55
1, XXIV, 2021
Economics
DOI: 10.15240/ ul/001/2021-1-004
QUANTIFYING THE ECONOMIC
DEVELOPMENT DYNAMICS OF A COUNTRY
BASED ON THE LORENZ CURVE
Romualdas Gine ičius1, Joanicjusz Naza ko2,
Daino a Ged ilai ė3, Zdzisława Dacko-Pikiewicz4
1 Vilnius Gediminas Technical Uni e si y, Facul y o Business Managemen , Ins i u e o Dynamic Managemen ,
Li huania, ORCID: 0000-0003-2067-4398, [email p o ec ed];
2 Bialys ok Uni e si y o Technology, Facul y o Enginee ing Managemen , Poland, ORCID: 0000-0002-0236-2334,
[email p o ec ed];
3 Vilniaus Kolegija/Uni e si y o Applied Sciences, Facul y o Economics, Depa men o Finance, Li huania, ORCID:
000-0001-5813-0607, [email p o ec ed];
4 WSB Uni e si y, Facul y o Applied Sciences, Poland, ORCID: 0000-0003-4542-5574, [email p o ec ed].
Abs ac : The wel a e o a coun y depends on i s economic de elopmen . In o de o ha e he
impac on i , we should ha e a possibili y o quan i a i ely assess i s si ua ion a he desi ed poin
in ime. Economic de elopmen , as a mul i ace ed and complex phenomenon, is e lec ed in wo
dimensions – in ensi y and uni o mi y. These men ioned abo e can be iewed as pa ial indica o s
o dynamics. Two main app oaches o measu ing de elopmen uni o mi y can be dis inguished.
In one o he cases, i is measu ed on he basis o an index ha includes he main esul s o he
coun y’s economic de elopmen . In he o he case, he alues o he indica o s e lec ing all he
essen ial de elopmen ac ions a e combined in one app op ia e way. F om a scien i ic poin o iew,
he second app oach is mo e accu a e as i allows o a be e assessmen o he complex na u e o
a coun y’s economic de elopmen . On he o he hand, i s applica ion oday is s ill p oblema ic due
o he ac ha he models o his di e in e ms o bo h he numbe and composi ion o indica o s.
Fo his eason, i is no possible o compa e coun ies. The e o e, in in e na ional p ac ice, he
economic de elopmen o coun ies is measu ed by g oss domes ic p oduc pe capi a (GDP).
Based on GDP indica o , he me hod o he measu emen o uni o mi y is p oposed and he essence
o which is he a io o he leng h o he ideal ajec o y o he de elopmen du ing he pe iod unde
e iew o he leng h o he ac ual ajec o y. Wi hou uling ou he app op ia eness o such an
app oach o assessing de elopmen uni o mi y, i makes sense o look o al e na i e me hods. In
his sense, me hods ha allow assessmen o he ex en o luc ua ions o he phenomenon unde
conside a ion as an essen ial ea u e o de elopmen dynamics a e sui able. These include he Gini
coe icien , which is de e mined om he Lo enz cu e.
Keywo ds: Coun y economic de elopmen , quan i ica ion o dynamics o economic de elopmen ,
Gini coe icien , Lo enz cu e.
JEL Classi ica ion: O1, O11.
APA S yle Ci a ion: Gine ičius, R., Naza ko, J., Ged ilai ė, D., & Dacko-Pikiewicz, Z. (2020).
Quan i ying he Economic De elopmen Dynamics o a Coun y Based on he Lo enz Cu e.
E&M Economics and Managemen , 24(1), 55–65. h ps://doi.o g/10.15240/ ul/001/2021-1-004
In oduc ion
Today ma ke playe s a all le els, om
business ope a o s o coun ies, s i e o be
compe i i e. This aspi a ion is no passi e, and
is no only abou main aining cu en posi ions.
As a esul o he global economic de elopmen
o he wo ld, he g ow h o global ma ke s o ces
pa icula coun ies o inc ease he economic
scale o he economic g ow h, o he wise
hey will lose hei posi ions. The pu sui o
56 2021, XXIV, 1
Economics
an inc easing sha e o in e na ional ma ke s
becomes he basis o a coun y’s compe i i e
capaci y. Only by keeping pace wi h he g ow h
o he single ma ke wa an y ha hey emain
compe i i e. The e o e, de elopmen becomes
a hallma k o compe i i eness.
In his con ex , i is impo an o bo h science
and p ac ice o ully analyse he phenomenon
o economic de elopmen .
The de elopmen o any p ocess can
be cha ac e ized by wo cha ac e is ics – i s
in ensi y and i s uni o mi y (Gine ičius e
al., 2018). In ensi y o de elopmen is he
a e o de elopmen g ow h o e he pe iod
conside ed, exp essed, o example, in yea s.
S eady de elopmen eads as luc ua ions in
he ex en o de elopmen o e he ime pe iods
conside ed. These wo dimensions can be seen
as sub-indica o s o he o e all de elopmen
p ocess. The i s indica o e lec s upon he
quan i a i e side o he de elopmen p ocess,
and he second indica o s e lec s upon he
quali a i e side. The de elopmen p ocess
as a whole is e lec ed by i s dynamics,
which combines bo h o he abo e men ioned
indica o s. I shows he ex en and ype o
de elopmen changes ha ha e aken place
o e he pe iod unde e iew.
The essence o dynamics as a phenomenon
is exp essed by he ollowing synonyms o his
e m in in e na ional dic iona ies: sus ainable,
s eady, esilien , s able, immo able, solid,
immu able, du able, pe manen , se led,
long-las ing, e en, e c. This abundance o
unde s anding o dynamics is in eg a ed by he
wo dimensions men ioned abo e – de elopmen
in ensi y and de elopmen con inui y.
In he scien i ic li e a u e a emp s o quan i y
his phenomenon can be me (Gine ičius
e al., 2018). Based on he me hodology
p esen ed in his s udy, he dynamics o
economic de elopmen in some EU coun ies
we e de e mined. This me hodology, which
is e e ed o as he MDD me hod, p o ides
o he in eg a ion o uni o mi y indica o o
he de elopmen wi h i s in ensi y indica o .
Uni o mi y is de ined as he a io o he alue
o he pe iod unde conside a ion o he o al
leng h o he ac ual de elopmen ajec o y:
(1)
whe e DT is he indica o o economic
de elopmen T o e he pe iod unde
conside a ion, N is he du a ion o he pe iod
unde conside a ion e.g. yea s; pi is he change
in de elopmen o e he ime pe iod i (yea ) o
he ime span unde e iew; n is he numbe
o ime pe iods (i = 1, n) o he pe iod unde
conside a ion T.
The in ensi y o he de elopmen shall be
de e mined as ollows:
(2)
whe e Kp
T is he in ensi y indica o o he coun y’s
economic de elopmen o e he pe iod T
unde conside a ion; Q
T is he signi icance o
economic de elopmen a he end o pe iod T,
Qb
T is he signi icance o economic de elopmen
a beginning o pe iod T.
The p ocess o economic de elopmen o
socio-economic sys ems (SES) is a he mul i-
ace ed, complex and ambiguous, and i makes
sense o look o o he ways o quan i y i s
dynamics.
1. Possibili ies o Quan i ying
Sus ainabili yo Coun y’s
EconomicDe elopmen
The challenge posed – o quan i y he dynamics
o he coun y’s economic de elopmen –
equi es wo ques ions o be answe ed. Fi s ,
which p incipled models a e sui able o
quan i ying he in ensi y and con inui y o he
SES de elopmen p ocess; secondly, wha
indica o o indica o s can be used o measu e
hese sub-indica o s. The measu e o he
in ensi y o economic de elopmen o a coun y
can be aken as he alue exp essed by o mula
(2). The de elopmen consis ency indica o can
be o mula ed wi h he ollowing equi emen s:
i mus be easy o unde s and, he in o ma ion
needed o he calcula ions mus be easily
accessible, and, mos impo an ly, i mus allow
compa isons be ween en i ies a any le el:
economy o coun ies, egions, coun ies as
a whole, g oup o coun ies, e c.
The economic de elopmen o he coun ies
will be used o es he adequacy o he
homogenei y indica o p oposed in his s udy.
The analysis o li e a u e sou ces ha examine,
measu e and compa e his phenomenon
highligh s wo p incipal possibili ies. In one
57
1, XXIV, 2021
Economics
case i e e s o GDP pe capi a in a e i o ial
uni . I is conside ed o be a eliable indica o o
success, e lec ing upon he le el o economic
de elopmen achie ed in he coun y (Cha san,
2013; Čiegis e al., 2010; Bolcá o á & Kološ a,
2015; Babu & Da a, 2015). O he wise, he
economic de elopmen o he analyzed objec s,
usually coun ies, o hei egions, is e alua ed
elying upon one agg ega e alue combining
a la ge numbe o pa ial indica o s, which in
he au ho s’ opinion e lec upon he essen ial
aspec s o economic de elopmen . Recen ly,
hese me hods ha e been inc easingly used
o quan i y phenomena o all kinds. Impo ance
o indica o s a ailable on an equal basis o
de e mined by pee e iew (Song e al., 2017;
Tu skis, 2018; Tu skis e al., 2017; Boggia &
Co ina, 2010; P asce ic & P asce ic, 2017).
We may conside ha mul i-c i e ia
assessmen o economic de elopmen is mo e
accu a e compa ed o GDP pe capi a, because
enla gemen i sel is a complex p ocess ha
mani es s i sel in many aspec s. Fo malizing
hem and exp essing hem as indica o s allows
a mo e complex e alua ion o he phenomenon
unde conside a ion. On he o he hand, he
possibili ies o applying his p omising me hod,
o example o compa ing coun ies, a e limi ed
by he a ie y o indica o s ha a e included in
he e alua ion model and hei composi ion. The
numbe o such indica o s a ies om se e al
o se e al dozen in indi idual cases (Gine ičius
e al., 2018; Čiegis e al., 2010). E en when
a model is composed o a small numbe o
indica o s, hei composi ion is usually di e en .
This depends on a numbe o ci cums ances:
wha aspec s o he phenomenon in ques ion
a e dis inguished by he au ho s; o om access
o in o ma ion needed o calcula e alues o an
indica o , e c. These ci cums ances make i
di icul o use he esul s o he mul i-c i e ia
assessmen o economic de elopmen o
compa ing coun ies. Meanwhile, in e na ional
da abases and s a is ical yea books o
coun ies p o ide all he necessa y in o ma ion
on hei economic de elopmen on he basis
o pe capi a GDP. The e o e, his s udy will be
based on his indica o ( u he – GDP).
Two di e en coun ies, Swi ze land and
I aly, ha e been selec ed o illus a e he
p oposed me hodology. Swi ze land has
expe ienced a s eady g ow h o GDP pe capi a
o e he pe iod unde e iew. On he o he hand,
his p ocess is accompanied by qui e no iceable
luc ua ions in indi idual ime pe iods (yea s) o
he pe iod unde conside a ion. In I aly, du ing
he pe iod unde e iew, he a e o economic
de elopmen has ha dly inc eased, bu he e is
no ma ked luc ua ion ei he (Tab. 1, Fig. 1).
To quan i y he con inui y o he SES
de elopmen p ocess me hods based on
dynamic analysis o p ocesses can be used,
which means ha hey can assess he
magni ude o he luc ua ion o he phenomenon
unde conside a ion, he essen ial a ibu e o
dynamics, and ha allow o be in e p e ed. To he
me hods o his kind we may asc ibe coe icien s
used o analyze he con inui y o economic and
social p ocesses. Fi s and o emos , hese would
be he me hods o assessmen o inequali ies
in income and economy (Amin, 2006; A kinson
e al., 2002; Čiule ičienė e al., 2006a, b;
Čiule ičius & Čiule ičienė, 2008; Ib agimo e
al., 2018; Kawachi & Kennedy, 1999; Kawachi e
al., 1999; Ku owska-Pysz e al., 2018).
Alongside he s uc u al coe icien s,
some special indica o s a e needed o assess
he economic and social inequali y o he
popula ion (Čiule ičienė, 2006a; Lazu ka,
2002). The mos commonly used is he Gini
coe icien (index). The a io a ies om 0 o 1
o om 0 o 100 pe cen . The la ge i is, he
g ea e is he inequali y. This indica o is widely
used in in e na ional compa isons. The Gini
coe icien is de e mined om he Lo enz cu e
(Dagum, 1980; G usky, 2001; Rudzkienė, 2005;
Coun y
Yea s
2008 2009 2010 2011 2012 2013 2014 2015 2016 2017
I aly 27.6 26.4 26.8 27.3 26.7 26.5 26.7 27.2 27.9 28.5
Swi ze land 49.3 50.2 56.1 63.7 65.0 64.1 65.3 74.0 72.4 72.6
Sou ce: www.eu os a .com
Tab. 1: G oss domes ic p oduc (GDP) pe capi a in EUR
58 2021, XXIV, 1
Economics
B a čiko ienė & De eiky ė, 2006; Jann, 2016).
The idea o his cu e is ha , i i is
possible o de e mine he o al inequali y o
he indica o unde conside a ion, he e comes
he oppo uni y o analyze he na u e o i s
dis ibu ion be ween he indi idual ime spans o
he pe iod unde conside a ion. Fo example, o
depic g aphically he dis ibu ion o households
by income, hey a e a anged in ascending o de
on he axis X o he o dina e sys em, indica ing
he pe cen age o he popula ion (households);
meanwhile depic ing he de e ed pe cen age
o hei income in axis Y (Fig. 2).
The na u e o he Lo enz cu e and he
Fig. 1: GDP pe capi a o he coun ies conce ned
Sou ce: own based on www.eu os a .com
Fig. 2: Lo enz cu e
Sou ce: own based on www.eu os a .com
59
1, XXIV, 2021
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o de in which i is cons uc ed allow i o be
applied o assess he homogenei y o he
coun y’s economic de elopmen . I s o de
would be as ollows.
Fi s , le us d aw a squa e wi h sides
100×100 and i s diagonal. The cumula i e
alues o he economic de elopmen du ing
he ime spans o he pe iod unde e iew,
exp essed as a pe cen age, a e plo ed on
he abscissa axis, and he pe cen age o he
ac ual de elopmen o he ime pe iods a e
plo ed on he o dina e axis. The in e sec ion
o each o hese deciles will gi e a pa icula
poin . Connec ing he poin s in a s aigh line
gi es a Lo enz cu e. I e lec s upon he ac ual
cou se o he p ocess o economic de elopmen
o e he pe iod unde conside a ion, and he
diagonal o he squa e ep esen s i s ideal
de elopmen .
The Lo enz cu e is e y widely applied in
ma hema ics, especially in economic s a is ics.
I is sui able o examining he inequali ies o
a wide a ie y o phenomena ep esen ed by
in e al lines. I s deg ee is ep esen ed by he
con exi y o he cu e, e.g. he dis ance om
he diagonal o he squa e; he s eepe he
magni ude cu e is, he lesse he dis ance om
he cu e i shows; he less i is dis an om
he diagonal, he lesse he di e si ica ion o
he dis ibu ion is, and ice e sa, as he cu e
mo es away om he diagonal, he dis ibu ion
i egula i y inc eases.
The Gini coe icien (Dagum, 1980; Gu sky,
2001; Rudzkienė, 2005) is used o es ima e
quan i a i e dis ibu ion compa isons based
on Lo enz cu e. I is de e mined as ollows
(Fig. 2):
(3)
The smalle he dis ibu ion i egula i y
is, which means he close he cu e is o he
diagonal o he squa e, he lowe he alue o
he ac o G we see.
I we equa e he a ea S1 + S2 o S, hen om
Fig. 2. ollows ha S1 = S – S2. Then
(4)
In ou case, when he case o quan i ying
he dynamics o SES economic de elopmen
is unde e iew, he smalle he a ia ion in
de elopmen and he close he Lo enz cu e
app oaches he squa e o he squa e, he mo e
e en he expansion is, so he Gini coe icien
mus be ans o med in his way.
(5)
He e TG is he coe icien desc ibing he
con inui y o SES economic de elopmen .
Fig. 1 and o mulas (4) and (5) show ha
in ideal case o de elopmen , S1 = 0, and
TG = 1.0.
Quan i a i e assessmen o he economic
de elopmen dynamics o he coun ies
conce ned was ca ied ou on he basis o he
abo e me hodology.
2. Quan i yingEconomic
De elopmen Dynamics
in Coun ies
As s a ed abo e, we quan i y he dynamics o he
coun y’s economic de elopmen on he basis o
pe capi a GDP. I s luc ua ions in he pa icula
ime spans o he pe iod unde conside a ion
may be e y une en (Tab. 1, Fig. 1). The same
ep esen a i e coun ies – I aly and Swi ze land
– a e aken o he calcula ions. F om Fig. 1 i
is seen ha Swi ze land is unde going in ense
de elopmen , whe eas I aly is expe iencing
s agna ion in i s de elopmen . On he o he hand,
in I aly, de elopmen is s able, wi h no majo
luc ua ions, while in Swi ze land, luc ua ions
a e signi ican ly highe . The e o e, i can be
assumed ha he e is a ela ionship be ween
de elopmen in ensi y and sus ainabili y,
which means ha he in ensi y in luences he
uni o mi y. F om a me hodological poin o
iew, i would be ad an ageous o elimina e
he in luence o in ensi y om he indica o o
sus ainable de elopmen and o e alua e i
la e , when a complex sus ainable de elopmen
indica o is c ea ed. O he wise, he low le el o
de elopmen may be o se by i s high in ensi y
and esul in an inadequa e pic u e.
The di e en economic de elopmen
si ua ions o he coun ies in Fig. 1 need o be
quan i ied. To ha end, i is i s o all necessa y
o de e mine he de elopmen si ua ion in ela ion
o which he ac ual si ua ion will be assessed. In
o he wo ds, we need o ha e a benchma k o
de elopmen con inui y o an ideal de elopmen
case. This can be said o be he case when he

60 2021, XXIV, 1
Economics
SES de elopmen inc emen o e he same
pe iod o ime, such as he yea , is he same. In
his case, he de elopmen ajec o y will coincide
wi h he diagonal o he squa e (Fig. 1), and he
de elopmen i sel will look like his (Fig. 3).
F om Fig. 3 i can be seen ha he
Fig. 3: Ideal case o SES dynamics o de elopmen
Sou ce: own
Fig. 4: Changes in de elopmen o e ime span i; i, i + 1 o he pe iod unde e iew
Sou ce: own based on Gine ičius e al., 2018
61
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Economics
uni o mi y o SES de elopmen uns ideally
∆qi = ∆qi + 1 (whe e ∆qi is he inc ease in
de elopmen o he ime pe iod i). In eal li e we
usually see a si ua ion whe e ∆qi≠∆qi + 1, which
means we see g ea e o lesse luc ua ions in
de elopmen o e pa icula ime pe iods o he
pe iod unde e iew. In his case, he uni o mi y
o de elopmen can be quan i ied based on he
de elopmen al di e ences o hese pe iods
∆∆q=qi – qi + 1 (Fig. 4).
We exp ess he a ia ions in he economic
de elopmen o he coun ies shown in Fig. 1
by he Lo enz cu e. Fo his pu pose, we will
use Tab. 1 o calcula e he absolu e magni udes
o de elopmen di e ences be ween adjacen
ime pe iods (Tab. 2).
We will p esen he di e ences in Tab. 2 in
ascending o de . Such a p ocedu e is possible
because i will no a ec he a ea S2 o he
Lo enz cu e S2, and he a ea S1 (Tab. 3).
Based on Tab. 3, we de e mine he ela i e
weigh o he di e ences as a pe cen age (Tab. 4).
Coun y Yea s
2008 2009 2010 2011 2012 2013 2014 2015 2016 2017
I aly 0.2 1.2 0.4 0.5 0.6 0.2 0.2 0.5 0.7 0.6
Swi ze land 3.1 0.9 5.9 7.6 1.3 0.9 1.2 8.7 1.6 0.2
Sou ce: own
Coun y
Yea s
To al
12345678910
I aly 0.2 0.2 0.2 0.4 0.5 0.5 0.6 0.6 0.7 1.2 5.1
Swi ze land 0.2 0.9 0.9 1.2 1.3 1.6 3.1 5.9 7.6 8.7 31.4
Sou ce: own
Coun y Yea s To al
12345678910
I aly 3.9 3.9 3.9 7.8 9.8 9.8 11.8 11.8 13.8 23.5 100
Swi ze land 0.6 2.9 2.9 3.8 4.1 5.1 9.9 18.8 24.2 27.7 100
Sou ce: own
Coun y
Yea s
12345678910
I aly 3.9 7.8 11.7 19.5 29.3 39.1 50.9 62.7 76.5 100
Swi ze land 0.6 3.5 6.4 10.2 14.3 19.4 29.3 48.1 72.3 100
Sou ce: own
Tab. 2: Di e ences in economic de elopmen o he coun ies conce ned
o e he pe iods conside ed
Tab. 3: Ascending o de o economic de elopmen di e ences be ween coun ies
o he pe iod unde e iew
Tab. 4: Compa a i e weigh s o di e ences in economic de elopmen o he coun ies
conce ned o e he pe iod o ime in pe cen
Tab. 5: Cumula i e alues o compa a i e weigh s o di e ences in economic
de elopmen o he coun ies unde e iew o he conside ed pe iod
62 2021, XXIV, 1
Economics
Based on Tab. 4, we de e mine he
cumula i e alues o he weigh s.
Based on Tab. 5, we can d aw he Lo enz
cu es o he economic de elopmen o he
coun ies unde conside a ion (Fig. 5–6).
The ques ion is how o assess he s a e o
SES de elopmen con inui y based on hese
esul s. The li e a u e p o ides he ollowing
Gini coe icien a ing scale (Lando e al., 2018;
Lyon e al., 2016) (Tab. 7).
Based on Tab. 7 and o mula (2), i is
possible o p o ide he ollowing scale o
assessing he sus ainabili y o economic
de elopmen in a coun y (Tab. 8).
Tab. 8 shows ha I aly has a mode a e
deg ee o economic de elopmen and
Swi ze land a low deg ee o sus ainabili y.
The TG ac o o coun ies’ economic
de elopmen unde es ima es i s in ensi y.
Meanwhile, om Fig. 1. i can be judged o
Fig. 5: Lo enz cu e o I alian economic de elopmen
Sou ce: own
Fig. 6: Lo enz cu e o Swiss economic de elopmen
Sou ce: own
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Economics
in luence uni o mi y. The in eg a i e size o
hese wo aspec s o de elopmen will gi e
a snapsho . I can be se up as ollows:
(6)
whe e is he index o economic de elopmen
dynamics o coun y i; Kpi is he a e o economic
de elopmen in ensi y o coun y i; and TGi is
he coe icien o homogeneous o economic
de elopmen o coun y.
I is ound ha I aly equals 1.03 and
Swi ze land 1.47. On his basis, an in eg al
index o hei economic de elopmen dynamics
can be calcula ed. I is 0.85 in I aly and 0.87 in
Swi ze land.
Such esul s o economic de elopmen
dynamics o he coun ies a e ecei ed by
conside ing ha he uni o mi y and in ensi y o
de elopmen a e equally signi ican in e ms
o dynamics. The si ua ion would change
subs an ially i we conside he impo ance
o he pa ial indica o s TG and Kp. I can
be assumed ha sho - e m luc ua ions in
economic de elopmen o e indi idual ime
pe iods (yea s) o he pe iod unde e iew
ha e a signi ican ly lesse impac on he s a e
o social de elopmen o he coun y compa ed
o he in ensi y o de elopmen co e ing he
whole pe iod unde e iew. Expe s es ima ed
he impo ance o de elopmen in ensi y o
i s dynamics a 70 pe cen ou o 100, and
luc ua ions in uni o mi y a 30 pe cen . In his
case, a gene alized indica o o he dynamics
o a coun y’s economic de elopmen can
be de e mined by a mul i-c i e ia e alua ion,
o example, by he SAW me hod, which is
exp essed as ollows (Hwang & Yoon, 1981):
(7)
he e Di is an index o coun y’s i dynamics
o economic g ow h; wi is an impo ance o
coun yʼs index o dynamics o economic g ow h
Coun y Coe icien
G TG
I aly 0.30 0.70
Swi ze land 0.49 0.51
Sou ce: own
Gini coe icien alue G Conclusion
G < 0.25 Ve y low le el
0.25 ≤ G ≤ 0.35 A e age le el
G > 0.35 High le el
Sou ce: own
Value o coe icien TG Conclusion
TG > 0.75 Ve y high deg ee o dynamics o de elopmen
0.75 > TG > 0.65 A e age deg ee o dynamics o de elopmen
TG < 0.65 Low deg ee o dynamics o de elopmen
Sou ce: own
Tab. 6: Resul s o he calcula ion o de elopmen dynamics o he analyzed coun ies
o 2008–2012
Tab. 7: G ading o Gini coe icien acco ding o uni o mi y o da a dis ibu ion
Tab. 8: Coe icien TG g ada ion by homogenei y