Theodosio, B uno Mille ; Webe , Jan
Wo king Pape
Back o he classics: R-e olu ion owa ds s a is ical
equilib ia
i so wo king pape , No. 28
P o ided in Coope a ion wi h:
Uni e si y o Duisbu g-Essen, Ins i u e o Socioeconomics (i so)
Sugges ed Ci a ion: Theodosio, B uno Mille ; Webe , Jan (2023) : Back o he classics: R-e olu ion
owa ds s a is ical equilib ia, i so wo king pape , No. 28, Uni e si y o Duisbu g-Essen, Ins i u e o
Socio-Economics (i so), Duisbu g
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i so wo king pape
B uno Mille Theodosio
Jan Da id Webe
2023 no.28
Back o he Classics:
R-E olu ion Towa ds
S a is ical Equilib ia
Back o he Classics: R-E olu ion Towa ds S a is ical
Equilib ia
Theodosio, B uno Mille
Uni e si y o U ah
Webe , Jan Da id
Ins i u e o Socio-Economics, Uni e si y o Duisbu g-Essen
Sep embe 12, 2023
Abs ac
Economic modeling s uggles o en wi h a lack o ealism. The eason o ha is ha
economic heo y o he las 100 yea s has ocused on simpli ying assump ions which
educed impo an aspec s o he economic eali y. Concen a ing on ix-poin solu ions
and ex e nal s a is ical shocks p e en ed he p o ession om accu a ely desc ibing he
economy as a complex sys em wi h cha ac e is ics like eedback mechanism, e olu ion,
and eme gence. We p opose a e-e alua ion o majo indings in he Classical economic
li e a u e. The classical li e a u e desc ibed he economic sys em as inhe en ly p ob-
abilis ic. In his spi i , we discuss he impo ance o s a is ical equilib ium models as
one way o model complex economic sys ems in a p obabilis ic way.
JEL Code: B12, B41, C18, D59
1
1 In oduc ion
Economics scien i ic ield open o dispu es and discussions abou he ma e ial ci cum-
s ances o li e. The di e en schools o hough compe e o academic a en ion, wi h
di e en assump ions and implica ions ega ding socio-economic ci cums ances. The
neoclassical pa adigm emains he leading ideology in he academic deba e. This hege-
mony is epea edly ques ioned when economic c ises occu (K ugman 2009, Va ou akis
e al. 2012, Cho a as 2013). Howe e , neoclassical hinking emains deep in he epe -
oi e o economis s.
We c i icize one aspec o he neoclassical pa adigm. We c i icize he use o he
neoclassical gene al equilib ium heo y as a me hodologically lawed concep . We dis-
cuss he p e ailing mindse in economics ha he ac ions and ci cums ances o agen s
a e de e minis ic in hei na u e. This leads o de e minis ic ix-poin solu ions which
qui e o en di e om economic eali y. We p opose a me hodological solu ion o his.
S a is ical equilib ia a e in hei na u e p obabilis ic and ins ead o ocusing on one
solu ion, i s solu ion is a dis ibu ion. The dis ibu ional o m has he ad an age o
assigning a p obabili y o ealiza ions and links he economic p o ession back o i s
oo s: The Poli ical Economy app oach o he Classical au ho s.
We decided o s uc u e his pape in a way ha opens he discussion abou equilib-
ium, especially o economis s who expe ienced neoclassical aining. In he i s s ep,
we mi o he eaching app oach in mos economic classes by ocusing on he neoclassi-
cal pe spec i e o discussing equilib ium. We discuss ou impo an cha ac e is ics ha
a e implied in he neoclassical equilib ium heo y. We con as hose ou cha ac e is-
ics o complexi y heo y and how hey ela e o economic ac i i y. In he second s ep,
we de i e a complex unde s anding o economic ac i i y independen o neoclassical
heo y. We gi e a de ailed desc ip ion o s a is ical equilib ia and p obabilis ic hink-
ing and accen ua e hose hough s wi h e e ences o classical poli ical economics. We
use he Quan al Response S a is ical Equilib ium (QRSE) model (Scha enake & Foley
2017) as one o many elabo a e models o analyze economic ac i i y in a p obabilis ic
amewo k.
Ou con ibu ion o he li e a u e is wo- old. On one hand, we p o ide an in oduc-
ion o p obabilis ic hinking as a way o be e unde s and economic ac i i y. To make
p obabilis ic economics accessible o a wide audience, we i s con as i o he es ab-
lished neoclassical amewo k and secondly, de i e i independen ly o he neoclassical
amewo k. On he o he hand, we show ha he s a is ical equilib ium app oach is an
adequa e way o combine Classical Poli ical Economics wi h p obabilis ic hinking.
2 The Neoclassical Equilib ium
The neoclassical heo y has a di e en unde s anding o equilib ium compa ed o he
classics. I b eaks in i s heo e ical unde s anding and concep ual amewo k wi h he
ea lie unde s anding o economics. Cou no (Mi owski 1991, p. 210) used op imiza ion
heo y, in he o m o Lag angian and Hamil onian, o de e mine he maximal ou come
2
in an economy. The ma hema ical app oach o Cou no h ew he shadows o he
ma ginalis e olu ion ahead. The “p emie oika o he ma ginalis e olu ion; Leon
Wal as, William S anley Je ons, and Ca l Menge ” (Mi owski 1991, p. 254) pushed he
use o ma hema ical equa ions u he . The e exis s a simila i y be ween he equa ions
in economics and hose used in New onian classical mechanics (Pa e o & P iuli 2012,
p. 16).
To bene i om di e en ial and in eg al calculus and u he de elopmen s om
New onian and Leibnizian me hods, neoclassical economics inco po a ed equilib ium
heo y deep in o i s me hodological amewo k and es ablished i sel as he dominan
pa adigm. The adi ional neoclassical implies se e al assump ions, a ew o which
necessa ily lead o speci ic ou comes. We ocus on pe ec a ionali y, homogeneous
agen s, he de e minis ic and s ochas ic na u e o he models, and he exis ence o ix-
poin equilib ia as ou o hese implici assump ions. We discuss hose ou assump ions
wi h espec o hei implici s a emen s and hei limi ing impac on he na u e o
modeling.
F om a philosophy o science s andpoin , he e a e some ea u es ha p o ide some
explana ion o he dominance and su i al o he neoclassical pa adigm, e en unde
e y un ealis ic assump ions. F iedman (1953, p. 14) highligh s how impo an i is o
a heo y ha “i ‘explains’ much by li le”. He desc ibes he necessi y o concep ualize
he complex na u e o he economy in simple e ms as he “wildly inaccu a e desc ip i e
ep esen a ions o eali y [ ha ] abs ac s he common and c ucial elemen s om he
mass o complex and de ailed ci cums ances”(p. 14).
Fou cha ac e is ics o he neoclassical heo y s and ou in hei impo ance o
he me hodological amewo k o he heo y. (1) Pe ec a ionali y o agen s and (2)
homogenei y o he agen s a e necessa y o achie e esul s in he applied ma hema ical
concep s. The (3) exogenous shocks h ough de e minis ic/s ochas ic e o e ms will
lead o (4) ix-poin solu ions o he model. These ou aspec s a e impo an aspec s
o he neoclassical heo y and cha ac e ize he unde s anding o equilib ium in ha
concep ual amewo k.
Pe ec a ionali y is exp essed by all agen s in he neoclassical amewo k. Pe ec
a ionali y implies ha agen s expe ience speci ic skills, ha allow he agen o make
decisions ha a e pe ec . E e y agen is pe ec ly awa e o hei own cos s and bene-
i s o each ansac ion. The agen s a e pe ec ly awa e o all he ac ions o po en ial
coun e pa s in ha ansac ion. This pe ec insigh in o he cu en ma ke en i on-
men allows he agen s o calcula e he ou come o each ac ion and choose he mos
bene icial. The pe ec a ionali y o agen s is no limi ed o he cu en ime in e al
bu s e ches in ini ely in o he u u e. Pe ec a ionali y excludes any su p ises o
he agen s as hey a e able o calcula e he op imal solu ion o hemsel es h oughou
ime. Fo economis s, i is possible o calcula e oday all in ini e payou s.
Homogeneous agen s a e ano he impo an concep in neoclassic economics. I is
assumed ha agen s ha e iden ical in e es s, cos s, and payou s. Such an assump ion
is jus i ied h ough he exis ence o pe ec compe i ion whe e any di e gence in he
ac ion o an agen would ei he assign he whole ma ke o a single agen (unde bidding
3

he compe i o s) o deny any ma ke sha e o ha agen (o e cha ging). These homo-
geneous agen s allow a esea che o accumula e he whole economy in o one single
agen . This ep esen a i e agen simpli ies he economic analysis as i makes i i el-
e an i a model consis s o one ep esen a i e agen o se e al iden ical agen s. The
idea o homogeneous agen s bypasses he necessi y o analyze dis ibu ional e ec s as
all agen s in he economy a e iden ical (Ki man 1992).
Shocks o c ises in he neoclassical amewo k a e empo a y exogenous and no a e-
sul o in e nal model dynamics. The neoclassical amewo k ules ou ha endogenous
dynamics lead o bubbles o mis alida ion o asse s o ansac ions. Any non- a ional
ac ion o agen s would lead o a bi age which o he agen s would immedia ely ealize
and use o inc ease hei own payou s eam. The exogenous shocks a e independen
o he ac i i y o he agen s. They mus be de ined by he esea che and a e no mally
cha ac e ized by no mal-dis ibu ed e o e ms. The ac i i y o agen s and how hey
adjus o ex e nal shocks become he ocus o economic analysis. A e an ex e nal
shock, he sys em ecalib a es o a new equilib ium and emains in ha s a e un il a
new shock dis up s he op imized plans o he agen s.
The neoclassical app oach elies on ixed-poin heo ems. The solu ions o a sys em
wi h a supply unc ion and a demand unc ion, q∗and p∗, a e conside ed ix-poin s o
quan i y and p ice equilib ia esul ing om he balance be ween supply and demand1.
The heo y s a es ha o any ma ke , p ices would be he same. Fo he p ice o
a commodi y ca ego y o ha e he same nume ical esul , he dis ibu ion o p ices
would be a Di ac Del a dis ibu ion. I is dispu ed i he Di ac Del a dis ibu ion
con ains he necessa y cha ac e is ics o be classi ied as a dis ibu ion. The Di ac
Del a dis ibu ion con ains all obse a ions a one single alue, like a his og am wi h
only one ba . The Di ac Del a dis ibu ion has ze o en opy. Ze o en opy implies ha
he e is no unce ain y o an agen . I an agen swi ches be ween s a es, hose swi ches
do no ha e any unce ain y o slow adjus men p ocess. The beha io al empe a u e
is ma ginally close o ze o (Foley 2020b).
Neoclassical economics has adap ed and e ol ed on hose co e ea u es. Mode n
app oaches o en include se e al agen s wi h di e en s a egies and limi ed o esigh .
These agen s can e as he e o e m is no applied o he ex e nal condi ions bu
o he in e nal calcula ion o he agen s. Those adap ions o he model can lead o
se e al di e en op imal s a egies o he agen s and solu ions o he model, including
bi u ca ing pa hs. This does no , howe e , change he unde lying c i icism o he
app oach. The app oach con inues o build on he old pa adigm as he assump ions
and modeling echniques a e closely ela ed. I emains ha he exogenous luc ua ion
o he model is he only sou ce o de i a ion om he ix-poin solu ion. The lack
o in e nal unce ain y in he model c ea es a wo ld ha is easy o analyze bu lacks
speci ic ea u es ha a e cha ac e is ic o social sys ems. The lack o unce ain y abou
he ou come c ea es a wo ld o ze o in o ma ional en opy.
1The poin whe e supply and demand in e sec de ines he ixed-poin equilib ium in ha ma ke , esul ing
in an o de ed pai o equilib ium quan i y and p ice. This single numbe would ep esen he p ice o e e y
ansac ion in ha ma ke . In eali y, howe e , p ices a y.
4
Bo h Wal as and Ma shall c ea ed an abs ac ep esen a ion o ma ke s (Foley
2020a). They subs i u ed he dis ibu ion o ansac ion p ices by he mean o ansac-
ion p ices. This c ea es he possibili y o disequilib ium p ices. As ma ke s a e de ices
ha gene a e equilib ium, no ansac ion would occu a disequilib ium p ices. One
me hodological solu ion o a oid disequilib ium p ices is a ic ional auc ionee . This
Wal asian auc ionee elimina es ansac ions a hose “disequilib ium p ices” and leads
o a gene al equilib ium. This Wal asian auc ionee calls he p ices un il he o al
excess o demand is equal o ze o. I he ela i e p ices ec o does no clea all ma -
ke s, ano he ound akes place. This is epea ed un il a p ice ec o is ound which
clea s all ma ke s a he same ime. The p ices a e single numbe s ha ep esen he
mean. In seeking an abs ac ep esen a ion o ma ke s, bo h Wal as and Ma shall
eso ed o he ma hema ical simpli ica ion o ep esen ing he s a is ical dis ibu ion
o ansac ion p ices by i s mean, p esumably wi h he idea ha he equilib ium s a-
is ical dis ibu ion o ansac ion p ices is highly concen a ed so ha he mean would
closely app oxima e he ac ual dis ibu ion (Pe i & Hahn 2002). The Wal asian un-
de s anding o ma ke s can be concep ualized in e ms o s a is ical mechanics. When
he ma ke is hough o a p obabili y ield o ansac ions2. In his p obabili y ield,
non-equilib ium p ices a e possible. The exis ence o non-equilib ium p ices c ea es he
possibili y ha iden ical agen s ex an e ecei e di e en bundles a di e en p ices in
he exchange (Scha enake & Yang 2020).
3 Complexi y Theo y
I is common o analyze complex dynamics in he con ex o biological o physical
sys ems. Complexi y esea ch has i s been adap ed o a i icial socie ies and la e
o eal-wo ld social sys ems (Mi chell 2011). The goal o Complexi y Economics is o
model and analyze social sys ems as complex sys ems o gain addi ional insigh in o he
unde lying sys em which is no able o be disco e ed in o he amewo ks. Complexi y
economics is o en c i icized o i s ‘ma hiness’ (Rome 2016), while i elies in i s model
se up on simpli ied desc ip ions o human beha io , ules o humb, o habi s (A hu
2021). Assump ions in a complex sys em amewo k can appea ad-hoc and mus
he e o e p ope ly de ined o main ain academic s anda ds. Simila o he p emise o
he neoclassical pa adigm, hese assump ions mus no be explici ly modeled bu hey
a e he unde lying ounda ion o complex models.
The me hods used in complexi y heo y a e mani old and di e se. They s e ch om
agen -based modeling, quali a i e easoning, ne wo k heo y, and in o ma ion heo y.
These app oaches ake di e en poin s o heo e ical depa u e. The agen -based ap-
p oach ollows a s ic mic o- ounda ion while ne wo k heo y comes om a mo e mac o
pe spec i e (Mille & Page 2007). Wha all o hese app oaches ha e in common is ha
hey a e able o model he sys em ha hey analyze as an (1) e ol ing and (2) eme ging
sys em wi h (3) he e ogeneous agen s ha ace (4) bounded a ionali y. While complex
2T ansac ions ha e a p obabili y o occu . This p obabili y can depend on endowmen s, locally obse ed
p ices, o o he ac o s (Eps ein e al. 1997).
5
sys ems can ha e a ious cha ac e is ics, much like neoclassic economics models, we o-
cus on hose ou aspec s. We discuss he concep s o e olu ion, eme gence, bounded
a ionali y, and he e ogeneous agen s. We con as how hey ela e o neoclassical
economics and a e by hemsel es ealis ic cha ac e is ics o socioeconomic sys ems.
E olu ion concep ualizes how agen s lea n by expe ience, bo h om o he s’ expe i-
ences and om hei own pa icipa ion in he sys em. The exposu e o hose expe iences
and in e ac ions makes agen s adjus hei cou se o ac ion o e ime. The lea ning p o-
cess o he complexi y app oach means ha he agen s endogenously adap o su i e.
This con as s he Neoclassical app oach whe e a game is played wi h p ede e mined
ules se by he esea che . This emo es all pedagogical possibili ies o he agen s o
adap o e ime. The s a egies o he agen s e lec he ules o he game, no hei
own lea ning p ocess.
Eme gence is an impo an aspec o complexi y heo y. I desc ibes when he
sys em exp esses cha ac e is ics ha a e unexpec ed om he o iginal se up o he
speci ica ions o he agen s. Feedback loops wi hin he sys em and sel -o ganiza ion
a e o en he eason o eme gence. Feedback loops can be posi i e o nega i e, whe e
a speci ic ac ion ein o ces o weakens he incen i e o such an ac ion. Such eedback
loops a e o in e es when agen s ac and a e analyzed in hei en i onmen . In ne wo k
heo y, hose loops a e named mo i s and hei la ge a ie y o ealiza ion can be isually
exp essed (Sho al & Alon 2010, Cimini e al. 2021). In he case o a nega i e eedback
loop, i all agen s pe o m he same ac ion, le ’s say buy, i becomes mo e in e es ing
o a single agen o swi ch hei s a egy o sell. An example o a posi i e eedback
loop could be, despi e i s nega i e e ec s on he sys em as a whole, he digging o a
well. Digging deepe han you neighbo gi es you mo e wa e bu a he same ime, i
incen i izes he neighbo o dig deepe hemsel es. In sel -o ganiza ion, local ules will
be c ea ed on a mac o-le el pa e n o unc ionali y ha was no in ended o o ganized
by an ex e nal planne . A lock o bi ds is he esul o sel -o ganiza ion as indi idual
bi ds wan o ly nex o hei pee s, bu no oo close o emain in hei own pe sonal
space (Reynolds 1987). Those wo es ic ions lead o he beha io o he lock ha
dances in he skies, wi hou p ede ined di ec ions o pa e ns whe e he beha io o he
lock is mo e han he sum o all bi ds (Ande son 1972). In social sciences, he di ision o
labo and he ise o socie y a e conside ed examples o such sel -o ganiza ion (G aebe
& Weng ow 2021).
Bounded a ionali y is a i aling concep o pe ec a ionali y. Ra he han al-
lowing agen s o make he pe ec decision ha maximizes hei payo as hey ha e
pe ec knowledge ac oss space and ime, he capabili ies o agen s o make decisions
a e es ic ed. I explici ly assumes he cogni i e, in o ma ional, and calcula o y lim-
i a ions o agen s in decision-making. Bounded a ionali y assumes ha agen s make
hei decisions based on hei gu s and he ule o humb. Repe i i ely occu ing deci-
sions ha a e o he same o simila na u e a e no op imized bu a he copied om
p e ious ins ances and, pe haps, mildly adap ed. The eason is ha he acquisi ion o
in o ma ion and he ( e-)calcula ion o decisions is oo cos ly compa ed o he expec ed
imp o emen o he ou come. In o he ci cums ances, agen s simply lack he necessa y
6
in o ma ion o make a well-in o med decision. This can be he unknown li e expec ancy
and e u n on an in es men , which o ces he agen o ake an educa ed guess and be
on possible ou comes.
The he e ogenei y o he agen s is an impo an aspec o modeling and analyzing
complex sys ems. I is he esul o he combina ion o e olu ion and bounded a ional-
i y. E en in a sys em whe e all agen s s a o wi h he same endowmen s, p e e ences,
and objec i e unc ions, bounded a ionali y will lead o di e en pe cep ions o he
en i onmen . Di e en pe cep ions o he en i onmen will lead o di e en s a egies
and payou s in he nex s ep. The di e en success o he agen s leads o di e en
o ms o adap ion and e olu ion, and he e o e he e ogeneous agen s. E en i agen s
s a in he beginning wi h andomly assigned endowmen s and skills, he he e ogenei y
will pe sis o e ime as simula ions wi h a i icial socie ies ha e shown (Eps ein e al.
1997).
The cons an and consis en e olu ion, adap ion, and eme gence o agen s and pa -
e ns make complex sys ems an in e es ing poin o s udy. These can be biological o
chemical sys ems, bu also in social sys ems, we obse e such cha ac e is ic beha io .
These cha ac e is ics lead o de elopmen s o he sys em ha a e no p edic able in ad-
ance i modeled co ec ly. This shi s he ocus o he analysis om he possible/exac
ou come o he mechanism ha d i es he sys em. This sys emic le el desc ibes inhe -
en phenomena and is called meso-le el (Schulz-Gebha d 2023). This meso-le el as he
linkage be ween he mic o and mac o le el ge s he spo ligh . In empi ical wo k whe e
indi idual agen s a e no obse able, i is impo an o ind ways o ex ac he mech-
anism on he meso-le el, as well as he indi idual beha io al ules o make s a emen s
abou he sys em as a whole.
4 The Classical Equilib ium
The s udy o he economy by he Classics is, in oday’s wo ding, complexi y heo y.
Wi hou naming i , Adam Smi h, Ka l Ma x, and o he s desc ibed he economic p ocess
using many ea u es de eloped by Complexi y Theo y, such as he unde s anding o he
economy as an e ol ing and eme ging sys em, consis ing o he e ogeneous agen s wi h
bounded a ionali y.
The classical heo y includes complex sys em ea u es. This inclusion is isible in he
examples o he di ision o labo (Smi h 1976) o he equaliza ion o p o i a e (Ma x
& Ko sc 2009). Despi e hei me hodological idiosync asies, Smi h, Ma x, and Rica do
we e able o desc ibe he economy in such ways because hei heo ies we e e lec ions
o eali y, no ideal c ea ions om hei minds. Ma x’s me hod s a s he in es iga ion
om he conc e e and e u ns o i a e he abs ac ion p ocess in which he disco e s
he de e mina ions o he concep s e ie ed om ma e ial eali y i sel . Thei heo y
was a mi o o he eal wo ld and an a emp o model how agen s make decisions
and all he ins i u ional cons ain s ha shape each his o ical phase. Fo ins ance, he
weal h o na ions a ises om p oduc i i y gains unlocked by he labo di ision, which
esul s om sel -in e es pu sui in Smi h. The d i ing o ce o economic decisions is
7
A no mal dis ibu ion esul s om maximizing en opy subjec o bo h a no maliza-
ion cons ain and cons ain s on he mean and second momen ( a iance). The e o e,
when cons ain s a e applied o complex sys ems hey al e he maximum en opy dis-
ibu ion, illus a ing he in luence o in o ma ion u iliza ion in he p ocess o in e ence.
The majo ask o esea ch is o igu e ou om a ailable in o ma ion wha he ele an
cons ain s a e. The “cons ain s ep esen all o he in o ma ion used ha mus be
sa is ied in he op imiza ion p ocess. They cap u e he ules ha go e n he sys em
modeled” (Golan & Foley 2022). This is a om being an easy ask, i is “ he mos
impo an issue in se ing up [cons ained maximum en opy] p oblems is he speci i-
ca ion o he cons ain s o e lec all o he in o ma ion we ha e in a comple e and
consis en way”(Golan & Foley 2022).
Foley (1994) de i es a s a is ical model o accoun o ma ke exchanges conside -
ing all easible Pa e o-imp o ing ma ke ansac ions. He compa es his model wi h he
Wal asian compe i i e equilib ium heo y and concludes ha (1) all easible Pa e o-
imp o ing mul ila e al ades can occu wi h some p obabili y in he s a is ical equi-
lib ium, (2) equal agen s will ha e di e en esul ing payo s as hey ade in di e en
p ice a ios, wha Foley called ”ho izon al inequali y” in o ma ke ou comes, and (3)
s a is ical equilib ium goes in he di ec ion bu does no achie e Pa e o-e iciency due
o he exis ence o mu ually ad an ageous ades. In conclusion, bo h unemploymen
and excess capaci y a e s a is ical equilib ium phenomena o he model. Those esul s
a ise because he s a is ical equilib ium does no equi e con exi y o p e e ences o
echnology which is bene icial o a model no conjec u ing un ealis ic assump ions.
This app oach is ex ended o a labo ma ke ha clea s a en opy p ices (Fo-
ley 1996). In his s a is ical equilib ium model, he equilib ium is achie ed wi h a
mean wage abo e he ese a ion wage. This makes he model compa ible wi h in-
olun a y unemploymen . The compa ibili y o he s a is ical equilib ium model wi h
unemploymen elimina es he unde s anding o unemploymen as an anomaly, p esen
in Wal asian-Ma shallian adi ion. This model has s ong assump ions such as no
lea ning, expe ience, o memo y p ocess and is a single- un model.
Many o he a iables can be analyzed in a s a is ical equilib ium amewo k. G ea
e o has been pu in o he analysis o compe i ion and how capi al and powe mo e
be ween i ms and sec o s. The a iables unde conside a ion ha e been p o i a es
Tobin’s q (Scha enake & dos San os 2015), (Scha enake & Semieniuk 2017), s ock
e u ns (Ci e a 2021) o he ma kup (Webe 2023).
5.4 Quan al Response S a is ical Equilib ium
The la es e olu ion o he applica ion o en opy-cons ained me hods in Poli ical Econ-
omy is he quan al esponse s a is ical equilib ium. The name o his class o models
adds “quan al esponse” o he s a is ical equilib ium me hods. The eby he ac ions o
he agen s a e no exp essed in s ep unc ions a ound he h eshold bu a he p ob-
abili ies o ac ion which change wi h he ealiza ion o he ele an a iable. So does
he low o capi al s abilize he p o i a e (Scha enake & Foley 2017). This is a di-
ec conclusion o he no ion o compe i ion in Smi h (1976). In he quan al esponse
14

model, wo mechanisms lead o his s abiliza ion. Fi s , capi al owne s a e sensi i e
and eac o p o i a e di e en ials. They en e in o sec o s o ades wi h highe - han-
a e age p o i a es and exi ing o lowe - han-a e age p o i abili y. Secondly, exi ing
a ma ke inc eases p o i abili y in ha ma ke while en e ing lowe s i . The quan al
esponse is condi ional on he decisions o agen s o en e ing and exi ing ma ke s o
ades depending on p o i a e le els.
Scha enake & Foley (2017) discuss how he impac o hese ac ions dependen on
p o i a es, (a|x), and he impac s on p o i abili y dependen on ac ions, (x|a). The
wo condi ional dis ibu ions ep esen he in e ac ion be ween p o i a es and ac ions
o en e ing and exi ing. This o malizes he mu ual dependency in e ac ion be ween
he quan al ac ions and he ou come. The s a is ical equilib ium is a join dis ibu ion
(a, x) o he wo condi ional p obabili ies, whe e (a, x) = (a|x) (x) = (x|a) (a).
The model can be exp essed ei he di ec ly, by he join dis ibu ion o indi ec ly, by
ma ginal and condi ional p obabili ies. To sol e he sys em, he au ho s wo k wi h
cons ained en opy:
max { (a|x)≥0}X
a
(a|x)u(a, x)
subjec o X
a
(a|x) = 1
−X
a
(a|x) log( (a|x)) ≥Hmin
(4)
The op imiza ion p oblem can be ew i en as a Lag angian op imiza ion p oblem, wi h
he wo Lag angian mul iplie s Tand λsuch as:
L=−X
a
(a|x)u(a, x)−λ X
a
(a|x)−1!
+T −X
a
(a|x) log( (a|x)) −Hmin!(5)
The Lag ange mul iplie Tis he beha io empe a u e ha measu es he a en ion o
sensi i i y o agen s o di e ences in payo . I limi s he deg ee o which he agen will
clus e a ound he highes payo ac ion. The lowe Tis, he mo e is he agen ale ed
o di e ences in he a e o p o i . The ou come a he mac o le el will be close o he
uncons ained payo -maximizing ou come (degene a e Di ac Del a dis ibu ion).
The duali y o he cons ained en opy op imiza ion can be ep esen ed by he
op imiza ion o he expec ed payo cons ained by a minimum en opy le el. The
payo -cons ained en opy maximiza ion is a gene al model when he assump ions a e
no known. This is a sui able app oach o sys ems ha change smoo hly o e ime.
The dual e sion o maximizing he expec ed payo subjec o a minimum en opy is a
ma hema ical equi alen closely ela ed o economic models. The en opy-cons ained
payo maximiza ion o he en opy-cons ained quan al esponse model is a gene aliza-
ion o a ional choice heo y. In a ional heo y, he agen s ha e well-de ined payo
15
unc ions which hey maximize by choosing a mixed s a egy. This p oblem esul s in
mo e equen choices o a highe -payo ac ion. The en opy cons ain ensu es a le el
o andomness o unp edic abili y in he sys em. This can be pa icula ly use ul in
si ua ions whe e he decision-make wan s o a oid concen a ing oo much equency
on he highes -payo ac ion.
The sec o p o i a e shi ed by a cons an µis a good app oxima ion o he payo .
The payo unc ion is u(a, x) = x−µleads o he quan al esponse unc ions:
(en e |x) = 1
1 + e−(x−µ
T)and (exi |x)=1− (en e |x) = e−(x−µ
T)
1 + e−(x−µ
T)(6)
These unc ions a e he logi quan al esponse o Gibb’s dis ibu ion. Fo T≫0,
(a|x) ge s la e which ansla es as a weak dependence be ween ac ions on p o i abil-
i y. This inc eases unce ain y and implies la ge luc ua ions in he agg ega e p o i
a e. The opposi e, (a|x)≈0 s eng hens he dependence o aon x. This educes
unce ain y and luc ua ions in he p o i a e. Wi h lowe impac s om en y-and-exi
dynamics on p o i a es (a|x), becomes a Hea iside s ep unc ion. In his unc ion,
small de ia ions o x om µ esul in a colla e al esponse om agen s in o de o elimi-
na e his de ia ion (see Figu e 1). The asymme y in he equency dis ibu ions o he
-1.0 -0.5 0.0 0.5 1.0
x
0.2
0.4
0.6
0.8
1.0
(x|x)
-1.0 -0.5 0.0 0.5 1.0
x
0.2
0.4
0.6
0.8
1.0
(x|x)
Figu e 1: The logi quan al esponse unc ion o di e en beha io al empe a u es T.
Blue: T= 0.01, Yellow: T= 0.1, G een: T= 1, Blue: T= 100.
Le : The p obabili y o en e ing. Righ : The p obabili y o exi ing.
a e o p o i comes om un ul illed expec a ions o he en opy-cons ained beha io o
he decision-make s (Scha enake 2020). When agen s make w ong p edic ions o hei
expec a ions emain un ul illed, he dis ibu ion o he ou come becomes asymme ic.
6 Conclusion
The s a is ical equilib ium amewo k is a new esea ch agenda o economis s look-
ing o mo e ealis ic models o economic beha io . I akes in o accoun he ole o
andomness and unce ain y in economic ac i i y. This is done in a p obabilis ic way
which does no ely on ex e nal shocks bu pu ely on he p obabilis ic modelling o
decisions and ac ions. This makes he s a is ical equilib ium amewo k a successo o
he Classic unde s anding o economics when i was s ill called Poli ical Economy.
16
We sugges hei applica ion o Classical Poli ical Economy. To me ge he s a is ical
equilib ium me hods wi h Poli ical Economy a ises om he ecogni ion ha Classical
hinking was inhe en ly s a is ical. P obabilis ic models educe un ealis ic assump ions
bo h heo e ically and me hodologically. When wo king wi h s a is ical equilib ium
models, we emphasize how c ucial i is o co ec ly se he cons ain s ha will shape
he dis ibu ion because o he ole o non-linea in e ac ions and eedback loops in
economic ac i i y.
The s a is ical equilib ium app oach elies hea ily on he concep o en opy. By
p o iding a o mal concep ha includes unce ain y abou he sys em and incomple e
in o ma ion in he modeling p ocess, we can cap u e he dynamics and di e si y o
eal-wo ld economic sys ems. We show ha quan al esponse models, as a subg oup
o s a is ical equilib ium models, p o ide a ealis ic and nuanced unde s anding o eco-
nomic beha io . Hence, s a is ical equilib ium models can be used o s udy p ope ies
om dis ibu ion, labo ma ke s, goods ma ke s, and o he ele an aspec o economic
sys ems. We show how hese cha ac e is ics can help o explain phenomena such as
inno a ion, adap a ion, lea ning, coo dina ion, coope a ion, con lic , and eme gence in
economic ac i i y.
By conside ing he complex eali y cha ac e is ics and modeling he economy h ough
he lenses o he s a is ical equilib ium app oach, we sugges ha he no ion o equi-
lib ium ha a ises om his me hodology is much mo e app op ia e o desc ibe he
economic eali y. The eme ging unde s anding o eali y is mo e nuanced and less e-
s ic i e. I conside s he equilib ium as he esul ing p obabili y dis ibu ion wi h i s
i s momen and a iabili y, ins ead o single momen equilib ium such as in ixed-poin
heo ems.
17
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