Knez, Klemen; Jaklič, And eja; S a e, Me ka
A icle
An ex ended app oach o alue chain analysis
Jou nal o Economic S uc u es
P o ided in Coope a ion wi h:
Pan-Paci ic Associa ion o Inpu -Ou pu S udies (PAPAIOS)
Sugges ed Ci a ion: Knez, Klemen; Jaklič, And eja; S a e, Me ka (2021) : An ex ended app oach o
alue chain analysis, Jou nal o Economic S uc u es, ISSN 2193-2409, Sp inge , Heidelbe g, Vol. 10,
pp. 1-37,
h ps://doi.o g/10.1186/s40008-021-00244-6
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An ex ended app oach o alue chain
analysis
Klemen Knez* , And eja Jaklič and Me ka S a e
1 In oduc ion
In ecen decades, he g owing complexi y o he di ision o labou has been e lec ed
in he ac ha e e mo e p oduc ion is occu ing wi hin alue chains, bo h a home
and ab oad. Theo e ical and empi ical app oaches o he analysis o alue chains ha e
ad anced apidly, ye a e e y eclec ic and he e ogeneous. The ea lies de ini ions o
commodi y chains1 da e back o he wo ld-sys ems2 heo y: “Wha we mean by such
chains is he ollowing: ake an ul ima e consumable i em and ace back he se o inpu s
ha culmina ed in his i em— he p io ans o ma ions, he aw ma e ials, he ans-
po a ion mechanisms, he labou inpu in o each o he ma e ial p ocesses, he ood
Abs ac
In he a icle, we p opose a comp ehensi e me hodology o alue chain analysis in
he in e na ional inpu –ou pu amewo k ha in oduces a new measu e o alue
chain pa icipa ion and an ex ended ypology o alue chains, wi h he no el inclusion
o domes ic alue chain o add ess he ex en o agmen a ion o pu ely domes ic
p oduc ion. This allows o he simul aneous analysis o bo h global and domes ic
p oduc ion agmen a ion, he complex pa e ns o hei e olu ion and hei impac
on economic de elopmen . The main con ibu ion o he p oposed me hodology is
concep ual: i pe mi s he measu emen o all alue chain pa hs ha pass h ough
each coun y-sec o om p oduc ion o inal consump ion, whe he he pa h includes
downs eam linkages, ups eam linkages o hei combina ion. Empi ical applica ion o
his me hodology shows he impo ance o including domes ic agmen a ion in alue
chain analysis: The agmen a ion o bo h global and domes ic le els o p oduc ion has
a signi ican posi i e co ela ion wi h economic g ow h. This implies ha he e ec s o
global p oduc ion agmen a ion mus be analysed oge he wi h he changing s uc-
u e o he agmen a ion o domes ic p oduc ion o ob ain he whole pic u e, one ha
migh p o ide impo an in o ma ion o policymaking and indus ial policy.
Keywo ds: Value chain ypology, Value chains, Global alue chain, Domes ic alue
chain, Pa icipa ion a e, Inpu –ou pu amewo k
JEL Classi ica ion: F1, F4, F6
Open Access
© The Au ho (s), 2021. Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s
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RESEARCH
Kneze al. Economic S uc u es (2021) 10:13
h ps://doi.o g/10.1186/s40008-021-00244-6
*Co espondence:
Klemen.Knez@ d .uni-lj.si
Cen e o In e na ional
Rela ions, Uni e si y
o Ljubljana, Ljubljana,
Slo enia
1 The e m global commodi y chain is a p edecesso o global alue chain.
2 Emb acing a his o ical and mac oeconomic app oach o he analysis o he global di ision o labou , he wo ld-sys ems
app oach examines he unequal pa e ns o exchange along global commodi y chains as well as di e en s uc u al pa -
e ns o he in e na ional in eg a ion o he co e, pe iphe y and semi-pe iphe y (A ighi and D angel 1986).
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Kneze al. Economic S uc u es (2021) 10:13
inpu s in o he labou . This linked se o p ocesses we call a commodi y chain (Hop-
kins and Walle s ein 1977)”. In he 1990s, he esea ch p og amme o global commodi y
chains was i s sys ema ically ou lined by Ge e i’s seminal con ibu ion (Ge e i 1994)
ha de ined h ee in e locking dimensions o he esea ch: he inpu –ou pu dimension,
he spa ial dimension, and he ques ion o commodi y chain go e nance3. This esea ch
pe iod was cha ac e ised by mo ing away om a his o ical and mac oeconomic pe -
spec i e owa ds a special ocus on indus ial chains and he in e - i m coope a ion pe -
spec i e, wi h nume ous case s udies on alue chains. The global alue chain amewo k
eme ged ea ly in he new cen u y wi h he exp ess aim o uni ying he p e ious he e o-
geneous esea ch (Ge e i 1999;Ge e i e al. 2001). On one hand, he global alue chain
app oach inc eased he ocus on he en e p ise le el and me ged wi h he li e a u e om
in e na ional business and managemen 4, while also d awing om he new ins i u ional
ansac ion cos app oach5. On he o he hand, he c ea ion o in e na ional inpu –ou -
pu ables6 led o a e i al o he agg ega ed mac oeconomic app oach o global alue
chains, albei wi h a di e en ocus han he wo ld-sys ems app oach.7
In his a icle, we p esen a new me hodology o measu ing di e en alue chain pa -
icipa ion a es in he in e na ional inpu –ou pu amewo k. Compa ed o he mos
widely used measu emen o alue chain pa icipa ion in oduced by Wang e al. (2017),
we make wo undamen al concep ual enhancemen s.
Fi s , ou me hodology c ea es a single and consis en measu emen o alue chain
pa icipa ion on he coun y-sec o le el, as opposed o he wo (ups eam and down-
s eam) pa icipa ion a es ha ea u e in Wang’s me hodology. The a gumen a ion and
logic used o de i e a single alue chain pa icipa ion sha e on he coun y-sec o le el is
e y simila o he app oach o A o e al. (2019), which combines he sou ce- and sink-
based app oaches o expo decomposi ion. The idea is ha decomposi ion based on
inal demand (sink-based decomposi ion) is independen o he decomposi ion o down-
s eam alue added (sou ce-based) and hus bo h can be linea ly combined o g asp bo h
he in o ma ion ega ding he sou ce o alue added as well as he pa h o inal demand
simul aneously. Me hodologies o expo decomposi ion ha e ecen ly seen signi i-
can imp o emen s (A o e al. 2019; Bo in and Mancini 2019; Mi oudo and Ye 2021).
Howe e , he alue chain pa icipa ion a e me hodologies ei he s ill chie ly ely on he
alue-added expo ma ix o desc ibe he alue lows be ween any wo coun y-sec o s
in he economy (Johnson and Nogue a 2012) and esul in sepa a e ups eam and down-
s eam pa icipa ion a e measu es o combine a sink- and a sou ce-based measu e in
3 Go e nance was concei ed as ei he consume -d i en (appa el sec o ) o p oduce -d i en (au omo i e sec o ).This
app oach was u he ex ended by Pon e and S u geon (2013).
4 Po e ’s (1985) concep o he in a- i m alue chain is o en used o discuss he specialisa ion o en e p ises, and co e
compe encies and business li e a u e on mul ina ional en e p ises o e lap wi h he global alue chain amewo k.
5 Which was used o ex end he p oduce -d i en and consume -d i en go e nance ypology o commodi y chain
esea ch o a mo e gene al ypology o alue chain linkages, om ansac ions in a comple ely ee ma ke o a s ic
hie a chy (Ge e i e al. 2005).
6 In in e na ional economics, use o he inpu –ou pu me hodology g ew in impo ance as esea che s o a ious in e -
na ional incen i es in eg a ed na ionally based inpu –ou pu ables in o ha monised global inpu –ou pu ables. The
mos p ominen a e he Wo ld Inpu –Ou pu Da abase (Timme e al. 2015), he OECD’s T ade in Value Added and he
EORA (Lenzen e al. 2013).
7 While all he e ogeneous app oaches o alue chains ocus on a de elopmen issue, he ecen GVC app oach has been
adop ed by in e na ional ins i u ions o highligh he gains om libe alisa ion and indus ial upg ading, while he wo ld-
sys ems app oach c i ically examines unequal ewa ds along he alue chain and di e en s uc u al in eg a ion pa e ns
ha may cause he pe pe ua ion o unequal de elopmen (Ge e i 2018; Taglioni and Winkle 2016).
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Kneze al. Economic S uc u es (2021) 10:13
me ely one-sided, o wa d-looking measu es. Ou app oach o alue chain decomposi-
ion no longe uses he alue-added expo ma ix and ins ead b eaks down he asym-
me ic alue chain s emming bo h downs eam and ups eam om each coun y-sec o
conce ned simul aneously. C ea ing a single consis en a iable on he coun y-sec o
le el ha measu es he o e all le el o pa icipa ion in alue chains enables he empi ical
es ing o many esea ch heses ha we e p e iously ei he limi ed o he agg ega e le el
o had o be a icula ed sepa a ely in e ms o measu ing he impac s o ups eam and
downs eam alue chain in eg a ion.
Second, ou me hodology allows ex ensions o he alue chain ypology ha a e no
possible wi h Wang’s app oach o he decomposi ion o p oduc ion ac i i ies o wi h
expo decomposi ions. We in oduce a no el measu e o he domes ic alue chain pa -
icipa ion a e o measu e he sha e o p oduc ion which ep esen s he ex en o he
agmen a ion o domes ic p oduc ion. In place o a single and undi e en ia ed domes ic
componen , we dis inguish domes ic p oduc ion, which is agmen ed (in ol ing meas-
u able coope a ion among domes ic i ms), and domes ic p oduc ion, which is no ag-
men ed (consis ing o p oducing di ec alue o consump ion wi hou he coope a ion
o domes ic i ms). This makes ou concep o he domes ic alue chain a comple ely new
and di e en concep compa ed o Wang’s domes ic componen , which does no dis in-
guish he wo and combines bo h ca ego ies wi hin a single undi e en ia ed concep .
While Wang’s sha e o he domes ic componen is only a simple esidual—a nega ion o
he sha e o he agmen a ion o global p oduc ion and he global Rica dian ade sha e
ha does no p o ide in o ma ion abou he na u e o he domes ic economy, ou no el
me hodology allows us o measu e he ex en o agmen a ion o domes ic p oduc ion
in addi ion o he usual s udy o he agmen a ion o in e na ional p oduc ion.
We aim o use ou app oach o p o ide me hodological ools ha acili a e explo a ion
o he complex in e ela ionship o global and domes ic alue chains and hei e olu ion
o e ime. We belie e his will add o unde s anding o he di e se pa e ns o he s uc-
u al in eg a ion o a ious coun ies/sec o s and he di e en e ec s o such pa e ns on
economic de elopmen . While his is p ima ily a me hodological con ibu ion, we shall
use elemen a y empi ical da a o y o show he possible link be ween he le el o ag-
men a ion o global and domes ic p oduc ion and o e all economic g ow h.
The a icle is s uc u ed as ollows: In Sec .2, we e iew he exis ing alue chain
indica o s and add ess hei sho comings. In Sec .3, we p esen ou me hodology. In
Sec .3.1, we p esen a new concep ualisa ion o alue chain in he in e na ional I–O
amewo k and de ine ou objec o disagg ega ion. A new alue chain ypology is p e-
sen ed in Sec .3.2 whe e we also de i e pa icipa ion sha es. In Sec .4, we p esen an
example o empi ical applica ion and some basic empi ical esul s o he new me hodol-
ogy o show he insigh s in o economic s uc u es ha can be gained by using he new
alue chain measu es and which links exis be ween alue chain in eg a ion pa e ns and
o e all economic g ow h. Finally, we discuss he con ibu ions o he pape , i s limi a-
ions and possibili ies o u he esea ch.
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Kneze al. Economic S uc u es (2021) 10:13
2 Backg ound
The mos ecen mac oeconomic analyses o global alue chains ely on he in e na ional
inpu –ou pu me hodology. As in e na ional I–O da a a e essen ially an in eg a ed
s anda d accoun ing da a se ha monised on he sec o al le el, in o ma ion is lacking
on he ypology o alue chain go e nance. This means he in e na ional I–O da abase
canno be he sole sou ce o he s udy o p oduc ion ne wo ks, which heo e ically di -
e om pu ely open ade ansac ions by including a leas some le el o hie a chy, and
which in es iga e he local embedding o p oduc ion linkages (Buckley2009;Hende -
son e al. 2002; Hess and Coe 2006; Ho açsu and Sy e son 2009). Howe e , he gene al
amewo k o global alue chains can unc ion wi hou such dis inc ions and his makes
he in e na ional I–O da a se one o i s mos impo an sou ces o in o ma ion. The key
bene i o applying he I–O me hodology in global alue chain analysis is ha agg e-
ga ed in o ma ion abou he s uc u e o alue chains can be ob ained, as opposed o
isola ed i m-speci ic case s udies ha can p o ide a mo e de ailed unde s anding o di -
e en aspec s o a gi en alue chain. Thus, o he h ee dimensions o commodi y chain
esea ch no ed by Ge e i (1994), bo h he I–O aspec and he spa ial dimension, can
be conside ed in he in e na ional I–O app oach, while he go e nance aspec canno .
Va ious agg ega ed and sec o al global alue chain indica o s, indices and measu es ha e
been p oposed, all de i ed om he in e na ional I–O amewo k. GVC indica o s may
be oughly di ided in o measu es o leng h8 and pa icipa ion a es, which we will dis-
cuss b ie ly.
Ea ly I–O measu es o he GVC s uc u e we e simple ups eam and downs eam indi-
ca o s ha co esponded o he measu e o dis ance o inal demand (ups eam) and he
Leon ie measu e o backwa d linkage (downs eam) and we e o en e e ed o as he
leng h o a alue chain (Ahmad e al. 2017). Fally (2011) and An às e al. (2012) de ined
he downs eam indica o o “ e lec how many plan s (s ages) a e in ol ed in p oduc ion
one a e he o he ” up o he poin obse ed and he ups eam indica o o “measu e how
many plan s his p oduc will pass h ough (e.g. by assembly wi h o he p oduc s) be o e i
eaches inal demand (Fally 2011, 10)”. Fally (2011) de ined hem as he numbe o e ical
s ages weigh ed by he alue added o each s age, wi h he dis ance be ween each s age se
o 1.9 Since hen, he a e age e ical dis ance has been he basic measu e o he leng h o
he alue chain in he in e na ional I–O amewo k. Mille and Temu shoe (2015) u -
he speci ied he exis ing measu es by p esen ing ups eam and downs eam indica o s in
a ma ix o mula ion using Ghosh’s o wa d and Leon ie ’s backwa d coe icien ma ices
(Ghosh 1958; Leon ie 1936). These ups eam and downs eam measu es a e simple meas-
u es o he ups eam and downs eam leng h o alue chains measu ed by he a e age e -
ical dis ance. Wi hin his amewo k, u he imp o emen s we e in oduced by Mu ado
(2016), who ocused on sepa a ing he domes ic om he global p oduc ion componen
while calcula ing he leng h o alue chains.
The exis ing dominan concep ualisa ion o GVC pa icipa ion measu es is la gely
based on he wo k o Johnson and Nogue a (2012), who p oduced a alue-added expo
ma ix ha cap u es in o ma ion on alue lows in he economy be ween any wo poin s
8 Rela i e posi ion indices can easily be de i ed om leng h measu es as simple a ios.
9 Using a me hod simila o ha used o calcula e he a e age p opaga ion leng h equi ed o he analysis o he
dynamic esponse o shocks, de ined by Die zenbache and Rome o (2016).

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Kneze al. Economic S uc u es (2021) 10:13
(coun y-sec o s) in he economy. This p o ides he basis o he disagg ega ion o alue
on he coun y-sec o le el, depending on whe he he alue was p oduced domes ically
o domes ic consump ion o in ol ed c oss-bo de ansac ions o ei he inal o p oduc-
i e consump ion (Koopman e al. 2014; Los e al. 2015;Wang e al. 2017). Since he alue-
added expo ma ix ells us abou he sou ce and des ina ion o alue added and co e s all
possible pa hs be ween any wo coun y-sec o s in he economy, he e a e wo indica o s
o he sha e o GVC pa icipa ion— he ups eam and downs eam sha e. The concep ion
o he ups eam pa icipa ion sha e o pa icipa ion s a s om he alue added o indi id-
ual indus ies (coun y-sec o s), disagg ega ing all possible pa hs leading o he ealisa ion
o hei alue, while he concep ion o he downs eam sha e o pa icipa ion s a s wi h
inal consump ion, disagg ega ing all possible pa hs o he downs eam p oduc ion link-
ages. Wi hin his amewo k, disagg ega ion is de ined on he domes ic pa , he “Rica d-
ian ade” in inished goods, he simple GVC and he complex GVC, which is cu en ly
he mos widely used accoun ing amewo k o GVC pa icipa ion and hus a has been
used by he bes -known esea ch on GVC ca ied ou join ly by he WTO, he WB g oup,
he OECD, IDE-JETRO, RCGVC-UIBE and he China De elopmen Resea ch Founda ion
(GVC De elopmen Repo s). Fu he imp o emen s and cla i ica ions o he amewo k
we e made by Bo in and Mancini (2019), who de i e hei own measu e o GVC- ela ed
bila e al ade lows by decomposing expo o ha a ibu able o adi ional ade and
GVC ade. Thei indica o is composed o sou ce-based backwa d and sink-based o -
wa d pa s o hei expo decomposi ion, which can be calcula ed in a bila e al, coun y
and wo ld se ing.
The de elopmen o I–O pa icipa ion sha e measu es o alue chains, which a e he
p ima y in e es o his a icle, e ol ed simul aneously wi h he de elopmen o me hod-
ologies o decomposing ade in alue added (Johnson and Nogue a 2012) as well as alue
added in ade (A o e al. 2019; Bo in and Mancini 2019; Mi oudo and Ye 2021). How-
e e , despi e simila i ies and some concep ual and o mal ma hema ical o e lapping, he
ields o alue chain pa icipa ion sha e measu es and alue added in ade a e d i en by
qui e dis inc esea ch ques ions and esea ch in e es s. On one hand, p incipal in e es in
decomposing expo s is he co ec e alua ion o c oss-bo de lows (p ope ly emo ing
double coun ing), assessing ade policy impac s and conduc ing o e all impac analysis,
ei he in a bila e al se ing o wi h a ocus on a speci ic coun y. On he o he hand, alue
chain pa icipa ion measu es a emp o g asp he s uc u e o an economy, sec o al and
coun y in e dependencies and he speci ic embeddedness o each p oduc ion uni in di -
e en alue chain s uc u es, bo h a home and ab oad. Value chain pa icipa ion sha e
measu es usually co espond o a sha e o p oduc ion, which s a is ically sa is ies ce ain a
p io i c i e ia, such as “a leas wo c oss-bo de ansac ions” o “a leas one c oss-bo de
p oduc ion sha ing ansac ion”. The e iewed li e a u e has con ibu ed o be e unde -
s anding o alue chains and hei I–O applied esea ch, bu s ill su e s wo sho comings
ha we y o add ess and imp o e wi h ou app oach.
The i s main sho coming o all cu en alue chain pa icipa ion sha e indica o s is
he lack o a single uni o m measu e o di e en alue chain pa icipa ion a es on he
coun y-sec o le el. Fi s , he alue chain decomposi ion o Wang e al. (2017) esul s in
downs eam and ups eam alue chain pa icipa ion a es, which p o ide wo di e en
ypes o in o ma ion a he coun y-sec o le el. This is ele an o some ypes o analysis
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Kneze al. Economic S uc u es (2021) 10:13
ha deal wi h he ela ionship be ween ups eam and downs eam pa icipa ion in GVCs,
bu he e is a a ie y o si ua ions whe e a common measu e o GVC pa icipa ion, de ined
uni o mly on he coun y-sec o le el, is equi ed ei he as he main objec o he analysis
o as a supplemen a y o con ol a iable.10 Second, GVC measu es based on he decom-
posi ion o expo s, e en hough hey o e come he sink- and sou ce-based decomposi-
ion in one uni ying amewo k o expo decomposi ion (A o e al. 2019; Bo in and
Mancini 2019), a e concep ually unable o o e a consis en solu ion o he ques ion o
a single coun y-sec o alue chain pa icipa ion measu e. Tha is because he c i e ia o
expo decomposi ion (sepa a ing domes ic alue added om o eign alue added and he
emo al o double coun ing) do no co espond wi h he gene al c i e ia o di e en alue
chains on he coun y-sec o le el ( he sha e o p oduc ion wi h a ce ain numbe o c oss-
bo de ansac ions). Al hough expo can be decomposed bo h wi h ega d o he o igin
o he alue added as well as he inal demand, he e y ac ha he objec o decomposi-
ion is expo means i has a one-sided, o wa d o ien a ion since expo decomposi ion
canno add ess he agmen a ion o p oduc ion o a coun y-sec o ha has li le o no
expo s (bu can s ill o m pa o he agmen a ion o a global alue chain downs eam).
In his sense, he a emp by Bo in and Mancini (2019) o p o ide a GVC measu e o bila -
e al ade by decomposing expo s canno iden i y he sha e o p oduc ion o a gi en coun-
y-sec o which sa is ies he c i e ion o a ce ain numbe o c oss-bo de ansac ions,
bu only examines i s o wa d pa and is hence concep ually simila o Wang’s o wa d
GVC measu e. Ou a emp o sol e his issue demands he decomposi ion o he g oss
ou pu ( o al ou pu ) o each coun y-sec o o simul aneously accoun o bo h down-
s eam and ups eam alue chain linkages.
The second majo sho coming o exis ing alue chain indica o s is he lack o a meas-
u e o domes ic alue chain agmen a ion. The decomposi ion pu o wa d by Wang e al.
(2017) includes a b oadly de ined “domes ic componen ”, which co e s all o he alue ha
does no comply wi h he GVC and Rica dian ade c i e ia. One o he majo con ibu-
ions o his a icle is o concep ually u he di ide his b oad domes ic componen in o a
i s pa which comp ises domes ic p oduc ion agmen a ion (in ol ing p oduc ion sha -
ing be ween a leas wo domes ic i ms) and he second pa which does no . This yields
new in o ma ion ega ding he sha e o p oduc ion no in ol ed in he agmen a ion o
global p oduc ion, bu is pa o he agmen a ion o domes ic p oduc ion and enables
esea ch in o he ole o domes ic p oduc ion agmen a ion, which was impossible wi h
he exis ing concep ualisa ions. As a esul o he p esen disagg ega ion o pa icipa ion
sha es in o he “domes ic componen ” and he GVC pa icipa ion a es (and he Rica d-
ian ade sha e) consis ing o a simple duali y ha in i s cons uc ion sums o 1, he sha e
o he domes ic componen is ne e used in eg essions (due o collinea i y) and ne e
e en examined as a heo e ical concep . I is simply a esidual, a sha e ha does no in e -
es esea che s gi en ha all he in o ma ion hey disagg ega e is included in hei GVC
10 I is also ob ious ha a simple solu ion, such as using he a e age o exis ing ups eam and downs eam indica o s,
canno be jus i ied in heo y. I , o example, a gi en coun y-sec o ’s sha e in he ups eam global alue chain is high
(close o 100%) and i s sha e in he downs eam global alue chain is ela i ely low (close o 0%), hen he a e age sha e
in he alue chain would be a ound 50%, which is misleading because he alue chain as a whole is almos en i ely global
(using he c i e ion ha he alue c osses a bo de a leas once). As a as alue chain pa hs a e conce ned, despi e he
small sha e o downs eam global alue chain pa hs, a high sha e (close o 100%) o he same pa hs con inues in he
ups eam global alue chain such ha p oduc ion as a whole has a e y high global sha e (close o 100%), while he use
o he a e age o he ups eam and downs eam indica o s does no co espond o he de ini ion o he global alue
chain.
Page 7 o 37
Kneze al. Economic S uc u es (2021) 10:13
pa icipa ion a es. The exis ing app oaches a e used by esea che s o ocus exclusi ely on
he in e na ional dimension o he agmen a ion o p oduc ion, neglec ing he po en ial
held by he in e na ional I–O me hodology ha allows analysis o domes ic p oduc ion
agmen a ion. Ou app oach is b eaks g ound in his a ea as i p oposes a new concep o
domes ic agmen a ion able o be measu ed on i s own and acco ding o i s own de ini-
ion and ha is no collinea wi h he sum o he GVC pa icipa ion a e.
Ou me hodological app oach s a s wi h he o mal c i e ia, which is common o
mos o he GVC li e a u e whe e alue chains a e de ined acco ding o ce ain ansac-
ion c i e ia (numbe o c oss-bo de p oduc ion-sha ing ansac ions o simila ). I is
impo an o no e ha any such c i e ia a e a bi a y and po en ial mul iplici y o such
c i e ia and hence alue chain ypologies can coexis and o e esea che s some leeway
in hei empi ical applica ions.11 Wi h a iew o c ea ing a uni o m alue chain measu e
on he coun y-sec o le el, we use he o al ou pu o each coun y-sec o as he s a -
ing poin o ou disagg ega ion. Decomposing o al ou pu (as opposed o expo o o al
alue added) enables us o simul aneously g asp bo h he downs eam and ups eam alue
chain pa hs as well as he s uc u e o he economy ha is en i ely domes ic. Ou decom-
posi ion begins wi h a se o he p esen ed alue chain ee ma ices (
τi
) which desc ibe
all o he alue chain pa hs, om any coun y-sec o o p ima y o igin o any coun y-
sec o o p oduc ion o inal consump ion ha passes h ough (include a p oduc ion s age
o ) a single pa icula coun y-sec o . The logic o ou app oach is e y simila o ha o
A o e al. (2019) o combining he sink- and sou ce-based decomposi ion o expo s:
because he decomposi ion o pa hs o inal demand is independen o he decomposi ion
o downs eam alue added, hese decomposi ions can be linea ly combined o cap u e
bo h ypes o in o ma ion in a single decomposi ion along wo di e en dimensions a he
same ime. The big dis inc ion wi h his app oach is ha objec o decomposi ion is di e -
en —in ou case, i is he o al ou pu (g oss ou pu ) o each coun y-sec o . Ou choice o
he objec o decomposi ion is a p e equisi e o p ope ly cap u ing downs eam linkages
and, mo e impo an ly, p ope ly accoun ing o he domes ic s uc u e o he economy.
This o mula ion is he i s a emp o cap u e in o ma ion conce ning he asymme ic
alue chain ee, which is a speci ic ea u e o each indi idual coun y-sec o (Fig.1). The
p oposed alue chain ee ma ices a e unique in ha hey allow us o simul aneously cap-
u e he s uc u e o he downs eam and ups eam alue chain pa hs and o de ine alue
chain pa icipa ion a es as a single measu e o each coun y-sec o . The c ucial poin o
he p oposed me hodology is o enable he disagg ega ion o alue chains based solely on
he s uc u e o alue chain pa hs— aking in o accoun whe he hese pa hs include only
domes ic p oduc ion agmen a ion, in e na ional p oduc ion agmen a ion o no p o-
duc ion agmen a ion a all. This allows us o in oduce he concep o domes ic alue
chain agmen a ion ha simply canno be c ea ed wi hin he exis ing amewo k o 2 sep-
a a e pa icipa ion indices. This mul iplies he esea ch oppo uni ies o e ed by he alue
chain me hodology based on he in e na ional inpu –ou pu s uc u e by pe mi ing gen-
e al analysis o he agmen a ion o bo h domes ic and global p oduc ion and hei in e -
dependence along wi h any mu ual e ec s o hei de elopmen .
11 Fo example, in a o hcoming a icle we explo e he decomposi ion o alue chains based on he c i e ion o he
numbe o domes ic ansac ions subjec o mee ing he usual global alue chain c i e ion o ha ing a leas one p oduc-
ion-sha ing c oss-bo de ansac ion. In his se ing, we decompose he global alue chain sha e in o a GVC wi h no
domes ic coope a ion, a GVC wi h simple domes ic coope a ion, and a GVC wi h complex domes ic coope a ion, o e -
ing in o ma ion on he speci ic pa e n o he EU pe iphe y’s in eg a ion.
Page 8 o 37
Kneze al. Economic S uc u es (2021) 10:13
Applying his me hodology, we show ha inc easing agmen a ion o global p oduc ion
in ecen decades has been a gene al end o mos coun ies (wi h a backlash in la e
yea s), bu di e en ins i u ional a angemen s and s uc u al economic posi ions led o
a ious ypes o global economic in eg a ion, b inging di e se e ec s o domes ic ag-
men a ion. Wi h ou me hodology, we shall empi ically demons a e ha in many coun-
ies wi h high g ow h and e e s onge global in eg a ion domes ic agmen a ion also
inc eased. Howe e , one can ind cases whe e domes ic agmen a ion s agna ed o e en
declined whe eas agmen a ion o he global alue chain inc eased. The di e en ypes o
in eg a ion in global alue chains a e he ou come o se e al s uc u al and ins i u ional
de elopmen s.12 On one hand, he simul aneous inc ease in domes ic and global ag-
men a ion migh only be a consequence o he g owing complexi y and di ision o labou .
Ye , on he o he hand, he simul aneous ise in global agmen a ion and d as ic decline
in domes ic in eg a ion migh be due o he ac u ing o domes ic e ically in eg a ed
companies, pa s o which a e in eg a ed in o global alue chains as subsidia ies, o due
VA
FC
VA
FC
Valueadded (VA)
Finalconsump ion (FC)
The uni in ocus
(Thesiphon)
Pa hso alue ealisa ion
(The lea es)
(The oo s)
Pa hso alue c ea ion
VA
FC
VA
FC
C oss-bo de
Domes ic
C oss-bo de
Domes ic
C oss-bo de
Domes ic
C oss-bo de
Domes ic
C oss-bo de
Domes ic
C oss-bo de
Domes ic
F. D. F. D. F. D. F. D.
F. D. F. D. F. D. F. D.
Fig. 1 Value chain ee. Sou ce: own concep ualisa ion and design. A ows ep esen p oduc ion-sha ing
ansac ions—buying and selling o in e media e p oduc s o p oduc ion. O ange colou deno es
p oduc ion ha does no in ol e any p oduc ion sha ing, while any combina ion o ed o o ange pa hs
deno es domes ic p oduc ion agmen a ion. Any alue chain pa h which includes a c oss-coun y
p oduc ion-sha ing ansac ion (a black a ow) is pa o a global alue chain om he pe spec i e o he
pa icula uni in ocus. The pa hs o alue c ea ing and alue ealisa ion in a gene al case con inue o b anch
ad in ini um ( h ee le els a e chosen only o demons a ion pu poses)
12 Fo example, he concep o in eg a ed pe iphe y was in oduced o desc ibe a speci ic ype o in eg a ion in he case
o he Slo ak and Czech ca indus ies, cha ac e ised by hei p oximi y o consume ma ke s, cheape labou o ce, he
absence o posi i e spillo e e ec s and lack o domes ic linkages (Oldřich and Vladan 2019; Pa línek 2018).
Page 15 o 37
Kneze al. Economic S uc u es (2021) 10:13
decomposi ion gi es in o ma ion abou how he p oduc o he i h coun y-sec o is con-
sumed ei he di ec ly o as pa o he inal p oduc o o he coun y-sec o s.
Two sepa a e ec o s which disagg ega e he alue chain pa hs o he downs eam (i h
column o Z) and ups eam alue chain (i h ow o W) hus span an en i e ma ix o o al
ou pu sha es ha cap u e he alue chain ee s uc u e o he i h coun y-sec o . We
combine hem wi h he di ec p oduc ha de ines he ma ix o he alue chain ee o
each coun y-sec o (i) by mul iplying each elemen o
Zei
( he i h column o Z) by each
elemen o 
ei
TW
( he i h ow o W).
De ini ion 3 Value chain ee ma ix
τi=Z�ei⊗�ei
TW
;
τi∈Rn×n
, whe e
�ei∈
R
n
ep esen s he s anda d o hono mal basis o
Rn.
This de ines each elemen o he alue chain ee ma ix
ijk ∈τi
as
ijk =wijzki
. Each
elemen o he alue chain ee ma ix
τi
hus ep esen s a sha e o he o al ou pu o
coun y-sec o i, which is p ima ily p oduced in coun y-sec o k and consumed as an end
p oduc o coun y-sec o j, along any ups eam and downs eam alue chain pa h.
The main poin o ou de i a ion is no he exp essed inal alue dis ibu ion o he o al
ou pu o each coun y-sec o along any o i s ups eam and downs eam alue chain
pa hs, bu he exp ession o he o al ou pu dis ibu ion (o he espec i e coun y-sec o )
along any alue chain pa h, be i a downs eam alue chain pa h, an ups eam alue chain
pa h o any combina ion o bo h pa hs a he same ime.
The s uc u e o he alue chain ee ma ices allows us o ocus ou disagg ega ion on
he composi ion o he alue chain pa hs co e ed by he wo global Leon ie in e ses in
he equa ion, he i s ep esen ing all downs eam pa s o he alue chain pa hs and he
second ep esen ing all ups eam pa s o he alue chain pa hs.
A single alue chain pa h is de e mined by a se ies o conc e e ansac ions be ween
companies: I is a unique pa h om p ima y alue c ea ion ( alue c ea ed in p oduc ion,
no ans e ed om in e media e p oduc s) o alue ealisa ion ( inal consump ion, no
p oduc i e consump ion o in e media e p oduc s), which passes h ough he p oduc ion
s age o he i h coun y-sec o . The o al ou pu o i is no only disagg ega ed along all pos-
sible pa hs leading om any coun y-sec o o o igin ia coun y-sec o i o any coun y-
sec o o inal s age p oduc ion (as de e mined by
τi
), bu is also disagg ega ed in much
ine de ail, along all he unique alue chain pa hs ha pass h ough i. Tha a conc e e
alue chain pa h only o ms pa o he alue chain ee ma ix can easily be ecognised i
bo h in e ses in
τi
a e eplaced by an in ini e se ies (
(I
−
A)−1
=
I
+
A
+
A2+···
). Such
disagg ega ion hen esul s in an in ini e numbe o alue chain pa hs, and he o al ou pu
o he i h coun y-sec o is dis ibu ed o e all o hese pa hs.
A ce ain alue chain pa h sha e o he o al ou pu o i is de e mined by he Leon ie
echnical coe icien s
aij ∈A
. Fo example, ake a alue chain pa h consis ing o alue
(3.11)
τi
=ˆ
C(I
−
A)−1
�
ei
⊗�
eiT
ˆ
x−1(I
−
A)−1ˆ

Page 16 o 37
Kneze al. Economic S uc u es (2021) 10:13
p ima ily p oduced in coun y-sec o
CS1
20, hen used as an in e media e in
CS2
, which
in u n is used as an in e media e in i ( he coun y-sec o whose alue chain is b oken
down), and hen sen as an in e media e o
CS3
, which is hen sen as an in e media e o
CS4
, whe e i is inished and sold o consump ion. This alue chain pa h has an o igin
(
CS1
), a midpoin (i) and a inal des ina ion o p oduc ion (
CS4
), as well as a conc e e pa h
wi h a leng h o 5 (5 coun y-sec o s con ibu e o p oduc ion om o igin o inal con-
sump ion). The sha e o he o al ou pu o he i h coun y-sec o ha may be a ibu ed o
his speci ic pa h is:
A speci ic unique alue chain pa h o he i h coun y-sec o ’s alue chain ee, ha has
i s o igin in k and inal s age in j, can be w i en as:
Such a pa h has a downs eam leng h o d and an ups eam leng h o
u−1−d
and he
pa h is de e mined by a unique se o p oduc ion-sha ing ansac ions om he o igin o
he inal s age ( om o igin
j=CS0
, o
CS1
, o
CS2
, ..., o
i=CSd
, and u he o
CSd+1
,
CSd+2
, ..., o
k=CSu
). Leon ie echnical coe icien s
aCSp−1CSp
de e mine each p oduc-
ion-sha ing ansac ion. The summa ion along he o al ou pu sha es o i a ibu ed
o all such unique alue chain pa hs, aking in o accoun all pe mu a ions o possible
ansac ion sequences and also all possible leng hs (all possible leng h combina ions o
downs eam and ups eam leng hs) as well as all possible o igins and inal s age des ina-
ions, esul s in a uni :
Ou concep ualisa ion allows us o de ine decomposi ion c i e ia applicable o each
alue chain pa h o he alue chain ee o he i h coun y-sec o . Based on his p ope y,
we will decompose he alue chain s uc u e o each coun y-sec o sepa a ely in he ol-
lowing sec ion.
3.2 The alue chain ypology
3.2.1 De ini ions
The amewo k o he in e na ional I–O analysis allows he sepa a e analysis o inal ans-
ac ions o consume s and ansac ions be ween companies. Based on his cha ac e is ic,
we p opose a ypology o alue chains based solely on he s uc u e o linkages be ween
en e p ises, while adding a u he decomposi ion wi h ega d o di e en possible
(3.12)
CS1
a
CS1CS2
a
CS2i
x
−1
i
a
iCS3
a
CS3CS4
CS4.
(3.13)
d
p=1 CS0aCSp−1CSpx
−1
i

u−1
p
=
d
aCSpCSp+1 CSu
.
(3.14)
1
Tτ
i
1=1Tˆ
C
(I−A)
−
1�e
i
⊗�e
i
Tˆ
x
−
1(I−A)
−
1
ˆ
1T=
1.
20
CSk
ep esen s an index o di e en coun y-sec o s.
aCS1CS2
hus ep esen s a single Leon ie echnical coe icien
indica ing ha he alue p oduced by
CS2
equi es a
aCS1CS2
sha e o
CS1
inpu .
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Kneze al. Economic S uc u es (2021) 10:13
ansac ions o each he inal consump ion pos es um.21 Each ma ix
τi
exp essed by
equa ion3.13 ep esen s he desegmen a ion o he o al p oduc o coun y-sec o i along
di e en downs eam and ups eam pa hs. When we e e o a alue chain, we e e o
he speci ic sha e o alue (sha e o ou pu ) ha co esponds o a pa icula alue chain
pa h. Pa h22 o each alue sha e gene ally includes any combina ion o domes ic and c oss-
bo de p oduc ion-sha ing ansac ions, which can ake place bo h downs eam and
ups eam ela i e o he espec i e coun y-sec o . Ou c i e ia o he alue chain ypol-
ogy hus e e o each speci ic alue sha e co esponding o a single pa h wi hin a alue
chain ee speci ic o each coun y-sec o .
De ini ion 4 Domes ic alue chain
Domes ic alue chain (DVC) is a alue ha in ol es a leas 1 domes ic p oduc ion-sha ing
ansac ion and in ol es only domes ic p oduc ion-sha ing ansac ions along i s pa h.
De ini ion 5 Global alue chain
Global alue chain (GVC) is a alue ha in ol es a leas 1 c oss-bo de p oduc ion-sha ing
ansac ion along i s pa h. We u he dis inguish wo ypes o global alue chains: simple
and complex.
De ini ion 5.1 Simple global alue chain
Simple global alue chain (SGVC) is a alue ha in ol es exac ly 1 c oss-bo de p oduc-
ion-sha ing ansac ion anywhe e along i s pa h.
De ini ion 5.2 Complex global alue chain
Complex global alue chain (CGVC) is a alue ha in ol es mo e han 1 c oss-bo de p o-
duc ion-sha ing ansac ion along i s pa h.
De ini ion 6 No alue chain
No alue chain (NVC) is a alue ha does no in ol e any p oduc ion-sha ing ansac ions
and has no alue chain pa h wi hin p oduc ion.
21 Fo example, Wang’s disagg ega ion in o simple and complex GVCs uses he numbe o c oss-bo de ansac ions,
ega dless o whe he he alue c ossed a bo de o p oduc ion o whe he i is only an expo o end use s. Such a c i-
e ion mixes wo concep ually di e en ansac ions, leading o unnecessa y calcula ion complexi y and he impossibili y
o u he concep ual disagg ega ion. Exis ing de ini ions o he ypology o alue chains, like all such de ini ions, a e
cons uc ed in a ela i ely a bi a y way. Mo e impo an han s ic adhe ence o he p e ailing de ini ions is he cla i y
o he p oposed e ision and he p esen a ion o he concep ual ela ionship o he new concep s wi h he old ones. Ou
p oposal acili a es a mo e de ailed decomposi ion ha will allow esea che s o cons uc an indica o be e sui ed o
hei esea ch ques ions. Since he e ised ypology is based on a mo e de ailed decomposi ion compa ed o he cu -
en ly p e ailing ypology, esea che s can (by simply adding componen s o he e ised decomposi ion) also eplica e
objec s ha co espond o exis ing s udies.
22 He e we examine he pa h o p oduc ion agmen a ion, while he pa h o inal consump ion, which ep esen s an
addi ional ansac ion, is analysed in Sec .3.2.5.
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Kneze al. Economic S uc u es (2021) 10:13
A ew b ie commen s a e app op ia e on ou de ini ions and hei in e p e a ion. No
ma e ial p oduc o se ice belongs o a single classi ica ion o alue chain, and no en e -
p ise can be conside ed pa o a single ype o alue chain. The ou pu o each en e p ise
belongs o a a ie y o alue chain pa hs. In gene al, one pa o he ou pu comp ises many
c oss-bo de ansac ions, ano he pa only domes ic ansac ions, and ye ano he pa
hei ela i ely complex in e ela ionship. Each p oduc (o coun y-sec o in ou case) can
be assigned di e en sha es o he alue chain pa hs. These sha es a e objec s ha p o ide
in o ma ion abou he s uc u e o he economy. Fo example, i ually no en e p ise could
be classi ied exclusi ely as pa o a no alue chain, bu some en e p ises ha p o ide se -
ices (e.g. domes ic se ices) ha e a ela i ely high sha e o ou pu ha has no alue chain
pa h, especially in se ices, whe e sala ies accoun o almos all o he en e p ise’s expend-
i u e and whe e hei p oduc di ec ly sa is ies inal demand. On one hand, en e p ises
ha specialise in in e media e goods a e always pa o a alue chain, whe he domes ic
o global. On he o he hand, e en mode n indus ies such as ood-p ocessing and pha -
maceu icals, also ha e a ce ain (usually small) sha e o alue added ha is no pa o any
alue chain (no alue chain sha e), co esponding o he sha e o domes ic alue added
in hese indus ies ha is also di ec ly consumed (pa o ou pu ha has no alue chain
pa h). The alue chain sha es and hei changes a e he objec ha p o ide in o ma ion
abou he s uc u e o he economy, whe he on he sec o o coun y le el. As he econ-
omy de elops, he di ision o labou also inc eases, which co esponds o he g owing
agmen a ion o p oduc ion, in pa icula in e na ional p oduc ion agmen a ion, and a
dec ease in sha es whe e he e is limi ed o no alue chain agmen a ion. Compa ed o
he exis ing ypology o alue chains, his e ised ypology allows o analysis o he ela-
ionship be ween global and domes ic agmen a ion, which migh p o e especially ele-
an o he policies o de eloping coun ies.
3.2.2 The decomposi ion o pa hs
Ou alue chain ypology is es ablished acco ding o c i e ia along he en i e alue chain.
Fo his eason, we disagg ega e he alue chain ee ma ices
τi
in e ms o c i e ia o
di e en ypes o alue chain pa hs. Ou decomposi ion consis s o he decomposi ion
o wo Leon ie in e ses, which may be in e p e ed as he decomposi ion o he down-
s eam pa and ups eam pa o each alue chain pa h, as de ined by equa ion3.11:
τi
=ˆ
C(I
−
A)−
1�
ei
⊗�
eiT
ˆ
x−
1
(I
−
A)−
1
ˆ
. The decomposi ion is cons uc ed based on o
he c i e ia o he numbe o c oss-bo de and domes ic p oduc ion-sha ing ansac ions
ha a e consis en wi h he e ised alue chain ypology.
Fi s , we in es iga e he decomposi ion o only a single Leon ie in e se (in e p e ed
symme ically wi h espec o ou c i e ia in he ups eam and downs eam alue chain)
and only hen do we analyse he decomposi ion o all alue chain pa hs cha ac e ised by
he wo Leon ie in e ses. The in e na ional I–O da a ha e a speci ic block ma ix s uc-
u e in which he block diagonal elemen s ep esen domes ic p oduc ion-sha ing ans-
ac ions and he block o -diagonal elemen s ep esen in e na ional p oduc ion-sha ing
ansac ions (
AD
deno es domes ic—block diagonal—and
ACB
c oss-bo de —block-o
diagonal—pa o A), which allows us o decompose he Leon ie in e se in he ollowing
way:
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Kneze al. Economic S uc u es (2021) 10:13
(1) I ob iously ep esen s ha pa o he ou pu which con ains no p oduc ion-sha ing
ansac ions—no alue chain linkages. In he ups eam pa , i ep esen s he sha e o
o al ou pu ha di ec ly sa is ies inal demand (i.e. no ups eam alue chain), while in
he downs eam pa i ep esen s he di ec alue added o he coun y-sec o whose
p oduc ion is being decomposed (i.e. no downs eam alue chain).
(2) A
D
(I−A
D
)
−1
=A
D
+A
2
D
+A
3
D
+
...
ep esen s ha pa o ou pu which con-
ains a leas 1 domes ic p oduc ion-sha ing ansac ion and con ains only domes ic
p oduc ion-sha ing ansac ions.
(3)
(I
−
AD)−1ACB(I
−
AD)−1
ep esen s ha pa o he ou pu which con ains a leas
1 p oduc ion-sha ing ansac ion and con ains exac ly one c oss-bo de p oduc ion-
sha ing ansac ion somewhe e along i s alue chain pa h. This can be demons a ed
by pa aph asing he pa as all possible combina ions o a single c oss-bo de ansac-
ion among any possible se o domes ic p oduc ion-sha ing ansac ions ha occu
be o e o a e he single c oss-bo de p oduc ion-sha ing ansac ion:
(4)
(I
−
A)−1
−
(I
−
AD)−1
−
(I
−
AD)−1ACB(I
−
AD)−1
ep esen s ha pa o he
ou pu which con ains a leas wo o mo e p oduc ion-sha ing ansac ions, o which
a leas wo a e c oss-bo de p oduc ion-sha ing ansac ions. This logically ollows
om he ac ha pa s (1), (2) and (3) co e he o al ou pu ha con ains less han
wo c oss-bo de ansac ions, and ha he ull Leon ie in e se co e s he o al ou -
pu .
3.2.3 Value chain ee ma ix decomposi ion
We p oceed by disagg ega ing all o he alue chain pa hs as hey a e s uc u ed in he
alue chain ee ma ices. Using he decomposi ion o he Leon ie in e se ha we disag-
g ega ed in he p e ious subsec ion and inse ing i in o Eq.3.11, we ob ain 16 componen s
(3.15)
(
I−A)−
1
=(I−AD)−
1
+(I−A)−
1
−(I−AD)−
1
=I+AD+A2
D+A3
D+···+(I−A)−1−(I−AD)−1
=I+AD(I+AD+A2
D+...)+(I−A)−1−(I−AD)−1
=I+AD(I−AD)−1+(I−A)−1−(I−AD)−1
=I

1.)
+AD(I−AD)−1
 
2.)
+(I−AD)−1ACB(I−AD)−1
  
3.)
+(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1
 

4.)
(
I−AD)
−1
ACB(I−AD)
−1
=ACB +ACBAD+ACBA
2
D···
+ADACB +ADACBAD+ADACBA2
D···
+A2
DACB +A2
DACBAD+A2
DACBA2
D
+···+
.
.
.
Page 20 o 37
Kneze al. Economic S uc u es (2021) 10:13
(
4×4
p oduc ) o each ma ix
τi
.23 This disagg ega ion along bo h he ups eam and
downs eam pa hs is he basis o de i ing alue chain sha es ha co espond o ou ypol-
ogy. We decompose each
τi
ma ix desc ibing all possible alue chain pa hs o he ou -
pu o he i h coun y-sec o in o a ma ix consis ing o domes ic alue chain pa hs only,
a ma ix con aining all possible global alue chain pa hs (as well as simple and complex
global alue chain pa hs sepa a ely), and a ma ix consis ing only o he alue ha has no
alue chain pa h.
De ini ion 7 Domes ic alue chain ee
τDVC
i
The domes ic alue chain ee ep esen s all alue chain pa hs o he ou pu o each
coun y-sec o which, acco ding o De ini ion 4, a e pa o he domes ic alue chains. In
Fig.1, he domes ic alue chain pa hs a e ma ked in ed. Domes ic alue chain pa hs a e
de ined as all pa hs ha con ain a leas one ed-colou ed linkage ( ep esen ing ansac-
ions be ween domes ic en e p ises) and include only ed-colou ed linkages and o ange
pa hs ( ep esen ing he alue c ea ion o ealisa ion in he espec i e coun y-sec o in
ocus). The i s pa ( ˆ
CAD(I
−
AD)−
1�
ei
⊗�
eiT
ˆ
x−
1
ˆ
) co e s he downs eam domes-
ic alue added (downs eam domes ic pa h), which ends as he i h coun y-sec o
inal s age (no ups eam pa h), he second pa ( ˆ
C
�
ei
⊗�
ei
Tˆ
x
−1
AD(I
−
AD)
−1
ˆ
) co -
e s he alue added o he i h coun y-sec o (no downs eam pa h) ha is ans e ed
ia he ups eam domes ic alue chain (ups eam domes ic pa h), and he hi d pa
( ˆ
CAD
(
I
−
AD
)
−
1�
ei
⊗�
eiT
ˆ
x−
1
AD
(
I
−
AD
)
−
1
ˆ
) comp ises he downs eam domes ic alue
added ha is used as an in e media e p oduc in he p oduc ion o i and hen used as an
in e media y u he in he ups eam domes ic alue chain un il i eaches inal demand
(bo h downs eam and ups eam domes ic pa hs). All h ee cases mee he de ini ion o a
domes ic alue chain.
De ini ion 8 Global alue chain ee
τGVC
i
The global alue chain ee ep esen s all pa hs o he ou pu o he indi idual coun y-
sec o , which o m pa o global alue chains acco ding o De ini ion 5. In Fig.1, he
global alue chain pa hs a e ep esen ed by all pa hs con aining a leas one black-col-
ou ed linkage ( ep esen ing c oss-bo de ansac ions be ween en e p ises). Global alue
chain pa hs can con ain any numbe o ed (domes ic) and o ange (no alue chain) link-
ages p o ided he e is a leas one black (c oss-bo de ) linkage along hei pa h. The i s
elemen ( ˆ
C
(I−A
D
)−1�e
i
⊗�e
i
Tˆ
x−1

(I−A)−1−(I−A
D
)−1
ˆ
) co e s he downs eam
τ
DVC
i=ˆ
CAD(I−AD)
−
1�ei⊗�eiTˆ
x
−
1
ˆ
+ˆ
C�ei⊗�eiTˆ
x
−
1AD(I−AD)
−
1
ˆ
+ˆ
CAD
(
I
−
AD
)−1�
ei
⊗�
ei
Tˆ
x
−1
AD
(
I
−
AD
)−1ˆ
.
τGVC
i=ˆ
C(I−AD)
−
1�ei⊗�ei
T
ˆ
x
−
1(I−A)
−
1−(I−AD)
−
1
ˆ
+ˆ
C(I−A)−1−(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ˆ
+ˆ
C
(I−A)−1−(I−A
D
)−1)�e
i
⊗�e
i
Tˆ
x−1

(I−A)−1−(I−A
D
)−1

ˆ
.
23 De ails o he disagg ega ion a e gi en in Appendix B.

Page 21 o 37
Kneze al. Economic S uc u es (2021) 10:13
domes ic and no alue chain pa hs, which ha e global ups eam linkages (simple o
complex), he second elemen (
ˆ
C
(I−A)
−
1−(I−A
D
)
−
1

�e
i
⊗�e
iT
ˆ
x
−
1(I−A
D
)
−
1
ˆ
)
co e s downs eam global linkages (simple o complex), which ha e
a ups eam domes ic o no alue chain pa h and he hi d elemen
( ˆ
C
(I−A)−1−(I−A
D
)−1

�e
i
⊗�e
i
Tˆ
x−1

(I−A)−1−(I−A
D
)−1
ˆ
) co e s he alue
ha has global pa hs bo h ups eam and downs eam. All o hese cases co espond o ou
de ini ion o a global alue chain.
De ini ion 8.1 Simple global alue chain ee
τSGVC
i
The simple global alue chain ee ep esen s all pa hs o he ou pu o each coun y-
sec o ha a e pa o simple global alue chains as de ined by 5.1 The i s elemen
( ˆ
C(I
−
AD)
−1�
ei
⊗�
ei
Tˆ
x
−1
(I
−
AD)
−1
ACB(I
−
AD)
−1
ˆ
) co e s a downs eam domes ic
and no alue chain pa h ha has simple global ups eam linkages and he second ele-
men ( ˆ
C
(
I
−
AD
)−1
ACB
(
I
−
AD
)−1�
ei
⊗�
ei
Tˆ
x
−1(
I
−
AD
)−1
ˆ
) co e s downs eam simple
global linkages ha ha e an ups eam domes ic o no alue chain pa h. These a e he
only cases ha i ou de ini ion o a simple global alue chain. A alue chain pa h co -
e ing bo h downs eam and ups eam simple global linkages al eady has mo e han 1
c oss-bo de ansac ion and is hence pa o a complex global alue chain.
De ini ion 8.2 Complex global alue chain ee
τCGVC
i
The complex global alue chain ee ep esen s all pa hs o he ou pu o indi idual coun-
y-sec o s ha o m pa o complex global alue chains as de ined in 5.2 The i s ele-
men ( ˆ
C
(I−A
D
)
−
1�e
i
⊗�e
iT
ˆ
x
−
1

(I−A)
−
1−(I−A
D
)
−
1−(I−A
D
)
−
1A
CB
(I−A
D
)
−
1
ˆ
)
co e s he downs eam domes ic and no alue chain pa h, ha -
ing complex global ups eam linkages, he second elemen
( ˆ
C

(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1

�ei⊗�eiTˆ
x−1(I−AD)−1
ˆ
)
comp ises downs eam complex global linkages, which ha e an
ups eam domes ic o no alue chain pa h, and he hi d elemen
( ˆ
C
(I−A)
−
1−(I−A
D
)
−
1

�e
i
⊗�e
iT
ˆ
x
−
1

(I−A)
−
1−(I−A
D
)
−
1
ˆ
) ep esen s combi-
na ions o global downs eam and ups eam pa hs (simple-simple, simple-complex, com-
plex-simple, complex-complex). All o hese elemen s mee ou de ini ion o a complex
global alue chain because he alue in all cases c osses bo de s o p oduc ion a leas
wice.
De ini ion 9 No alue chain ee
τNVC
i
τ
SGVC
i=ˆ
C(I−AD)
−
1�ei⊗�ei
T
ˆ
x
−
1(I−AD)
−
1ACB(I−AD)
−
1
ˆ
+ˆ
C
(
I
−
AD
)−1
ACB
(
I
−
AD
)−1�
ei
⊗�
ei
Tˆ
x
−1(
I
−
AD
)−1ˆ
.
τCGVC
i=ˆ
C(I−AD)
−1
�ei⊗�ei
T
ˆ
x
−1
(I−A)
−1
−(I−AD)
−1
−(I−AD)
−1
ACB(I−AD)
−1

ˆ
+ˆ
C(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ˆ
+ˆ
C
(I−A)−1−(I−A
D
)−1

�e
i
⊗�e
i
Tˆ
x−1

(I−A)−1−(I−A
D
)−1

ˆ
.
τNVC
i
=ˆ
C
�e
i
⊗�e
iT
ˆ
x
−1ˆ
.
Page 22 o 37
Kneze al. Economic S uc u es (2021) 10:13
A no alue chain ee ep esen s ha pa o he ou pu o each coun y-sec o which
is no pa o a alue chain acco ding o De ini ion 6. In Fig.1, a no alue chain pa h is
ep esen ed by he o ange colou only (any o he linkage ep esen s a alue chain pa h).
Solely he sha e o alue added p oduced in he espec i e coun y-sec o in ocus (no
downs eam s ages) and also comple ed o inal consump ion (no ups eam s ages) in
he same p oduc ion phase sa is ies his c i e ion. Since he I–O me hod dis inguishes
be ween a p oduc used as an in e media e p oduc wi hin he same sec o 24 and he p od-
uc manu ac u ed o inal consump ion, he use o his de ini ion as no alue chain does
no depend on he le el o de ail o I–O da a disagg ega ion. The cyclical e ec o he p o-
duc ion o in e media e goods wi hin he same coun y-sec o is al eady included in he
domes ic alue chain ee and, a e aking in o accoun all o he de ined alue chain pa hs
(domes ic, simple and complex global alue chain pa hs), a alue sha e emains wi hou a
alue chain pa h and wi h a simple ep esen a ion as he alue added o he coun y-sec o
which is also di ec ly consumed. This ep esen s a alue ha has no pa h in e ms o ans-
ac ions ha ep esen he agmen a ion o p oduc ion.
This concludes he alue chain ee decomposi ion, which can be w i en as:
3.2.4 The alue chain pa icipa ion a es
In Sec .3.1, we showed ha a se o alue chain ee ma ices
τi
ep esen s all possible alue
chain pa hs o he ou pu o each coun y-sec o and ha he summa ion along all sha es
o o al ou pu assigned o all such unique alue chain pa hs yields a uni y o each alue
chain ee (Eq.3.14). Namely, we p esen ed a unique disagg ega ion o he ou pu o each
coun y-sec o along all o i s alue chain pa hs. In he same way, he summa ion along he
wo disagg ega ing dimensions o ou decomposed se o ma ices (global, domes ic and
no alue chain ee ma ices) cap u es he o e all sha e o he o al ou pu o each coun y-
sec o i ha mee s he c i e ia by which he alue chain pa hs we e decomposed by includ-
ing ei he only domes ic alue chain pa hs, only global alue chain pa hs, o only alues
ha ha e no alue chain pa hs a all. In o he wo ds, he summa ion o he disagg ega ed
alue chain ma ices along any o igin and end s age ep esen s he sha e o ou pu o each
coun y-sec o ha has ei he a domes ic, a global o a no alue chain.
De ini ion 10 Domes ic alue chain sha e DVCs
DVCs ∈IR n
; DVCsi=
n
j
=
1n
k
=
1
DVC
ijk
;
DVCs
=





1
T
τ
DVC
11
1TτDVC
21
.
.
.
1TτDVC
n
1





.
(3.16)
τi
=τ
DVC
i
+τ
GVC
i
+τ
NVC
i,
(3.17)
τGVC
i
=τ
SGVC
i
+τ
CGVC
i.
24 This is de e mined by he pu e diagonal elemen s o he Leon ie echnical ma ix A. Each
aii
ep esen s he po ion
o he o al p oduc o he i h coun y-sec o ha equi es he use o he in e media e p oduc o he same coun y-
sec o in he p oduc ion p ocess, he eby co e ing cyclical ansac ions wi hin a sec o . These cyclical ansac ions a e o
cou se included in he decomposi ion o he domes ic alue chain and no he no alue chain since cyclical ansac ions
ep esen he agmen a ion o a domes ic alue chain.
Page 23 o 37
Kneze al. Economic S uc u es (2021) 10:13
Domes ic alue chain sha e ep esen s he sha e o each coun y-sec o ’s ou pu ha has a
domes ic alue chain pa h.
De ini ion 11 Global alue chain sha e GVCs
GVCs ∈IR n
;
GVCs
i=
n
j=1
n
k=1
GVC
ijk
;
GVCs
=





1
T
τ
GVC
11
1TτGVC
21
.
.
.
1TτGVC
n
1





.
Global alue chain sha e ep esen s he sha e o each coun y-sec o ’s ou pu ha has a
global alue chain pa h.
De ini ion 11.1 Simple global alue chain sha e SGVCs
SGVCs ∈IR n
;
SGVCs
i=
n
j=1
n
k=1
SGVC
ijk
;
SGVCs
=





1
T
τ
SGVC
11
1TτSGVC
21
.
.
.
1TτSGVC
n
1





.
Simple global alue chain sha e ep esen s he sha e o each coun y-sec o ’s ou pu ha
has a simple global alue chain pa h.
De ini ion 11.2 Complex global alue chain sha e CGVCs
CGVCs ∈IR n
;
CGVCs
i=
n
j
=1
n
k
=1
CGVC
ijk
;
CGVCs
=





1
T
τ
CGVC
11
1TτCGVC
21
.
.
.
1T
τ
CGVC
n
1





.
Complex global alue chain sha e ep esen s he sha e o each coun y-sec o ’s ou pu ha
has a complex global alue chain pa h.
De ini ion 12 No alue chain sha e NVCs
NVCs ∈IR n
; NVCsi=
n
j
=1
n
k
=1
NVC
ijk
;
NVCs
=





1
T
τ
NVC
11
1TτNVC
21
.
.
.
1TτNVC
n
1





.
A no alue chain sha e ep esen s he sha e o each coun y-sec o ’s ou pu ha has a no
alue chain pa h.
(3.18)
DVCs
i+GVCsi+NVCsi=
n

j
=
1
n

k
=
1
DVC
ijk + GVC
ijk + NVC
ijk

=
n

j
=
1
n

k
=
1
ijk =
1,
(3.19)
SGVCs
i+CGVCsi=
n

j
=1
n

k
=
1
SGVC
ijk + CGVC
ijk

=
n

j
=1
n

k
=
1
GVC
ijk =GVCsi
.
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Kneze al. Economic S uc u es (2021) 10:13
Wi h his, we conclude ou disagg ega ion o each coun y-sec o ’s o al ou pu wi h
espec o i s speci ic alue chain in eg a ion based on p oduc ion-sha ing linkages. We
can summa ise ou decomposi ion in he simple ec o o m:
3.2.5 Decomposi ion o  he ansac ion o he inal consume
Since all alue chain pa hs wi hin p oduc ion a e co e ed and decomposed, we s ill ha e
one las ansac ion o he consume o comple e he alue chain pa h om p oduc ion o
consump ion. We can decompose he inal ansac ion o he consume upon he c i e ion
o whe he i is a ansac ion o domes ic consume s o a c oss-bo de ansac ion (expo
o he inal p oduc o consump ion). Domes ic consump ion he e e e s o he coun y-
sec o in which he las s age o p oduc ion ook place and no he coun y-sec o whose
alue chain we a e analysing. Each coun y-sec o has a unique alue chain and a speci ic
s uc u e o alue chain pa hs. The comple ion o each alue chain pa h by a ansac ion
o he consume can be achie ed by an addi ional c oss-bo de ansac ion o expo ing
he inal p oduc o consump ion in he coun y whe e he p oduc was inalised. Such a
u he decomposi ion o he alue chain pa hs allows a mo e de ailed analysis o he alue
chains.
The I–O da a include in o ma ion on he ansac ion o inal consume s wi hin ma ix
F, which can be decomposed in o i s c oss-bo de and domes ic lows o inal consume s
(
F=FCB +FD
) due o i s block ec o s uc u e. We cons uc a ma ix o all c oss-bo -
de inal consump ion lows and a ma ix o all domes ic consump ion lows:
E e y alue chain pa h wi hin p oduc ion can hus be u he decomposed wi h an addi-
ional c i e ion o a ansac ion o inal consume s. Each se o disagg ega ed alue chain
ma ices, de ined by Eqs.3.16 and 3.17, can be sepa a ed on wo ma ices, one co e ing
all o he p oduc ion pa hs ha end in domes ic inal consump ion (no expo -
τNE
i
) and
he o he all o he p oduc ion alue chain pa hs ha end wi h expo ing o inal con-
sump ion (
τE
i
).
Due o hei simple addi i e p ope ies o ope a ion, all o he decomposed alue chain
ee ma ices a e simila ly decomposed o ones wi h expo ing o wi h no expo ing as
he inal ansac ion.
(3.20)
DVCs +GVCs +NVCs =1,
(3.21)
GVCs =SGVCs +CGVCs.
(3.22)
ˆ
=
ˆ
D
+
ˆ
CB.
(3.23)
τi
=ˆ
C(I
−
A)−1
�
ei
⊗�
eiT
ˆ
x−1(I
−
A)−1ˆ
D
+ˆ
C(I
−
A)−1
�
ei
⊗�
eiT
ˆ
x−1(I
−
A)−1ˆ
CB
=
τ
NE
i
+
τ
E
i
(3.24)
τi
=τ
NE
i
+τ
E
i
=τ
GVC−NE
i
+τ
DVC−NE
i
+τ
NVC−NE
i
+τ
GVC−E
i
+τ
DVC−E
i
+τ
NVC−E
i
(3.25)
τGVC
i
=τ
GVC−NE
i
+τ
GVC−E
i
=τ
SGVC−NE
i
+τ
CGVC−NE
i
+τ
SGVC−E
i
+τ
CGVC−E
i
Page 31 o 37
Kneze al. Economic S uc u es (2021) 10:13
o p oduc ion, ha is chie ly co ela ed wi h g ow h, i espec i e o i s global o domes-
ic na u e. Acco dingly, he p oposed measu e and he new ypology o alue chains, in
pa icula he no el concep ualisa ion o domes ic alue chain agmen a ion, could b ing
Fig. 7 USA manu ac u ing pa icipa ion a es. c edi Sou ce: WIOD 2016; own calcula ions
Table 1 Reg ession esul s Sou ce: WIOD, 2016; WB; own calcula ions
*
p
<
0.05
, **
p
<
0.01
, ***
p
<
0.001
(1) (2) (3)
Yea ly g ow h Yea ly g ow h Yea ly g ow h
logGDP − 0.013*** − 0.013*** − 0.013***
(0.00) (0.00) (0.00)
DVC sha e 0.169*** 0.181*** 0.183***
(0.03) (0.04) (0.04)
GVC sha e 0.163*** 0.160*** 0.162***
(0.03) (0.03) (0.03)
logPOP − 0.001 − 0.001
(0.00) (0.00)
EU − 0.003
(0.00)
Cons an 0.036 0.049 0.055
(0.03) (0.03) (0.04)
R2
0.819 0.821 0.824
F57.126 42.314 33.806

Page 32 o 37
Kneze al. Economic S uc u es (2021) 10:13
o ligh impo an in o ma ion ha has been concealed in he exis ing ypology, which
concep ualises he domes ic componen only as a nega ion o he global alue chain and
hus did no allow esea ch wi h explici ques ions conce ning domes ic in eg a ion. The
complex de elopmen o globalisa ion in ecen decades and he shi s o la e owa ds he
localisa ion and egionalisa ion o economic in eg a ion caused by poli ical, economic and
ex e nal ac o s make his new app oach inc easingly ele an . The p oposed measu e,
pa icula ly in conjunc ion wi h da a om o he sou ces, could u he deepen he heo-
e ical discussion and empi ical in es iga ions.
In conclusion, we belie e ha ou new me hodological app oach and he new ex ended
ypology o alue chains associa ed wi h i p o ide e ile g ounds o ob aining deepe
insigh s in o di e en ypes o alue chains as well as a b oade se o ools o use o a i-
ous ex ensions o esea ch.
Appendix A: No a ions
nS∈IN
Numbe o sec o s.
nC∈IN
Numbe o coun ies.
n∈IN
;
n=nS∗nC
Numbe o coun y-sec o s.
1∈IR n
Vec o o ones.
�
1∈IR
n
C
ec o o ones.
�
ei
∈
IR n
;
eij=δij
S anda d o hono mal basis o
IR n
.
I∈IR n×n
Iden i y ma ix.
x∈IR n
To al ou pu ec o .
ˆ
x∈IR n×n
;
ˆ
x=diag(x)
To al ou pu ma ix.
C∈IR n×n
In e media e consump ion ma ix.
F∈IR n×nC
Final consump ion ma ix on he coun y le el.27
∈IR n
;
=
F�
1
To al inal consump ion ec o .
ˆ
∈IR n×
n
;
ˆ
=
diag
(
)
To al inal consump ion ma ix.
A∈IR n×n
;
A=Cˆ
x−1
Leon ie echnical coe icien ma ix.
G∈IR n×n
;
G=ˆ
x−1C
Ghosh echnical coe icien ma ix.
∈IR n
;
T=xT−
1
TC=
1
(
ˆ
x−A
ˆ
x)=
1
T(I−A)ˆ
x
Vec o o o al alue added.
ˆ
∈IR n×n
;
ˆ
=diag( )
To al alue-added ma ix.
C∈IR n
;
T
C
=
T
ˆ
x
−1
=1
T
(I−A
)
Vec o o alue-added coe icien s – alue-added
sha e in o al ou pu .
ˆ
C∈IR n×n
;
ˆ
C=diag
(
C)
Value-added coe icien s ma ix.
C, A and G ha e a block-ma ix s uc u e
IR (n
S
×n
S
)×(n
C
×n
C
)
, while F has a block ec-
o s uc u e
IR n
S
×(n
C
×n
C
)
. Diagonal block elemen s wi h espec o coun ies ep esen
domes ic in e media e ans e s and domes ic consump ion and o diagonal block ele-
men s ep esen ansac ions ha c oss a bo de ei he o in e media e use o inal
consump ion.
27 In he in e na ional I–O amewo k, F is usually disagg ega ed on he coun y le el as well as in an addi ional dimen-
sion o inal consump ion (household, go e nmen and non-p o i consump ion, ixed capi al o ma ion and changes in
in en o ies), which in ou de i a ion is i ele an and le ou . Disagg ega ion by coun ies is ele an o enabling he
sepa a ion o domes ic inal consump ion and expo .
Page 33 o 37
Kneze al. Economic S uc u es (2021) 10:13
C=CCB +CD
A=ACB +AD
G=GCB +GD
F=FCB +FD
CB ∈IR n
;
CB
=
FCB�
1
To al inal consump ion by expo ing.
D∈
IR
n
;
D
=
FD�
1
To al inal consump ion by domes ic ansac ions.
ˆ
CB
∈IR
n×n
;
ˆ
CB
=
diag( CB)
To al inal consump ion by expo ing ma ix.
ˆ
D
∈
IR
n
×n
;
ˆ
D
=
diag( D)
To al inal consump ion by domes ic ansac ions ma ix.
Appendix B:
τi
decomposi ion
τ
i=ˆ
C(I−A)
−1
�ei⊗�ei
T
ˆ
x
−1
(I−A)
−1ˆ
=ˆ
CI+AD(I−AD)−1+(I−AD)−1ACB(I−AD)−1
+(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei
⊗�eiTˆ
x−1I+AD(I−AD)−1+(I−AD)−1ACB(I−AD)−1
+(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ
=ˆ
C�ei⊗�eiTˆ
x−1ˆ
+ˆ
CAD(I−AD)−1�ei⊗�eiTˆ
x−1ˆ
+ˆ
C�ei⊗�eiTˆ
x−1AD(I−AD)−1ˆ
+ˆ
CAD(I−AD)−1�ei⊗�eiTˆ
x−1AD(I−AD)−1ˆ
+ˆ
C�ei⊗�eiTˆ
x−1(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
CAD(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1ˆ
+ˆ
C(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1AD(I−AD)−1ˆ
+ˆ
C(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
CAD(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1AD(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei
⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ
=ˆ
C�ei⊗�eiTˆ
x−1ˆ
+ˆ
CAD(I−AD)−1�ei⊗�eiTˆ
x−1ˆ
+ˆ
C�ei⊗�eiTˆ
x−1AD(I−AD)−1ˆ
+ˆ
CAD(I−AD)−1�ei⊗�eiTˆ
x−1AD(I−AD)−1ˆ
+ˆ
C(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1ˆ
=τNVC
i
+τDVC
i
+τGVC
i
=τ
i
.
Page 34 o 37
Kneze al. Economic S uc u es (2021) 10:13
Appendix C
We make a demons a ion o he me hodology on a simple 2 sec o 2 coun ies nume i-
cal example.28 This simple case o in e na ional economy has ollowing in e media e
consump ion ma ix and inal demand:
To al ou pu is he sum o all he in e media e and inal demand:
Calcula ion o alue added coe icien s and Leon ie echnical coe icien s:
We con inue wi h sepa a e ups eam and downs eam decomposi ions, W and Z:
τ
GVC
i=ˆ
C(I−AD)
−
1�ei⊗�eiTˆ
x
−
1(I−A)
−
1−(I−AD)
−
1
ˆ
+ˆ
C(I−A)−1−(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1ˆ
=ˆ
C(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ˆ
+ˆ
C(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ
x−1(I−AD)−1ˆ
+ˆ
C(I−A)−1−(I−AD)−1�ei⊗�eiTˆ
x−1(I−A)−1−(I−AD)−1ˆ
=τSGVC
i
+τCGVC
i
=τGVC
i
.
C
=



2112
0.5 0.33 0.33 0.33
0.5 0.5 0.167 0.33
1 1.5 2 2



=



4
2
3
5



.
x
=C1+ x=



10
3.5
4.5
11.5



.
T
C=1
T
(I−A)=�0.6 0.0476 0.2222 0.5942�
,
A=Cˆ
x−1=


0.2 0.2857 0.2222 0.1739
0.05 0.09523 0.0740 0.0289
0.05 0.1428 0.0370 0.0289
0.1 0.4285 0.4444 0.1739


,
ADom =


0.2 0.2857 0 0
0.05 0.09523 0 0
0 0 0.0370 0.0289
0 0 0.4444 0.1739



ACB =


0 0 0.2222 0.1739
0 0 0.0740 0.0289
0.05 0.1428 0 0
0.1 0.4285 0 0



.
28 The decimal numbe s a e unca ed on he ou h digi .
Page 35 o 37
Kneze al. Economic S uc u es (2021) 10:13
Value chain ee ma ices a e calcula ed o each coun y-sec o in he ollowing manne :
Fo each ype o alue chain (DVC, GVC, NVC,...) we ha e 4 ma ices, each co e ing all
he alue chain pa hs o each coun y-sec o (we ha e 4 in ou example) ha con o m o
ou alue chain c i e ia.
W
=ˆ
x−1(I−A)−1ˆ
=



0.5461 0.1337 0.1555 0.1645
0.1044 0.6788 0.1224 0.0942
0.0820 0.1047 0.7391 0.0740
0.09126 0.1433 0.1913 0.5740



,
Z=ˆ
C(I−A)−1=


0.8192 0.4013 0.3110 0.1974
0.0043 0.0565 0.0068 0.0031
0.0205 0.0523 0.2463 0.0148
0.1559 0.4896 0.4357 0.7845



.
τi=Z�ei⊗�ei
T
W
τ
1=


0.4474 0.1095 0.1274 0.1348
0.0023 0.0005 0.0006 0.0007
0.0112 0.0027 0.0031 0.0033
0.0851 0.0208 0.0242 0.0256


,τ2=


0.0419 0.2724 0.0491 0.0378
0.0059 0.0383 0.0069 0.0053
0.0054 0.03556 0.0064 0.0049
0.0511 0.3324 0.0599 0.0461


,
τ
3=


0.0255 0.0325 0.2299 0.0230
0.0005 0.0007 0.0050 0.0005
0.0202 0.0258 0.1820 0.0182
0.0357 0.0456 0.3220 0.0322



,τ4=


0.0180 0.0283 0.0377 0.1133
0.0002 0.0004 0.0006 0.0018
0.0013 0.0021 0.0028 0.0084
0.0716 0.1124 0.1501 0.4504


.
τ
DVC
1=



0.1502 0.0616 0 0
0.0017 0.0002 0 0
0 0 00
0 0 00



τDVC
2=



0.0194 0.1556 0 0
0.0043 0.0073 0 0
0 0 00
0 0 00



τ
DVC
3=



00 0 0
00 0 0
0 0 0.0169 0.0096
0 0 0.2374 0.0138



τDVC
4=



00 0 0
00 0 0
0 0 0.0012 0.0044
0 0 0.1083 0.1327



τ
GVC
1=



0.0571 0.0479 0.1274 0.1348
0.0006 0.0003 0.0006 0.0007
0.0112 0.0027 0.0031 0.0033
0.0851 0.0208 0.0242 0.0256



τGVC
2=



0.0224 0.1167 0.0491 0.0378
0.0015 90.0038 0.0069 0.0053
0.0054 0.0355 0.0064 0.0049
0.0511 0.3324 0.0599 0.0461



τ
GVC
3=



0.0255 0.0325 0.2299 0.0230
0.0005 0.0007 0.0050 0.0005
0.0202 0.0258 0.0170 0.0085
0.0357 0.0456 0.0846 0.0183



τGVC
4=



0.0180 0.0283 0.0377 0.1133
0.0002 0.0004 0.0006 0.0018
0.0013 0.0021 0.0016 0.0040
0.0716 0.1124 0.0417 0.0592



τ
NVC
1=



0.24 0 0 0
0 000
0 000
0 000



τNVC
1=



0000
0 0.0272 0 0
0000
0000



τ
NVC
3=



00 0 0
00 0 0
0 0 0.1481 0
00 0 0



τNVC
4=



000 0
000 0
000 0
0 0 0 0.2583



Page 36 o 37
Kneze al. Economic S uc u es (2021) 10:13
The alue chain pa icipa ion sha es a e ob ained by summa ion o all elemen s o he
alue chain ee ma ices:
Acknowledgemen s
The au ho s hank he edi o and all e iewe s o hei commen s and sugges ions ha helped imp o e his a icle.
Au ho s’ Con ibu ions
KK con ibu ed he me hodological de i a ions and empi ical esul s, all h ee au ho s con ibu ed o he li e a u e
e iew, he discussion o he esul s and ex ensi e p oo eading.
Funding
The au ho s o his a icle acknowledge he inancial suppo ecei ed om he Slo enian Resea ch Agency ( esea ch
co e unding No. P5-0177 and No. 52075).
A ailabili y o da a and ma e ials
The da ase s analysed du ing he cu en s udy a e a ailable a h p:// www. wiod. o g.
Decla a ions
E hics app o al and consen o pa icipa e
No applicable.
Consen o publica ion
No applicable.
Compe ing in e es s
The au ho s decla e ha hey ha e no compe ing in e es s.
Recei ed: 30 No embe 2020 Re ised: 14 June 2021 Accep ed: 18 July 2021
Re e ences
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DVCs
=




1
T
τ
DVC
11
1TτDVC
21
1TτDVC
31
1TτDVC
41




=



0.2138
0.1868
0.2779
0.2467



,
GVCs
=




1TτGVC
11
1TτGVC
21
1TτGVC
31
1TτGVC
41




=



0.5461
0.7859
0.5739
0.4949



,
NVCs
=




1TτNVC
11
1TτNVC
21
1TτNVC
31
1TτNVC
4
1




=



0.24
0.0272
0.1481
0.2583


.

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