomes

unstable.

Any

perturbationscause divergentoscillations that areeventually

bounded by

DM308

13

capitalstock,producing a limit cycle.

The

long

wave

appearstobea self-sustaining oscillation

that, although influencedby shocks andperturbations, doesnot require external excitation to persist. In contrast,the short-term business cycle appearsto bea stable,

damped mode

that

requires externalexcitation, as inFrisch (1933).

One

ofthe most fundamental self-reinforcingfeedbacks isthe capital investmentmultiplier, or

'capital self-ordering', the fact thatin theaggregate the capitalproducing sectorofthe

economy

ordersandacquiresplantand equipment

fiDm

itself. Ifthe

demand

for

consumer

goods and servicesincreases,the

consumer

goods industrymust

expand

itscapacityand so places ordersfor

new

factories, machinery,vehicles,etc.

To

supplythe high

volume

oforders, the capital

producingsector

must

also

expand

itscapitalstock and henceplaces orders for

more

buildings,

machines,rollingstock, trucks, etc.,causing thetotal

demand

for capital to risestill furtherin a

self-reinforcing spiralof increasingorders,a greaterneedforexpansion, and still

more

orders.

Inequilibrium, themultiplier effectofcapital self-orderingis

modest

(Sterman 1985).

However,

the long

wave

isaninherently disequilibrium

phenomenon,

andduring transientadjustmentsthe

strengthofself-ordering

becomes

much

greaterthaninequilibrium. Thisispartly aconsequence

ofthe classical investmentaccelerator. Duringdisequilibriuma varietyofadditional positive

feed-backloopsfurther

augment

the

demand

forcapital. Theseinclude:

(i)Amplification caused byinventory and backlogadjustments: Rising orders deplete the

inven-toriesandswell thebacklogs ofcapital-sector firms, leadingtofurtherpressure to

expand

andstill

more

orders. During thedownturn,

low

backlogs andinvoluntary inventory accumulation further

depress

demand,

leadingtostill

more

excess inventory.

(ii)Amplificationcaused byrisinglead timeforcapital: Duringthelong

wave

expansion,the

demand

for capital outstrips capacity. Capital producersfindittakeslonger than anticipatedto

acquire

new

capacity, causing capacityto lag furtherbehinddesiredlevels, creatingstill

more

EM308

14

(iii)Amplificationcausedby growthexpectations:

Growing demand,

risingbacklogs,andlong lead times during thelong

wave

expansionall encourageexpectationsofadditionalgrowth in

de-mand

forcapital. Expectationsofgrowth leadtoadditionalinvestments,furtherswelling

demand

in aself-fulfillingprophecy. During thedownturn, pessimismfurtherundercuts investment.

Sterman (1985)developeda behavioral

model

capturingthedestabilizingpositivefeedbackcaused

by

capitalself-wdering.

The

model

isdesignedtoisolatethe

minimum

structure sufficientto

gen-eratethelong

wave

withrealisticparametervalues. Itdoesnot includethefullrangeoffeedbacks

includedinthe

MIT

model.

However

simulations with

more

comprehensiveversionshave

shown

thatthe characteristic behavicwproduced

by

the simple

model

isrobustto structuralelaborationof

the model. Itisalsopossibleto find

more

complicated

modes

of behavioras the

model

isextended

(Mosekildeetal. 1992)

and

disaggregated

(Kampmann

1984).

The

model

creates atwo-sector

economy

with acapitalproducing and goods producingsector.

The

focusisthe capitalinvestment accelerator.

Goodwin

(1951,4)notesthatthe traditional

acceleration principleassumes

...thatactual,realizedcapitalstockismaintainedat thedesiredrelation withoutput

We

knowin reality thatitis

seldomso,th^ebeingnowtoomuchandnowtoolittlecapitalstock. Forthistherearetwo goodreasons. Therate

of investmentislimitedbythecapacityoftheinvestmentgoodsindustry....Attheotherextremethereisan even

moreinescapableandeffectivelimit Machines,oncemade,cannot beunmade,sothatnegativeinvestmentis

limitedto attritionfromwear....Thereforecapitalstockcannotbeincreasedfastenoughintheupswing, nor decreased

fastenoughinthedownswing,sothat atonetimewehaveshortagesandrationingoforden andattheotherexcess capacitywithidleplantsandmachines.

A

singlefactw ofproduction (capitalplantand equipment) isconsidered.

The model

includes,

however, anexplicitrepresentationofthe capital acquisitiondelay (construction lag)andthe

capac-ityoftheinvestmentgoodsseaor.

As

aresult,ordersfor

and

acquisitionofcapitalarenot

neces-sarilyequal,

and

atany

moment

there will typicallybea supplylineofcapitalunderconstruction.

For

simplicity,the

demand

for capitalofthegoods-producing sectorisexogenous, and thereis

no

CM

308 15

Wc

firstdescribe the equations for the capital producer, then thecouplings between sectors.

The

model

allowsfor variable utilizationofthe capital stock.

Thus

production

P

depends on utilization

of production capacity C. Utilization isanonlinear function oftheratioof desired production P*

tocapacity. Desired output

P*

isdetermined by thetotal backlogofunfilledorders

B

and the

nor-mal deliverydelay A*. Capacityis proportional tothe capitalstock K, withcapital/output ratiotc

P

=

u{P*/C)C

, u(0)

=

0, u{1)

=

1. u

^

0, u"

<

0,

u{«}

=

uniax

q)

P*

= B/A*

(2)

C

=

K/K. (3)

The

capital stockofthe capital sectoris

augmented

by acquisitions

A

and diminished bydiscards

D. Discardsarcexponential withaveragelifetimex:

(d/dt)K

=

A

-

D

, (4)

D

=

K/T. (5)

The

acquisitionofcapital

depends

on

the firm'ssupply lineofunfilledordersfor capital S andthe

capital acquisition lagA:

A

= S/A

(6)

The

supplylineofcapitalunderconstruction representstheordersfor capitalplantand equipment,

C\, thefirmhas placed but notyet received:

(d/dt)S

=

Ok-A

(7)

Thus

farthe

model

describesthe stock and flowstructureofthefirmandthephysical limitson capacityutilization.

The

key behavioral formulation isthedecision rule for capitalordersOk:

D-4308 16

Ok*

=

D

+

ak(K*

-

K)

+

_as(S* - _{S) (9)}

herethe actualorderratedependsnonlinearlyon dieindicatedorderrate

Ok*

asafractionperyear

ofexisting capitalstock K,ensuringthatordersremain nonnegativeevenifthereisa largesurplus

ofcapital

Due

to limits

on

e.g.financing, absorptioncapacity,etc.,orders arelimited to a

maxi-mum

fractionofexistingcapacity

P"^,

asin

Goodwin

(1951). Threemotivationsforinvestment

are assumed: (1)toreplacediscards;(2)to correctanydiscrepancybetweenthedesiredcapital

stock

K*

and the actualstockK; and (3)tooxrect anydiscrepancybetween thedesiredsupplyline

ofcapitalunderconstruction

S*

andthe actual supplylineS.

The

adjustment parametersttkand

cxsdeterminethe aggressivenessoftheresponse todiscrepancies.

To

ensurean appropriate

acquisition rateof

new

capital,firms

must

maintainasupplylineproportionalto thedelay theyface

inacquiringcapital.

Thus

thedesiredsujl^lylineisproportionaltothe capital acquisition lag

A

and

thecurrentcapitaldiscardrate

D

(see Sterman 1989a and

1989b

for detailsandexperimental evidence supportingthisformulation):

S*

= A-D

(10)

The

desiredcapitalstock

K*

isa nonlinear function of desired output P*:

K*

= Kog{KP*/Ko),

g{0)=0,g{l)

=

l,g'>0,g"<0

(11)

Desiredcapitalstockis

assumed

toriseproportionatelywith desired outputforsmall deviations

from

theequilibriumvalue Ko, but diminishingreturnstocapitalare

assumed

to limit capital

expansion

when

kP*/Ko becomes

large.

Finally,the backlogofthefirmis

augmented

by

customerorders

O

and reduced

by

outputP:

(d/dt)B

=

0-P

(12)

D-1308 17

inthebacklog,

A

= B/P. (13)

Equations (1)-₍₁₃₎ describe asimple

model

ofa firm.

The model

includes anexplicitdelayin

acquiringcapital stock andrealistic nonlincaritiesrepresentingbasic physical processessuch as

nonnegativityofgrossinvestmentandlimits toutilizationofexisting capacity. Sterman (1985)

shows

theindividual decisionrulesofthe

model

areintendedlyrational,andinvestigates its

sensi-tivityto parameters.

With

realisticparametersfor a capital producingfirm (k=3,

A

=

A

=

1.5,t

=

20,

Ok

=

3,and Os

=

3)and

exogenous

ordersO, the transientresponseofthe

model

to shocksis

a highlydamjjcdoscillation with aperiod ofabout

20

years.

As

describedabove,the cyclearises

from

thenegativefeedback loop by whichoutputisregiilatedthrough changesin production

capacity,wath a lagcausedby the capital acquisition delay.

The model

doesnotproduce the

short-term business cyclebecause labo»isnotexplicitly treated;production

P

instantly adjusts tothe

desiredrate

P*

as long as thefirmisnotcapacity constrained.

To

see

how

the long

wave

mightarisethroughcapitalself-ordering,

we

now

modify the

model

to

representthe entirecapital-producingsectorofan

economy.

In theaggregate the capital sector

orderscapital

from

itself,so thetotalrateatwhich

new

ordersfor capital are received

O

is

now

the

sum

ofthe capital sector's

oiden

forcapital,(\, andordersfor capital placed bythe goodssector,

Og,

which

representsall other purchasers ofcapitalplantand equipment:

O

=

_Ok

-fOg (14)

The

backlog ofunfilledorders for capitalis

now

the

sum

ofthe supplylinesofthe capitaland

goods

sectors:

B

=

S

+

Sg

(12)

D^308

18

(d/dt)Sg

=

(Og

-

Ag)

₍₁₅₎

The

raleat

which

the goodssectoracquirescapitaldepends

on

thegoodssector's supplyline Sg andthedeliverydelayofthe capital sector

A

Ag=Sg/A

(16)

Likewise, sincethe capital sectoracquirescapital

from

itself,thecapital acquisitionlag.A,itfaces

is its

own

delivery delay. A:

A

=

A

(17)

Finally, the

demand

for capitalderived

from

the

goods

sectorofthe

economy

Og

isexogenous.

The

full

model

isathirdorder nonlineardifferentialequation system(thestatevariablesareK,S,

andSg). Itcaptures

some

ofthepositivefeedbackscreated

by

thedependenceofthe capital sector

ofany

economy

on

its

own

output

As shown

in Sterman (1985) and

Br0ns

and Stuns(1991),

due

to thesepositivefeedbackstheequilibriumofthe

model

isunstable.

With

the

same

parameters

as

above

andconstant orders

from

the

goods

sector, Og,asmall perturbationproducesexpanding

oscillations

which

areultimately

bounded

by thenonlinearconstraintsassociated with the

invest-ment

function_g{

•}

and

capacityutilizationfunction u

{

•).

The

steady statebehaviorofthe

model

is

alimitcyclewith a period ofapproximately

50

years(Figure5).

The

long

wave

generated

by

die

model

has

many

ofthefeaturesofthelong

wave

generated

by

thefull

MIT

model,includingphase

relationshipsandrelativeamplitudesforoutput, capital stocks, capital orders, acquisitionsand

discards,delivery delay,

and

capacityutilization.

A

fullequationlisting,explanationof

formula-tions,

and

sensitivity testsarcfoundin

Sterman

(1985).

pit.

^

^

_The

_cycle arisesviathe laggednegativefeedbackloop describedinthediscussionofthe

construc-tion cycle.

To

understand

how

the oscillationsustainsitself, consider theprocesses thatproduce

EM308

19

small increasein the

demand

derivedfrom the goods sector.

The

capital producing sector findsit

hasinsufficientcapacityand thereforeincreases its

own

ordersabovethereplacementrate.

The

total

demand

for capitalthus increases still furtherabovecapacity,stimulatingorders still more.

Total ordersrise faster thancapacitydue totheconstructiondelay, sothebacklogofunfilled orders

rises,andcapital producers find theirattempts toexpand areslowed byrisingdeliverydelays.

The

gap berween desired andacnial capital widens further,causing still

more

orderstobe placed.

These feedbacksgenerate a self-reinforcing spiralof increasingorders, a greaterneedfor capital

and still

more

orders. Evenniaily,the variousnonlinearitieslimitthe increasein

demand.

Production capacity gradually overtakesorders.

The

backlog then stanstofall.

Now

the

same

positive loopsthat

powered

theexpansiondrive the

economy

into depression.

With

decreasing

backlogs, desiredproduction capacitystartsto fall, leadingto areductioninorders. Falling

deliv-erydelaysreduceordersby acceleratingacquisitionsandreducingtherequiredsupply line.

Thus

the capital sector findsitselfwith^xcesscapacity andcutsitsordersforcapital, funherdecreasing

the

demand

for capital andleadingtostill nrorecutbacksin orders.

At

theendofthe upswing,the

capital producing sectorhas severeexcess capacity andcutsits

own

ordersto zero. Capital

pro-duction

must

remain below the levelrequired forreplacements until theexcess capacity depreciates

-

aprocess

which

may

takeadecade or

more

duetothelong lifetimeofthe capital stock.

The

lowerturning point andinitiationofthenextcycleare directconsequencesof

bounded

rationality.

The

model

assumescapitalproducers buildcapacityto

meet

theorderratetheyforecast

and

do

notuiKJerstandorinvest to satisfythegeneralequibbriumofthefull

economy.

Specifically,duringthedepression phaseofthe long cycle

demand

for capitalislessthan the

sys-tem'sequilibrium becausethe capital sectoritselfisordering lessthandiscards. Evenniaily

capac-ityapproachesthe level required to

meet

the

demand

ofthe goodssector. Capital producers then

increase theirordersinorder tooffset discards.

However,

theincrease inorders boosts the total

demand

for capital abovecapacity, and backlogsbegintorise. Faced

now

with capacity toolow to

D^308

20

own

OTdersfurther

above

replacementneeds, andthenextexpansionbegins.

Thus

the long

wave

isgeneratedendogenously

by

theinvestmentbehaviorofthe capitalproducing

sectOT,

and

persistswithout

exogenous

excitation.

Changing

theparametersofthe

model

suchas,

for instance, the capital/outputratioorthe

maxima

ofthenonlinear functions

may

changethe

amplitude

and

periodofthewave.

However,

the characteristic self-sustainedoscillationwith a period

on

theorderof

50

yearsisrobustover

most

oftherealisticparameterrange.

Beyond

this

rangevarious bifurcations(i.e.changesinthe steady-statebehaviorofthemodel) occur

(Rasmussen

etal. 1985,

Szymkat

and Mosekilde 1989,Brons and Stuns 1991).

The

model,particularlythe criticaldecisionrulefor capitalinvestment, hasbeentestedboth econometricallyandexperimentally. Senge (1980)

showed

thatadisequilibriuminvestment

func-tionsimilar totherulehereprovides abetteraccountof post-war

US

dataforavarietyofindustries

than the neoclassicalinvestmentfunction. Sterman (1987, 1989a) convertedthe

model

intoan experimentin

which

subjects,including

some

experiencedmanagers,

made

the capitalinvestment decisionfor the capitalproducingsector. Despitefullinformation,thevastmajorityofthe subjects

generatedlong

wave

cyclesccHrespondingclosely tothoseofthemodel. Econometricestimation

ofthesubjects'decisions

showed

they

conformed

well tothe

assumed

decisionrule for capital

orders. Simulation

showed

thattheestimated decisionrulesforabout

40%

ofthe subjects

producedthe limitcycle behavicw,and about

25%

yieldeddeterministicchaos (Sterman 1989c). Subsequent experiments

have

shown

theseeffects toberobustto financial incentives, training,

experience,

and

thepresenceofmaricet institutions (Ehehl 1992,

Kampmann

and Sterman 1992).

5. Interacting cycles:

Nonlinear

entrainment

and

mode

locking

The

discussion

above

provides a disequilibrium, behavioral foundationforeachofthe three

main

cyclical

modes

inthe

economy.

Thus

far,each

mode

hasbeendiscussed separately. Ifthe

ecwi-omy

werelinear,the cyclesgenerated

by

eachfirm

would

evolveindependentlyofoneanother,

rM308

21

a characteristic

power

spectrumin response tovariousdisturbancesin theenvironment. Buttothe

extentsuchvariations were imperfectlycorrelated acrossfirms, the cyclical

movements

of

inde-pendentfirms

would

tend toaverage outatthe industryand

macroeconomic

levels. Insucha

world theonly

way

a coherent aggregate business cyclecould

come

aboutis through

common

sourcesof

exogenous

variation, suchas

government

monetary andfiscalpoliciesor highly

corre-latedshocksor expectations,and indeed there are

many

suchtheoriesofbusiness cycles

(Bums

1969, Mitchell 1927;

Zamowiu

1985provides a survey).

However,

there arestrongtheoreticalargumentstosuggestthatnonlincarityplaysa crucial rolein

bringingaboutinteraction between the

modes

andthereby shapingtheoverall behavior.

Even

at

the levelofthe individual firm, the nonlinearlimitson the

workweek

and

work

forceadjustment

processes tend tocouple the

shon

andloag term

modes

tooneanother. Othernonlinearities arise

from

nonnegativityconstraintson gross investment, shipmentsofgoods

from

inventory,etc.;

from

upperlimits tocapacity utilization, hiringand investmentrates;and because thesedecisions

depend

nonlinearly

on

multiplecues.

The

empiricalevidencefornonlinearinteractionsbetween thevarious

modes

isalso strong.

As

an

example

figure6a

shows

the variationinoil-tankerspot rates

from

1950through 1991. Spotrates arecharacterized byseriesofsharppeaksand deepvalleys occurring at3 to 5 yearintervals,

sepa-ratedby periodsof 10-15yearsin

which

ratesandtheirvarianceare low. During thef)eaks, which

often lastforonly a

few

nwnths, ratesof

more

than

400

are attainedwhileduring diedepression periodsratesareas

low

as40.

The

altematingpatternof

calm

punctuated by wild swingsreflects

thenonlinearinteraction ofthetanker construction cycle with business cycle variationsin the

demand

foroil transportation.

The

constructioncyclein thiscase arises

from

the long delaysin the

ordering and buildingof

new

tankers.

Pl(5. t

^^^

_{Econometric, experimental and}

field studies

show

that ship-owner's decisionstoorder

new

tankers

D-4308 22

Suppose

demand

foroilshipmentishighrelativetothecapacityoftheworldfleet

Tanker

rates

willbehigh.

The

resultinghighprofitsinduceexistingoperatorsto

expand

their fleets and cause

entryof

new

playersintothe

market

Ordersfor

new

ships swell.

However, due

tothe long

con-structiondelay (2-4years),

demand

willremainhighforseveral years,during

which

rime

addi-tional

new

orders areplaced

by

existingplayersand

new

entrants.

When

these shipsare

commis-sionedexcesscapacitydevelopsandtankerratesfall.

New

ordersdrop

below

scraprates(often

nearlyreachingzero), but sincetheservicelifeoftypicaltankersis 15-25 years,spotratesand

new

construction remain depressedfor years,untilcapacityonceagaindrops

below

demand,

ratesrise,

and

thenext cycle begins. Consistentwith thetheory of

bounded

rationality, thisdescription

assumes

shipowners

do

nothave completeinformation abouttheglobal shipbuildingmarketor understanding of long-termmarket dynamics,butrelyprimarilyoncurrent profitpotential (spot

rates relative tocostsof

new

ships) inplacing orders (Zannetos 1966,

Bakken

1992).

The

nonlinearinteractionofthe businessandconstructioncyclesis

shown

by

comparing

figure6a

to figure 6b. Spotratesare

low

andinsensitive to thebusiness cycleinperiodsof surplus tanker

capacity,since

demand

fluctuations are easily

accommodated by

higherutilization(the short

mn

elasticity ofsupplyishigh). Conversely, ratesarchigh andvolatile

when

capacityutilizationfor

thewcffldfleetishigh.

High

utilization

means

supplyisquiteinelastic inthe short run; small

variationsin

demand

caused

by

the business cycleorby geopoliticalshocksyielddramaticchanges

in spwtrates.

The

parametersgoverning theresponseofthe maricet toshorttermvariationsin

demand

includingthebusiness cycle

depend on

thephase ofthe long constructioncycle.

Thus

the

Suez

crisis,

coming

atatimeof highfleet utilization,causedsurgesin rates,while theIran-Iraq

war,

coming

during atime ofexcesscapacity,isbarelyvisible inthe data.

Nonlineardynamicaltheoryalsosuggeststhatthe different cyclic

modes

may

entrainone another throughtheprocessof mode-locking. Specifically, oscillatory

modes

innonlinearsystemswith

similarfrequenciestendtoadjust tooneanothersuchthattheirperiods

become

precisely thesame.

mo-D-4308 23

tion, so that the

same

hemisphereofthe

moon

perpetually faces theearth. Other well-known ex-amplesarc the synchronization ofthecircadianrhythmof

many

organisms tothe 24 hourcycle of

night andday, thesynchronization of (mechanical) clocks

hangmg

on the

same

wall,and the

syn-chronization ofmenstrual cyclesbetween

women

living inclosecontaa. Nonlinearcoupling of

different oscillatorscan thusexplain

why

thereareaggregate business cycles

when

thediffering

parameters andinitialstatesofdifferentfirmsmightcause

them

tooscillatewith different

frequen-ciesandphases,averaging out atthe

macrocconomic

level. Couplings between firms cause the

cyclesgeneratedbydifferentfirms tobe

drawn

togetherinto acoherent aggregate cycle with stable

phase relations(Forrester 1977).

Homer

(1980)

shows

how

basicmarketprocesses such as

con-sumer

responsetorelativepriceand availability provide sufficientcouplingto synchronize firms

withdifferentparametersand initialphases.

Synchronization isonlyone manifestation ofthe

more

general

phenomenon

offrequency-locking ornonlinearentrainment

(Amol'd

1965,Glassetal. 1984,Jensenetal. 1983, 1984,

Rand

etal.

1982,Mosekildcetal. 1990). Innonlinear systems, anoscillatory

mode

containsvarious

harmon-ics, and

two

modes

may

synchronize

whenever

aharmonicofone

mode

isclose toaharmonic of

the other.

As

aresult, nonlinearoscillatorstendtolocktooneanothersuch thatone oscillator

completespreciselypcycleseachtimetheotheroscillatorcompletes qcycles, wherepand qare

integers.

Such

mode

lockingmightexplain Schumf)eter's (1939) observationthattheperiod ofthe

construction cycle

was

approximately threetimestheperiodofthe businesscycles,andtheperiod ofthelong

wave

was

approximatelythree timestheperiodoftheconstructioncycle.

To

illustrate nonlinearentrainment andexplore

how

the different cyclical

modes

mightinteract,

we

nxxlify thelong

wave

model

sothatordersfor capitalderived fromthegoods sector fluctuate

sinu-soidallywith period

T

andfractional amplitude

A

aroundaconstantlevelOg*:

Og

= Og*(l

-t- Asin(27Ct/D). (18)

econ-D^308

24

omy. Faced

withthisfwcing, thefrequencyofthelong

wave

will adjustina

manner

thatdepends both

on

theamplitudeand frequency oftheexternal forcing.

The

adjustmentwilltendtolockthe

two

cyclesintoanoverallperiodicmotionin

which

the long

wave

completesprecisely_pcycles eachtimetheforcingsignalcompletes

q

cycles,

where

p and

q

are integers.

As

an

example

figure7a

shows

the resultsobtained

when

the

model

isperturbed

by

a

20

percent

(A

=

0.20) sinusoidalmodulationwith a forcing period

T

=

22.2years.

Here

theforcing

fre-quency

isrepresentativeoftheconstructioncycle. Relativetotheunforcedlimitcyclebehavior

(figure 5),thelong

wave

has increaseditsperiodbycloseto

40%

soas to

accommodate

precisely

3 periodsof thefastercycle. Moreover, withinthe interval 19.9years

<

T

<

24.8 years,achange

intheperiodoftheforcing signal willcauseapreciselyproportionalshift in thelong

wave

such

thatthe 1:3entrainmentismaintained. ^

pit.

^A,^t

H€^^

A

clearillustrationoftheperiodio natureofthe

mode-locked

solutionis

shown

inphasespace

pro-jectionsofthe steady-statebehaviorofthesystem. Figure

7b shows

the phasef)ortrait

correspond-ingtothe time-domain behaviorinfigure 7a. Here,

we

haveplotted simultaneous values ofthe

capitalsector capital

K

andthe goods

goods

sector capitalorders

Og

over

many

cycles.

The

hori-zontal axisrepresentstheexternalforcing, andthe vertical axis theresponseofthemodel.

Production capacityofthe coitalsectorbuilds

up

and decayspreciselyonceforeachthreeswings oftheexternal signal.

Figure8

shows

theresultsobtainedwith the

same

amplitudeoftheforcing signal

(A

=

0.20),but withthe

fwcing

period

T

=

4.6years. Thiscase,

which

couldrepresentthe interactionbetween the

eccMiomiclong

wave

andtheshort-termbusinesscycle, produces 1:10 entrainment.

The

long

wave

completespreciselyoneoscillation foreach 10 businesscycles.

The

1:10

mode-locked

solu-tion existsinthe internal4.47

<

T

<

4.70years.

Near

thisinterval

we

findintervalswith

entrain-ment

ratiosof 1:9, 1:11, 2:19, 2:21,etc.

D-4308 25

the long

wave

model, figure9

shows

the results obtained with

A

=

0.20 and

T

=

19.4 years. For

the first

200

years, thenxxlel runswitha constant

demand

for capital tothegoodssector, showing

the undisturbed long

wave

oscillation. In year 200, theexternal forcing begins. Aftera short

transient the nxxicl locks into a2:6 solution,with 2 long

waves

foreach 6cyclesoftheexternal

forcing. This isaresultofa period-doubling oftheabove 1:3 solution.

The

true periodis

now

116.4 years,and the half-period(which

we

may

still identify as the long

wave

period)is58.2

years.

v":^^

_A

_more

_complete pictureoftheentrainmcnt process isobtained

by

plotting theobserved

mode-locking ratioas a functionoftheforcing period. Figure 10

shows

an

example

ofsucha

construc-tion, aso-called devil'sstaircase (Mandelbrot 1977).

The

periodoftheexternalforcing has here beenvaried

from

5to

54

years whilekeepingthe amplitude constantat

A

=

0.025.

We

observea

seriesofintervals with l:n

mode-locked

solutions.

Between

these,intervals with other

commen-suratewinding

numbers

areobserved. In theregion

from

27

<

T

<

37 years, forexample,

we

find

intervalswith3:5, 2:3, 3:4,4:5and 5:6entrainment.

\'(^~^^

_By

refining thecalculationsone finds

more

and

more

resonances coveringnarrowerand narrower

intervals. Forsmallvaluesof

A

the

phenomenon

has aself-similarstructure thatcausesittorepeat

adinfmitumon a smallerandsmallerscale.

The

fractalnatureofthedevil'sstaircase isillustrated in theinsertoffigure 10. Here,

we

haveplotted

some

oftheprincipal

mode-locked

solutions

betweenthe 1:3 andthe 1:2steps. Inpractice, the finer details will be

washed

outby noise

-

the

random

shocksthatcontinuously

bombard

the

economy

willnotallowthe trajectory to settle inthe

neighborhoodofone ofthe

more

complicatedsolutions.

However,

the

more

fundamentalratios,

such as,for instance, 1:3 and 1:4are stable over

much

broaderintervals.

Mode

locking canthus

berobusttoperturbations andnoisethatcause individualcycleshape andtimingto vary.

If the amplitudeoftheforcing signal ischanged, the intervalsof entrainment alsochange. Figure

pife- »1

0^308

26

therecan,of course, be

no

entrainmentatall.

As

A

isincreased, however, widerand wider

intervalsof

mode-locked

behaviordevelop,andtheregionsof

mode

locking,

known

as

Arnold

tongues,broaden. Forsmallamplitudesquasiperiodicbehaviorexists

between

thetongues.

The

tcmguescannotcontinuetogrow,however.

As

theamplitudeoftheforcing signal

grows

the

tongues begintooverlap,andquasiperiodicbehaviorthen vanishes. Inour

model

thisoccursat

A

=>_0.025.

Above

thecriticalvaluethe trajectoryiseitherperiodicorchaotic.

Figure 12

shows

an

example

ofchaoticbehaviorinthemodel.

The

periodand amplitudeofeach

long

wave

are

now

different.

The

periodandtheamplitudeoftheperturbing signal are

T

=

16.1

years and

A

=

0.20, respectively. Chaoticbehaviorischaracterized

by

itssensitivity toinitial

conditionssuchthat

two

simulations withinitialconditionsdifferingonly slightly willdiverge exponentiallyuntiltheposition ofonebears

no

relation to thatofthe other.

H€»^

A

varietyof

complex

nonlinear

phenomena

arise

where

the

Amol'd

tonguesoverlap,including

perioddoubling, intermittency, andfrustration. Figure 13

shows

abifurcation

diagram

in

which

the 1:2

mode-locked

solution istransformedinto 2:4, 4:8,8:16,... solutions asthe forcing ampli-tude increases

from

0.0475 to0.0625 whilemaintaining

T

=

19.6years.

The

variableplotted

alongthe vertical axis inthisdiagramisthe

maximal

productioncapitalreachedatthepeakofeach longwave.

When

the forcingamplitudeislessthan 0.048 all

maxima

are equal. Forslighdy higher amplitudes,however,the

model

bifurcates intoabehavior

where

low

andhigh

maxima

alternate.

At

about

A

=

0.0552a

new

bifurcationoccurs sothatthe

model

now

shifts

between

4

different

maxima.

The

period-doublingcascadecontinuesuntil at

A

=

0.0570thebehavior

becomes

chaotic.

As

A

isincreasedfurther

we

observethe characteristic

windows

ofperiodic

behaviOT

(Feigenbaum

1978)until finally, atabout

A

=

0.0597, a sudden expansion ofthechaotic

attractoroccurs. Thisrepresentsa so-calledcrisis(Grebogietal. 1982),

where

the

model

now

generates acomplicatedbehaviorinwhichintermittencychaosdueto the interactionofthe 1:3

CM308

27

Inotherregionsofthephasediagram,

two

ornx)rcperiodic solutions coexist, andinitial

condi-tions (or subsequentpcnurbations)determine which solution thesystemchooses. Thisis, for

in-stance, the case in theregion around

T

=

29.4 yearsand

A

=

0.05,

where

the 2:3 and 3:5 tongues

cross. Figure 14

shows

a

200

x

200

point scanovertheplaneofinitial conditions forthe capital

sector capital stock

K

andthe capitalsectorsupply line S. Blackpoints indicate thoseinitial

condi-tionsthat lead tothe2:3 [jcriodicsolution, andwhite pointsindicatethoseconditionsthat leadto

the3:5 solution.

The

boundary between thebasins ofattractionforthe

two

simultaneously

exist-ingperiodic solutionsisclearly fractal.

Minor

changesininitialconditionscauseunpredictable,

qualitativechangesin the steadystate behavior.

6.

Conclusion

Recent developments in nonlineardynamics,behavioral decision theory,andexperimental eco-nomics havejoined to

form

the basis forempiricallytestable,nonlinear, disequilibriumtheoriesof

economic dynamics

groundedinexperimentaltestandfieldstudyof

economic

decisionmaking.

The

integrationof thesedisciplinesshedssignificantlightontheoriginof aggregatecyclical

movements

atdifferentfrequencies, aswellas the interactionofthese nxxles. Inparticular,

cyclical

movements

ofdifferent periodicitiescanarisethroughthe interactionofboundedlyrational

decision

making

with thetimedelays,stockand flowstructure, andnonlinearitiesfundamentalto

the structureof

economic

activity.

Behavioral nxxicls of disequilibrium

dynamics

show

how

firmscan generate cyclesthatclosely

resemblethe shorttermbusiness cycle andthe 15-25 year construction cycle. Incorporating

posi-tivefeedbackprocessesarising

from

macroeconomic

couplingsbetweenfirmsand

among

the

pro-duction, consumption, financial,and

government

sectorsexplains

how

the long

wave

can arise.

Unlike the shorttermbusinesscycle, thelong

wave

appearstobe a self-organized cyclethatdoes

notrequire continuous

exogenous

excitation topersist.

D^308

28

systematiccoincidenceofdifferent cyclical

modes

in

economic

dynamics

was

suggested longago

by Schumpeter

(1939),andForrester (1977)proposednonlinearentrainmentas theexplanation for theapparentmode-locking

among

macroeconomic

cycles.

However,

formalinvestigationof such

macroeconomic

entrainment processes with

modem

nonlineartheorydoesnotappeartohave been

attemptedbefore.

Though

the

model

investigatedhereis highlysimplified,

we

have

shown

how

entrainment

may

ariseinasystemthatcapturesbasic

macroeconomic

feedbackprocessesand fundamentalnonlinearitiessuchasnonnegativityandci^acityconstraints.

More

generally,entrainmentcancausedifferent oscillatoryprocesseswithapproximately similar

periodsto

move

inphaseatasinglefrequency,producingaggregate businessfluctuations.

Nonlinearentrainmentalsoaccountsfortheexistenceofa small

number

ofrelatively well-defined

periodicities: oscillatorytendenciesofsigiilarperiodicity in different partsofthe

economy

are

drawn

togetherin 1 : 1 synchronyto

form

a single

mode,

and eachof these

modes

isseparated

from

thenext

by

a

wide

enough

margintoavoidentrainmentatthe

same

period.

Hence

the

econ-omy

exhibits clearly distingiiishable

modes

economic

historianshave

dubbed

thebusinesscycle,

the Kuznetscycle,

and

the

economic

longwave,ratherthanfluctuationsequallydistributed atall

frequencies

and

phases,fluctuations that

would

wash

outinthe aggregate.

Even

withrelatively

wide

separationinperiodicity,the interaction

between

modes

may

bestrong

enough

tolock

them

togethersuchthattheyhave

commensurate

periods. Nonlinearinteractions

may

thuspulltheKuznetscycleand businesscycleintophasewiththelong

wave

andaccentuate

itspeaksotdownturns. AdditionaUy,sincemode-lockingata givenrationalwinding

number

is

stableoverafiniterange ofindividualcycleperiods,mode-locking isrobust with respectto

varia-ticms intheparametersgoverningtheindividualcycles,allowingentrainmentto persistoverlong

time periods despite technologicalandinstitutionalchange,perturbations,

and

other sourcesof