Næ dal, E ic; Wagne , Ma in
A icle
A no e on he op imal speed o ansi ion: Aghion and
Blancha d e isi ed
Ge man Economic Re iew (GER)
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Ve ein ü Socialpoli ik / Ge man Economic Associa ion
Sugges ed Ci a ion: Næ dal, E ic; Wagne , Ma in (2025) : A no e on he op imal speed o ansi ion:
Aghion and Blancha d e isi ed, Ge man Economic Re iew (GER), ISSN 1468-0475, De G uy e ,
Be lin, Vol. 26, Iss. 1, pp. 1-14,
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ge 2025; 26(1): 1–14
E ic Næ dal and Ma in Wagne *
A No e on he Op imal Speed o T ansi ion:
Aghion and Blancha d Re isi ed
h ps://doi.o g/10.1515/ge -2024-0054
Recei ed May 21, 2024; accep ed Sep embe 15, 2024; published online Oc obe 17, 2024
Abs ac :This no e illus a es, by econside ing he seminal op imal speed-o -
ansi ion model o Aghion, P., and O. J. Blancha d. (1994. “On he Speed o T ansi ion
in Cen al Eu ope.” NBER Mac oeconomics Annual 9: 283–319), ha op imal ansi-
ion pa hs, in gene al, exhibi nonlinea i ies and discon inui ies. Aghion and Blan-
cha d conside only an app oxima e solu ion wi h a cons an unemploymen a e
o e he ansi ion p ocess. The exac solu ion ea u es an inc easing unemploy-
men a e wi h a discon inui y when he s a e sec o is closed down a he op imally
chosen endpoin o ansi ion. Economic ansi ion p oblems bea many simila i-
ies o sc ap alue p oblems wi h ee e minal ime, o en encoun e ed in esou ce
economics. In ela ion o he ansi ion o a g een economy, he discussion in his
no e he e o e cas s doub on he op imali y o a g een ansi ion discussed in, e.g.,
he Eu opean Union in e ms o poli ically specified a he han op imally designed
miles ones o emissions educ ions, i.e., by −55 % compa ed o 1990 le els un il
2030 and ne ze o un il 2050.
Keywo ds: dynamic op imiza ion; end-o - ansi ion; op imal unemploymen a e;
ansi ion
JEL Classi ica ion: C61; E61; P20
1 In oduc ion
The his o y o economic de elopmen has epea edly seen and con inues o see un-
damen al ansi ions be ween pa ly d as ically di e en economic (and poli ical
*Co esponding au ho : Ma in Wagne , Depa men o Economics, Uni e si y o Klagen u ,
Uni e si ä ss asse 65–67, 9020 Klagen u , Aus ia; Bank o Slo enia, Ljubljana, Slo enia; and Ins i u e
o Ad anced S udies, Vienna, Aus ia, E-mail: [email p o ec ed].h ps://o cid.o g/0000-0002-
6123-4797
E ic Næ dal, HVL Business School, Wes e n No way Uni e si y o Applied Sciences, Sogndal, No way.
h ps://o cid.o g/0000-0002-7325-6811
Open Access. ©2024 he au ho (s), published by De G uy e . This wo k is licensed unde he C ea i e
Commons A ibu ion 4.0 In e na ional License.
2—E. Næ dal and M. Wagne
o echnical) egimes. In his pe spec i e, he fi s majo economic ansi ion was
p obably he ansi ion om nomadic socie ies o ag a ian, non-nomadic socie ies.
Impo an majo ansi ions in mo e ecen cen u ies include, o cou se, he (fi s )
indus ial e olu ion and mo e ecen ly, s a ing in he ea ly 1990s, he ansi ion
o ma ke economies in (mos o ) he o me cen ally planned communis coun-
ies. This p ocess s a ed wi h he demise o he So ie Union and has b ough qui e
apid (no only) economic change o nume ous coun ies.1Howe e , his o y has no
come o an end. Fu he ansi ion p ocesses a e ongoing and can be expec ed o
exe majo impac s. These include, e.g., new wa es o echnological change labelled
as “digi aliza ion”, “indus y 4.0” o “in e ne o hings” (see, e.g. B ynjol sson and
McA ee 2011), which can be conside ed a 21s -cen u y e sion o an indus ial e -
olu ion. Ano he impo an ansi ion p ocess – e y likely he mos impo an
ansi ion o ou ime – is he ansi ion owa ds an (essen ially g eenhouse gas)
emissions- ee economy necessa y o limi he de imen al e ec s o an h opogenic
clima e change. This so-called g een ansi ion equi es, in pa icula , undamen-
al changes in he p oduc ion (and consump ion) o ene gy, i.e., a eplacemen o
ca bon-based ene gy sou ces by ca bon-neu al ene gy sou ces. In his p ocess,
ene gy usage will shi s ongly owa ds elec ici y o be gene a ed om emissions-
ee sou ces. Clea ly, he edesign o he global ene gy in as uc u e and sys em
(see, e.g., In e na ional Ene gy Agency 2023) will ha e p o ound impac s on all sec-
o s and po en ially also on he composi ion o he global economy.2
No wi hs anding he he e ogenei y o he scopes and impac s o he ansi ion
p ocesses men ioned, all hese ansi ions necessi a e o imply majo ealloca ions
o p oduc ion ac o s, in pa icula also o labo , om old sec o s o new sec o s. The
impo ance and magni ude o hese p ocesses makes an e icien design impe a i e.
Concep ually, an op imal ansi ion policy is he eby defined in e ms o bo h an op i-
mal speed o ansi ion and an op imal endpoin o , equi alen ly, an op imal du a-
ion o a ansi ion p ocess. We illus a e hese wo dimensions by de i ing in de ail
bo h he op imal speed as well as he op imal endpoin o a ansi ion p ocess by
econside ing he well-known speed-o - ansi ion model o Aghion and Blancha d
(1994) ha deals wi h he ansi ion om a cen ally planned owa ds a ma ke -
based economy. Mo e specifically, Aghion and Blancha d (1994, Sec ion 6.4) p esen
a dynamic op imiza ion model o de e mine he op imal speed o ansi ion, which
1Fo a ecen discussion conce ning he pa ly ongoing ansi ion p ocesses om cen ally
planned o ma ke economies, see, e.g., Dab owski (2023).
2A pi o al con ibu ion desc ibing policy needs o comba ing clima e change is he epo o Si
Nicholas S e n, see S e n (2007) and, o an assessmen o he de elopmen s since he o iginal pub-
lica ion, S e n (2015).Hassle , K usell, and Smi h (2016) p o ide an o e iew o e mac oeconomic
modelling o clima e change and esou ce sca ci y.
A No e on he Op imal Speed o T ansi ion —3
in hei model co esponds o finding he op imal pa h o he unemploymen a e
(see also he discussion in Roland 2000).3When sol ing he dynamic op imiza ion
p oblem, Aghion and Blancha d (1994) do no , howe e , de i e he exac solu ion,
bu only an “app oxima e” solu ion ha neglec s he beha io o he economy a e
he s a e sec o has been closed down. Due o he dynamic na u e o he economy,
howe e , he pos - ansi ion economic pe o mance influences he op imal beha -
io al eady du ing he ansi ion p ocess and hus influences he op imal pa h also
whils he s a e sec o s ill employs people.4In his espec , Aghion and Blancha d
(1994, Foo no e 33, p. 305) s a e ha hey a e “chea ing” by se ing ce ain quan i ies
cons an , which hey label “ u npike” app oxima ion.
The exac solu ion di e s om he app oxima e solu ion in wo ela ed ways:
Fi s , he op imal unemploymen a e is no cons an , bu inc eases o e ime and
exhibi s a discon inui y when he s a e sec o is closed down a an op imally cho-
sen endpoin o he ansi ion. Second, we find a highe op imal unemploymen a e
han Aghion and Blancha d, which implies a sho e op imal du a ion o he an-
si ion p ocess, i.e., an ea lie endpoin . In he Aghion and Blancha d (1994) model,
he ine iciency c ea ed by he app oxima e solu ion wi h a cons an unemploymen
a e is ha a oo low unemploymen a e educes he a e o job c ea ion in he new
sec o , which slows down ou pu g ow h in his mo e p oduc i e sec o and ex ends
he du a ion o he ansi ion p ocess.5
The ansi ion mechanism desc ibed he e – o in Aghion and Blancha d
(1994) – is concep ually closely ela ed o impo an aspec s ( ha need o be
3Fo a de ailed discussion conce ning labo ma ke dynamics in ansi ion economies see, e.g.,
Boe i (2000). Ou con ibu ion he e is o a concep ual o me hodological na u e and, hus, se e al
aspec s o labo ma ke dynamics ha a e ound o be ele an om a labo economics pe spec i e
a e neglec ed, as in he model o Aghion and Blancha d (1994). Also, o cou se, his ype o model
has o be in e p e ed in a s ylized ashion wi h espec o he ole o he s a e in an economy. In he
model he s a e sec o is closed down en i ely, i.e., he non- i ial ole o s a e sec o s also in ma ke
economies is, o simplici y and o ocus on one aspec , abs ac ed om. Fo he same eason, i.e.,
o ocus on one aspec , we also abs ac om e o m unce ain y and po en ial e o m e e sal,
issues discussed in Fe nandez and Rod ik (1991) o Dewa ipon and Roland (1995).
4The beha io o he economy a he poin in ime when he s a e sec o is closed (and he e-
a e ) is neglec ed also in o he speed o ansi ion models: B ixio a and Youse (2000) assume a
cons an closu e a e o he s a e sec o , which may also lead o di e en dynamic beha io and
wel a e losses compa ed o op imal closu e. Bu da (1993) also finds a cons an op imal unemploy-
men a e, whe e again he e ec o s a e sec o closu e is no analyzed in de ail. Cas anhei a and
Roland (2000) a oid he p oblem by assuming ha he e is no unemploymen and ha capi al can
be mo ed eely om he old o he new sec o .
5As al eady men ioned, ou analysis is o a me ely concep ual and quali a i e na u e, bu a is-
ing unemploymen a e du ing an op imal ansi ion p ocess inc eases he isk o cos ly e o m
e e sals and backlashes. This is an issue ha , howe e , canno be add essed when conside ing a
cen al planning solu ion only.
4—E. Næ dal and M. Wagne
adequa ely de ailed in ully specified models) o he o he ansi ion p ocesses men-
ioned abo e. This is ob ious, e.g., o he men ioned 21s -cen u y indus ial e o-
lu ion, by simply eplacing he e minology s a e and p i a e sec o wi h old and
new sec o . The e a e also close links o he g een ansi ion, when eplacing he
labels s a e and p i a e sec o wi h di y and clean sec o . In he con ex o he g een
ansi ion, he go e nmen , once i in e nalizes he nega i e clima e ex e nali y o
he di y sec o (wi h a highe p i a e bu lowe social ma ginal p oduc han he
clean sec o ), aces he p oblem o managing an op imal ansi ion o an e ec i ely
ca bon emissions ee economy.6
This sho pape is o ganized as ollows: In Sec ion 2, we se up and analyze he
Aghion and Blancha d (1994) model in de ail and Sec ion 3 d aws some conclusions.
2 The Aghion–Blancha d Model: Exac Solu ion
o No ma i e Analysis
We ocus on he dynamic op imiza ion p oblem used o a no ma i e analysis o
a ansi ion p ocess in Aghion and Blancha d (1994, Sec ion 6.4) and p esen only
hose pa s o he model p esen ed in hei pape in de ail ha a e o immedia e
ele ance he e.
Deno e wi h E( ) he numbe o people employed in he s a e sec o (wi h con-
s an ma ginal p oduc i i y x), wi h N( ) he numbe o people employed in he
eme ging p i a e sec o (wi h cons an ma ginal p oduc i i y y>x>0) and wi h
U( ) he numbe o unemployed people a ime . Popula ion is no malized o one,
i.e., E( )+N( )+U( )=1, which implies ha U( )isbo h henumbe o unem-
ployed people and he unemploymen a e. Aghion and Blancha d (1994) de elop
an e iciency wage-based explana ion o cos ly labo adjus men be ween he old
s a e sec o and he new p i a e sec o . In pa icula , hey de i e he ollowing
6The analogy o he g een ansi ion has o be conside ed mo e ca e ully also in e ms o mod-
elling s ocks and flows: A key aspec o be included in clima e-economy ansi ion models is ha
hey need o ake in o accoun he (unce ain) nega i e impac s o he accumula ed s ock o emis-
sions on all sec o s in he economy ia a damage unc ion o some so , see, e.g., Hassle , K usell,
and Olo sson (2024) o a ecen policy-o ien ed discussion. Fu he mo e, he di e en ansi ions
lis ed a e, o cou se, in e wined, wi h, e.g., he di ec ion and speed o echnical change no inde-
penden o clima e policies, see, e.g., Hassle , K usell, and Olo sson (2022). Fu he mo e, in he
con ex o en i onmen al p oblems he e is a well-known discussion abou ins umen choice in
en i onmen aleconomics(quan i ycons ain s, axa ion,...). Whiche e ins umen chosen, he
esul will be a sh inking o he di y sec o ha has o be managed by choosing op imal policy
pa hs.
A No e on he Op imal Speed o T ansi ion —5
ela ionship o he speed o job c ea ion in he new p i a e sec o (see hei
equa ion (9) on page 298):7
N= (U)=a[U
U+ca][y− c −(b
1−U)],(1)
wi h a,b,cand posi i e cons an s. He e, aindica es he impac o pe -wo ke
p ofi s in he p i a e sec o on he speed o p i a e-sec o job c ea ion, ba e unem-
ploymen benefi s and cis a cons an ela ed o he “wage-p emium”, de i ed using
an e iciency-wage se ing, ha p i a e fi ms a e willing o pay (o e and abo e
unemploymen benefi s as ou side op ion). Fu he mo e, is he in e es a e and
he cos o job c ea ion in he p i a e sec o is gi en by 1
2a ( (U))2.8The go e nmen
chooses he op imal speed o closu e o he ine icien s a e sec o and, he eby,
unemploymen .
Rema k 1. I may be in e es ing o discuss he building blocks leading o p i a e-
sec o job c ea ion as gi en in (1) in a bi mo e de ail: This ela ionship is based on
he combina ion o h ee modelling assump ions wi h an e iciency-wage conside -
a ion. Fi s , Aghion and Blancha d (1994) assume ha he a e o p i a e-sec o job
c ea ion is p opo ional o p ofi s pe wo ke , i.e.,
N=a(y−z−𝑤)wi hybeing
he (cons an ) a e age p oduc o labo in he p i a e sec o , z axes pe wo ke , 𝑤
he p i a e-sec o wage and he p opo ionali y cons an a, compa e (2) in Aghion
and Blancha d (1994). Second, p i a e-sec o wages 𝑤depend upon labo ma ke
condi ions as ollows 𝑤=b+c( +
N
U), wi h unemploymen benefi s b, hein e -
es a e ,
N
U he a io o new jobs o unemploymen and a cons an c ha scales
he wage p emium o e unemploymen benefi s ha fi ms a e willing o pay, com-
pa e (3) in Aghion and Blancha d (1994).9The hi d elemen is ha unemploymen
benefi s a e financed by labo axes unde he condi ion o a balanced budge , i.e.,
7To a oid o e loaded no a ion we some imes skip he ime index . Fu he mo e,
N( ) deno es
he de i a i e o N( ) wi h espec o , wi h his no a ion also used o o he a iables.
8Ano he es ic ion on he pa ame e s is y−b− c >0. In his case, (U) is posi i e o alues o
Ula ge han ze o and smalle han y−b− c
y− c . Clea ly, i canno be op imal o conside unemploymen
pa hs ha include alues la ge han his alue wi h nega i e a es o job c ea ion in he mo e
p oduc i e p i a e sec o .
9The ela ionship o p i a e-sec o wages is de eloped in Aghion and Blancha d (1994) in an
e iciency-wage amewo k unde he assump ions ha all hi es a e om unemploymen and
ha once employed in he p i a e sec o he e is no isk o u u e unemploymen . Thus, he
alue o being unemployed, VU, is gi en, see (4) and (5) in Aghion and Blancha d (1994),by
VU=
N
U(VN−VU)+dVU
d ,wi hVNbeing he alue o being employed in he p i a e sec o , i sel
gi en by VN=𝑤+dVN
d . The e iciency-wage a gumen en e s he conside a ions by pos ula ing
ha p i a e-sec o fi ms will se a wage such ha he VN=VU+c, o somec≥0. This in u n
implies dVN
d =dVU
d and he wo equa ions gi en in his oo no e can be easily ea anged – by
simply aking he di e ence – o lead o he equa ion o p i a e-sec o wages gi en in he ema k.
6—E. Næ dal and M. Wagne
Ub =(1 −U)z,see(6) in Aghion and Blancha d (1994). The ela ionship (1) now
ollows om inse ing 𝑤=b+c( +
N
U)andUb =(1 −U)zin o
N=a(y−z−𝑤).
Aghion and Blancha d (1994, Sec ion 3.1) assume ha a he ou se o ansi ion,
employmen in he s a e sec o d ops om 1 o some E(0) =E0<1, which implies
an ini ial unemploymen a e equal o U(0) =1−E0. This alue o U(0) will, in
gene al, no co espond o he op imal choice o he unemploymen pa h ha max-
imizes he ne p esen alue o ou pu . Consequen ly, he op imal unemploymen
pa h will ha e a discon inui y a =0 and jump o he op imal alue om U(0)
immedia ely. Only, when he ini ial unemploymen a e co esponds o he op imal
choice will he op imal pa h o he unemploymen a e be as illus a ed below in
Figu e 1. Since ou ocus he e is on he du a ion and end o a ansi ion p ocess,
we abs ac om he possibili y o a discon inui y a =0 by assuming ha U(0)
is op imally chosen as well, o equi alen ly ha s a e sec o employmen d ops o
E(0) =1−U(0)∗,wi hU(0)∗deno ing he op imal choice o ini ial unemploymen .
The go e nmen is only conce ned wi h e iciency and chooses employmen
in he s a e sec o o maximize he p esen discoun ed alue o ou pu . The
go e nmen ’s op imiza ion p oblem is hus gi en by:
max
E( )
∞
∫
0[E( )x+N( )y−1
2a ( (U( )))2]e− d ,(2)
subjec o:
N( )= (U( )),(3)
N(0) =0,(4)
E( )+N( )+U( )=1(5)
and non-nega i i y o E( ), N( )andU( ).
Based on he iden i y E( )+N( )+U( )=1, one immedia ely obse es ha he
p oblem can equi alen ly be o mula ed using U( ) as con ol a iable, he eby
elimina ing E( ), which lea es only U( )andN( ) in bo h he objec i e unc ion and
he cons ain s.10 This equi alen o mula ion o he p oblem is gi en by:
max
U( )
∞
∫
0[(1 −N( )−U( ))x+N( )y−1
2a ( (U( )))2]e− d ,(6)
10 We pe o m his subs i u ion o ha e U( ), pos ula ed o be cons an along op imal pa hs by
Aghion and Blancha d (1994), as he con ol a iable. The benefi o his e o mula ion is ha i
allows us o highligh he di e ences be ween he app oxima e and he exac solu ions.
A No e on he Op imal Speed o T ansi ion —7
subjec o:
N( )= (U( )),(7)
N(0) =0,(8)
N( )∈[0,1],(9)
N( )+U( )≤1(10)
and non-nega i i y o U( ).
No e fi s ha an op imal, in ac any, pa h mus necessa ily ulfill exac ly one
o he ollowing wo p ope ies: The e exis s a 𝜏<∞such ha 𝜏=in ≥0(N( )+
U( )=1) o condi ion (10) is no binding o any fini e . These wo cases will be
discussed sepa a ely below. Be o e doing so, an impo an p ope y o he model is
de i ed in P oposi ion 1.
P oposi ion 1. Along any pa h, i holds ha N( )<1 o all <∞.
P oo : Fo alueso N( ) su icien ly close o 1, he la ges possible alue o
N( )
is gi en by se ing U( )=1−N( ). The o dina y di e en ial equa ion
N( )= (1 −
N( )) has a s able s eady s a e a N=1, since (0) =0andd (1−N)
dN =− ′(1 −N)<0
o N=1. Gi en ha N(0) =0, i ollows ha N( )<1 o <∞.□
Le us now in es iga e po en ial op imal pa hs, s a ing wi h he case ha he
cons ain (10) becomes binding o he fi s ime a some 𝜏<∞. Gi en ha s a e
sec o employmen is mono onically non-inc easing, i ollows ha o ≥𝜏 he
con ol p oblem has a i ial op imal solu ion. Deno e wi h N( ,N𝜏) he solu ion o
he di e en ial equa ion
N( )= (1 −N( )), sol ed o e (𝜏,∞), wi h ini ial condi-
ion N(𝜏)=N𝜏. No e nex ha i i ially holds ha 𝜕N(𝜏,N𝜏)
𝜕N𝜏
=1 and also no e ha
up o now bo h 𝜏and N𝜏a e unspecified. The objec i e unc ion o he op imiza ion
p oblem om 𝜏onwa ds is gi en by:
V(𝜏,N𝜏)=
∞
∫
𝜏[N( ,N𝜏)y−1
2a ( (1−N( ,N𝜏)))2]e− d . (11)
No e he ollowing ela ionships o he pa ial de i a i es o he objec i e
unc ion (11):
𝜕V(𝜏,N𝜏)
𝜕𝜏 =−
[N(𝜏,N𝜏)y−1
2a ( (1−N(𝜏,N𝜏)))2]e− 𝜏,(12)
8—E. Næ dal and M. Wagne
𝜕V(𝜏,N𝜏)
𝜕N𝜏
=
∞
∫
𝜏[y+1
a (1−N( ,N𝜏)) ′(1−N( ,N𝜏))]e− d
=y
e− 𝜏+
∞
∫
𝜏[1
a (1−N( ,N𝜏)) ′(1−N( ,N𝜏))]e− d .(13)
The op imiza ion p oblem co esponding o he case conside ed can be ew i -
en as a sc ap alue p oblem wi h ee e minal ime, i.e., as a p oblem whe e 𝜏is o
be chosen op imally as well:
max
U( )∈[0,1],𝜏∈[0,∞)⎡⎢⎢⎣
𝜏
∫
0[(1 −N( )−U( ))x+N( )y−1
2a ( (U( )))2]e− d +V(𝜏,N𝜏)⎤⎥⎥⎦
,
(14)
subjec o (7),(9) and (10).
P oblems o his ype a e s udied in Seie s ad and Sydsæ e (1987,Theo em3
and No e 2, p. 182–184), which p o ide necessa y condi ions o op imali y.11
The (cu en - alue) Hamil onian co esponding o his p oblem is gi en by
H(N,U,𝜇,𝜏)=(1 −N−U)x+Ny−1
2a ( (U))2+𝜇 (U),whe eweigno e, o
b e i y, he o he cons ain s, (9) and (10), and he associa ed mul iplie s. I is
s aigh o wa d bu cumbe some o p esen he solu ion including hese addi ional
e ms in he Lag angean.12
Necessa y condi ions o op imali y a e gi en by:
−x−1
a (U) ′(U)+𝜇 ′(U)=0,(15)
𝜇 = 𝜇+x−y.(16)
Fu he mo e, he ollowing ans e sali y condi ion has o hold:
𝜇(𝜏)e− 𝜏=𝜕V(𝜏,N𝜏)
𝜕N𝜏
.(17)
11 To be p ecise we a i e a his ype o p oblem only a e e i ying ha he addi ional con-
s ain s – see he ollowing Foo no e12 – a e no binding. P oblems wi h hese addi ional con-
s ain s conside ed, i.e., wi h mixed and pu e s a e cons ain s a e discussed in Seie s ad and
Sydsæ e (1987, Chap e 6).
12 We use he e minology o Seie s ad and Sydsæ e (1987) and e e o he Hamil onian aug-
men ed by he addi ional cons ain s as Lag angean. I can be shown ha hese cons ain s a e no
binding, excep possibly a =0and =∞. Mo e specifically, i can be shown ha he only pos-
sible case whe e any o he cons ain han U( )+N( )≤1 is binding o <∞is he case whe e
U(0) =1, in which case 𝜏=0.
