Cat.rise.and.fall

7/26/2019 Cat.rise.and.fall

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THE RISE AND FALL OF CATASTROPHE THEORY APPLICATIONS IN

ECONOMICS: WAS THE BABY THROWN OUT WITH THE BATHWATER?

J. Barkley Rosser, Jr.1

September, 2004

1Program in Economics, James Madison ni!ersity. "#e a$t#or is grate%$l to &illiam Brock, 'ee 'ec#ert,(#ristop#e 'eissenberg, )$sta! *eic#tinger, Ste!en )$astello, (ars +ommes, J$dy lein, +ans-&alteroren/, Erick Mosekilde, "n$ P$$, &illi Semmler, *loriaan &agener, and especially Roy &eintra$b %oreit#er materials or ad!ice. one o% t#e abo!e are responsible %or any errors or $estionable interpretationsin t#is paper.

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Introduction

"#e science 3riter, Jo#n +organ 1556, 15578, #as ridic$led 3#at #e labels

9c#aople:ology,; a combination o% c#aos t#eory and comple:ity t#eory. < central c#arge

against t#is alleged monstrosity is t#at it, or more precisely its t3o component parts

separately, are or 3ere8 %ads, intellect$al b$bbles o% little conse$ence. "#ey 3o$ld

soon disappear and deser!edly so, once sc#olars and intellects reali/ed 3#at 3ort#less

dross t#ey tr$ly 3ere or are8.


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Ren? "#om. bit$aries and ot#er reports in media o$tlets disc$ssed #o3 #e #ad been

some3#at depressed and 3it#dra3n in #is latter years beca$se o% t#e decline in stat$s o%

#is most %amo$s intellect$al progeny. S$c# reports 3o$ld not be made i% t#ere 3ere

not#ing to t#em.


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t#at +organ 3as %$lly @$sti%ied to #old it $p as t#e prime e:ample o% a ridic$lo$s and

$ltimately 3ort#less intellect$al %ad.

"#is paper 3ill s$ggest t#at t#is !ie3point may need some reconsideration. "#e

re%erence to 9t#e baby being t#ro3n o$t 3it# t#e bat#3ater; 3as %irst applied to t#e

$estion o% catastrop#e t#eory d$ring a debate o!er its $se more t#an t3o decades ago by

li!a and (apde!ille 15H08 in t#e @o$rnalBehavioral Science. "#is s$ggests t#at indeed

t#ere 3as some bat#3ater t#at needed to be t#ro3n o$t, b$t t#at catastrop#e t#eory itsel%

3as not t#at bat#3ater, t#at it 3as in %act an at least not totally $nlo!eable baby t#at

deser!ed to be preser!ed and raised in a proper #o$se#old. "#e position $ltimately taken

in t#is paper 3ill partly agree 3it# t#is perspecti!e. Sins o% intellect$al #ype and

e:aggeration 3ere committed as 3ere inappropriate applications o% t#e t#eory. At is not

as general in application as its original proponents claimed, and most certainly it is not a

general intellect$al panacea. "#ere 3as a %ad and an intellect$al b$bble, and it 3as

per%ectly reasonable t#at t#ere s#o$ld #a!e been a disco$nting and a do3ngrading to

some e:tent. +o3e!er, it 3o$ld appear t#at t#is #as been o!erdone, t#at t#e intellect$al

marketplace #as ine%%iciently o!ers#ot on t#e do3nside. < rat#er telling sign o% t#is is

t#at probably t#e %ield in 3#ic# t#e attit$de to3ards catastrop#e t#eory did not %all so

lo3, partly beca$se it did not rise so #ig# in t#e %irst place, is mat#ematics. At is indeed

time t#at non-mat#ematicians became cogni/ant o% t#is %act and ree!al$ated t#e %ormer

%ad in order to mo!e it to a more proper !al$ation. "#is incl$des in economics as 3ell,

3#ere t#e baby s#o$ld be bro$g#t back in %rom t#e o$tside to be gi!en a proper #ig#

c#air, i% not t#e #ig#est one in t#e #o$se.

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"#is paper 3ill proceed by %irst re!ie3ing t#e mat#ematical origins and

backgro$nd o% catastrop#e t#eory. "#en it 3ill re!ie3 t#e details o% some o% its

applications in economics. "#is 3ill be %ollo3ed by an e:amination o% t#e contro!ersy

and debate regarding t#e $se o% catastrop#e t#eory, 3it# a special %oc$s on disc$ssion o%

t#e eeman 15748 model o% %inancial market dynamics. *inally, 3e s#all note some

alternati!e approac#es t#at are being $sed no3 in order to model t#e p#enomenon o%

dynamic discontin$ity in economics.


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di%%erent kinds o% bi%$rcations, 3it# )$cken#eimer and +olmes 15HF8 pro!iding a good

s$mmary o% t#e !ario$s types.

"#e distinction bet3een critical points o% %$nctions t#at are non-degenerate

associated 3it# e:trema8 and degenerate ones sing$lar, non-e:tremal8 3as %$rt#er

st$died by )eorge Birk#o%% 15278, a %ollo3er o% Poincar?, and also by Birk#o%%=s

%ollo3er, Marston Morse 15F18. An partic$lar, Morse s#o3ed #o3 a %$nction 3it# a

degenerate sing$larity co$ld be slig#tly pert$rbed to a ne3 %$nction t#at 3o$ld no3

e:#ibit t3o distinct non-degenerate critical points instead o% t#e sing$larity. "#is 3as a

bi%$rcation o% t#e degenerate e$ilibri$m and indicates t#e close connection bet3een t#e

sing$larity o% a mapping and its str$ct$ral stability see *ig$re 18.

Oinsert *ig$re 1

+assler itney 15668 %ollo3ed Morse by st$dying di%%erent kinds o%

sing$larities and t#eir stabilities. "#is 3as t#e sort o% approac# t#at 3as essentially t#e

origin o% catastrop#e t#eory, alt#o$g# nobody 3as $sing t#at term yet. An %act, itney

disco!eredCin!ented t#e t3o sing$larities associated 3it# t#e t3o most commonly st$died

kinds o% elementary catastrop#es, and t#e only ones t#at are stable in all t#eir %orms, t#e

%old and t#e c$sp see *ig$re 28. +e s#o3ed t#at t#ese 3ere t#e only t3o kinds o%

str$ct$rally stable sing$larities %or di%%erentiable mappings bet3een t3o planar s$r%aces.

"#$s itney can be !ie3ed as t#e real in!entorCdisco!erer o% catastrop#e t#eory.

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Oinsert *ig$re 2

*ollo3ing #is disco!ery o% trans!ersality 156D8, Ren? "#om de!eloped %$rt#er

t#e classi%ication o% sing$larities, or o% elementary catastrop#es, alt#o$g# a more

complete categori/ation 3o$ld e!ent$ally be carried o$t by


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9slo3,; 3#ereas t#e state !ariables are c#aracteri/ed as being 9%ast.; "#e $s$al

pres$mption #as been t#at t#e state !ariables ad@$st $ickly to be on t#e e$ilibri$m

mani%old 3#ile t#e control !ariables mo!e t#e system aro$nd on t#e mani%old. An t$rn,

t#e catastrop#e %$nction is t#e pro@ection o% t#e e$ilibri$m mani%old into t#e r-

dimensional control !ariable space, 3it# its sing$larities t#e main %oc$s o% catastrop#e

t#eory not to be con%$sed 3it# pro@ection %$nctions %o$nd in game t#eory8..

"#om=s "#eorem, 3#ic# 3as rigoro$sly demonstrated by Malgrange 15DD8 and

Mat#er 15DH8, states t#at i% t#e $nderlying %$nctions %@ are generic $alitati!ely stable

$nder slig#t pert$rbations8, i% r D, and i% n is %inite 3it# all b$t t3o state !ariables being

represented by linear and non-degenerate $adratic terms, t#en any sing$larity o% a

catastrop#e %$nction 3ill be str$ct$rally stable generic8 $nder slig#t pert$rbations and

can be classi%ied into ele!en di%%erent types. "#ere are se!en s$c# types %or r 6, and

"#om 15728 pro!ided color%$l names %or eac# o% t#ese, along 3it# detailed disc$ssions

o% t#eir !ario$s c#aracteristics, 3it# %$rt#er disc$ssion carried o$t by "rotman and

eeman 157D8.4 *or r 6 and more t#an t3o control parameters, t#e set o% possible

catastrop#es is in%inite.


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applications #a!e in!ol!ed t#e t3o simplest %orms already kno3n to +assler itney in

1566, t#e %old and t#e c$sp depicted in *ig$re 2. An order to analy/e a partic$lar model

$sing one o% t#ese one m$st make ass$mptions regarding #o3 t#e system mo!es bet3een

e$ilibria in sit$ations o% m$ltiple e$ilibria. *or t#e simplest case o% t#e %old

catastrop#e %o$r kinds o% be#a!ior can occ$rL hysteresis, bimodality, inaccessibility, and

sudden jumps.


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"#e earliest p$blis#ed application 3as d$e to eeman 15748 and 3as an e%%ort to

model b$bbles and cras#es in stock markets. "#is e:ample #as been m$c# critici/ed

a#ler and S$ssman, 1577N &eintra$b,15HF8. &e s#all ret$rn later to disc$ss t#e

criticisms o% t#is partic$lar model a%ter 3e disc$ss t#e $estion o% t#e broader criticisms

o% catastrop#e t#eory and t#e debate t#at arose aro$nd it.


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Bonanno 15H78 st$died a model o% monopoly in 3#ic# t#ere 3ere non-

monotonic marginal re!en$e c$r!es d$e to market segmentation. M$ltiple e$ilibria can

arise 3it# smoot#ly s#i%ting cost c$r!es, 3#ic# #e analy/ed $sing catastrop#e t#eory.

Per#aps t#e most in%l$ential application o% catastrop#e t#eory in economics 3as to

t#e analysis o% b$siness cycles in a paper by arian 15758. +e adopted a nonlinear

in!estment %$nction o% aldor 15408 as modi%ied by (#ang and Smyt# 15718 to

constr$ct t#e %ollo3ing model.

dyCdt s(y88 T Ay,k8 U y 68

dkCdtg Ay,k8 U A0 D8

(y8 cy T ', 78

3it# y as national income, k as capital stock meas$red against a long-r$n trend, (y8 as

t#e cons$mption %$nction, Ay,k8 as t#e gross in!estment %$nction 3it# A0an a$tonomo$s

le!el o% replacement in!estment, and s a speed o% ad@$stment parameter ass$med to be

rapid relati!e to t#e mo!ements o% t#e capital stock. "#e nonlinear in!estment %$nction

3as ass$med to #a!e a sigmoid s#ape and 3o$ld s#i%t 3it# t#e capital stock as depicted

in *ig$re 4, 3it# S A being t#e e$ilibri$m condition.

Oinsert *ig$re 4

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&it#in t#is model a #ysteresis cycle 3it# discontin$ities can arise as t#e

in!estment %$nction s#i%ts back and %ort# d$ring t#e co$rse o% a b$siness cycle, as

depicted in *ig$re 6.

Oinsert *ig$re 6

arian t#en e:tended t#is model by allo3ing t#e cons$mption %$nction to incl$de

3ealt#, 3, as a control !ariable as %ollo3sL

(y,38 c38y T '38, H8

3it# c=38 0 and '=38 0. "#is %orm$lation allo3s %or a tilting o% t#e sa!ings

%$nction s$c# t#at t#ere are no longer any m$ltiple e$ilibria. "#is allo3ed arian to

disting$is# bet3een simple recessions and longer term depressions. "#is 3as depicted by

a c$sp catastrop#e in 3#ic# 3ealt# is t#e splitting %actor, as depicted in *ig$re D.

Oinsert *ig$re D

ne o% t#e %e3 e%%orts to empirically estimate a catastrop#e t#eory model in

economics 3as o% a model o% in%lationary #ysteresis in!ol!ing a pres$mably s#i%ting

P#illips ($r!e. "#is 3as d$e to *isc#er and Jammernegg 15HD8. "#e met#od t#ey $sed

3as a m$lti-modal density %$nction d$e to (obb 157H, 15H18.H *or .S. data %or t#e

period o% J$ne 1567 to J$ne 15HF, t#ey %o$nd a c$sp point in t#e space o% t#e

$nemployment rate and in%lationary e:pectations o% abo$t 7 percent %or eac# !ariable.

H*or %$rt#er disc$ssion see (obb, oppstein, and (#en 15HF8 and (obb and acks 15H6, 15HH8. At isstriking t#at %e3 o% t#e critics o% catastrop#e t#eory 3#o c#arge t#at it cannot be $sed practically orempirically seem to be a3are o% t#is 3ork.

1F


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"#is dre3 on an ad #oc model s$ggested by &oodcock and 'a!is 157H8, and in e%%ect

arg$ed t#at t#is system co$ld be !ie3ed as a c$sp catastrop#e, 3it# t#e economy @$mping

to t#e 9#ig#er; s#eet o% t#e e$ilibri$m mani%old d$ring 157F-74 and t#en back do3n

again, b$t t#en @$mping $p again at t#e end o% 1577 only to grad$ally come back do3n

by going aro$nd t#e c$sp point a%ter 15H0.

'ra3ing on models d$e to Br$no 15D78 and Magill 15778, Rosser 15HF8

analy/ed dynamic discontin$ities in an optimal control t#eoretic gro3t# t#eory model

t#at contained capital t#eoretic parado:es. +o and Sa$nders 15H08 de!eloped a

catastrop#e t#eory model o% bank %ail$re 3#en risk %actors go beyond critical le!els.

"#e areas o% $rban and regional economics sa3 especially large n$mbers o%

applications o% catastrop#e t#eory, incl$ding t#e $se o% catastrop#e t#eory models o%

#ig#er dimensionality t#an t#e t#ree dimensional c$sp catastrop#e seen abo!e, alt#o$g#

some o% t#is 3ork 3as done by geograp#ers rat#er t#an economists.


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$mbilic catastrop#es by P$$ 1575, 15H1a, b85and by Beckmann and P$$ 15H68.


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many models are no3 st$died o% m$ltiple e$ilibria in %oreign e:c#ange rate models,

3it# many o% t#ese taken !ery serio$sly gi!en t#e n$mero$s %oreign e:c#ange crises t#at

#a!e occ$rred in recent years.

Th D+$t $nd Do,n#$)) o# C$t$%tro&h Thor'

"#e ma@or contro!ersy and debate s$rro$nding catastrop#e t#eory er$pted $ite

early in t#e process d$ring t#e late 1570s, prior to t#e %$ll 3orking o$t o% many o% t#e

applications noted in t#e pre!io$s section. B$t t#e o$tcome o% t#is debate 3o$ld be a

resid$e t#at 3o$ld grad$ally corrode t#e s$pport %or $sing catastrop#e t#eory and

c$lminated in t#e 3idespread disdain t#at 3e see today.


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Responses appeared in ScienceandNaturein 1577, 3it# a more !igoro$s and e:tended

set o% de%enses appearing inBehavioral Scienceli!a and (apde!ille, 15H0N )$astello,

15H18,12

3it# t#e %irst o% t#ese being t#e so$rce o% t#e line abo$t 9t#e baby 3as t#ro3n

o$t 3it# t#e bat#3ater.; More balanced o!er!ie3s came %rom mat#ematicians

)$cken#eimer, 157HN


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R$ssia s$c# as


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S$ssman and a#ler 157Ha,b8 3ent %$rt#er to arg$e t#at any s$r%ace can be %it to

a set o% points and t#$s one can ne!er !eri%y t#at a global %orm is correct %rom a local

estimate. (ertainly one s#o$ld be ca$tio$s abo$t e:trapolating a partic$lar mat#ematical

%$nction beyond a narro3 range o% obser!ation, b$t t#is arg$ment smacks indeed o%

9t#ro3ing t#e baby o$t 3it# t#e bat#3ater.; At 3o$ld seem to deny t#e possibility o% any

kind o% con%idence testing %or nonlinear econometric models. (ertainly t#ere are critics

o% econometrics 3#o #old s$c# positions, b$t t#ey are generally #eld %or all econometric

models, not merely t#e nonlinear ones. "o t#e e:tent t#at t#is is a !alid criticism, it is not

one abo$t empirical estimates o% catastrop#e t#eory models, per se.


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< c$rio$s o$tcome res$lting %rom t#is debate o!er $alitati!e met#ods and

sp$rio$s $anti/ation 3as a split bet3een t#e t3o main prom$lgators o% catastrop#e

t#eory, "#om and eeman. ereas eeman 3as t#e main a$t#or o% t#e $antitati!e

st$dies t#at came $nder criticism, "#om #ad al3ays been more t#e abstract t#eoretician

and p#ilosop#er o% catastrop#e t#eory.14 +e e!ent$ally came to agree 3it# eeman=s

critics "#om, 15HF8, arg$ing t#at 9"#ere is little do$bt t#at t#e main criticism o% t#e

pragmatic inade$acy o% (.". Ocatastrop#e t#eory models #as been in essence 3ell

%o$nded; 15HF, c#ap. 78. More broadly #e de%ended t#e strictly $alitati!e approac#,

critici/ing 3#at #e labeled as 9neo-positi!e epistemology.; (atastrop#e t#eory 3as to be

$sed %or 9$nderstanding reality; and %or t#e 9classi%ication o% analogo$s sit$ations.;

E!en be%ore t#e contro!ersy broke #e #ad declared 1576, p. FH28L

9n t#e plane o% p#ilosop#y properly speaking, o% metap#ysics,

catastrop#e t#eory cannot, to be s$re, s$pply any ans3er to t#e great problems

3#ic# torment mankind. B$t it %a!ors a dialectical, +eraclitean !ie3 o% t#e

$ni!erse, o% a 3orld 3#ic# is t#e contin$al t#eatre o% t#e battle bet3een Vlogoi,=

bet3een arc#etypes.;16

S$c# remarks 3o$ld lead


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Regarding t#e %irst o% t#e arg$ments regarding strict mat#ematical ass$mptions,

t#e need %or a potential %$nction to e:ist is one t#at is a serio$s problem %or many

economics applications. "#ere are ones t#at clearly satis%y t#is ass$mption, 3it# t#e

general e$ilibri$m ones by Rand 157D8 and Balasko 157H8 %$l%illing t#is, as 3ell as t#e

regional models o% P$$ 1575, 15H1a, b8 and Beckmann and P$$ 15H68. At 3as t#is

re$irement t#at led Balasko to arg$e t#at proper applications o% catastrop#e t#eory to

economics 3o$ld necessarily be limited, and t#is is a strong arg$ment. ne response d$e

to oren/ 15H58 is t#at t#e e:istence o% a stable yap$no! %$nction may be a s$%%icient

alternati!e, 3#ic# 3ill #old %or many models, alt#o$g# s$c# cannot in general be

demonstrated %or p$rely $alitati!e models.

"#e second mat#ematical limitation in!ol!es t#e %act t#at gradient dynamics do

not allo3 %or time to be an independent !ariable, a point especially emp#asi/ed by

)$cken#eimer 157F8 3ell be%ore t#e noisier critics took t#e stage. "#om 15HF, pp.

107-10H8 responded to t#is point by arg$ing t#at an elementary catastrop#e %orm may be

embedded in a larger system 3it# a time !ariable. A% t#e larger system is trans!ersal to

t#e catastrop#e set in t#e enlarged space, t#en t#ere 3ill be no problem. % co$rse, it 3ill

be !ery di%%ic$lt to determine t#is in practice. (ertainly 9catastrop#e t#eory; models t#at

$se time as an e:plicit independent !ariable are not really catastrop#e t#eory models.

*inally t#ere is t#e arg$ment t#at t#e elementary catastrop#es are only a limited

s$bset o% t#e possible range o% bi%$rcations and discontin$ities.


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An t#e c$rrent sit$ation 3#ere it is 3idely denied t#at it #as any rele!ance or $se%$lness,

s$c# a criticism #as rat#er o$tli!ed its day. "#is is not a !alid criticism o% $sing

catastrop#e t#eory in sit$ations %or 3#ic# it is appropriate.

Critici%! o# $n Econo!ic A&&)ic$tion

% co$rse it is not m$c# o% a de%ense o% catastrop#e t#eory to point o$t t#at

criticisms made o% speci%ic models #appen to #a!e been misg$ided. B$t in regard to

economic applications, t#ere 3as relati!ely little speci%ic disc$ssion d$ring t#e main

debates in t#e late 1570s. "#e main economic application t#at 3as disc$ssed 3as

probably t#e %irst one e!er made, t#e model o% stock market cras#es d$e to eeman

15748.


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%$ndamentalists, t#en t#e market 3ill be 3ell-be#a!ed and stable, 3it# a $ni$e

e$ilibri$m, keeping in mind t#at t#is e$ilibri$m is act$ally a rate o% c#ange o% price,

alt#o$g# i% t#e e$ilibri$m %or t#e midpoint e$als /ero, t#en t#at 3ill essentially

coincide 3it# t#e random 3alk model.


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in many models. *inancial economists in 1577 3ere especially enamored o% e%%icient

markets models o% rational, #omogeneo$s agents, arg$ably e!en more so t#an most ot#er

economists. At only began to become intellect$ally respectable to allo3 %or

#eterogeneo$s agents in %inancial market models a%ter *isc#er Black 15HD8 ga!e #is

%amo$s presidential speec# to t#e

17M$c# o% t#e immediate intellect$al response to t#at e!ent 3as %or many economists to try to e:plain it$sing t#e sensiti!e dependence on initial conditions idea o% c#aos t#eory, 3#ic# 3as !ery m$c# near t#e

peak o% its intellect$al b$bble at t#at time.


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< less 3ell-kno3n criticism 3as made later by &eintra$b 15HF8, alt#o$g# #e did

not critici/e $sing catastrop#e t#eory in economics in general. (oming %rom a disc$ssion

o% t#e tWtonnement process in general e$ilibri$m t#eory and t#e stability conditions %or

t#at process, #e concl$ded t#at eeman=s model implied t#at c#artist traders m$st #a!e

$p3ard-sloping demand c$r!es. +e t#en cited Stigler 154H8 3#o arg$ed t#at t#ere ne!er

#as been a tr$e )i%%en good 3it# an $p3ard-sloping demand c$r!e. Regarding t#e idea

t#at a stock market participant mig#t belie!e t#at tomorro3=s price 3ill depend on 3#at

t#e price #as been, &eintra$b declared in italics 15HF, p. H08, 9There is no evidence

whatsoever to support such an hypothesis,; citing Malkiel 15768 and t#e random 3alkmodel.


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catastrop#e t#eory in economics is simply abs$rd.


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inter3o!en 3it# eac# ot#er as in oren/ 15528. Xet anot#er in!ol!es sel%-organi/ing

criticality 3#erein small e:ogeno$s s#ocks can trigger m$c# larger endogeno$s reactions

Bak, (#en, Sc#einkman, and &ood%ord, 155F8. Still anot#er $ses t#e idea o%

synergetics, especially as in!ol!ing t#e $se o% t#e master e$ation approac# &eidlic#

and Bra$n, 15528. An its emp#asis $pon disting$is#ing bet3een slo3ly c#anging control

!ariables and more rapidly c#anging sla!ed !ariables, t#esynergeticsapproac# #as a

m$c# stronger s$per%icial similarity to catastrop#e t#eory.

*inally 3e m$st note t#e increasing spread t#ro$g#o$t economics o% models t#at

posit m$ltiple e$ilibria. An many cases t#ese models generate possible e$ilibri$m

s$r%aces t#at #a!e similarities to t#e e$ilibri$m mani%olds o% catastrop#e t#eory,

alt#o$g# t#ey o%ten %ail to %$l%ill all o% t#e mat#ematical c#aracteristics o% tr$e

catastrop#e t#eory. e!ert#eless, t#ese models can prod$ce dynamic discontin$ities as

control parameters are !aried in 3ays t#at ca$se t#e system to cross bi%$rcation points

t#at separate one e$ilibri$m /one %rom a discretely di%%erent e$ilibri$m /one. S$c#

models are so 3idespread and $bi$ito$s t#at it is not 3ort# e!en beginning to list t#em,

alt#o$g# t#ey #a!e been kno3s as long as economists #a!e been a3are o% t#e possibility

o% m$ltiple e$ilibria. "#is is no3 approac#ing a cent$ry and a #al% !on Mangoldt,

1HDFN &alras, 1H74N Mars#all, 1H508. en s$c# models generate e$ilibri$m s$r%aces

t#at resemble t#ose o% tr$e catastrop#e t#eory, 3e are dealing 3it# somet#ing t#at is

rat#er like a close sibling o% catastrop#e t#eory. "#is sibling no3 sits e!er taller in t#e

#ig# c#airs o% t#e #o$se o% economics, and its close relations#ip 3it# catastrop#e t#eory

mig#t as 3ell be more generally recogni/ed.

2H


29/39


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s$dden, and re!ol$tionary c#anges mig#t #appen #ad considerable 3idespread appeal,

especially among more dissident intellect$als. B$t 3idespread applications o% t#e t#eory

t#at 3ere inappropriate eit#er t#eoretically or met#odologically $ndermined its

credibility. < co$nterattack came in t#e late 1570s, and as t#e 15H0s 3ore on, %e3er and

%e3er applications o% catastrop#e t#eory 3ere seen, especially in economics, alt#o$g#

catastrop#e t#eory al3ays retained more respectability among mat#ematicians as a

special case o% bi%$rcation t#eory. e!ert#eless, t#ere 3ere many applications o%

catastrop#e t#eory in economics t#at 3ere properly done be%ore t#e co$nterattack=s

in%l$ence 3as %$lly %elt.

(riticisms o% applications o% catastrop#e t#eory incl$ded t#at it in!ol!ed

e:cessi!e reliance on $alitati!e met#ods, t#at many applications in!ol!ed sp$rio$s

$anti/ations or improper statistical met#ods, and t#e general %ail$re o% many models to

%$l%ill certain mat#ematical conditions s$c# as possessing a tr$e potential %$nction or by

incl$ding time as an independent !ariable in t#e analysis.


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economists #a!e s#ied a3ay %rom $sing catastrop#e t#eory beca$se o% t#ose criti$es,

t#ey s#o$ld no longer do so.

&it# t#e decline o% catastrop#e t#eory, a !ariety o% alternati!e met#ods o%

modeling dynamic discontin$ities in economic models #a!e appeared, alt#o$g# some o%

t#em #a!e been aro$nd %or m$c# longer t#an catastrop#e t#eory #as, and some #a!e close

connections 3it# catastrop#e t#eory, especially t#e analysis o% Skiba points in m$ltiple

e$ilibria dynamical systems.

An s$m, it 3o$ld appear t#at indeed t#e baby o% catastrop#e t#eory 3as largely

t#ro3n o$t 3it# t#e bat#3ater o% its inappropriate applications to a large e:tent.


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(#ang. &.&. and '.J. Smyt#. 1571. 9"#e E:istence and Persistence o% (ycles in aonlinear ModelL aldor=s 1540 Model Re-e:amined,;eview of Economic StudiesFHL1, pp. F7-44.

(lark, (olin &. 157D.'athematical Bioeconomics. e3 XorkL &iley-Anterscience.

(obb, oren. 157H. 9Stoc#astic (atastrop#e Models and M$ltimodal 'istrib$tions,;Behavioral Science2F, pp. FD0-F74.

(obb, oren. 15H1. 9Parameter Estimation %or t#e ($sp (atastrop#e Model,; BehavioralScience2DL1, pp. 76-7H.

(obb, oren, P. oppstein, and .+. (#en. 15HF. 9Estimation and Moment Rec$rsionRelations#ips %or M$ltimodal 'istrib$tions o% t#e E:ponential *amily,;-ournal of the!merican Statistical !ssociation7HLFH1, pp. 124-1F0.

(obb, oren and S#elemya#$ acks. 15H6. 9


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*isc#er, Ed3in . and &erner Jammernegg. 15HD. 9Empirical An!estigation o% a(atastrop#e "#eory E:tension o% t#e P#illips ($r!e,;eview of Economics and StatisticsDHL1, 5-17.

*llmer, +ans. 1574. 9Random Economies 3it# Many Anteracting


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aldor, ic#olas. 1540. 9< Model o% t#e "rade (ycle,;Economic -ournal60L157, pp.7H-52.

indelberger, (#arles P. 1555.'anias, Panics, and &rashes, Frdedn. e3 XorkL Basic

Books.

olata, )ina Bari. 1577. 9(atastrop#e "#eoryL "#e Emperor #as no (lot#es.; Science,15DL42HD, pp. 2H7, F60-F61.

r$gman, Pa$l. 15H4. 9"#e Anternational Role o% t#e 'ollarL "#eory and Prospect,; inE*change rate theory and practice. Jo#n *.. Bilson and Ric#ard (. Marston, eds.(#icagoL ni!ersity o% (#icago Press, pp. 2D1-27H.

r$gman, Pa$l. 1551. 9Ancreasing Ret$rns and Economic )eograp#y,;-ournal ofPolitical Economy55LF, 4HF-455.

r$gman, Pa$l. 155D. The Self#$rgani4ing Economy. :%ordL Black3ell.

$/netso!a,


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Malkiel, B$rton. 1576.! andom 2al3 %own 2all Street. e3 XorkL orton.

Malgrange, Bernard. 15DD.)deals of %ifferentiable unctions. :%ordL :%ord ni!ersityPress.

Mars#all,


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Rosser, J. Barkley, Jr. 15HF. 9Res3itc#ing as a ($sp (atastrop#e,;-ournal of EconomicTheoryF1L1, 1H2-15F.

Rosser, J. Barkley, Jr. 1551.rom &atastrophe to &haos+ ! .eneral Theory of Economic

%iscontinuities. BostonL l$3er. Second edn. 2000a. (olume )+ 'athematics,'icroeconomics, 'acroeconomics, and inance. BostonL l$3er8

Rosser, J. Barkley, Jr. 1557. 9Spec$lations on onlinear Spec$lati!e B$bbles,;Nonlinear%ynamics, Psychology, and 0ife Sciences1L4, pp. 276-F00.

Rosser, J. Barkley, Jr. 1555. 9n t#e (omple:ities o% (omple: Economic 'ynamics,;-ournal of Economic Perspectives1FL4, pp. 1D5-152.Rosser, J. Barkley, Jr. 2000b. 9


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Structural Stability and 'orphogenesis+ !n $utline of a Theory of 'odels. ReadingLBen@amin8.

"#om, Ren?, 3it# response by E. (#ristop#er eeman. 1576. 9(atastrop#e"t#eoryL AtsPresent State and *$t$re Perspecti!es,; in%ynamical systems#2arwic3 ;. ect$re

otes in Mat#ematics o. 4DH.


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&ilson,

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